Matching Impatient and Heterogeneous Demand and Supply
MMatching Impatient and Heterogeneous Demand and Supply
Angelos Aveklouris , Levi DeValve , Amy R. Ward , and Xiaofan Wu The University of Chicago Booth School of BusinessJanuary 29, 2021
Abstract
Service platforms must determine rules for matching heterogeneous demand (customers) andsupply (workers) that arrive randomly over time and may be lost if forced to wait too long fora match. We show how to balance the trade-off between making a less good match quickly andwaiting for a better match, at the risk of losing impatient customers and/or workers. Whenthe objective is to maximize the cumulative value of matches over a finite-time horizon, wepropose discrete-review matching policies, both for the case in which the platform has accessto arrival rate parameter information and the case in which the platform does not. We showthat both the blind and nonblind policies are asymptotically optimal in a high-volume setting.However, the blind policy requires frequent re-solving of a linear program. For that reason, wealso investigate a blind policy that makes decisions in a greedy manner, and we are able toestablish an asymptotic lower bound for the greedy, blind policy that depends on the matchingvalues and is always higher than half of the value of an optimal policy. Next, we develop afluid model that approximates the evolution of the stochastic model and captures explicitly thenonlinear dependence between the amount of demand and supply waiting and the distributionof their patience times. We use the fluid model to propose a policy for a more general objectivethat additionally penalizes queue build-up. We run numerous simulations to investigate theperformance of the aforementioned proposed matching policies.
Keywords: matching; two-sided platforms; bipartite graph; discrete-review policy; high-volumesetting; fluid model; reneging
Service platforms exist to match demand and supply (see, for example, for a broader perspectiveon the intermediary role of platforms in the sharing economy [33]). The challenge is that demandand supply often come from heterogeneous customers and workers that arrive randomly over timeand may be lost if forced to wait too long for a match. For example, in ride-sharing applications,riders and drivers can cancel their requests; in online tutoring, both students and tutors leave ifthey wait too long; and in transplant applications, there is the unfortunate event of a death of apatient. This creates a trade-off between making valuable matches quickly and waiting in order toenable better matches, at the risk of losing impatient customers and/or workers. Our objective isto develop matching policies that balance this trade-off to optimize the value of matches made.1 a r X i v : . [ m a t h . O C ] F e b a) The network. (b) v = 0 . . (c) λ = 50 . Figure 1: Performance comparison between FCFS and HV matching policy. Customers are willingto wait for 0.1 time units and workers wait forever to be matched.With impatient customers, this trade-off between speed and value of matches requires platformsto carefully consider the way they match supply and demand, i.e., their matching policy. The issuesat play can be illustrated in a simple example as shown in Figure 1a by comparing a naive first-come,first-served (FCFS) matching policy with a policy that reserves all supply for the highest value (HV)match and ignores completely the edge with the lowest value. When the platform’s objective is tomaximize the value of matches made, the FCFS policy performs arbitrary poorly compared withthe HV policy as we notice in Figures 1b and 1c. This can be explained intuitively as follows: Whenthe arrival rate of the first demand is relatively large and its corresponding matching value is small,the FCFS policy matches this low-value demand quickly upon arrival and there is not supply left forthe high-value demand node (second node). Meanwhile, the HV policy achieves a higher cumulativematching value by waiting for high-value demand arrivals to make a match, trading off speed forvalue.The FCFS policy in this example trivially coincides with selfish customer behavior in a platformin which customers choose their match (i.e., the platform does not control the matchings). Fromthat perspective, Figure 1 illustrates the potentially poor performance of decentralized policiesin matching platforms. The relevance of this observation stems from the fact that in practiceplatforms with both centralized and decentralized decision-making exist. The understanding ofwhen decentralized decision-making can result in good performance requires first understanding howto optimize the decisions when the platform can choose the matches (i.e., the optimal centralizedpolicy). However, developing an optimal centralized policy is a hard problem in its own right.Therefore, we limit our focus in this paper to centralized policies and view our work as a potentialbenchmark to use when studying decentralized policy performance.The matching model we consider has an arbitrary number of demand and supply types, eachcharacterized by their arrival processes and patience time distributions (also known in the literatureas reneging or abandonment distributions). The patience time distribution can be any generaldistribution with value 0 at 0. The value of a match can differ based on the demand and supplytypes being matched. The platform knows the matching values but may not know the arrivalprocesses and patience time distributions. The objective is to maximize the total value of thematches made. We also consider an extension which subtracts a type-dependent holding cost.
This paper makes two main contributions. First, we develop matching policies and we show thattheir cumulative matching value is asymptotically optimal in a high-volume setting. Second, we2emonstrate the effect of the patience time distributions on the holding costs and generalize ourmatching policies to account for that.
Development of asymptotically optimal matching policies . We develop matching policiesin two basic settings. First, we consider a setting where the platform has access to or can estimatewell enough the demand and supply arrival rates. Second, we consider a setting where the platformmay not know the system’s arrival rates possibly due to error in estimating arrival rates, or a lackof reliable data. We call these settings nonblind and blind , respectively, and we discuss each settingnext.In the nonblind setting, we develop a matching policy that bases its matching decisions on thesolution of a linear program (LP), which we term the static matching problem (SMP). The SMPignores the stochasticity of the demand and supply and uses the known arrival rates of each customerand worker type to optimize the matching rates between types. The issue for the stochastic systemis to ensure those rates are achieved. We do this using a discrete-review matching policy that basesmatching decisions at review time points on the SMP solution, modified to ensure feasibility. Thisdelicate procedure requires the review points to be far enough apart to ensure sufficient customerand worker arrivals of each type but also close enough together to prevent losing customers orworkers impatient to be matched. We strike this balance in high volume when a large number ofcustomers and workers arrive and show that the implemented matching rates achieve those of theSMP (Theorem 2). We further demonstrate that the SMP provides an asymptotic upper bound onthe matches achievable under any feasible policy (Theorem 1), and hence our proposed policy isasymptotically optimal.In the blind setting, we develop matching policies that must make decisions without knowledgeof the arrival rate parameters. Our first proposed blind policy is a discrete-review policy thatmakes the matches at each review time point in a greedy fashion according to the highest matchingvalue. The advantage of this policy is that it is intuitive and very computationally lightweight. Theplatform does not need to solve any optimization problem and only needs to know the matchingvalues and current queue lengths for implementation. Remarkably, we show that this simple policyachieves a cumulative matching value better than a lower bound that depends on the matchingvalues and is at least half of the optimal cumulative matching value in high volume (Theorem 5).We then move to a more sophisticated blind policy that decides on matches by solving an LP at eachdiscrete-review point. We show that affording the platform the computational resources required tofrequently resolve the LP allows the blind, LP-based policy to again achieve the optimal cumulativematching value in high volume (Theorem 3). Further, when the solution of the SMP is unique, theLP-based policy also achieves the optimal matching rates (Theorem 4).
The effect of the patience time distributions . Having proposed our matching policies, wemove to analyzing the demand and supply waiting to be matched, i.e., the queue lengths. Since anexact analysis of the queue lengths seems intractable, we develop a fluid model (Definition 3). Thisis a deterministic set of dynamical equations that is rich enough to explicitly show the impact ofthe patience time distributions. Even though these equations are still rather complicated, we areable to find and characterize their corresponding invariant points (Theorem 6). These points canbe seen as approximations of the mean demand and supply waiting to be matched and depend onthe patience time distributions through a nonlinear relationship.This fluid model allows us to consider a more general objective that maximizes the total valueof matches made minus holding costs. The problem is complicated due to the nonlinear relationshipbetween the queue lengths and the patience time distributions. Based on the fluid model and its3nvariant points, we generalize the SMP and we propose a matching policy that bases the matchingdecisions on the solution to the general SMP. Even though the SMP generally is not convex and itmay be NP-hard, we are able to characterize a wide class of patience time distributions for which theSMP becomes a convex optimization problem (Theorem 7). Then, we explore the performance ofthe aforementioned matching policy and investigate the impact of the cost on the matching decisionsusing simulation examples.The remainder of the paper is organized as follows. We end Section 1 by reviewing relatedliterature. In Section 2, we provide a detailed model description. In Section 3, we propose both ourblind and our parameter-dependent matching policies. In Section 4, we introduce our high-volumesetting and we study the asymptotic behavior of our proposed matching policies. A fluid modelis presented in Section 5. In particular, we present asymptotic approximations for the stationarymean queue lengths, and we show the impact of the patience time distributions. In Section 6, weadd holding costs in the system and we show how the aforementioned matching policies behave inthis case with the help of the fluid model. All the proofs are gathered in Appendices A–C.
We focus on on-demand service platforms that aim to facilitate matching. We refer the reader to[16], [22], and [34] for excellent higher-level perspectives on how such platforms fit into the sharingeconomy and to [33] for a survey of recent sharing-economy research in operations management.Three important research questions for such platforms identified in [16] are how to price services,pay workers, and match requests. These decisions are ideally made jointly; however, because thejoint problem is difficult, the questions are often attacked separately. In this paper, we focus on thematching question and make progress on the research opportunity identified at the end of Section2 in [34] regarding how to make matching decisions when arrival rate information is unknown.Our basic matching model is a bipartite graph with demand on one side and supply on theother side. There is a long history of studying two-sided matching problems described by bipartitegraphs, beginning with the stable matching problem introduced in the groundbreaking work of [29]and continuing to this day; see [49] and [1] for later surveys and [9] for more recent work. In theaforementioned literature, much attention is paid to eliciting agent preferences because the outcomesof the matching decisions, made at one prearranged point in time, can be life-changing events (forexample, the matches between medical schools and potential residents). In contrast, many platformmatching applications are not life-changing events, so there is less need to focus on eliciting agentpreference. Moreover, supply and demand often arrive randomly and continuously over time, andmust be matched dynamically over time, as the arrivals occur (as opposed to at one prearrangedpoint in time).Recent work has used dynamic two-sided matching models in the context of organ allocation[26, 39], online dating and labor markets [8, 36, 38], and ridesharing [14, 44]. Simultaneously andmotivated by the aforementioned applications, there have been many works that begin with a moreloosely motivated modeling abstraction, as ours does [6, 7, 13, 17, 18, 35, 41]. The aforementionedworks focus on a range of issues; however, most of them assume that agents’ waiting times are eitherinfinite, deterministic, or exponentially distributed. Allowing for more general willingness-to-waitdistributions is important because in practice the time an agent is willing to wait to be matchedmay depend on how long that agent has already waited. The difficulty is that tracking how thesystem evolves over time requires tracking the remaining time each agent present in the systemwill continue to wait to be matched, resulting in an infinite-dimensional state space. Our ability to4andle more general willingness-to-wait distributions relies on finding appropriate bounds for thenumber of customers that will be lost in a small time interval in a high-volume asymptotic regime.Taking an approach similar in spirit to how stochastic processing networks are controlled in thequeueing literature developed in [31, 32], we use a high-volume asymptotic regime to prove thatboth our matching-rate-based and blind policies are asymptotically optimal, and also to prove anasymptotic performance bound for our blind, greedy policy. In particular, we begin by solving aSMP and we focus on fluid-scale asymptotic optimality results. Adopting an approach reminiscentof the one [30] use in a dynamic multipartite matching model that assumes agents will wait foreverto be matched, we use a discrete-review policy to balance the trade-off between having agents waitlong enough to build up matching flexibility but not so long that many will leave.We consider both a finite-time horizon (when our goal is to optimize the cumulative number ofmatches) and an infinite horizon objective (when we add holding costs). In the latter case, we needto consider the steady-state performance of the system. This is the focus of the infinite bipartitematching queueing models considered for example in [2, 3, 4, 5, 20, 21, 25, 27].
In this section, we give a detailed model description and we define the essential quantities we needin the rest of the paper. All vectors are denoted by bold letters and their dimension is clear fromthe context.
Primitive inputs:
There is a set of demand nodes J := { , . . . , J } and a set of supply nodes K := { , . . . , K } , as shown in Figure 2. Furthermore, there are matching values v jk ≥ for anypossible matching between j ∈ J and k ∈ K . We assume that demand of type j ∈ J (customers)and supply of type k ∈ K (workers) arrive according to renewal processes, denoted by D j ( · ) and S k ( · ) , having respective rates λ j > and µ k > for all j ∈ J and k ∈ K . The arrival time of the h th type j customer and the arrival time of the h th type k worker can be expressed as e Djh = inf { t ≥ D j ( t ) ≥ h } and e Skh = inf { t ≥ S k ( t ) ≥ h } , where it is assumed that E (cid:104) | e Dj | (cid:105) < ∞ and E (cid:2) | e Sk | (cid:3) < ∞ for all j ∈ J , k ∈ K .The customers and workers have some patience time. The patience time is the maximum amountof time a customer or worker will wait in the system to get matched. Upon arrival, each type j ∈ J customer independently samples from the distribution determined by cumulative distributionfunction (cdf) G Dj ( · ) to determine his patience time. Similarly, upon arrival, each type k ∈ K workerindependently samples from the distribution determined by cdf G Sk ( · ) to determine his patience time.We refer to G Dj ( · ) and G Sk ( · ) as the patience time distributions .Patience time draws are visible to customers and workers but are unknown to the system con-troller. In other words, the system controller does not know how long each arriving customer/workerwill wait and only knows the distributions of the patience times. We denote the patience time ofthe h th type j customer and the patience time of the h th type k worker by r Djh and r Skh , respectively.We assume that the patience time distributions satisfy lim x → + G Dj ( x ) = lim x → + G Sk ( x ) = 0 for all j ∈ J , k ∈ K . Further, the demand and supply are used in FCFS order for each queue separately.In other words, the head-of-the-line (HL) customer/worker of each node is matched first. State space and queue evolution equations:
Let H Dj ∈ [0 , ∞ ] and H Sk ∈ [0 , ∞ ] be the rightedges of the support of the cumulative patience time (or abandonment) distribution functions for5igure 2: The two-sided matching model with general patience time distributions.any j ∈ J and k ∈ K . Further, for H ∈ [0 , ∞ ] , let M [0 , H ) denote the set of finite, nonnegativeBorel measures on [0 , H ) endowed with the topology of weak convergence. A state of our model attime t ≥ can be described by a vector ( Q ( t ) , I ( t ) , η D ( t ) , η S ( t )) where Q j ( t ) ∈ Z + is the numberof type j ∈ J demand (customers) and I k ( t ) ∈ Z + is the number of type k ∈ K supply (workers) inthe system waiting to be matched. Moreover, the measure η Dj ( t ) ∈ M [0 , H Dj ) tracks type j potentialcustomers in the system. The measure η Dj ( t ) ∈ M [0 , H Dj ) stores the amount of time that has passedbetween each type j customer’s arrival time up until that customer’s potential abandonment time(the arrival time plus the sampled patience time) for every type j customer that has arrived beforetime t ≥ and whose potential abandonment time is after t ; the measure η Sk ( t ) ∈ M [0 , H Sk ) storestime similarly for every type k worker. These measures do not consider whether that customer orworker has been matched, meaning these are the potential customers in queue. The FCFS matchingassumption implies that all potential customers that have waited longer than the customer at thehead of the line have already been matched, and all potential customers that have waited less thanthat customer are in queue. The waiting time of the type j ∈ J ( k ∈ K ) HL customer (worker) attime t ≥ , χ Dj ( t ) ( χ Sk ( t ) ) is χ Dj ( t ) := inf (cid:40) x ∈ R + : (cid:90) [0 ,x ] η Dj ( t )( dy ) ≥ Q j ( t ) (cid:41) , where inf ∅ = ∞ . Thus, Q j ( t ) = (cid:82) [0 ,χ Dj ( t )] η Dj ( t )( dx ) ( I k ( t ) = (cid:82) [0 ,χ Sk ( t )] η Sk ( t )( dx ) ) is the number oftype j ∈ J ( k ∈ K ) customers (workers) in queue. The fact that the potential queue measures areindependent of the scheduling policy is helpful for analytic tractability.We next show how to define the reneging processes so that we can define the queue lengthprocesses for any given matching policy M ( · ) := { M jk ( · ) , j ∈ J , k ∈ K } , where M jk ( t ) for any j ∈ J and k ∈ K denotes the cumulative matches between demand j and supply k in [0 , t ] . We refer to M ( · ) as the matching process . For a given matching policy M ( · ) , the demand and supply queuelengths are uniquely determined by Q j ( t ) := Q j (0) + D j ( t ) − R Dj ( t ) − K (cid:88) k =1 M jk ( t ) ≥ (1) We focus on defining the essential stochastic processes we shall use later. We do not provide the full system dynamicsrequired to determine how the state evolves over time here because that involves excessive methodological overhead. I k ( t ) := I k (0) + S k ( t ) − R Sk ( t ) − J (cid:88) j =1 M jk ( t ) ≥ , (2)for all j ∈ J and k ∈ K where R Dj ( t ) and R Sk ( t ) denote the number of type j customers and type k workers that left the system due to reneging until time t ≥ . We refer to R D ( · ) and R S ( · ) as the reneging processes . Note that (1) (and (2) analogously) says that the number of type j customersthat are waiting to be matched at time t ≥ is equal to the initial queue at time zero plus thecumulative demand minus the cumulative reneging customers minus the cumulative matches.In order to define the reneging processes, we first define the potential waiting time of the h thtype j potential customer (demand) at time t ≥ , w Djh ( t ) := min (cid:8) [ t − e Djh ] + , r Djh (cid:9) . In an analogous way, we define the potential waiting time of the h th type k potential worker (supply)at time t ≥ , w Skh ( t ) := min (cid:8) [ t − e Skh ] + , r Skh (cid:9) . Let m Djh and m Skh denote the matching time of the h th type j customer and the matching timeof the h th type k worker, respectively . If a customer (worker) abandons the queue, then we set m Djh = ∞ ( m Skh = ∞ ). Furthermore, we require that if m Djh < ∞ , then e Djh ≤ m Djh < e
Djh + r Djh (3)and if m Skh < ∞ , e Skh ≤ m Dkh < e
Skh + r Skh . (4)Note that (3) and (4) arise naturally. They enforce that customers and workers cannot be matchedprior to arriving in the system or once they abandon the system. Now, the number of reneging type j customers and type k workers at time t ≥ are given as follows: R Dj ( t ) := D j ( t ) (cid:88) h = − Q j (0)+1 (cid:88) s ∈ [0 ,t ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) (5)and R Sk ( t ) := S k ( t ) (cid:88) h = − I k (0)+1 (cid:88) s ∈ [0 ,t ] (cid:26) s ≤ m Sjh , dwSkhdt ( s − ) > , dwSkhdt ( s +)=0 (cid:27) . (6)The reneging processes clearly depend on the control policy as a customer (worker) at time s canleave the system if he is not matched, i.e., s ≤ m Djh . Further, a customer leaves the system at thetime his potential waiting time becomes constant, i.e., when his waiting time exceeds his patiencetime. This motivates the definition of the reneging processes since the event dw Dkh dt ( s − ) > meansthat the h th type j customer is still in the system just before time s and the event dw Skh dt ( s +) = 0 means that he abandons the system exactly after time s . Admissible policy:
The class of matching policies over which we optimize satisfies some naturalrestrictions. First, if a match occurs, then it cannot be taken back. This leads to the nondecreasing7roperty of a matching process, and hence sometimes it may be better to wait a bit rather than tomatch demand and supply at their arrival time. Further, we cannot match customers and workerswho are not present in the system and so a matching policy ensures that the queue lengths arenonnegative, i.e., the inequalities in (1) and (2) are satisfied. Last, a matching policy is independentof the future, so it is nonanticipating , and further, we do not allow a matching policy to use theknowledge of the patience times. We summarize these properties in the following definition. Definition 1.
A matching policy given by a stochastic process M ( · ) is called an admissible policy if M jk ( t ) is nondecreasing, nonanticipating, M jk ( t ) ∈ Z + , and (1) , (2) hold almost surely for all j ∈ J and k ∈ K . Moreover, M jk ( t ) does not take into account the patience time draws. Notice that reneging makes the queue lengths stable, in the sense that they do not build up,regardless of the matching policy used. Hence our matching policies do not need to enforce stabilityand we are able to consider a wider class of policies than, for example, [5] in which the class ofmatching policies is restricted to ensure stability.
Objective function:
Our goal is to choose an admissible matching policy M ( · ) to maximizethe finite-horizon cumulative matching value. In other words, we want to maximize the followingfunction V M ( T ) := J (cid:88) j =1 K (cid:88) k =1 v jk M jk ( T ) , (7)for a given finite time horizon T > . The trade-off of the objective function is as follows: One couldmatch currently available demand and supply to gain the resulting matching value immediately.Otherwise, one could wait to potentially obtain a better match, but at the risk of the currentlyavailable demand or supply reneging and losing a valuable match. An upper bound:
We define a performance metric through an upper bound on the cumulativevalue of matches that will be useful for our analysis. Suppose we do not make any matches untiltime t ∈ [0 , T ] . We then decide on the matches by solving max (cid:88) j ∈ J ,k ∈ K v jk Y jk s.t. (cid:88) j ∈ J Y jk ≤ S k ( t ) + I k (0) , k ∈ K , (cid:88) k ∈ K Y jk ≤ D j ( t ) + Q j (0) , j ∈ J ,Y jk ∈ R + , for all j ∈ J , k ∈ K . (8)Note that (8) does not take reneging into account. It assumes that demand (customers) and supply(workers) will wait forever to be matched. However, this is not true for our model as they mayabandon the system if they wait too long. In a rigorous mathematical way, nonanticipating means a matching policy that isadapted with respect to a filtration. In our case, the filtration is generated by (cid:8) σ (cid:0) D j ( s ) , D k ( s ) , R Dj ( s − ) , R Sk ( s − ) , Q j (0) , I k (0) , s ∈ [0 , t ] , j ∈ J , k ∈ K (cid:1) , t ∈ R + (cid:9) . A similar filtration is definedin [44, Section 2.1]. Y (cid:63) ( t ) denote a solution to (8) for t ∈ [0 , T ] , which exists because Y (cid:63)jk ( t ) = 0 for all j ∈ J , k ∈ K is feasible. This provides an almost sure upper bound on the cumulative value for any admissiblematching policy M ( · ) because V M ( t ) V Y (cid:63) ( t ) ≤ , (9)for any t ∈ [0 , T ] . To see (9), we only need to observe that the matches made by an admissiblepolicy up to time t are feasible for (8). For that, by (1) and (2), we have that (cid:88) j ∈ J M jk ( t ) ≤ S k ( t ) + I k (0) and (cid:88) k ∈ K M jk ( t ) ≤ D j ( t ) + Q j (0) , and by the definition of an admissible policy M jk ( t ) ∈ R + . In other worlds, any admissible policy M ( · ) provides a feasible solution for (8) and by the optimality of Y (cid:63) ( t ) , we obtain (9).The comparison in (9) is ambitious since it is not clear if an admissible policy that maximizes(7) achieves the upper bound. In particular, the impatience of customers and workers makes itchallenging to capture the full value of matches available in the system. Looking ahead, in Section 3we propose matching policies, and, in Section 4 we show they can achieve the upper bound in ahigh-volume asymptotic regime. We propose a number of matching policies, all of which are discrete review policies. A discrete-review policy decides on matches at review time points and does nothing at all other times [see, forexample, 42, 43]. Longer review periods allow more flexibility in making matches but risk losingimpatient customers. Shorter review periods prevent customer loss but may not have sufficientnumbers of customers of each type to ensure the most valuable matches can be made. Sections 3.1–3.3 introduce our proposed discrete review matching policies and Section 3.4 contains some numericobservations on their performance (that motivate the more rigorous analysis in Section 4). Further,without loss of generality, we assume that the system is initially empty in Sections 3 and 4.We let l, l, l, . . . be the discrete-review time points, where l ∈ [0 , T ] is the review-period length.Moreover, let for j ∈ J , k ∈ K , and i ≥ , Q j ( il − ) := Q j (( i − l ) + D j ( il ) − D j (( i − l ) − R Dj ( il ) + R Dj (( i − l ) and I k ( il − ) := I k (( i − l ) + S k ( il ) − S k (( i − l ) − R Sk ( il ) + R Sk (( i − l ) . The quantities Q j ( il − ) and I k ( il − ) are the maximum type j demand and type k supply that can bematched at time il . These follow by (1) and (2) and the fact that we make matches at discrete-reviewperiods and no matches are made between ( i − l and il . There are situations in which the arrival rates are completely unknown and matches need to bemade quickly, with minimal computational power. An easy and natural way to make matches inthis case is to prioritize matches in an order that reflects the matching value. This is the topic ofthe current section where we define a blind, greedy policy which matches the demand and supply9ccording to the highest value at each discrete-review period. We refer to the matching policy as blind to highlight the fact that it does not require the platform to have knowledge of the arrivalrates.Before we move to the main definition of the matching policy we first prioritize the edges of thebipartite graph according to their matching values. In particular, we sort the edges in decreasingorder of matching value, breaking ties arbitrarily. Formally, let (cid:31) denote the chosen order, so thatfor j, j (cid:48) ∈ J and k, k (cid:48) ∈ K , if v jk > v j (cid:48) k (cid:48) then v jk (cid:31) v j (cid:48) k (cid:48) , while if v jk = v j (cid:48) k (cid:48) then we arbitrarily breakthe tie to set v jk (cid:31) v j (cid:48) k (cid:48) (and this order will remain fixed throughout the policy). To summarize,we write v jk (cid:31) v j (cid:48) k (cid:48) ⇔ (cid:40) if v jk > v j (cid:48) k (cid:48) , if v jk = v j (cid:48) k (cid:48) and we give priority to edge ( j, k ) . This order says that if two values are different the higher one gets matching priority and if theyare the same the user chooses one according to his preference. To streamline the implementation ofthe greedy policy we may batch sets of high value edges that do not share endpoints since matchescan be made on these edges independently. To this end, define the following sets of priority edgesrecursively: P := ( j, k ) ∈ J × K : (cid:88) k (cid:48) (cid:54) = k :( j,k (cid:48) ) ∈ J × K { v jk (cid:48) (cid:31) v jk } = (cid:88) j (cid:48) (cid:54) = j :( j (cid:48) ,k ) ∈ J × K { v j (cid:48) k (cid:31) v jk } = 0 and for h ∈ { , . . . , J K } , P h := ( j, k ) ∈ C h − : (cid:88) k (cid:48) (cid:54) = k :( j,k (cid:48) ) ∈ C h − { v jk (cid:48) (cid:31) v jk } = (cid:88) j (cid:48) (cid:54) = j :( j (cid:48) ,k ) ∈ C h − { v j (cid:48) k (cid:31) v jk } = 0 , where C h := ( J × K ) \ ∪ hm =0 P m . The sets P h form a partition of the edges of the bipartite graphthat prioritize the edges in decreasing order of value. In particular, P is the set of all edges thathave the highest value among all adjacent edges, while P h for h > is defined similarly on thegraph with all higher priority edges (i.e., edges in P m for which m < h ) removed. An illustrationof these sets is shown in Figure 3. Demand Supply (a) P = { (1 , } , P = { (2 , } , P = { (2 , } , P = { (3 , } . Demand Supply (b) P = { (1 , , (3 , } , P = { (2 , } , P = { (2 , } . Figure 3: Examples of partitions of the edges of the bipartite graph. In the second example, there isnot a unique partition and we choose to give priority to edge (1 , instead of (2 , (i.e., v (cid:31) v ).10e are ready now to define recursively the greedy blind matching policy. The amount of type j ∈ J demand we match with type k ∈ K supply for ( j, k ) ∈ P at the end of review period i ≥ is M gijk := min ( Q j ( il − ) , I k ( il − )) . (10)For ( j, k ) ∈ P h , h = 1 , . . . , J K and i ≥ , define recursively M gijk := min Q j ( il − ) − h − (cid:88) m =0 (cid:88) k (cid:48) :( j,k (cid:48) ) ∈P m M gijk (cid:48) , I k ( il − ) − h − (cid:88) m =0 (cid:88) j :( j (cid:48) ,k ) ∈P m M gij (cid:48) k . (11)The aggregate number of matches at time t ≥ is given by M gjk ( t ) = (cid:98) t/l (cid:99) (cid:88) i =1 M gijk . (12)The main advantages of the greedy blind matching policy given in (10)–(12) are that it doesnot require knowledge of the arrival rates, it takes into account only the matching values and thecurrent queue lengths, and it does not require solving an optimization problem. Thus, this policycan be implemented in a wide range of settings, including when there is little data available on thearrival processes or when computational resources are constrained. However, one may expect thispolicy to not always perform well, especially when the matching values are similar. The reasonis that a high-value match on one edge may disallow matches on two of its adjacent edges, whichmay have had a higher combined matching value. To address this issue, we next propose anotherblind policy that makes matches in a more sophisticated way, by solving an optimization problemat every review time period. When sufficient computational power is available, the number of matches made at each discrete-review time point can be decided by solving an optimization problem with an objective to maximizethe matching value and constraints that respect the amount of demand and supply available. Forany i ≥ , let M bi be given by a solution of the following optimization problem max (cid:88) j ∈ J ,k ∈ K v jk M bijk s.t. (cid:88) j ∈ J M bijk ≤ I k ( il − ) , k ∈ K , (cid:88) k ∈ K M bijk ≤ Q j ( il − ) , j ∈ J ,M bijk ∈ Z + , for all j ∈ J , k ∈ K . (13)The quantity M bijk is the number of matches between type j demand and type k supply and thecumulative number of matches for j ∈ J , k ∈ K , and t ≥ , is given by M bjk ( t ) = (cid:98) t/l (cid:99) (cid:88) i =1 M bijk . (14)11he difference between (13) and (8) is that the right-hand sides of the constraints in (8) arereplaced by the queue lengths Q j ( il − ) ∈ N and I k ( il − ) ∈ N . The proposed policy is not greedybut myopic in the sense that the matching decisions made at each review time point are optimalgiven the available demand and supply but disregard the impact of future arrivals. The hope is thatwhen discrete-review points are well-placed, the aforementioned myopicity will not have too muchnegative impact, and the resulting total value of matches made can be close to the solution of theupper bound in (8).The implementation of the matching policy M bjk ( · ) does not require the knowledge of the arrivalrates λ and µ but it requires the solution of (13) at each review point. This problem may becomputationally intractable when the length of the discrete-review period is small and impractical insituations where the matching has to occur quickly (e.g., in milliseconds). Still, from the perspectiveof computational efficiency, it is helpful to observe that (13) can effectively be solved as an LP onany sample path, since the queue lengths are always integer valued. This solution arises becausethe constraint matrix of (13) is totally unimodular, which implies that an extreme point solutionis integer valued if the right-hand sides of the constraints are integer valued; see Lemma 6 in theappendix.One potential problem with the blind policies introduced so far (the greedy policy in Section 3.1and the LP-based policy in this section) is that neither is forward looking. However, when arrivalrate information is known, then the matching policy can use that information to be more forward-looking, as our next proposed matching policy does. The first step in order to define the matching-rate-based policy in this section is to formulatethe static matching problem (SMP). The SMP ignores stochasticity in order to find the maximuminstantaneous matching values by solving: max (cid:88) j ∈ J ,k ∈ K v jk y jk s.t. (cid:88) j ∈ J y jk ≤ µ k , k ∈ K , (cid:88) k ∈ K y jk ≤ λ j , j ∈ J ,y jk ≥ , j ∈ J , k ∈ K . (15)The quantity y jk can be interpreted as the optimal instantaneous rate of matches between j ∈ J and k ∈ K in time interval [0 , T ] if the arrival processes were replaced with their means. The first twoconstraints make sure that the total number of matches does not exceed the demand and supplycapacity.The issue with (15), of course, is that the stochastic nature of demand and supply does notallow the optimal matching rates to be maintained on every sample path. The challenge, then, isto define a policy that can mimic the matching rates given by (15) as closely as possible, subjectto the available supply and demand. To that end, let y be a feasible point of (15). Then, to mimicthe matching rates defined by y , let the amount of type j ∈ J demand we match with type k ∈ K i ∈ { , . . . , (cid:98) T /l (cid:99)} be M ijk := (cid:22) y jk min (cid:18) Q j ( il − ) λ j , I k ( il − ) µ k (cid:19)(cid:23) ≥ , (16)which implies the cumulative number of matches made for j ∈ J , k ∈ K , and t ≥ is M jk ( t ) := (cid:98) t/l (cid:99) (cid:88) i =1 M ijk . (17)The admissibility of the blind matching policies is straightforward by their definition. However,the matching-rate-based policy is based on a feasible point of (15). Hence its admissability is notobvious and depends on the properties of a feasible point. Proposition 1.
The proposed matching policy given by (16) and (17) is admissible for any feasiblepoint y of (15) . Before we move to the performance analysis of the matching policies, we motivate the results withsome simulations on the behavior of the three different matching policies compared to the upperbound given by (8). Figures 4–6 show in the vertical axis ratio (9) (i.e., V M ( t ) V Y (cid:63) ( t ) ) for all three policiesas a function of the arrival rate (horizontal axis) for various values of discrete-review length l , fortwo different sets of matching values, and for two patience time distributions. Below, we summarizethe main observations from the simulations. The performance of the matching policies improves for smaller review period:
We noticethat all the policies behave better for small values of the discrete-review length. We need to decreasethe review period l , especially when the mean patience times are smaller, in order to see a betterbehavior from the matching policies. This can be explained intuitively as follows: When customersand workers have small patience times, we need to decide on matches in a faster manner in ordernot to lose the most valuable matches. On the other hand, in order to make matches, a reviewperiod should allow enough demand and supply to enter into the system. In other words, thediscrete-review length should strike a balance between the arrival and reneging rates. The matching-rate-based and blind, LP-based policy perform close to the upper bound (8) for large arrival rates and small review period:
This suggests that these policies can berigorously shown to attain the upper bound (8) in a high-volume setting, with an appropriatelydefined relationship between the arrival rates and the review period length. Importantly, thesepolicies look to attain the upper bound (8) over a wide range of patience time distributions, includingboth a finite mean distribution (the exponential), a finite mean and infinite variance distribution(the first Pareto), and an infinite mean distribution (the second Pareto).
The greedy policy may or may not coincide with the blind, LP-based one:
Anotherobservation is that the greedy and the blind, LP-based policies coincide and perform better thanthe matching-rate-based policy for a set of matching values, in particular for ( v , v , v , v ) =(1 , , , . However, this is not always the case. We see that the greedy policy has the oppositebehavior and does not coincide with the blind, LP-based policy for the other set of matching values.Even though the greedy policy seems to be decreasing as arrival rates increase, it appears to do no13orse than some lower bound that depends on the matching values. We shall see in Section 4 howthis lower bound depends on the matching values.Figure 4: A two demand and two supply model with matching values ( v , v , v , v ) = (1 , , , , T = 100 , and unit mean patience times. The Pareto distribution has scale parameter . andinfinitive variance.Figure 5: A two demand and two supply model with same arrival rates, matching values ( v , v , v , v ) = (0 . , , , . , T = 100 , and unit mean patience times. The Pareto dis-tribution has scale parameter . and infinitive variance.14igure 6: A two demand and two supply model with same arrival rates, and Pareto distributedpatience times with infinite mean. The shape parameter is . , the scale parameter is . , T = 100 . In this section, we study the performance of the three proposed matching policies in a high-volumesetting when the goal is to maximize the value of matches made. We first derive an upper bound onthe cumulative matching value in a high-volume setting in Section 4.1, then show that the matching-rate-based and blind, LP-based policies achieve this upper bound asymptotically in Section 4.2. Weprove that the blind, greedy policy achieves at least half of the upper bound asymptotically, andthis bound can be improved when we take into account the matching values in Section 4.3.
Consider a family of systems indexed by n ∈ N , where n tends to infinity, with the same basicstructure as that of the system described in Section 2. To indicate the position in the sequence ofsystems, a superscript n will be appended to the system parameters and processes.We start by describing our high-volume setting (or asymptotic regime). In particular, we scalethe arrival rates in such a way that they increase linearly with n ∈ N ; namely, λ nj = nλ j for all j ∈ J and µ nk = nµ k for all k ∈ K . In other words, we study the system when a large number of customers and workers is expectedto arrive. Note that we do not scale the reneging parameters of the system, which is consistentwith the existing literature, for example see [37]. The high-volume scaled processes are defined bymultiplying the stochastic processes by a factor of /n ; for example, given a policy M n ( · ) for the n th system, the scaled number of matches is defined as M n ( · ) /n . In the high-volume setting, thecumulative value for any admissible policy can not be higher than the optimal value of the SMP.15 heorem 1. For any admissible matching policy in the n th system, M n ( · ) , we have that for any t ∈ [0 , T ] , lim sup n →∞ n V M n ( t ) ≤ (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk t, (18) almost surely. The main idea of the proof is to show that a scaled solution of the upper bound on cumulativevalue (8) approaches a solution of the SMP in the high-volume setting. This will allow us to comparethe value of our matching policies directly with the SMP. To this end, consider the upper bound(8) under the high-volume setting, i.e., max (cid:88) j ∈ J ,k ∈ K v jk Y njk s.t. (cid:88) j ∈ J Y njk ≤ S nk ( t ) , k ∈ K , (cid:88) k ∈ K Y njk ≤ D nj ( t ) , j ∈ J ,Y njk ∈ R + , for all j ∈ J , k ∈ K . (19) Proposition 2.
There exists a solution of (19) , Y (cid:63),n ( · ) , such that for any j ∈ J and k ∈ K , (cid:88) j ∈ J ,k ∈ K v jk Y (cid:63),njk ( t ) n → (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk t, almost surely for any t ∈ [0 , T ] as n → ∞ . The proof of the last proposition makes use of the fact that a small perturbation of the vectors λ and µ does not change the solution of the SMP very much as established by the next result. Lemma 1.
Let λ ∈ R J + and µ ∈ R K + . There exists a Lipschitz continuous mapping y (cid:63) : R J + × R K + → R J + × R K + such that y (cid:63) ( λ , µ ) is a solution of (15) . Having established an upper bound for the cumulative value, we investigate the existence of apolicy that achieves this upper bound in the high-volume setting.
Definition 2.
An admissible policy M n ( · ) is asymptotically optimal if lim n →∞ V M n ( T ) V Y (cid:63),n ( T ) = 1 , in probability for T > . In the remainder of Section 4, we study the asymptotic behavior of our proposed discrete-reviewmatching policies. To do so, we also need to scale the review period length as follows l n = 1 n / l. The intuition behind the scaling of the length of the review period is that it should be long enoughto allow us to make the most valuable matches as dictated by the SMP solution, but also shortenough that the reneging does not hurt. Lemma 1 implies that there exists a solution of (15) that is Lipschitz continuous. The later holds for any norm (cid:107) · (cid:107) in R J + × R K + as all norms in this space (and in general in any vector space with finite dimension) are equivalent. Inthe sequel, we freely use the norm that suits our approach. .2 Asymptotic optimality of matching-rate-based and blind, LP-based policies In this section, we study the asymptotic performance of the matching-rate-based policy defined inSection 3.3 and the blind, LP-based policy introduced in Section 3.2.Given a solution y (cid:63) to (15), the number of matches for the matching-rate-based policy in the n th system at discrete-review period i ∈ { , . . . , (cid:98) T /l n (cid:99)} is given by M nijk = (cid:36) ny (cid:63)jk min (cid:32) Q nj ( il n − ) λ nj , I nk ( il n − ) µ nk (cid:33)(cid:37) = (cid:22) y (cid:63)jk min (cid:18) Q nj ( il n − ) n λ j , I nk ( il n − ) µ k (cid:19)(cid:23) . (20)Note that the instantaneous matching rates y (cid:63)jk are scaled by n in (20) because this solution remainsoptimal to (15) when the arrival rates are scaled by a factor of n . The cumulative number of matchesuntil time t in the n th system is given by M njk ( t ) = (cid:98) t/l n (cid:99) (cid:88) i =1 M nijk . (21)The goal of this section is to show that the matching-rate-based policy defined by (16) and (17)is asymptotically optimal. The following result is even stronger as it shows that the scaled numberof matches approaches asymptotically a solution of the SMP. Theorem 2.
Let y (cid:63) be a solution of (15) . Under the matching-rate-based policy (20) and (21) , as n → ∞ , sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M njk ( t ) n − y (cid:63)jk t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → , in probability, for all j ∈ J and k ∈ K . The last theorem suggests that the quantity y (cid:63)jk t can be seen as an approximation of the meannumber of matches at time t for j ∈ J and k ∈ K . Knowing the optimal matching rates is helpfulfor the system manager for two main reasons. First, it provides a fairness property in the sensethat the platform has a consistent target for matching rates in the system so that users can come toknow what to expect in terms of matching priorities. Second, a solution of the SMP can be used toidentify the most valuable matches in the network, which is useful for forecasting purposes. Now,the asymptotic optimality of the matching-rate-based policy is a direct consequence of Theorem 2. Corollary 1.
The matching-rate-based policy is asymptotically optimal, i.e., lim n →∞ V M n ( T ) V Y (cid:63),n ( T ) = 1 , in probability. Although the detailed proof is given in the appendix, we provide the reader with a brief outlinehere to establish the basic template used in the proofs of this section. The proof proceeds in threebasic steps: i) establish an upper bound on the amount of reneging during a review period, ii) usethe reneging upper bound to derive a lower bound on the number of matches made during a reviewperiod, and iii) show that this lower bound on matches made approaches the solution of the SMP.17ext we consider the asymptotic behavior of the blind, LP-based policy introduced in Section 3.2.The number of matches in the n th system, M b,ni , at a discrete-review period is given by a solution of(13) replacing the right-hand sides of the constraints by Q nj ( il n − ) and I nk ( il n − ) . The next theoremstates that the matching policy is asymptotically optimal. Theorem 3.
The blind, LP-based matching policy given by (13) and (14) in the n th system isasymptotically optimal. That is, as n → ∞ , lim n →∞ V M b,n ( T ) V Y (cid:63),n ( T ) = 1 , in probability, for all j ∈ J and k ∈ K . Both policies we have studied in this section are asymptotically optimal. An advantage of theblind, LP-based matching policy is that it does not use any information about the arrival rates.However, there is a trade-off with how often we have to solve an optimization problem. Moreover,the matching-rate-based policy provides insight for the individual number of matches in contrast tothe LP-based policy.We prove Theorem 3 using the same three basic steps as Theorem 2. However establishing a lowerbound on the matches made during a review period (step ii) requires more effort since we cannotcompare directly to the SMP solution. We overcome this challenge by leveraging a monotonicityproperty of the SMP (15).Having shown that the LP-based policy is asymptotically optimal, a question that arises iswhether the matches made under this policy also achieve the optimal matching rates. The answeris nuanced in the case when there are multiple optimal solutions to the SMP, since the limit of M b,n ( T ) n may oscillate between them. However, we are able to show that the matching rates mustapproach the set of optimal SMP solutions asymptotically. To this end, for a real vector x and aset A in Euclidean space, denote the distance between them by d ( x , A ) , e.g., one could consider d ( x , A ) := inf z ∈ A (cid:107) x − z (cid:107) . Theorem 4.
Let S be the set of all optimal solutions of (15) . In other words, we have that S := { y (cid:63) : y (cid:63) is a solution of (15) } . Then, as n → ∞ , d (cid:18) M b,n ( T ) nT , S (cid:19) → , in probability. A consequence of the last theorem is that an analogous result to Theorem 2 holds for theLP-based policy in the special case that (15) has a unique solution.
Corollary 2.
Assume LP (15) has a unique solution y (cid:63) . We have that as n → ∞ , M b,njk ( T ) n → y (cid:63)jk T, in probability, for all j ∈ J and k ∈ K . .3 An asymptotic lower bound for the blind, greedy policy It is clear (even from the simulations) that the blind, greedy policy generally is not asymptoticallyoptimal. A natural question now is how far it can be from optimal. We show that the answerdepends on the matching values, and that in the worst case the asymptotic performance relative tooptimal is bounded by a constant. Define γ v := min j,k (cid:16) v jk v j ( k )+ v k ( j ) (cid:17) with v j ( k ) = max k (cid:48) (cid:54) = k { v jk (cid:48) : v jk (cid:48) ≤ v jk } and v k ( j ) = max j (cid:48) (cid:54) = j { v j (cid:48) k : v j (cid:48) k ≤ v jk } , i.e., v j ( k ) denotes the value of the next highest match for demand node j ∈ J after v jk , and v k ( j ) denotes the value of the next highest match for supply node k ∈ K after v jk . Clearly, the factor γ v depends on the matching values, and γ v ≥ / since v jk ≥ v j ( k ) and v jk ≥ v k ( j ) by definition.We can interpret the factor γ v as follows: the opportunity cost of making a match between j ∈ J and k ∈ K is at most the sum of the next highest value matches that both j and k could make,and γ v captures the worst case relative opportunity cost for any edge in the system. The nextproposition gives a lower bound for a greedy solution to the SMP. We define the greedy solution to(15) recursively for ( j, k ) ∈ P h as y gjk = min λ j − (cid:88) k (cid:48) (cid:54) = k :( j,k (cid:48) ) ∈C h − y gjk (cid:48) , µ k − (cid:88) j (cid:48) (cid:54) = j :( j (cid:48) ,k ) ∈C h − y gj (cid:48) k . Proposition 3.
Given a greedy solution y g to (15) , we have that (cid:88) j ∈ J ,k ∈ K v jk y gjk ≥ min (1 , γ v ) (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk . This proposition establishes that a greedy solution approximates the SMP (15) by a factor min(1 , γ v ) ≥ / . Based on our proof strategy of comparing to the SMP upper bound, this suggeststhat our blind, greedy policy may also enjoy a similar bound asymptotically, and this indeed is whatwe show next.Given the priority sets P h of the edges of the bipartite graph defined in Section 3.1, the blind,greedy policy matches are made recursively as follows. In the n th system, we have that for any h = 0 , , . . . , J K and i ≥ , M g,nijk = min Q nj ( il n − ) − h − (cid:88) m =0 (cid:88) k (cid:48) :( j,k (cid:48) ) ∈P m M g,nijk (cid:48) , I nk ( il n − ) − h − (cid:88) m =0 (cid:88) j (cid:48) :( j (cid:48) ,k ) ∈P m M g,nij (cid:48) k . (22)Given Proposition 3, the next theorem, which is our main result in this section, is intuitive. Theorem 5.
Under the blind, greedy matching policy (22) , as n → ∞ , lim n →∞ V M g,n ( T ) V Y (cid:63),n ( T ) ≥ min (1 , γ v ) ≥ , in probability. min(1 , γ v ) ≥ / of the optimal cumulative matching value. This is similar to results in the literature establishinga / greedy approximation in matching and allocation settings distinct from our model (e.g., 28,Theorem 1; 40, Theorem 18), but our characterization of the guarantee in terms of the matchingvalues allows further insight depending on the problem instance. In particular, when γ v ≥ theblind, greedy policy is asymptotically optimal. This condition is similar to the so-called Mongeconditions [19, 23] but it does not take into account the cycles of the graph. When γ v ≥ (this isthe case in the simulation of Figure 4), then the greedy solution coincides with the solution of theLP-based policy [35, Proposition 2]. However, Theorem 5 provides a stronger result giving a lowerbound on the cumulative matching value for any problem instance. Furthermore, our blind, greedypolicy is computationally tractable as its computation only requires knowledge of the order of thevalues, and does not require solving an optimization problem.We use the same basic proof strategy for Theorem 5 as for Theorems 2 and 3. As we did withTheorem 3, we establish a monotonicity property of the greedy solution to the SMP to derive alower bound on the number of matches made during a review period (step ii). This allows us to useProposition 3 to establish the result.Now that we have proposed and analyzed our matching policies, the reader may have noticedthat we are able to show that our matching policies behave well for any patience time distributionand we do not even need to approximate the queue lengths (bounds are enough). Still, our proposedmatching policies depend on the whole reneging processes because the queue lengths depend on thereneging processes.The approximation of the queue lengths is important to understand system behavior and requiredif one wants to consider a more general objective function that penalizes long queues. Hence weare motivated to develop a fluid model that allows us to develop such an approximation (Section 5)and to use that fluid model to develop matching policies for a more general objective function thatincludes holding costs (Section 6). We will find that the queue length approximation strongly de-pends on the patience time distribution, and so the matching policies for the more general objectivefunction will also. The analysis in the previous section does not explicitly show the connection between the matchingpolicy used and the resulting queue lengths. However, queue size is an important and well-studiedperformance metric. For example, a matching policy with smaller queue sizes in targeted classesmay be preferable to a matching policy that strictly maximizes the total value of the matchesmade. Hence we are motivated to develop a queue length approximation. We do this by firstspecifying a continuous fluid model in Section 5.1, and second determining its invariant states inSection 5.2. The invariant states are fixed points of the fluid model equations, and can be used toapproximate the queue lengths. Section 5.3 provides numeric evidence that the invariant states are good approximations.
Recall that H Dj , H Sk are the right edges of the support of the cumulative patience time distributionfunctions for any j ∈ J and k ∈ K and define Y := R J + × R K + × (cid:16) × Jj =1 M [0 , H Dj ) (cid:17) × (cid:16) × Kj =1 M [0 , H Sk ) (cid:17) .20urther, let h Dj ( · ) and h Sk ( · ) be the hazard rate (or failure) functions of the demand and supplypatience time distributions, respectively. Moreover, in the rest of the paper we assume that thepatience times are absolutely continuous random variables.Roughly speaking, a fluid model solution is represented by a vector ( Q , I , η D , η S ) ∈ C ( R + , Y ) .This is the analogue of the state descriptor introduced in Section 2. The first two functions representthe fluid queue lengths and the measure-valued functions η D ( · ) , η S ( · ) approximate the potential fluidqueue measures at any time. Below, we define the auxiliary functions that a fluid model solutionhas, present the conditions and the evolution equations that a fluid model solution satisfies, andgive a rigorous definition of the fluid model.Let χ Dj ( t ) ( χ Sk ( t ) ) denote the fluid analogues to the waiting time of the type j ∈ J ( k ∈ K ) HLcustomer (worker) at t ≥ introduced in Section 2. This can be written as follows for any t ≥ , χ Dj ( t ) = inf (cid:40) x ∈ R + : (cid:90) [0 ,x ] η Dj ( t )( dy ) ≥ Q j ( t ) (cid:41) , where inf ∅ = ∞ . Further, assuming that the potential measures do not have atoms at zero, thefluid queue lengths and the potential measures are connected through the following relations for t ≥ , (cid:90) [0 ,χ Dj ( t )] η Dj ( t )( dx ) = Q j ( t ) ≤ (cid:90) [0 ,H Dj ] η Dj ( t )( dx ) , (cid:90) [0 ,χ Sk ( t )] η Sk ( t )( dx ) = I k ( t ) ≤ (cid:90) [0 ,H Sk ] η Sk ( t )( dx ) , (23)where these are similar relations to what we have seen in Section 2.We now move to define the fluid reneging functions. Observe that h Dj ( x ) with x ∈ [0 , χ Dj ( t )] represents the probability that fluid of type j demand that has waited x time units reneges in thenext time instant. Additionally, the measure η Dj ( u )( dx ) encodes how much fluid has waited x timeunits until time u ≥ . Thus, (cid:82) [0 ,χ Dj ( t )] h Dj ( x ) η Dj ( u )( dx ) is the instantaneous reneging rate at time u ≥ . In a similar way, (cid:82) [0 ,χ Sk ( t )] h Dk ( x ) η Sk ( u )( dx ) is the instantaneous reneging rate of fluid oftype k supply at time u ≥ . To derive the cumulative reneging fluid at t ≥ , one integrates theinstantaneous reneging rates in time interval [0 , t ] ; namely, R Dj ( t ) = (cid:90) t (cid:90) χ Dj ( t )0 h Dj ( x ) η Dj ( u )( dx ) du and R Sk ( t ) = (cid:90) t (cid:90) χ Sk ( t )0 h Sk ( x ) η Sk ( u )( dx ) du. (24)When the patience times are exponentially distributed with rates θ Dj and θ Sk , we have that R Dj ( t ) = (cid:90) t θ Dj Q j ( u ) du and R Sk ( t ) = (cid:90) t θ Sj I k ( u ) du, where these last formulas agree with the known results in the literature, e.g., see [45]. The conditions (cid:90) t (cid:90) [0 ,H Dj ] h Dj ( x ) η Dj ( u )( dx ) du < ∞ and (cid:90) t (cid:90) [0 ,H Sk ] h Sk ( x ) η Sk ( u )( dx ) du < ∞ (25) The rigorous connection between the fluid reneging functions and (5) and (6) follows from the compensator functionused to define an appropriate martingale, similar to [37, Proposition 5.1]. Q j ( t ) = Q j (0) + λ j t − R Dj ( t ) − (cid:88) k ∈ K M jk ( t ) , (26) I k ( t ) = I k (0) + µ k t − R Sk ( t ) − (cid:88) j ∈ J M jk ( t ) , (27)where the functions M jk ( · ) : R + → R + are the fluid analogues of the matching processes. Definition 3.
Given λ ∈ R J and µ ∈ R K , we say that ( Q , I , η D , η S ) ∈ C ( R + , Y ) is a fluid modelsolution if there exist nondecreasing and absolutely continuous functions M jk ( · ) : R + → R + with M jk (0) = 0 , such that (23) , (25) , (26) , and (27) hold for all j ∈ J and k ∈ K . Further, for anycontinuous and bounded function f ∈ C b ( R + , R + ) the following evolution equations hold for any j ∈ J , k ∈ K , and t ≥ , (cid:90) [0 ,H Dj ] f ( y ) η Dj ( t )( dy ) = (cid:90) H Dj f ( x + t ) 1 − G Dj ( x + t )1 − G Dj ( x ) η Dj (0)( dx ) + λ j (cid:90) t f ( t − u )(1 − G Dj ( t − u )) du, (cid:90) [0 ,H Sk ] f ( y ) η Sk ( t )( dy ) = (cid:90) H Sk f ( x + t ) 1 − G Sk ( x + t )1 − G Sk ( x ) η Sk (0)( dx ) + µ k (cid:90) t f ( t − u )(1 − G Sk ( t − u )) du. The first term of the evolution equations for measures η Dj ( · ) and η Sk ( · ) in the last definition trackswhen the patience time of fluid demand/supply initially present in the system at time zero expires.The second term has an analogous meaning for the newly arriving fluid.A fluid model solution associated with a particular initial state is not unique. Uniquenessrequires a more precise specification of the matching policy. Since we conjecture that a fluid modelsolution arises as a weak limit of the fluid-scaled state descriptor, Q j ( t ) and I k ( t ) can be seen asapproximations of the mean demand and supply queue lengths at time t for large enough arrivalrates and for a fixed matching policy.Based on the fluid model we can write a continuous fluid control problem and this can be helpfulin providing an upper bound of an objective under a finite horizon. For example, if we consider thematching value objective (7), then using similar arguments as in [44, Theorem 4], we can show that (cid:80) j ∈ J ,k ∈ K v jk M (cid:63)jk ( T ) = (cid:80) j ∈ J ,k ∈ K v jk y (cid:63)jk T , where M (cid:63) ( T ) is the optimal solution of the followingproblem max (cid:88) j ∈ J ,k ∈ K v jk M jk ( T ) s.t. ( Q , I , η D , η S ) is a fluid model solution.Thus, like the SMP solution, the fluid model solution also provides an asymptotic upper bound forthe cumulative matching value by Theorem 1 and could be used to derive asymptotic optimalityresults for the matching value objective as in Theorem 2.We conjecture that a similar property holds even for more general objective functions involvingthe queue lengths. However, the assumption of the general patience time distributions makes thefluid model equations rather complicated and hence the corresponding continuous fluid control prob-lem becomes intractable. To overcome this difficulty, one could consider infinite horizon objectivefunctions. This requires an invariant analysis of the fluid model, the topic of the next section.22 .2 Invariant states Having written a fluid model for the matching model in Section 2, we now move to characterizeits invariant states. The invariant states provide an approximation for the mean stationary queuelengths in the matching model.
Definition 4.
A point ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) ∈ Y is said to be an invariant state for given λ and µ if the constant function ( Q , I , η D , η S ) given by ( Q ( t ) , I ( t ) , η D ( t ) , η S ( t )) = ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) forall t ≥ is a fluid model solution. We denote the set of invariant states by I λ,µ . In the sequel, we let /θ Dj and /θ Sk be the means of the patience time distributions and definethe following set: M := m ∈ R J × K + : (cid:88) k ∈ K m jk ≤ λ j and (cid:88) j ∈ J m jk ≤ µ k . The set M contains all possible instantaneous matching rates m jk , j ∈ J , k ∈ K . The two constraintsmake sure that the instantaneous matching rates do not exceed the demand and supply rates, andare analogous to the constraints of the SMP (15). Let the excess life distributions of the patiencetimes be G De,j ( x ) = (cid:90) x θ Dj (1 − G Dj ( u )) du and G Se,k ( x ) = (cid:90) x θ Sk (1 − G Sk ( u )) du for j ∈ J , k ∈ K , x ∈ R + . Theorem 6 (Characterization of the invariant states) . Suppose that the patience time distributionsare invertible and let λ ∈ R J + and µ ∈ R K + . A point ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) lies in the in invariantmanifold I λ,µ if and only if it satisfies the following relations for j ∈ J and k ∈ K with m ∈ M : η D,(cid:63)j ( dx ) = λ j (1 − G Dj ( x )) dx, (28) η S,(cid:63)k ( dx ) = µ k (1 − G Sk ( x )) dx, (29) q (cid:63)j ( m ) = λ j θ Dj , if (cid:80) k ∈ K m jk = 0 , λ j θ Dj G De,j (cid:16) ( G Dj ) − (cid:16) − (cid:80) k ∈ K m jk λ j (cid:17)(cid:17) , if (cid:80) k ∈ K m jk ∈ (0 , λ j ] , (30) i (cid:63)k ( m ) = µ k θ Sk , if (cid:80) j ∈ J m jk = 0 , µ k θ Sk G Se,k (cid:16) ( G Sk ) − (cid:16) − (cid:80) j ∈ J m jk µ k (cid:17) (cid:17) , if (cid:80) j ∈ J m jk ∈ (0 , µ k ] . (31)There is in general a nonlinear relationship between the invariant states and the vector of thematching rates m that depends on the patience time distributions.Since the invariant states are functions of the matching rates, and optimal matching rates aregiven by a solution of the SMP (15) when the goal is to maximize the value of matches made,the properties of the SMP influence the invariant states associated with an asymptotically optimalpolicy (recall Theorem 2 and Corollary 2). To state those properties, we require the followingpreliminaries. Given a feasible solution, y jk , j ∈ J , k ∈ K , for (15), let the induced graph of thissolution refer to the graph ( { J , K } , { ( j, k ) : y jk > } ) , i.e., the bipartite graph between J and K consisting of those edges with positive flow in the feasible solution. We say a node j ∈ J ( k ∈ K ) isslack if (cid:80) k ∈ K y jk < λ j ( (cid:80) j ∈ J y jk < µ k ), and tight if (cid:80) k ∈ K y jk = λ j ( (cid:80) j ∈ J y jk = µ k ). Note that wecan assume an optimal solution of (15) is an extreme point solution without loss of generality.23 emma 2. The induced graph of an extreme point solution to (15) has the following properties:1. It has no cycles, i.e., it is a forest, or collection of trees;2. Each distinct tree has at most one node with a slack constraint in (15) ;3. Any node with a slack constraint is connected only to nodes with tight constraints in (15) . A consequence of the third property of Lemma 2, together with the invariant queue lengthdefinitions (30) and (31), is that for an edge ( j, k ) in the induced graph at least one (exactly one orboth) of q (cid:63)j and i (cid:63)k is zero (exactly the one for which the corresponding constraint is tight). In otherwords, an optimal solution of (15) identifies the queues that are zero. Further, Lemma 2 implies thatthe matching-rate-based policy can decompose the system into separate trees and make matches oneach tree independently, and that each tree will have at most one node with a queue that tends tobuild up over time. We have stated that the invariant points can be seen as approximations of the mean queue lengthsin steady state. Here, we provide an example in order to investigate the approximation error and thedependence on the patience time distribution. We consider one demand and one supply node andlet the arrival rates be nλ and nµ , respectively. We make every possible match between customersand workers and we define m := min( λ, µ ) . Tables 1 and 2 and Figure 7 present simulations andnumerics for the fraction of reneging customers and workers and also for the approximation errorbetween the simulated queue lengths and the invariant points given by (30) and (31). When thepatience times are exponential, then the distribution of queue lengths can be found explicitly as weshow in Appendix C.Table 1 shows the fraction of reneging customers and workers obtained by the simulations andthe corresponding values calculated by the fluid model equations (denoted by r (cid:63)S and r (cid:63)D ). Weobserve that for exponential and gamma patience times the fraction of reneging customers andworkers remains almost the same, which suggests an insensitivity property, similar to ones thatappear in [48] and [11].Table 1: The fraction of reneging population for exponential (right) and gamma (left, rate param-eter=0.7) distributions with unit means. The parameters are T = 100 , λ = 1 , and n = 100 . R D ( T ) /D ( T ) r (cid:63)D R S ( T ) /S ( T ) r (cid:63)S R D ( T ) /D ( T ) r (cid:63)D R S ( T ) /S ( T ) r (cid:63)S µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 1 . The invariant queue lengths are good approximations:
We notice in Figure 7 and in Table 2that the invariant queue lengths are quite close to the simulated queue lengths. That is true forexponential patience times, where we can calculate the mean queue lengths exactly, and for gammapatience times as well, where we simulate the mean queue lengths numerically. This suggeststhat the invariant queue lengths can help in extending and analyzing our model since they can becalculated by (30) and (31). This is helpful when we consider a more general objective functionthat includes the queue lengths in the next section.
The higher variance, the better performance:
We see in Figure 7 that when we increase thevariance the queue lengths become smaller. In other words, it seems that if we fix a patience timedistribution, the variance plays an important role. This can be explained intuitively as follows:When the variability of the patience times is high, there will be customers/workers with relativelysmall and large patience times. We expect to lose the customers/workers with a small patience timequickly and to match the ones with a large patience time, so the queue lengths will remain small.Table 2: Exact calculations for exponential patience times with λ = 1 , θ = 1 (left), θ = 2 (right),and n = 100 . E [ Q ( ∞ ) /n ] q (cid:63) E [ I ( ∞ ) /n ] i (cid:63) E [ Q ( ∞ ) /n ] q (cid:63) E [ I ( ∞ ) /n ] i (cid:63) µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 1 . The patience time distributions affect the queue sizes, i.e., the amount of demand and supplywaiting to be matched. As a result, in contrast to the results in Section 4, when the objectivefunction penalizes customer waiting we expect that matching policies that incorporate patience timedistribution information will have better performance. In this section, we include type dependentholding costs c Dj ≥ and c Sk ≥ for each j ∈ J , k ∈ K , incurred per customer or worker, per unittime, to capture the disutility associated with waiting. In deference to the fluid model analysis inSection 5, in which we developed approximations for the stationary mean queue lengths, we considerthe average cost objective: V c M := lim inf T →∞ T (cid:88) j ∈ J ,k ∈ K (cid:90) T v jk dM jk ( s ) − (cid:88) j ∈ J (cid:90) T c Dj Q j ( s ) ds − (cid:88) k ∈ K (cid:90) T c Sk I k ( s ) ds . Now, optimizing the trade-off between waiting to make a better match and risking customer lossrequires considering both the matching values and the waiting costs. For example, if the waiting25igure 7: Gamma distributed patience times with unit mean and variance /x , x = 0 . , , . Thesolid lines represent the simulated queue lengths and the dotted lines represent the invariant queuelengths.costs are high, then even patient customers should not wait too long to be matched.We propose modified matching policies in Section 6.1 that also account for waiting costs. Weprovide numeric evidence that our modified matching policies can have good performance in Sec-tion 6.2. We have proposed three different matching policies when the goal is to maximize the total value ofthe matches made. However, it is not clear how all of them can be extended naturally in the caseof holding costs. For example, the objective function of the LP-based matching policy cannot bemodified to include the costs in a convenient way. In the following, we focus on the greedy and thematching-rate-based policies and we explain how they are extended in this context.
A greedy policy:
When the goal of a platform is to maximize the total value of matches, a blind,greedy policy is defined by giving priority according to the highest value; see Section 3.1. However,when there are holding costs in the system, this is not enough because holding costs can significantlyhurt the total profit of a platform. For example, making a match between j ∈ J and k ∈ K not onlygains the matching value v jk , but also avoids incurring further holding costs of c Dj + c Sk , implyingthat nodes with higher holding costs should get higher matching priority. Accounting for thesenecessities changing the way we prioritize the edges of the network. In particular, we now definethe priority sets based on the weights v jk + c Dj + c Sk . The policy remains blind since knowledge ofthe arrival and reneging rate parameters is not required. A matching-rate-based policy:
A natural way to extend the matching policy proposed in Sec-tion 3.3 is to extend the objective of the SMP (15) to include holding costs by making use of the26pproximate queue lengths in Theorem 6: max (cid:88) j ∈ J ,k ∈ K v jk m jk − (cid:88) j ∈ J c Dj q (cid:63)j ( m ) − (cid:88) k ∈ K c Sk i (cid:63)k ( m ) s.t. (cid:88) j ∈ J m jk ≤ µ k , k ∈ K , (cid:88) k ∈ K m jk ≤ λ j , j ∈ J ,m jk ≥ , j ∈ J , k ∈ K . (32)We refer to (32) as the general static matching problem . Now, the matching-rate-based policy is de-fined by (16) and (17) for any feasible point of (32). Note that it makes sense to consider (32) sincewe expect that its objective function is approached by the scaled profit V c M n in the high-volume set-ting, where n is as introduced in Section 4.1. Intuitively, we conjecture that T (cid:82) T Q nj ( s ) n ≈ E (cid:104) Q nj ( ∞ ) n (cid:105) for large enough T and E (cid:104) Q nj ( ∞ ) n (cid:105) ≈ q (cid:63)j for large enough n in the high-volume setting, and thatthe same holds for the supply queue lengths. Hence the optimal value of (32) is a candidate upperbound of the total profit and the matching-rate-based policy is conjectured to be asymptoticallyoptimal (i.e., to achieve the upper bound in the high-volume setting).A key difference in the case of holding costs is that the general static matching problem generallyis non-convex because the invariant queue lengths depend on the patience time distributions throughthe nonlinear relationship shown in Theorem 6. The only time the relationship is linear is when thepatience times follow an exponential distribution. The relationship is quadratic when the patiencetimes are uniformly distributed in [0 , θ Dj ] and [0 , θ Sk ] , and is q (cid:63)j ( m ) = λ j θ Dj (cid:32) − (cid:18) (cid:80) k ∈ K m jk λ j (cid:19) (cid:33) and i (cid:63)k ( m ) = µ k θ Sk (cid:32) − (cid:18) (cid:80) j ∈ J m jk µ k (cid:19) (cid:33) . In some cases, the invariant states cannot even be written in a closed form, for instance when thepatience times follow a gamma distribution.Even though the analysis now becomes much harder, we are able to characterize the structureof the objective function of (32) for a rather wide class of patience time distributions.
Theorem 7.
If the hazard rate functions of the patience time distributions are (strictly) increas-ing (decreasing), then q (cid:63)j ( m ) and i (cid:63)k ( m ) are (strictly) concave (convex) functions of the vector ofmatching rates m , for all j ∈ J and k ∈ K . In this section, we present simulation results on the behavior of the aforementioned two matchingpolicies. We investigate the approximation error between the average queue lengths and the corre-sponding fluid invariant points for each policy. We further compare the total profit of the matchingpolicies with the candidate upper bound given by the optimal value of (32) plotting their ratio.For our simulation, we fix discrete review period l = 0 . and T = 100 . We consider a modelwith two demand and two supply nodes and Poisson arrivals with rates ( λ , λ , µ , µ ) = (1 , , , .The values at each edge are given by ( v , v , v , v ) = (1 , , , . and the holding costs are27 c D , c D , c S , c S ) = (1 , c, , with c > . We consider two different distributions of the patiencetimes (uniform and gamma) with unit means and variance 1/3. Figures 8 and 9 show the sampleaverage queue lengths (denoted by hat) and their corresponding invariant points (denoted by star)as a function of the cost c . Figurer 10–12 show the ratio of average profit of the matching policy tothe optimal value of (32) for the two policies as a function of the cost c and the scaling parameter n , respectively. Below, we summarize the main observations from the simulations. The matching-rate-based policy is more responsive to changes in c : In Figures 8 and 9, wesee that the invariant points are a good approximation of the mean queue lengths for both matchingpolicies and both patience time distributions since the error is relatively small. We note that forsmall holding cost ( c < . ) the second demand queue is high and the other queues are almost zero.However, for ( c ≥ . ), the picture for the matching-rate-based policy changes. The second demandqueue is close to zero and at the same time the first demand queue increases. This intuitively canbe explained as follows. Both policies use the non-diagonal edges until c = 1 . and thus the seconddemand queue is not exhausted. Now, for large cost ( c ≥ . ) the matching-rate-based policy sendsall the supply capacity to the second demand queue to minimize the cost and hence this becomesalmost zero, and the first demand queue increases. Note that in this case, the policy makes matchesat edge (2 , even if its value is v = 0 . On the other hand, for this particular example and for c ∈ [0 . , . the prioritization of the edges remains the same for all c and so the greedy policy doesnot change. The performance of the greedy policy becomes much worse for large holding cost:
Weobserve in Figure 10 that the performance of both policies becomes worse as the cost increases. Thisis a bit surprising since as we observed above, the invariant points (and hence the matching rates)are good approximations of the mean queue lengths and the matching processes. However, whencalculating the profit, even a small error in approximating the queues can hurt a lot because thiserror is multiplied by the cost and hence for relatively large cost, the difference between the profitand the optimal value of (32) can be large. This explains the picture in Figure 10.Moreover, we note that the greedy policy performs slightly better than the matching-rate-basedpolicy for c < . but then it becomes much worse. This sudden change can be explained as follows.The optimal solution of the general SMP and its greedy solution give the same optimal value for c < . . Now motivated by Theorem 5, we expect both policies to have similar behavior in thisregion. On the other hand, this is not the case in the region c ≥ . where the value of (32) andits greedy solution do not coincide. Moreover, the greedy policy uses much less information thanthe matching-rate-based policy, and hence this explains why the performance of the greedy policyis worse in general. The matching-rate-based policy seems to approach the upper bound:
Figures 11 and 12suggest that the matching-rate-based policy approaches optimal as the arrival rates increase. When c = 1 , we expect the two policies to have similar behavior as we saw earlier. Hence both policiesseem to be close to the upper bound for c = 1 (in general for c < . ) for both patience timedistributions. On the other hand, for c = 1 . (in general for c ≥ . ) only the matching-rate-basedpolicy approaches optimal. However, the rate of convergence depends on the distribution and thecost as we would naturally expect since the queue lengths depend on the patience time distributions;see Theorem 6. Further, we need to increase the scaling parameter n to reduce the approximationerror. 28 a) Matching rate-based policy. (b) Greedy policy. Figure 8: Comparison of the simulated sample average queue lengths with the invariant points for n = 100 and uniformly distributed patience times. (a) Matching rate-based policy. (b) Greedy policy. Figure 9: Comparison of the simulated average queue lengths with the invariant points for n = 100 and gamma distributed patience times. (a) Uniform patience times. (b) Gamma patience times. Figure 10: Performance comparison for n = 100 .29 a) c = 1 . (b) c = 1 . . Figure 11: Performance comparison for fixed cost and uniform patience times (a) c = 1 . (b) c = 1 . . Figure 12: Performance comparison for fixed cost and gamma patience times.
In this paper, we proposed and analyzed a model that takes into account three main features ofservice platforms: (i) demand and supply heterogeneity, (ii) the random unfolding of arrivals overtime, (iii) the non-Markovian impatience of demand and supply. These features result in a trade-offbetween making a less good match quickly and waiting for a better match. We proposed a greedyblind policy and two LP-based policies, one blind and one parameter-dependent. We established anasymptotic lower bound on the performance of the greedy blind policy based on the matching valuesand established that the two LP-based policies are asymptotically optimal when the objective is tomaximize the cumulative value of matches made. We used the invariant points of a fluid model toapproximate steady-state mean queue lengths. That allowed us to develop a parameter-dependentpolicy based on the solution to an optimization problem (which is not in general an LP) that weconjecture is asymptotically optimal when holding costs are included. We find that our proposedpolicies are sensitive to the patience time distribution only when holding costs are included.30 cknowledgments
We thank Baris Ata for offering valuable comments that help us to extend our manuscript, andJames Kiselik for his editorial assistance.
References [1] A. Abdulkadiroglu and T. Sönmez. Matching markets: Theory and practice. Acemoglu D,Arello M, Dekel E, eds. , Advances in Economics and Econometrics , 1:3–47, 2013.[2] I. Adan, A. Bušić, J. Mairesse, and G. Weiss. Reversibility and further properties of FCFSinfinite bipartite matching.
Mathematics of Operations Research , 43(2):598–621, 2018.[3] I. Adan and G. Weiss. Exact FCFS matching rates for two infinite multitype sequences.
Op-erations research , 60(2):475–489, 2012.[4] I. Adan and G. Weiss. A skill based parallel service system under FCFS-ALIS-steady state,overloads, and abandonments.
Stochastic Systems , 4(1):250–299, 2014.[5] P. Afeche, R. Caldentey, and V. Gupta. On the optimal design of a bipartite matching queueingsystem, 2019. SSRN 3345302.[6] M. Akbarpour, S. Li, and S. O. Gharan. Thickness and information in dynamic matchingmarkets.
Journal of Political Economy , 128(3):783–815, 2020.[7] N. Arnosti. Greedy matching in bipartite random graphs, 2020. Working Paper.[8] N. Arnosti, R. Johari, and Y. Kanoria. Managing congestion in matching markets, 2020.Forthcoming in
Manufacturing & Service Operations Management .[9] I. Ashlagi, M. Braverman, Y. Kanoria, and P. Shi. Communication requirements and informa-tive signaling in matching markets., 2020. Forthcoming in
Management Science .[10] B. Ata and S. Kumar. Heavy traffic analysis of open processing networks with complete resourcepooling: Asymptotic optimality of discrete review policies.
The Annals of Applied Probability ,15(1A):331–391, 2005.[11] R. Atar, A. Budhiraja, P. Dupuis, and R. Wu. Large deviations for the single server queue andthe reneging paradox, 2019. arXiv:1903.06870.[12] J.-P. Aubin and H. Frankowska.
Set-valued analysis . Birkhäuser, Boston, MA, 1990.[13] M. Baccara, S. Lee, and L. Yariv. Optimal dynamic matching, 2015. SSRN 2641670.[14] S. Banerjee, Y. Kanoria, and P. Qian. State dependent control of closed queueing networkswith application to ride-hailing, 2020. Working Paper.[15] A. Bassamboo, J. M. Harrison, and A. Zeevi. Design and control of a large call center: Asymp-totic analysis of an lp-based method.
Operations Research , 54(3):419–435, 2006.3116] S. Benjaafar and M. Hu. Operations management in the age of the sharing economy: what isold and what is new?
Manufacturing & Service Operations Management , 22(1):93–101, 2020.[17] J. H. Blanchet, M. I. Reiman, V. Shah, and L. M. Wein. Asymptotically optimal control of acentralized dynamic matching market with general utilities, 2020. arXiv:2002.03205.[18] B. Büke and H. Chen. Fluid and diffusion approximations of probabilistic matching systems.
Queueing Systems , 86(1-2):1–33, 2017.[19] R. E. Burkard, B. Klinz, and R. Rudolf. Perspectives of monge properties in optimization.
Discrete Applied Mathematics , 70(2):95–161, 1996.[20] A. Bušić, V. Gupta, and J. Mairesse. Stability of the bipartite matching model.
Advances inApplied Probability , 45(2):351–378, 2013.[21] R. Caldentey, E. H. Kaplan, and G. Weiss. FCFS infinite bipartite matching of servers andcustomers.
Advances in Applied Probability , 41(3):695–730, 2009.[22] Y.-J. Chen, T. Dai, C. G. Korpeoglu, E. Körpeoğlu, O. Sahin, C. S. Tang, and S. Xiao.Innovative online platforms: Research opportunities, 2020. Forthcoming in
Manufacturing &Service Operations Management .[23] U. Derigs, O. Goecke, and R. Schrader. Monge sequences and a simple assignment algorithm.
Discrete applied mathematics , 15(2-3):241–248, 1986.[24] L. DeValve, S. Pekeč, and Y. Wei. Matching supply and demand in a resource constrainedservice network, 2020. Working Paper.[25] A. Diamant and O. Baron. Double-sided matching queues: Priority and impatient customers.
Operations Research Letters , 47(3):219–224, 2019.[26] Y. Ding, S. McCormick, and M. Nagarajan. A fluid model for an overloaded bipartite queueingsystem with heterogeneous matching utility, 2018. Forthcoming in
Operations Research .[27] M. M. Fazel-Zarandi and E. H. Kaplan. Approximating the First-Come, First-Served stochasticmatching model with ohm’s law.
Operations Research , 66(5):1423–1432, 2018.[28] J. Feldman, N. Korula, V. Mirrokni, S. Muthukrishnan, and M. Pál. Online ad assignment withfree disposal. In
International workshop on internet and network economics , pages 374–385.Springer, 2009.[29] D. Gale and L. S. Shapley. College admissions and the stability of marriage.
The AmericanMathematical Monthly , 69(1):9–15, 1962.[30] I. Gurvich and A. Ward. On the dynamic control of matching queues.
Stochastic Systems ,4(2):479–523, 2014.[31] J. M. Harrison. The BIGSTEP approach to flow management in stochastic processing networks.Kelly F, Zachary S, Ziedins I, eds.,
Stochastic Networks: Theory and Applications , 4:147–186,1996. 3232] J. M. Harrison. Correction: Brownian models of open processing networks: canonical repre-sentation of workload.
The Annals of Applied Probability , 16(3):1703–1732, 2006.[33] M. Hu, editor.
Sharing economy: making supply meet demand . Springer Series in Supply ChainManagement, 2019.[34] M. Hu. From the classics to new tunes: A neoclassical view on sharing economy and innovativemarketplaces, 2020. Forthcoming in
Production Oper. Management .[35] M. Hu and Y. Zhou. Dynamic type matching.
Rotman School of Management Working Paper ,(2592622), 2018.[36] R. Johari, V. Kamble, and Y. Kanoria. Matching while learning, 2019. Working Paper.[37] W. Kang and K. Ramanan. Fluid limits of many-server queues with reneging.
The Annals ofApplied Probability , 20(6):2204–2260, 2010.[38] Y. Kanoria and D. Saban. Facilitating the search for partners on matching platforms, 2020.Working Paper.[39] A. Khademi and X. Liu. Asymptotically optimal allocation policies for transplant queueingsystems, 2020. Working Paper.[40] B. Lehmann, D. Lehmann, and N. Nisan. Combinatorial auctions with decreasing marginalutilities.
Games and Economic Behavior , 55(2):270–296, 2006.[41] J. Leshno. Dynamic matching in overloaded waiting lists, 2020. Working Paper.[42] C. Maglaras. Dynamic scheduling in multiclass queueing networks: Stability under discrete-review policies.
Queueing Systems , 31(3-4):171–206, 1999.[43] C. Maglaras. Discrete-review policies for scheduling stochastic networks: Trajectory trackingand fluid-scale asymptotic optimality.
The Annals of Applied Probability , 10(3):897–929, 2000.[44] E. Özkan and A. R. Ward. Dynamic matching for real-time ride sharing.
Stochastic Systems ,10(1):29–70, 2020.[45] G. Pang, R. Talreja, and W. Whitt. Martingale proofs of many-server heavy-traffic limits formarkovian queues.
Probability Surveys , 4:193–267, 2007.[46] C. H. Papadimitriou and K. Steiglitz.
Combinatorial optimization: algorithms and complexity .Courier Corporation, 1998.[47] E. L. Plambeck and A. R. Ward. Optimal control of a high volume assemble-to-order system.
Mathematics of Operations Research , 31(3):453–477, 2006.[48] A. L. Puha and A. R. Ward. Scheduling an overloaded multiclass many-server queue withimpatient customers. In
Operations Research & Management Science in the Age of Analytics ,pages 189–217. INFORMS, 2019.[49] A. E. Roth and M. Sotomayor.
Two-Sided Matching: A Study in Game-Theoretic Modelingand Analysis . Cambridge Univeristy Press, 1990.3350] A. Schrijver.
Theory of linear and integer programming . John Wiley and Sons, New York, 1986.[51] K. Sydsaeter and P. J. Hammond.
Mathematics for economic analysis.
Number HB135 S98.1995.[52] A. Tamir. On totally unimodular matrices.
Networks , 6(4):373–382, 1976.[53] A. R. Ward and P. Glynn. A diffusion approximation for a markovian queue with reneging.
Queueing Systems , 43(1):103–128, 2003.
A Proofs
A.1 Proofs for Section 3.3
Proof of Proposition 1.
We shall show that the stochastic process M jk ( · ) defined in (17) satisfiesthe properties of Definition 1.By definition of the matching process M jk ( · ) is nonanticipating. Next, we prove that the queuelengths are nonnegative. We only show it for the process Q j ( · ) , the proof for I k ( · ) follows in thesame way. We have that for j ∈ J and m ∈ { , . . . , (cid:98) T /l (cid:99)} , Q j ( ml ) = Q j (( m − l ) + D j ( ml ) − D j (( m − l ) − R Dj ( ml ) + R Dj (( m − l ) − K (cid:88) k =1 M mjk = Q j ( ml − ) − K (cid:88) k =1 M mjk . Now, using the inequalities M mjk ≤ (cid:106) y jk Q j ( ml − ) λ j (cid:107) ≤ y jk Q j ( ml − ) λ j and (cid:80) Kk =1 y jk λ j ≤ , we have that Q j ( ml ) ≥ Q j ( ml − ) − K (cid:88) k =1 M mjk ≥ Q j ( ml − ) − Q j ( ml − ) = 0 . The fact that M jk ( · ) is nondecreasing follows by (17) and the fact that Q j ( · ) ≥ and I k ( · ) ≥ . A.2 Proofs for Section 4.1
Proof of Proposition 2.
Let (cid:15) > . Let Y (cid:63),n ( D n ( t ) , S n ( t )) denote an optimal solution of (19) as afunction of the arrival processes at time t ∈ [0 , T ] in the n th system. By the functional law of largenumbers we know that for any j ∈ J , and k ∈ K , sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12) D nj ( t ) n − λ j t (cid:12)(cid:12)(cid:12)(cid:12) → and sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12) S nk ( t ) n − µ k t (cid:12)(cid:12)(cid:12)(cid:12) → , (33)almost surely as n → ∞ . Now, fix a sample path such that (33) holds. By Lemma 1, there exist Y (cid:63),n ( · ) and C > (independent of the arrival processes) such that (cid:107) Y (cid:63),n ( D n ( t ) , S n ( t )) − Y (cid:63),n ( D n ( t ) , S n ( t )) (cid:107) ∞ ≤ C (cid:107) ( D n ( t ) , S n ( t )) − ( D n ( t ) , S n ( t )) (cid:107) ∞ . (34)34y (33), it follows that for any j ∈ J , k ∈ K and large enough n : sup ≤ t ≤ T n (cid:12)(cid:12) D nj ( t ) − nλ j t (cid:12)(cid:12) ≤ (cid:15)CJ K max( v jk ) and sup ≤ t ≤ T n | S nk ( t ) − nµ k t | ≤ (cid:15)CJ K max( v jk ) , which implies sup ≤ t ≤ T n (cid:107) ( D n ( t ) , S n ( t )) − ( n λ t, n µ t ) (cid:107) ∞ ≤ (cid:15)CJ K max( v jk ) . (35)Fix t ∈ [0 , T ] . By the triangle inequality, we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( D nj ( t ) , S nk ( t )) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( D nj ( t ) , S nk ( t )) n − Y (cid:63),njk ( nλ j t, nµ k t ) n (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( nλ j t, nµ k t ) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( D nj ( t ) , S nk ( t )) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ J K max( v jk ) n (cid:107) Y (cid:63),n ( D n ( t ) , S n ( t )) − Y (cid:63),n ( n λ t, n µ t ) (cid:107) ∞ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( nλ j t, nµ k t ) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now, by (34) and (35), we derive (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( D nj ( t ) , S nk ( t )) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CJ K max( v jk ) n (cid:107) ( D n ( t ) , S n ( t )) − ( n λ t, n µ t ) (cid:107) ∞ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( nλ j t, nµ k t ) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J ,k ∈ K v jk (cid:32) Y (cid:63),njk ( nλ j t, nµ k t ) n − y (cid:63)jk t (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It remains to show that the last term in the last inequality is also small for large n . To this end,observe that Y (cid:63),n ( nλ j t, nµ k t ) = arg max (cid:88) j ∈ J ,k ∈ K v jk Y njk n s.t. (cid:88) j ∈ J Y njk n ≤ µ k t, k ∈ K , (cid:88) k ∈ K Y njk n ≤ λ j t, j ∈ J ,Y njk n ∈ R + , for all k ∈ K , j ∈ J . (cid:80) j ∈ J ,k ∈ K v jk Y (cid:63),njk ( nλ j t,nµ k t ) n = (cid:80) j ∈ J ,k ∈ K v jk y (cid:63)jk t which completes the proof. Proof of Theorem 1.
By Proposition 2, we have that (cid:88) j ∈ J ,k ∈ K v jk Y (cid:63),njk ( t ) n → (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk t, (36)almost surely for any t ∈ [0 , T ] as n → ∞ . Applying (36), leads to lim sup n →∞ n (cid:88) j ∈ J ,k ∈ K v jk Y n,(cid:63)jk ( t ) ≤ (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk t, for any t ∈ [0 , T ] . Together the latter and (9) imply lim sup n →∞ n (cid:88) j ∈ J ,k ∈ K v jk M njk ( t ) ≤ (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk t, for all sample paths such that (9) and (36). Proof of Lemma 1.
We begin by observing that the feasible region of (15) is nonempty, closed,and convex. The remainder of the proof follows by using the Lipschitz selection theorem [12,Theorem 9.4.3] and [50, Theorem 10.5] exactly as in [15, Proposition 2].
A.3 Proofs for Section 4.2
Before we present the proof of the main results, we show some preliminary results that are also usedin the proofs in Section 4.3.
Proposition 4.
For any j ∈ J , k ∈ K , and i ≥ , the following inequalities hold almost surely R Dj ( il ) − R Dj (( i − l ) ≤ Q j (( i − l ) + D j ( il ) (cid:88) m = D j (( i − l )+1 { r Djm ≤ l } ,R Sk ( il ) − R Sk (( i − l ) ≤ I k (( i − l ) + S k ( il ) (cid:88) m = S k (( i − l )+1 { r Smk ≤ l } . Proof.
We show only the first inequality, the second one follows in the same way using (6). By (5),we have that R Dj ( il ) − R Dj (( i − l ) = D j ( il ) (cid:88) h = D j (( i − l )+1 (cid:88) s ∈ [( i − l,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) + D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [( i − l,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) . D j ( il ) (cid:88) h = D j (( i − l )+1 (cid:88) s ∈ [( i − l,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) ≤ D j ( il ) (cid:88) h = D j (( i − l )+1 { r Djh ≤ l } . To see this, observe that if (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) = 1 , then dw Djh dt ( s +) = 0 . That is, s − e Djh > r
Djh . This yields l ≥ r Djh by the fact that s ≤ il and e Djh ≥ ( i − l . In other words, { r Djh ≤ l } = 1 .To finish the proof it remains to show that D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [( i − l,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) ≤ Q j (( i − l ) . To this end, the left hand side of the last inequality equals to D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [0 ,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) − R j (( i − l )= D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [0 ,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) − D j (( i − l ) + K (cid:88) k =1 M jk (( i − l ) + Q j (( i − l ) . Now observe that D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [0 ,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) + K (cid:88) k =1 M jk (( i − l )= D j (( i − l ) (cid:88) h =1 (cid:88) s ∈ [0 , ( i − l ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) + (cid:88) s ∈ (( i − l,il ] (cid:40) s ≤ m Djh , dwDjhdt ( s − ) > , dwDjhdt ( s +)=0 (cid:41) + K (cid:88) k =1 M jk (( i − l ) ≤ D j (( i − l ) . To see the last inequality, note that the left-hand side counts the number of customers than arrivein time interval [0 , ( i − l ] that depart (due to matching or reneging) until time ( i − l and renegein time interval (( i − l, il ] . Hence it can not exceed the number of customers than arrive untiltime ( i − l which is D j (( i − l ) .Moreover, we need the following lemma, which is taken from [10], and adapted to the belowform found in [47]. 37 emma 3. (Ata and Kumar) For any finite constant α > , there exists β > such that P (cid:18) max i ∈{ ,..., (cid:98) T/l n (cid:99)} max j ∈ J (cid:12)(cid:12) D nj ( il n ) − D nj (( i − l n ) − nλ j l n (cid:12)(cid:12) < αn / (cid:19) ≥ − βn − / and P (cid:18) max i ∈{ ,..., (cid:98) T/l n (cid:99)} max k ∈ K | S nk ( il n ) − S nk (( i − l n ) − nµ k l n | < αn / (cid:19) ≥ − βn − / . A consequence of Lemma 3 is that with high probability the following hold for each j ∈ J , k ∈ K ,and i ∈ { , . . . , (cid:98) T /l n (cid:99)} , nλ j l n − αn / < D nj ( il n ) − D nj (( i − l n ) < nλ j l n + αn / (37)and nµ k l n − αn / < S nk ( il n ) − S nk (( i − l n ) < nµ k l n + αn / . (38) A.3.1 Proof of Theorem 2
Proof of Theorem 2.
Let (cid:15) > . Fix n large enough so that n / > j,k ( y jk ) l/(cid:15) . Denote by Ω n the events such that (37) and (38) hold and by Ω n the events such both inequalities in Proposition 4hold. Let ω ∈ Ω n ∩ Ω n and α := min j,k ( λ j ,µ k ) l T min j,k ( y jk ) (cid:15) . We first derive a lower bound for the number ofreneging customers and workers at any discrete-review period. By Proposition 4, the followingupper bound of the reneging customers in the discrete-review period (( i − l n , il n ) holds almostsurely: for any j ∈ J , R D,nj ( il n ) − R D,nj (( i − l n ) ≤ Q nj (( i − l n ) + D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n } , (39)where r Djm denotes the patience time of the m th customer arriving at node j ∈ J . Further, for thereneging workers in the discrete-review period (( i − l n , il n ) , we have that for any k ∈ K , R S,nk ( il n ) − R S,nk (( i − l n ) ≤ I nk (( i − l n ) + S nk ( il n ) (cid:88) m = S nk (( i − l n )+1 { r Smk ≤ l n } , (40)where r Skm denotes the patience time of the m th worker arriving at node k ∈ K . In the sequel,we show that the second term of the right-hand side of (39) and (40) converge to zero under thehigh-volume setting. Denote by Ω n the events such that (37) holds and by (Ω n ) c its complement.For simplicity, let Γ n = nl n (cid:80) D nj ( il n ) m = D nj (( i − l n )+1 { r Djm ≤ l n } . For any δ > , we have that P (Γ n ≥ δ ) ≤ P ( { Γ n ≥ δ } ∩ Ω n ) + P ( { Γ n ≥ δ } ∩ (Ω n ) c ) ≤ P (Γ n ≥ δ | Ω n ) + P ((Ω n ) c ) ≤ E [Γ n | Ω n ] δ + βn − / ≤ E (cid:2) Γ n Ω n (cid:3) δ (1 − βn − / ) + βn − / , (41)38here the third inequality follows by the conditional Markov’s inequality and Lemma 3. To completethe proof that Γ n converges to zero in probability, we show that E (cid:2) Γ n Ω n (cid:3) goes to zero. To thisend, ≤ E nl n D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n } Ω n ≤ E (cid:104)(cid:16) nλ j l n + αn / (cid:17) { r Dj ≤ l n } (cid:105) = (cid:0) nλ j l n + αn / (cid:1) nl n P (cid:0) r Dj ≤ l n (cid:1) = (cid:16) λ j + αl (cid:17) P (cid:0) r Dj ≤ l n (cid:1) . It follows that ≤ E nl n D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n } Ω n → , as n → ∞ . Now, by (41), we have that lim n →∞ P (Γ n ≥ δ ) = 0 , for any δ > . By the lastconvergence and for large enough n , we have that nl n D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n } < λ j T max j,k ( y jk ) (cid:15). (42)In a similar way using (38), we obtain for large enough n , nl n S nk ( il n ) (cid:88) m = S nk (( i − l n )+1 { r Skm ≤ l n } < µ k T max j,k ( y jk ) (cid:15). (43)Define Ω n the events such that (42) and (43) hold and note that lim n →∞ P (Ω n ∩ Ω n ∩ Ω n ) = 1 . Inthe sequel, we take ω ∈ Ω n ∩ Ω n ∩ Ω n .Now, we move to the second step of proof of Theorem 2. We derive a desirable lower bound onthe number of matches at any discrete-review period. To this end, by (37), (39), and (42), we havethat for i ∈ { , . . . , (cid:98) T /l (cid:99)} , Q nj ( il n − ) = Q nj (( i − l n ) + D nj ( il n ) − D nj (( i − l ) − R n,Dj ( il n ) + R n,Dj (( i − l n ) ≥ D nj ( il n ) − D nj (( i − l ) − D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n }≥ D nj ( il n ) − D nj (( i − l ) − λ j T max j,k ( y jk ) nl n (cid:15) ≥ nλ j l n − αn / − λ j T max j,k ( y jk ) nl n (cid:15) = nλ j l n (cid:18) − αlλ j − (cid:15) T max j,k ( y jk ) (cid:19) ≥ nλ j l n (cid:18) − αl min j,k ( λ j , µ k ) − (cid:15) T max j,k ( y jk ) (cid:19) , I nk ( il n − ) = I nk (( i − l n ) + S nk ( il n ) − S nk (( i − l n ) − R n,Sk ( il n ) + R n,Sk (( i − l n ) ≥ nµ k l n (cid:18) − αl min j,k ( λ j , µ k ) − (cid:15) T max j,k ( y jk ) (cid:19) . Now, the inequality (cid:98) x (cid:99) ≥ x − and the bounds for the quantities Q j ( il n − ) n , I k ( il n − ) n yield M nijk nl n = 1 nl n (cid:22) y jk min (cid:18) Q nj ( il n − ) λ j , I nk ( il n − ) µ k (cid:19)(cid:23) ≥ y jk (cid:18) − αl min j,k ( λ j , µ k ) − (cid:15) T max j,k ( y jk ) (cid:19) − nl n ≥ y jk − (cid:15) T − (cid:15) T − (cid:15) T = y jk − (cid:15) T , for large enough n . The second inequality from the definition of α and the fact that nl n = n / l → , n → ∞ . That is, for any t ≥ and any feasible point y , M njk ( t ) n = 1 n (cid:98) t/l n (cid:99) (cid:88) i =1 M nijk ≥ (cid:98) t/l n (cid:99) l n ( y jk − (cid:15) T ) ≥ (cid:98) t/l n (cid:99) l n y jk − (cid:15) ≥ y jk t − y jk l n − (cid:15) ≥ y jk t − (cid:15), where the last inequality follows by n / > j,k ( y jk ) l/(cid:15) . Applying the last inequality for asolution of LP (15), we have that M njk ( t ) n − y (cid:63)jk t ≥ − (cid:15), (44)for large enough n .Having derived a lower bound for the number of matches, we now move to the last step of theproof which is to obtain a corresponding upper bound. To this end, by Theorem 1, we have thatfor large enough n , (cid:88) j ∈ J ,k ∈ K v jk (cid:32) M njk ( t ) n − y (cid:63)jk t (cid:33) ≤ (cid:15). By (44) and the last inequality, we obtain for large enough n , M njk ( t ) n − y (cid:63)jk t ≤ ( J K −
1) max ij ( y (cid:63)jk ) + 1 v jk (cid:15), and so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M njk ( t ) n − y (cid:63)jk t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max (cid:32) ( J K −
1) max ij ( y (cid:63)jk ) + 1 v jk , (cid:33) (cid:15), for any ≤ t ≤ T and (cid:15) is arbitrary small independent of time t . This concludes the proof.40 .3.2 Proof of Theorems 3 and 4 Before we move to the main part of the proof of Theorem 3 and 4, we show some helpful propertiesof the value of static matching problem (15).Let ¯ v := max jk ( v jk ) . For two vectors x ∈ R J and b ∈ R K , let y (cid:63)jk ( x , b ) be an optimal solutionof (15) if we replace λ and µ by x and b , respectively. Define the value of the static matchingproblem F ( x , b ) := (cid:88) j,k v jk y (cid:63)jk ( x , b ) . Lemma 4.
The value of (15) is nondecreasing in each of its arguments and the following inequalitieshold, for any α ≥ , F ( x + α e j , b ) − F ( x , b ) ∈ [0 , ¯ vJ KCα ] , ∀ j ∈ J ,F ( x , b + α e k ) − F ( λ , µ ) ∈ [0 , ¯ vJ KCα ] , ∀ k ∈ K , where the constant C is given in Lemma 1.Proof. Clearly, F ( · , · ) is nondecreasing in each of its arguments by the definition if the static match-ing problem. We shall show that the rate of change is bounded. To this end, observe that byLemma 1 there exists a Lipschitz continuous solution of (15) such that (cid:107) y (cid:63) ( x + α e j , b ) − y (cid:63) ( x , b ) (cid:107) ∞ ≤ C (cid:107) ( x + α e j , b ) − ( x , b ) (cid:107) ∞ ≤ Cα.
The latter leads to (cid:12)(cid:12) y (cid:63)jk ( x + α e j , b ) − y (cid:63)jk ( x , b ) (cid:12)(cid:12) ≤ Cα, for any j ∈ J and k ∈ K . Now, using the last inequality, we have that ≤ F ( x + α e j , b ) − F ( x , b ) = (cid:88) j,k v jk ( y (cid:63)jk ( x + α e j , b ) − y (cid:63)jk ( x , b )) ≤ (cid:88) j,k v jk (cid:12)(cid:12) y (cid:63)jk ( x + α e j , b ) − y (cid:63)jk ( x , b ) (cid:12)(cid:12) ≤ ¯ vJ KCα. The second inequality follows analogously. Note that both inequalities do not depend on the par-ticular solution y (cid:63) ( · , · ) , that is Lipschitz continuous, as any solution achieves the same value. Lemma 5.
For any (cid:15) > , and any A ≥ x − (cid:15) e and B ≥ b − (cid:15) e , F ( A , B ) ≥ F ( x , b ) − C (cid:15), where C := ( J + K ) J KC ¯ v .Proof. First, by a telescoping sum (and assuming an empty sum evaluates to zero), we observe that F ( x , b ) − F ( x − (cid:15) e , b − (cid:15) e ) = J (cid:88) j =1 (cid:104) F ( x − (cid:15) j − (cid:88) s =1 e s , b ) − F ( x − (cid:15) j (cid:88) s =1 e s , b ) (cid:105) + K (cid:88) k =1 (cid:104) F ( x − (cid:15) e , b − (cid:15) k − (cid:88) t =1 e t ) − F ( x − (cid:15) e , b − (cid:15) k (cid:88) t =1 e t ) (cid:105) ≤ ( J + K ) J KC ¯ v(cid:15), F ( · , · ) is monotone nondecreasing,we have F ( A , B ) ≥ F ( x − (cid:15) e , b − (cid:15) e ) ≥ F ( x , b ) − ( J + K ) J KC ¯ v(cid:15), completing the proof. Lemma 6.
The constraint matrix of (13) is totally unimodular, which implies that an extreme pointsolution is integer valued if the right hand side constraints of (13) are integer valued.Proof.
Totally unimodular matrices with elements a il are characterized by [52] as those where everysubset of rows R can be partitioned into two subsets, R ∪ R = R , such that (cid:80) i ∈ R a il − (cid:80) i ∈ L a il ∈{− , , } for all columns l . The constraint matrix of (13) has a row for each k ∈ K and each j ∈ J and a column for every variable y jk . For a given j ∈ J representing a row and j (cid:48) ∈ J , k (cid:48) ∈ K representing a column, the corresponding element of the constraint matrix is given by a ( j );( j (cid:48) k (cid:48) ) = (cid:40) , j (cid:48) = j , j (cid:48) (cid:54) = j , representing that only those y j (cid:48) k (cid:48) with j (cid:48) = j are included in the sum for the constraint correspondingto j . Similarly, for a given k ∈ K and j (cid:48) ∈ J , k (cid:48) ∈ K , the corresponding constraint matrix element is a ( k );( j (cid:48) k (cid:48) ) = (cid:40) , k (cid:48) = k , k (cid:48) (cid:54) = k . Any subset of the rows of the constraint matrix corresponds to the union of some subsets of J and K , i.e. J (cid:48) ∪ K (cid:48) where J (cid:48) ⊆ J and K (cid:48) ⊆ K . Therefore, given a subset of rows J (cid:48) ∪ K (cid:48) , define therequired partition as R = J (cid:48) and R = K (cid:48) . Then for a given variable y j (cid:48) k (cid:48) , (cid:88) j ∈ J (cid:48) a ( j );( j (cid:48) k (cid:48) ) − (cid:88) k ∈ K (cid:48) a ( k );( j (cid:48) k (cid:48) ) = − , j (cid:48) / ∈ J (cid:48) and k (cid:48) ∈ K (cid:48) , , ( j (cid:48) ∈ K (cid:48) and k (cid:48) ∈ K (cid:48) ) or ( j (cid:48) / ∈ J (cid:48) and k / ∈ K (cid:48) ) , , j (cid:48) ∈ J (cid:48) and k (cid:48) / ∈ K (cid:48) , This completes the proof that the constraint matrix of (13) is totally unimodular. Then by [52], ifthe right hand side constraints of (13) are integer valued, any extreme point solution of (13) is alsointeger valued. A similar unimodularity property is proven in [24].
Proof of Theorem 3.
Let (cid:15) > and α = l(cid:15)C T , where C is given in Lemma 5. Adapting the proof ofTheorem 2, we choose large enough n , such that nl n D nj ( il n ) (cid:88) m = D nj (( i − l n )+1 { r Djm ≤ l n } < (cid:15)C T (45)and nl n S nk ( il n ) (cid:88) m = S nk (( i − l n )+1 { r Skm ≤ l n } < (cid:15)C T . (46)42efine Ω n the events such that (45) and (46) hold. In the sequel, we take ω ∈ Ω n ∩ Ω n ∩ Ω n .The next step is to show the following inequality for large enough n , V M b,n ( T ) ≥ V Y (cid:63),n ( T ) − (cid:15). (47)Follow similar steps as in proof of Theorem 2, we obtain for i ∈ { , . . . , (cid:98) T /l (cid:99)} , Q nj ( il n − ) nl n ≥ λ j − (cid:15)C T and I nk ( il n − ) nl n ≥ µ k − (cid:15)C T .
Applying now Lemma 5, we have that for i ∈ { , . . . , (cid:98) T /l n (cid:99)} , F (cid:18) Q n ( il n − ) nl n , I n ( il n − ) nl n (cid:19) ≥ F ( λ , µ ) − (cid:15)T . By the properties of linear programming, we have that F ( · , · ) is homogenous of degree one. Thatis, F (cid:18) Q n ( il n − ) nl n , I n ( il n − ) nl n (cid:19) = 1 nl n F ( Q n ( il n − ) , I n ( il n − )) . Now observe that the optimization problem (13), which decides the matches for the blind, LP-basedpolicy, and the SMP (15) have the same objective function and the same constraint matrix, which istotally unimodular by in Lemma 6. This implies the well known fact that these programs have thesame optimal value if they have the same right hand side constraints, and if these constraints areinteger valued (i.e., the objective value of the integer program (13) is equal to the objective valueof its LP relaxation (15); see for example [46, Theorem 13.2]. Combining all the above together, wederive (cid:88) j ∈ J ,k ∈ K v jk M b,njk ( T ) n = (cid:98) T/l n (cid:99) (cid:88) i =1 F ( Q n ( il n − ) , I n ( il n − )) n ≥ (cid:98) T/l n (cid:99) (cid:88) i =1 F ( Q n ( il n − ) , I n ( il n − )) n ≥ l n (cid:98) T /l n (cid:99) ( F ( λ , µ ) − (cid:15)T ) ≥ T F ( λ , µ ) − F ( λ , µ ) l n . By definition of F ( · , · ) and for n such that n / > F ( λ , µ ) l/(cid:15) , we have that V M b,n ( T ) n = (cid:88) j ∈ J ,k ∈ K v jk M b,njk ( T ) n ≥ (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk T − (cid:15). Now, by Proposition 2, we have that (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk T ≥ V Y (cid:63),n ( T ) n − (cid:15), M b ( T ) isclearly an admissible policy. The desirable inequality follows by Theorem 1 and Proposition 2. Theproof is completed by applied the squeeze theorem. Proof of Theorem 4.
Without loss of generality assume that T = 1 otherwise fix a T > andappropriately scale (15) by T . First, we note that the set of optimal solutions S is a closed set.To see this observe that if it is not, then we can always increase one of the components of y (cid:63) andachieve a better optimal value.In the sequel, we proceed to the proof of the result by contradiction. Fix a sample path such thatTheorem 3 holds and assume that d (cid:16) M b,n (1) n , S (cid:17) (cid:57) . First, observe that for all j ∈ J and k ∈ K ,we have that M b,n (1) n ≤ D n (1) n and D n (1) n → λ almost surely. In other words, M b,n (1) n is a boundedsequence in R J × K . Now, by Bolzano–Weierstrass theorem there exists a convergent subsequence,which with abuse of notation we denote by M b,n (1) n , such that M b,n (1) n → x , with x / ∈ S . Notethat we can assume the last without loss of generality since if all the subsequential limits lies in theclose set S , then the distance of the whole sequence from S must convergence to zero which yieldsa contradiction.Now, consider M b,n (1) n along the previous subsequence. First, note that by the admissibility ofthe blind, LP-based policy any limit should be feasible for LP (15). To see this, observe that for all j ∈ J and k ∈ K the following hold almost surely by the nonnegativity of the queue lengths (cid:88) k ∈ K M b,njk (1) n ≤ D j (1) n and (cid:88) j ∈ J M b,njk (1) n ≤ S k (1) n , and by law of large numbers lim n →∞ D j (1) n = λ j , lim n →∞ S k (1) n = µ k . Now, we have that (cid:88) j ∈ J ,k ∈ K v jk M b,njk (1) n → (cid:88) j ∈ J ,k ∈ K v jk x jk , and by Theorem 3 (cid:80) j ∈ J ,k ∈ K v jk x jk = (cid:80) j ∈ J ,k ∈ K v jk y (cid:63)jk . In other words, the vectors x and y (cid:63) achievethe same optimal value of (15) and so x is a solution of (15). On the other hand, x / ∈ S whichyields a contradiction. A.4 Proofs for Section 4.3
Recall that we can define the greedy solution to (15) recursively for ( j, k ) ∈ P h as y gjk = min λ j − (cid:88) k (cid:48) (cid:54) = k :( j,k (cid:48) ) ∈C h − y gjk (cid:48) , µ k − (cid:88) j (cid:48) (cid:54) = j :( j (cid:48) ,k ) ∈C h − y gj (cid:48) k . (48)Note that y gjk is implicitly a function of the arrival rates λ , µ , but we suppress this dependence fornotational brevity. Then, define the value of the greedy solution as a function of the arrival rates, λ and µ , as follows G ( λ , µ ) := (cid:88) j,k v jk y gjk . roof of Proposition 3. By construction of the dual solution ( α , β ) in Algorithm 1, we have thatif α j > then (cid:88) k y gjk = λ j , and if β k > then (cid:88) j y gjk = µ k . (49)Furthermore by Lemma 10, if y gjk > then v jk ≤ α j + β k ≤ max( v jk , v j ( k ) + v k ( j )) ≤ max (cid:18) , v j ( k ) + v k ( j ) v jk (cid:19) v jk , where the last inequality leads to v jk ≥ (cid:18) max (cid:18) , v j ( k ) + v k ( j ) v jk (cid:19)(cid:19) − ( α j + β k )= min (cid:18) , v jk v j ( k ) + v k ( j ) (cid:19) ( α j + β k ) ≥ min (1 , γ ) ( α j + β k ) , (50)if v j ( k ) > , v k ( j ) > and α j + β k = v jk if v j ( k ) = 0 and v k ( j ) = 0 . By (49) and (50), we have that (cid:88) j ∈ J ,k ∈ K v jk y gjk ≥ (cid:88) j ∈ J ,k ∈ K min (1 , γ ) ( α j + β k ) y gjk , = min (1 , γ ) (cid:88) j ∈ J α gj (cid:88) k ∈ K y gjk + (cid:88) k ∈ K β gk (cid:88) j y gjk , = min (1 , γ ) (cid:88) j ∈ J λ j α j + (cid:88) k ∈ K µ k β k ≥ min (1 , γ ) (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jk , where the last inequality follows by weak duality and by Lemma 9 which states that the dual solutionconstructed by Algorithm 1 is feasible to (54). Proof of Theorem 5.
Let (cid:15) > . Consider a sample path such Lemma 3 and Proposition 4 hold andlet α = l(cid:15) ¯ v ( J + K ) T with ¯ v := max jk ( v jk ) . Adapting the proof of Theorem 3 and applying Lemma 8,we have that for i ∈ { , . . . , (cid:98) T /l n (cid:99)} , G (cid:18) Q n ( il n − ) nl n , I n ( il n − ) nl n (cid:19) ≥ G ( λ , µ ) − (cid:15)T . The definition of the greedy matching policy (11) implies that G ( · , · ) is homogenous of degree one.That is, G (cid:18) Q n ( il n − ) nl n , I n ( il n − ) nl n (cid:19) = 1 nl n G ( Q n ( il n − ) , I n ( il n − )) . (cid:88) j ∈ J ,k ∈ K v jk M g,njk ( T ) n = (cid:98) T/l n (cid:99) (cid:88) i =1 G ( Q n ( il n − ) , I n ( il n − )) n ≥ l n (cid:98) T /l n (cid:99) ( G ( λ , µ ) − (cid:15)T ) ≥ T F ( λ , µ ) − G ( λ , µ ) l n . By definition of G ( · , · ) and taking in the last equation large enough n such that n / > G ( λ , µ ) l/(cid:15) ,we have that (cid:88) j ∈ J ,k ∈ K v jk M g,njk ( T ) n ≥ (cid:88) j ∈ J ,k ∈ K v jk y gjk T − (cid:15). Applying Proposition 3, yields (cid:88) j ∈ J ,k ∈ K v jk M g,njk ( T ) n ≥ (cid:88) j ∈ J ,k ∈ K v jk y (cid:63)jkz T − (cid:15). The proof is completed by combining the last inequality and Propositions 2.
A.5 Proofs for Section 5
Proof of Theorem 6.
We first show the forward direction. Let ( Q ( t ) , I ( t ) , η D ( t ) , η S ( t )) = ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) ∈ I λ , µ for all t ≥ . We shall show that ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) satisfies (28)–(31) with m ∈ M . Relations (28)and (29) follow in the same way as in the proof of [48, Theorem 1]. Below we show (30) and (31)in the same way. Adapting [48, Theorem 1], let χ Dj ∈ [0 , H Dj ] be the unique (because the patiencetime distributions are invertible) solution to q (cid:63)j = (cid:90) [0 ,χ Dj ] η D,(cid:63)j ( dx ) . (51)If χ Dj = H Dj , then by (28) and (51), we obtain q (cid:63)j = λ j (cid:90) H Dj (1 − G Dj ( u )) du = λ j θ Dj . Further, by (24), (28), and the definition of the hazard rate, we have that R Dj ( t ) = (cid:90) t (cid:90) χ Dj h Dj ( x ) η D,(cid:63)j ( dx ) du = λ j (cid:90) t (cid:90) H Dj h Dj ( x )(1 − G Dj ( x )) dxdu = λ j t. By (26), we obtain for any t ≥ , (cid:88) k ∈ K M jk ( t ) = λ j t − R Dj ( t ) = λ j t − λ j t = 0 , (cid:80) k ∈ K m Djk = 0 for all j ∈ J . Hence, (30) is satisfied with (cid:80) k ∈ K m Djk = 0 .If χ Dj < H Dj , then by (28) and (51), we have that q (cid:63)j = λ j θ Dj G De,j ( χ Dj ) , or equivalently χ Dj = ( G De,j ) − (cid:32) θ Dj λ j q (cid:63)j (cid:33) . In this case, (24) and (28) lead to R Dj ( t ) = λ j (cid:90) t (cid:90) χ Dj h Dj ( x )(1 − G Dj ( x )) dxdu = λ j G Dj (cid:32) ( G De,j ) − (cid:32) θ Dj λ j q (cid:63)j (cid:33)(cid:33) t. By last equation and (26), we obtain for any t ≥ , (cid:88) k ∈ K M jk ( t ) = λ j t − R Dj ( t ) = λ j (cid:32) − G Dj (cid:32) ( G De,j ) − (cid:32) θ Dj λ j q (cid:63)j (cid:33)(cid:33)(cid:33) t. (52)Set (cid:80) k ∈ K m jk = M jk ( t ) t and observe that (cid:80) k ∈ K m jk ∈ (0 , λ j ] because ≤ G Dj ( · ) < . By (52), weobtain q (cid:63)j = λ j θ Dj G De,j (cid:16) ( G Dj ) − (cid:16) − (cid:80) k ∈ K m jk λ j (cid:17)(cid:17) , and so (30) is satisfied with (cid:80) k ∈ K m Djk ∈ (0 , λ j ] .We now move to the proof of the converse direction. Let m ∈ M and ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) ∈ Y thatsatisfies (28)–(31). For all t ≥ let ( Q ( t ) , I ( t ) , η D ( t ) , η S ( t )) = ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) . We shall showthat ( q (cid:63) , i (cid:63) , η D,(cid:63) , η S,(cid:63) ) is a fluid model solution. We do it for ( q (cid:63) , η D,(cid:63) ) and the proof is similar for ( i (cid:63) , η S,(cid:63) ) . By (28), we have that (cid:90) [0 ,H Dj ] h Dj ( x ) η D,(cid:63)j ( dx ) = λ j (cid:90) H Dj h Dj ( x )(1 − G Dj ( x )) dx = λ j < ∞ , and so (25) is satisfied. Now define M jk ( t ) := m jk t for all t ≥ , j ∈ J , and k ∈ K . Clearly, M jk ( · ) are nondecreasing and absolutely continuous for all j ∈ J and k ∈ K . Further, by (24), (28), and(30), we obtain R Dj ( t ) = (cid:40) λ j t, if (cid:80) k ∈ K m jk = 0 , ( λ j − (cid:80) k ∈ K m jk ) t, if (cid:80) k ∈ K m jk ∈ (0 , λ j ] , which leads to q (cid:63)j = q (cid:63)j + λ j t − R Dj ( t ) − (cid:88) k ∈ K m jk t. In other words, q (cid:63)j satisfies (26). Last, continuity, (23), and the two equations in Definition 3 followin the same way as in the proof of [48, Theorem 1]. Proof of Lemma 2.
Let y (cid:48) jk , j ∈ J , k ∈ K denote the extreme point solution. By definition, anextreme point solution cannot be written as a convex combination of any two distinct feasiblesolutions. 47 roperty 1. Assume the induced graph has a cycle, and renumber the nodes involved inthe cycle so that it can be represented by the set of edges C = { (1 , , (1 , , . . . , ( i, i ) , ( i, i +1) , . . . , ( m, m ) , ( m, } , where the length of the cycle is assumed to be m for some m withoutloss of generality because the underlying graph is bipartite. Then we will show that solution y (cid:48) jk , j ∈ J , k ∈ K can be written as a convex combination of two distinct feasible solutions, contra-dicting the fact that it is an extreme point.First, for edges in the cycle, let y ii = y (cid:48) ii + (cid:15), i = 1 , . . . , m and y ii +1 = y (cid:48) ii +1 − (cid:15), i = 1 , . . . , m − and y m = y (cid:48) m − (cid:15) for (cid:15) > , while for edges not in the cycle, i.e., ( j, k ) / ∈ C , let y jk = y (cid:48) jk . For (cid:15) ≤ min ( j,k ) ∈ C { y (cid:48) jk } , all variables remain nonnegative, and we next argue that this solution remainsfeasible for the other constraints as well. Consider a demand node i = 1 , . . . , m − involved in thecycle C . We have y ii + y ii +1 = y (cid:48) ii + (cid:15) + y ii +1 − (cid:15) = y (cid:48) ii + y ii +1 , so that the total contribution of edges ( i, i ) and ( i, i + 1) to the flow entering node i is unchanged, hence the associated demand constraintfor i is satisfied (since y (cid:48) was assumed feasible). Similarly, for demand node m in the cycle C , wehave y mm + y m = y (cid:48) mm + (cid:15) + y m − (cid:15) = y (cid:48) mm + y m so that the associated demand constraint for m is satisfied. An identical argument demonstrates that the supply constraints are also satisfied.Next, using the same (cid:15) , for edges in the cycle let y ii = y (cid:48) ii − (cid:15), i = 1 , . . . , m and y ii +1 = y (cid:48) ii +1 + (cid:15), i = 1 , . . . , m − and y m = y (cid:48) m + (cid:15) for (cid:15) > , while for edges not in the cycle, i.e., ( j, k ) / ∈ C , let y jk = y (cid:48) jk . Again, this solution is feasible, and we have y (cid:48) = 1 / y + 1 / y , hence y (cid:48) is not an extreme point of (15). Property 2.
Assume that in the induced graph, a given tree has more than one node witha slack constraint. Then pick any two of these slack nodes, consider the path between them,and renumber the nodes involved in the path so that it can be represented by the set of edges P = { (1 , , (1 , , . . . , ( i, i ) , ( i, i + 1) , . . . , ( m, m ) } . (This representation implicitly assumes that thepath begins on a demand node and ends on a supply node, but the following argument appliesfor the analogous path representation for any combination of begining/ending node classification).By the assumption that both endpoints of the path have slack constraints, we have λ i > (cid:80) k y (cid:48) ik and µ m > (cid:80) j y (cid:48) jm . Then we will show that solution y (cid:48) jk , j ∈ J , k ∈ K can be written as a convexcombination of two distinct feasible solutions, contradicting the fact that it is an extreme point.First, for edges in the path, let y ii = y (cid:48) ii + (cid:15), i = 1 , . . . , m and y ii +1 = y (cid:48) ii +1 − (cid:15), i = 1 , . . . , m − for (cid:15) > , while for edges not in the path, i.e., ( j, k ) / ∈ P , let y jk = y (cid:48) jk . For (cid:15) ≤ min min ( j,k ) ∈ P { y (cid:48) jk } , λ i − (cid:88) k y (cid:48) ik , µ m − (cid:88) j y (cid:48) jm , all variables remain nonnegative, we do not violate the constraints on the endpoint nodes of thepaths, and feasibility of this solution for the remaining constraints along the path follows from anidentical argument to that posed in the proof of Property 1.Next, using the same (cid:15) , for edges in the path let y ii = y (cid:48) ii − (cid:15), i = 1 , . . . , m and y ii +1 = y (cid:48) ii +1 + (cid:15), i =1 , . . . , m − , while for edges not in the path, i.e., ( j, k ) / ∈ P , let y jk = y (cid:48) jk . Again, this solution isfeasible, and we have y (cid:48) = 1 / y + 1 / y , hence y (cid:48) is not an extreme point of (15). Property 3.
This follows as a direct consequence of Property 2, since if it were false then therewould be more than one node in the same tree with a slack constraint.
A.6 Proofs for Section 6
Proof of Theorem 7.
We show the result when the hazard rate functions are increasing. The48ase of decreasing hazard rate functions shares the same machinery.Assume that the hazard rate functions are increasing. By [48, Lemma 1], we know that q (cid:63)j ( m ) and i (cid:63)k ( m ) are concave functions of m Dj := (cid:80) j ∈ J m jk ≤ λ j and m Sk := (cid:80) k ∈ K m jk ≤ µ k , respectively.By [51, Theorem 17.7], we have that for any x j = (cid:80) j ∈ J x jk ≤ λ j , q (cid:63)j ( m Dj ) − q (cid:63)j ( x j ) ≤ dq (cid:63)j ( x j ) dx j ( m Dj − x j ) . (53)Furthermore, taking into account that ddx j (1 − x j λ j ) = ∂dx jk (1 − x j λ j ) = − λ j for all k ∈ K , we have that dq (cid:63)j ( x j ) dx j = − h Dj (cid:16) ( G Dj ) − (cid:16) − x j λ j (cid:17)(cid:17) = ∂q (cid:63)j ( x ) ∂x jk Now, by (53), we obtain q (cid:63)j ( m ) − q (cid:63)j ( x ) = q (cid:63)j ( m Dj ) − q (cid:63)j ( x j ) ≤ dq (cid:63)j ( x j ) dx j ( m Dj − x j ) ≤ (cid:88) k ∈ K ∂q (cid:63)j ( x ) ∂x jk ( m Djk − x jk ) , and hence q (cid:63)j ( m ) is a concave function by [51, Theorem 17.7]. The proof for i (cid:63)k ( m ) follows the samemachinery. B Properties of a greedy solution
Let e j denote the j th unit vector, and e the vector of all ones. With a slight abuse of notation, weimplicitly assume these vectors to be the appropriate length given the context. Let ¯ v = max j,k ( v jk ) denote the maximum value. In order to prove a lower bound on G ( · , · ) , we first show that it ismonotone, and its rate of change is bounded by ¯ v . Lemma 7.
The greedy value is nondecreasing in each of its arguments and has rate of changebounded by ¯ v , i.e., for any α ≥ , G ( λ + α e j , µ ) − G ( λ , µ ) ∈ [0 , α ¯ v ] , ∀ j ∈ J ,G ( λ , µ + α e k ) − G ( λ , µ ) ∈ [0 , α ¯ v ] , ∀ k ∈ K . Proof.
We will prove the result by backwards induction on the sets P h , for which we first need todefine the function G ( · , · ) with a backwards recursion. Let m denote the maximum h for which P h is nonempty, and for any vectors λ , µ , define the following functions recursively: G m +1 ( λ , µ ) = 0 ,G h ( λ , µ ) = (cid:88) ( j,k ) ∈P h v jk min( λ j , µ k ) + G h +1 (Λ h ( λ , µ ) , M h ( µ , λ )) , ∀ h ∈ { . . . m } , where Λ h is a vector valued function the same length as λ , which updates the remaining demandas follows: Λ hj ( λ , µ ) = (cid:40) ( λ j − µ k ) + , for k s.t. ( j, k ) ∈ P h λ j , (cid:64) k s.t. ( j, k ) ∈ P h M h M hk ( λ , µ ) = (cid:40) ( µ k − λ j ) + , for j s.t. ( j, k ) ∈ P h µ k , (cid:64) j s.t. ( j, k ) ∈ P h , where we note that both are well defined since, by the definition of P h , for a given j , there can beat most one k such that ( j, k ) ∈ P h (and similarly for k ). Note that, with these functions defined,we can write G ( λ , µ ) = G ( λ , µ ) . We also recursively define a value for each node for each h as follows: p m +1 j = p m +1 k = 0 , ∀ j, kp hj = (cid:40) v jk , for k s.t. ( j, k ) ∈ P h p h +1 j , (cid:64) k s.t. ( j, k ) ∈ P h , ∀ j, h ∈ { . . . m } ,p hk = (cid:40) v jk , for j s.t. ( j, k ) ∈ P h p h +1 j , (cid:64) j s.t. ( j, k ) ∈ P h , ∀ k, h ∈ { . . . m } . Note that, by the definition of the priority classes P h , we have p hj ≥ p h +1 j for all j and h , andalso p hj ≥ p h +1 k for all ( j, k ) ∈ P h . Now we are ready to state our induction hypothesis: for each h ∈ { . . . m } and any α ≥ we have that for any j ∈ J and k ∈ K , G h ( λ + α e j , µ ) − G h ( λ , µ ) ∈ [0 , αp hj ] , and G h ( λ , µ + α e k ) − G h ( λ , µ ) ∈ [0 , αp hk ] . Clearly this holds for the constant function G m +1 , for which each difference is zero. Then assumethe hypothesis holds for h + 1 and we verify the claim for h . We will fix a given j , and show thecorresponding result (and a symmetric argument establishes the same for a fixed k ). First, weobserve that since only λ j changes, by the definition of G h we have G h ( λ + α e j , µ ) − G h ( λ , µ )= (cid:88) k :( j,k ) ∈P h v jk (min( λ j + α, µ k ) − min( λ j , µ k ))+ G h +1 (Λ h ( λ + α e j , µ ) , M h ( λ + α e j , µ )) − G h +1 (Λ h ( λ , µ ) , M h ( λ , µ )) . If there is no k such that ( j, k ) ∈ P h , then we have Λ h ( λ + α e j , µ ) = Λ h ( λ , µ ) + α e j and M h ( λ + α e j , µ ) = M h ( λ , µ ) , and thus by the definition of G h we have G h ( λ + α e j , µ ) − G h ( λ , µ )= G h +1 ( Λ h ( λ , µ ) + α e j , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ∈ [0 , αp h +1 j ] by the induction hypothesis, and since p h +1 j ≤ p hj we have [0 , αp h +1 j ] ⊆ [0 , αp hj ] which establishesthe claim.Otherwise, when there is a k such that ( j, k ) ∈ P h (recall that there is at most one such k ),we consider three cases. First, consider the case when λ j ≥ µ k . Then, we have min( λ j , µ k ) =min( λ j + α, µ k ) , and ( λ j + α − µ k ) + = ( λ j − µ k ) + + α and ( µ k − λ j − α ) + = ( µ k − λ j ) + , implying50gain that Λ h ( λ + α e j , µ ) = Λ h ( λ , µ ) + α e j and M h ( λ + α e j , µ ) = M h ( λ , µ ) . Together, theseimply that we again have G h ( λ + α e j , µ ) − G h ( λ , µ )= G h +1 ( Λ h ( λ , µ ) + α e j , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ∈ [0 , αp h +1 j ] which establishes the claim.Second, consider the case when λ j < µ k and λ j + α ≤ µ k . Then we have min( λ j , µ k ) = λ j and min( λ j + α, µ k ) = λ j + α , and also ( λ j + α − µ ) + = ( λ j − µ k ) + , and ( µ k − λ j − α ) + = ( µ k − λ j ) + − α ,which imply Λ h ( λ + α e j , µ ) = Λ h ( λ , µ ) and M h ( λ + α e j , µ ) = M h ( λ , µ ) − α e k . Together, theseimply that G h ( λ + α e j , µ ) − G h ( λ , µ )= αp hj + G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ ) − α e k ) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ ))= αp hj − ( G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ ) − α e k )) . By the induction hypothesis, we have G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ ) − α e k ) ∈ [0 , αp h +1 k ] ⊆ [0 , αp hj ] , since p h +1 k ≤ p hj . Thus, we have αp hj − ( G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) ∈ [0 , αp hj ] , establishing the claim.Finally, consider the case when λ j < µ k and λ j + α ≥ µ k . In this case, let β = µ k − λ j > , andnote that β ≤ α . Then we have min( λ j + α, µ k ) − min( λ j , µ k ) = µ k − λ j = β . Also, ( λ j + α − µ ) + =( λ j − µ k ) + + λ j + α − µ = ( λ j − µ k ) + + α − β , and ( µ k − λ j − α ) + = ( µ k − λ j ) + − β , which imply Λ h ( λ + α e j , µ ) = Λ h ( λ , µ ) + ( α − β ) e j and M h ( λ + α e j , µ ) = M h ( λ , µ ) − β e k . Together, theseimply that G h ( λ + α e j , µ ) − G h ( λ , µ )= βp hj + G h +1 (Λ h ( λ , µ ) + ( α − β ) e j , M h ( λ , µ ) − β e k ) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) . Since the induction hypothesis implies that G h +1 is nondecreasing in each argument, we have βp hj + G h +1 ( Λ h ( λ , µ ) + ( α − β ) e j , M h ( λ , µ ) − β e k ) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ≤ βp hj + G h +1 ( Λ h ( λ , µ ) + ( α − β ) e j , M h ( λ , µ )) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ≤ βp hj + ( α − β ) p h +1 j ≤ αp hj , where the second inequality again follows from the induction hypothesis, and the third from p h +1 j ≤ p hj . Also because G h +1 is nondecreasing, we have βp hj + G h +1 ( Λ h ( λ , µ ) + ( α − β ) e j , M h ( λ , µ ) − β e k ) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ≥ βp hj + G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ ) − β e k ) − G h +1 ( Λ h ( λ , µ ) , M h ( λ , µ )) ≥ βp hj − βp h +1 k ≥ , where the second inequality again follows from the induction hypothesis, and the third from p h +1 k ≤ p hj . Together, these inequalities imply that G h ( λ + α e j , µ ) − G h ( λ , µ ) ∈ [0 , αp hj ] , establishingthe claim and finishing the induction step. Then the lemma follows by the bounds for G since [0 , αp j ] ⊆ [0 , α ¯ v ] . 51 emma 8. For any (cid:15) > , and any A ≥ λ − (cid:15) e and B ≥ µ − (cid:15) e , G ( A , B ) ≥ G ( λ , µ ) − ( J + K )¯ v(cid:15). Proof.
First, by a telescoping sum (and assuming an empty sum evaluates to zero), we observethat G ( λ , µ ) − G ( λ − (cid:15) e , µ − (cid:15) e ) = J (cid:88) j =1 (cid:104) G ( λ − (cid:15) j − (cid:88) s =1 e s , µ ) − G ( λ − (cid:15) j (cid:88) s =1 e s , µ ) (cid:105) + K (cid:88) k =1 (cid:104) G ( λ − (cid:15) e , µ − (cid:15) k − (cid:88) t =1 e t ) − G ( λ − (cid:15) e , µ − (cid:15) k (cid:88) t =1 e t ) (cid:105) ≤ ( J + K )¯ v(cid:15), where the inequality follows from the upper bound of Lemma 7 for each incremental change in theobjective value. Then, since the lower bound in Lemma 7 implies G ( · , · ) is monotone nondecreasing,we have G ( A , B ) ≥ G ( λ − (cid:15) e , µ − (cid:15) e ) ≥ G ( λ , µ ) − ( J + K )¯ v(cid:15), completing the proof. B.1 Approximation of the greedy solution
We want to compare the greedy solution to optimal. To this end, consider the dual of (15): min (cid:88) j ∈ J λ j α j + (cid:88) k ∈ K µ k β k s.t. α j + β k ≥ v jk , ( j, k ) ∈ J × K ,α j , β k ≥ j ∈ J , k ∈ K . (54)For a given edge ( j, k ) , define the next highest value edge connected to j as: v j ( k ) = max k (cid:48) (cid:54) = k { v jk (cid:48) : v jk (cid:48) ≤ v jk } , where we let v j ( k ) = 0 if the set { k (cid:48) (cid:54) = k, v jk (cid:48) ≤ v jk } is empty (i.e., v jk is the lowest value edgeconnected to j ). Similarly, for edge ( j, k ) , define the next highest value edge connected to k as: v k ( j ) = max j (cid:48) (cid:54) = j { v j (cid:48) k : v j (cid:48) k ≤ v jk } , with a similar default value of zero if the relevant set is empty.In the construction of the greedy solution in (48), we say that an edge ( j, k ) ∈ P h “made j tight”if y gjk > and (cid:80) l ≤ h (cid:80) k (cid:48) :( j,k (cid:48) ) ∈P l y gjk (cid:48) = λ j , since ( j, k ) was the last edge set positive by greedy beforethe capacity λ j was exhausted, i.e., ( j, k ) made the primal constraint for j tight. Similarly, we saythat ( j, k ) ∈ P h “made k tight” if y gjk > and (cid:80) l ≤ h (cid:80) j (cid:48) :( j (cid:48) ,k ) ∈P l y gj (cid:48) k = µ k . Note that each ( j, k ) such that y gjk > makes at least one of j or k tight, since the greedy solution always sends flow onan edge until either the supply or demand is exhausted.52 lgorithm 1 Dual Reverse Greedy Input sets P h for ≤ h ≤ m and v jk and y gjk ∀ j, k Initialize α j = 0 ∀ j , β k = 0 ∀ k , for h = m, m − , . . . , do for ( j, k ) ∈ P h do if y gjk > then if ( j, k ) made j and k tight then Set α j = v j ( k ) + ( v jk − v j ( k ) − v k ( j )) + , and β k = v k ( j ) + ( v jk − v j ( k ) − v k ( j )) + else if ( j, k ) made j tight then Set α j = max( v jk − β k , v j ( k )) else if ( j, k ) made k tight then Set β k = max( v jk − α j , v k ( j )) end if end if end for end forLemma 9. The dual solution constructed by Algorithm 1 is feasible to (54) .Proof.
By construction in Algorithm 1, the dual variables are nonnegative, and so satisfy the secondconstraint of (54). For the first constraint of (54), given an adge ( j, k ) we consider two cases. Case 1.
First, if y gjk > , then ( j, k ) either made both j and k tight, or it made just one of j or k tight (it had to make at least one node tight, since greedy sends flow along an adge until one ofthe capacities is exhausted). If ( j, k ) made both j and k tight, then by Algorithm 1 we have α gj + β gk = v j ( k ) + v k ( j ) + ( v jk − v j ( k ) − v k ( j )) + = max( v jk , v j ( k ) + v k ( j )) ≥ v jk , so the first constraint of (54) is satisfied. Otherwise, if ( j, k ) made just one of j or k tight, then weassume it made j tight (a symmetric argument holds when it made k tight). Then by Algorithm 1we have α gj + β gk = max( v jk , v j ( k ) + β gk ) ≥ v jk , so again the first constraint of (54) is satisfied. Case 2.
Next consider the case when y gjk = 0 . For this to be true, at least one of the primalconstraints for j or k must bind at the end of the primal greedy algorithm, since otherwise we couldhave made y gjk > and greedy would not have terminated. If the primal constraint for j bindsat the end of the greedy algorithm, we say that j is tight (and similarly for k ). We consider twoalternatives. First, consider if just one of j or k is tight, then we assume it is j that is tight while k is not tight, i.e., at the end of the primal greedy algorithm there is still capacity available at k to bematched (a symmetric argument holds when k is tight and j is not tight). To derive a contradiction,assume that the first constraint of (54) is violated for ( j, k ) , so that α gj + β gk < v jk . This furtherimplies that α gj < v jk . Now, consider the edge ( j, k (cid:48) ) that made j tight with y gjk (cid:48) > (so k (cid:48) (cid:54) = k ).From Algorithm 1, we set the dual variable for j to either α gj = v j ( k (cid:48) ) + ( v jk (cid:48) − v j ( k (cid:48) ) − v k (cid:48) ( j )) + or α j = max( v jk (cid:48) − β k (cid:48) , v j ( k (cid:48) )) , and in either case we have α gj ≥ v j ( k (cid:48) ) . Thus, we have v jk > v j ( k (cid:48) ) ,which further implies that v jk > v jk (cid:48) (since otherwise v j ( k (cid:48) ) = v jk by definition). However, this is53 contradiction, because when the primal greedy algorithm assigned y gjk (cid:48) > , j was not yet tight,and k also was not tight (by assumption, since we are considering the case when it is not tight atthe end of the algorithm), so the primal greedy algorithm could have sent flow along edge ( j, k ) forhigher value.Next consider if both j and k are tight. Again assume that the first constraint of (54) is violatedfor ( j, k ) , so that α gj + β gk < v jk , which implies both α gj < v jk and β gk < v jk . By the same argumentgiven for the first alternative, for the edge ( j, k (cid:48) ) that made j tight, we must have v jk > v jk (cid:48) . Asymmetric argument shows that for the edge ( j (cid:48) , k ) that made k tight, we must have v jk > v j (cid:48) k . Thisagain is a contradiction to the construction of the primal greedy algorithm. To see this, considerthe iteration of the primal greedy algorithm when the first of nodes j or k were made to be tight.At the beginning of this iteration, both j and k were not yet tight, so the primal greedy algorithmcould have sent flow along edge ( j, k ) for higher value than either ( j, k (cid:48) ) or ( j (cid:48) , k ) . Lemma 10.
For the dual solution constructed by Algorithm 1, if y gjk > then v jk ≤ α gj + β gk ≤ max( v jk , v j ( k ) + v k ( j )) . Further, if α gj > then (cid:80) k y gjk = λ j , and if β gk > then (cid:80) j y gjk = µ k .Proof. The second statement, that α gj > implies (cid:80) k y gjk = λ j and β gk > implies (cid:80) j y gjk = µ k ,follows by the construction of the dual solution in Algorithm 1, since we only set the dual variablespositive for j and k that have tight primal constraints.For the first statement, given ( j, k ) with y gjk > , we consider two cases. First if ( j, k ) madeboth j and k tight, then by Algorithm 1 we have α gj + β gk = v j ( k ) + v k ( j ) + ( v jk − v j ( k ) − v k ( j )) + = max( v jk , v j ( k ) + v k ( j )) . Otherwise, if ( j, k ) made just one of j or k tight, then we assume it made j tight (a symmetricargument holds when it made k tight). Then, we claim that β gk ≤ v k ( j ) . To see this, first note thatif k is not tight at the end of the primal greedy algorithm, then Algorithm 1 sets β gk = 0 , so theclaim holds. Otherwise, if k is tight at the end of the primal greedy algorithm, we observe thatit must have gone tight after j , since otherwise there would have been no capacity left at k to set y gjk > . Consider the edge ( j (cid:48) , k ) that made k tight. By Algorithm 1 we set the dual variable for k to either β gk = v k ( j (cid:48) ) + ( v j (cid:48) k − v j (cid:48) ( k ) − v k ( j (cid:48) )) + = max( ( v j (cid:48) k + v k ( j (cid:48) ) − v j (cid:48) ( k )) , v k ( j (cid:48) )) ≤ v j (cid:48) k , or β gk = max( v j (cid:48) k − α gj , v k ( j (cid:48) )) ≤ v j (cid:48) k , where both inequalities follows from v k ( j (cid:48) ) ≤ v j (cid:48) k . Thus, we have β gk ≤ v j (cid:48) k Then, since k went tightafter j in the primal greedy algorithm and since ( j (cid:48) , k ) made k tight and ( j, k ) made j tight, wemust have v j (cid:48) k ≤ v jk by the definition of the primal greedy algorithm. By definition, this impliesthat v k ( j ) ≥ v j (cid:48) k ≥ β gk , completing the proof of the claim. Finally, by the construction of the dualsolution in Algorithm 1 we have α gj + β gk = max( v jk , v j ( k ) + β gk ) ≤ max( v jk , v j ( k ) + v k ( j )) . Exact calculations of the mean queue lengths in Section 5.3
Let X ( ∞ ) be a Markov chain with transition rates are shown in Figure 13. ( 1) x x x x x ( 1) x x x x x Figure 13: Transition rates of the Markov chain.Its probability distribution π ( · ) is the (unique) solution of the balance equations that are writtenas follows: for x > λ + µ + xθ ) π ( x ) = λπ ( x −
1) + ( µ + ( x + 1) θ ) π ( x + 1) , ( λ + µ + xθ ) π ( − x ) = µπ ( − x + 1) + ( λ + ( x + 1) θ ) π ( − x − , and x = 0 , ( λ + µ ) π (0) = ( λ + θ ) π ( −
1) + ( µ + θ ) π (1) . The solution of the balance equations in given by π ( x ) = λ x (cid:81) xj =1 ( µ + jθ ) π (0) if x > , µ − x (cid:81) − xj =1 ( λ + jθ ) π (0) if x < , (55)where π (0) = (cid:32) ∞ (cid:88) x =1 λ x (cid:81) xj =1 ( µ + jθ ) + ∞ (cid:88) x =1 µ x (cid:81) xj =1 ( λ + jθ ) (cid:33) − . To see this, observe that for x > , λπ ( x −
1) + ( µ + ( x + 1) θ ) π ( x + 1) = (cid:32) λ λ x − (cid:81) x − j =1 ( µ + jθ ) + ( µ + ( x + 1) θ ) λ x +1 (cid:81) x +1 j =1 ( µ + jθ ) (cid:33) π (0)= (cid:32) ( µ + xθ ) λ x (cid:81) xj =1 ( µ + jθ ) + λ λ x (cid:81) xj =1 ( µ + jθ ) (cid:33) π (0)= ( λ + µ + xθ ) π ( x ) . In a similar way, for x > we have that µπ ( − x + 1) + ( λ + ( x + 1) θ ) π ( − x −
1) = (cid:32) µ µ x − (cid:81) x − j =1 ( λ + jθ ) + ( λ + ( x + 1) θ ) µ x +1 (cid:81) x +1 j =1 ( λ + jθ ) (cid:33) π (0)= (cid:32) ( λ + xθ ) µ x (cid:81) xj =1 ( λ + jθ ) + µ µ x (cid:81) xj =1 ( λ + jθ ) (cid:33) π (0)= ( λ + µ + xθ ) π ( − x ) . ( λ + θ ) π ( −
1) + ( µ + θ ) π (1) = (cid:18) ( λ + θ ) µλ + θ + ( λ + θ ) λµ + θ (cid:19) π (0)= ( λ + µ ) π (0) . In other words, (55) satisfies the balance equations. Now, adapting [53], we derive the followingformulas x (cid:89) j =1 ( µ + jθ ) = 1 µ x (cid:89) j =0 ( µ + jθ ) = θ x +1 Γ( µθ + x + 1) µ Γ( µθ ) , x (cid:89) j =1 ( λ + jθ ) = 1 λ x (cid:89) j =0 ( λ + jθ ) = θ x +1 Γ( λθ + x + 1) λ Γ( λθ ) , ∞ (cid:88) x =1 λ x (cid:81) xj =1 ( µ + jθ ) = γ (cid:18) λθ , µθ (cid:19) (cid:18) λθ (cid:19) − µ/θ µθ e λ/θ − , ∞ (cid:88) x =1 µ x (cid:81) xj =1 ( λ + jθ ) = γ (cid:18) µθ , λθ (cid:19) (cid:16) µθ (cid:17) − λ/θ λθ e µ/θ − , where γ ( x, y ) := (cid:82) x t y − e − t dt and Γ( y ) := γ ( ∞ , y ) . The first two equations follow directly by [53].We show the third equation and the forth one follows by symmetry. By the power series expansionof the incomplete gamma function we have that γ (cid:18) λθ , µθ (cid:19) = (cid:18) λθ (cid:19) µ/θ e − λ/θ ∞ (cid:88) x =0 ( λ/θ ) x (cid:81) xj =0 ( µ/θ + j )= (cid:18) λθ (cid:19) µ/θ e − λ/θ ∞ (cid:88) x =0 λ x θ (cid:81) xj =0 ( µ + jθ )= (cid:18) λθ (cid:19) µ/θ e − λ/θ θµ ∞ (cid:88) x =0 λ x (cid:81) xj =1 ( µ + jθ )= (cid:18) λθ (cid:19) µ/θ e − λ/θ θµ (cid:32) ∞ (cid:88) x =1 λ x (cid:81) xj =1 ( µ + jθ ) + 1 (cid:33) , where we define an empty product to be one.Let π n ( · ) denote the distribution of X n ( ∞ ) with rates nλ and nµ . The means of X + ,n ( ∞ ) n and X − ,n ( ∞ ) n are given by the following expressions n E (cid:2) X + ,n ( ∞ ) (cid:3) = 1 n ∞ (cid:88) x = −∞ max( x, π n ( x ) = 1 n ∞ (cid:88) x =1 x ( nλ ) x (cid:81) xj =1 ( nµ + jθ ) π n (0)= µθ Γ (cid:16) nµθ (cid:17) π n (0) ∞ (cid:88) x =1 x ( nλ/θ ) x Γ (cid:0) nµθ + x + 1 (cid:1) and n E (cid:2) X − ,n ( ∞ ) (cid:3) = λθ Γ (cid:18) nλθ (cid:19) π n (0) ∞ (cid:88) x =1 x ( nµ/θ ) x Γ (cid:0) nλθ + x + 1 (cid:1) , π n (0) = (cid:32) γ (cid:18) nλθ , nµθ (cid:19) (cid:18) nλθ (cid:19) − nµ/θ nµθ e nλ/θ + γ (cid:18) nµθ , nλθ (cid:19) (cid:16) nµθ (cid:17) − nλ/θ nλθ e nµ/θ − (cid:33) − . When µ/θ := a ∈ N and λ/θ := b ∈ N , we have that ∞ (cid:88) x =1 x ( nµ/θ ) x Γ (cid:0) nλθ + x + 1 (cid:1) = b − an n − an +1 (cid:0) − ae bn γ ( an + 1 , bn ) + b bn e bn γ ( an + 1 , bn ) + b an +1 n an (cid:1) ( an )! . In the special case where λ = µ = θ , by observing that Γ ( n ) = ( n − and (cid:80) ∞ x =1 x n x Γ( n + x +1) = n − ,the above expressions can be simplified further as follows n E (cid:2) X + ,n ( ∞ ) (cid:3) = 1 n E (cid:2) X − ,n ( ∞ ) (cid:3) = ( n − π n (0) ∞ (cid:88) x =1 xn x ( n + x )! = π n (0) , where π n (0) = (cid:0) γ ( n, n ) n − n +1 e n − (cid:1) − and γ ( x, n ) := (cid:82) x t n − e − t dt = ( n − (cid:16) − e − x (cid:80) n − k =0 x k k ! (cid:17) .Moreover, the mean queue lengths are exactly n E [ Q n ( ∞ )] = n E [ X + ,n ( ∞ )] and n E [ I n ( ∞ )] = n E [ X − ,n ( ∞ )] ..