Stationary Distribution Convergence of the Offered Waiting Processes in Heavy Traffic under General Patience Time Scaling
aa r X i v : . [ m a t h . P R ] F e b Stationary Distribution Convergence of the Offered WaitingProcesses in Heavy Traffic under General Patience TimeScaling
Chihoon Lee ∗ School of BusinessStevens Institute of TechnologyHoboken, NJ 07030 Amy R. Ward † Booth School of BusinessThe University of ChicagoChicago, IL 60637Heng-Qing Ye ‡ Dept. of Logistics and Maritime StudiesHong Kong Polytechnic UniversityHong KongFebruary 10, 2021
Abstract
We study a sequence of single server queues with customer abandonment (
GI/GI/ GI ) under heavy traffic. The patience time distributions vary with the sequence, whichallows for a wider scope of applications. It is known ([20, 18]) that the sequence ofscaled offered waiting time processes converges weakly to a reflecting diffusion processwith non-linear drift, as the traffic intensity approaches one. In this paper, we furthershow that the sequence of stationary distributions and moments of the offered waitingtimes, with diffusion scaling, converge to those of the limit diffusion process. Thisjustifies the stationary performance of the diffusion limit as a valid approximation forthe stationary performance of the GI/GI/ GI queue. Consequently, we also derivethe approximation for the abandonment probability for the GI/GI/ GI queue inthe stationary state. Keywords: Customer Abandonment; Heavy Traffic; Stationary Distribution Convergence ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] Introduction
In a recent paper, Lee et al. [17], we have studied a sequence of single server systems withabandonment (
GI/GI/ GI ). In each system, the customers arrive following a renewalprocess and have a general service time distribution. We have shown that under appropri-ate conditions the stationary distributions of suitably scaled offered waiting time processesconverge to the stationary distribution of the associated reflected Ornstein-Uhlenbeck pro-cess in the positive real line, as the traffic intensity approaches one (that is, as the heavytraffic scaling parameter n approaches infinity). One of the key assumptions made in theanalysis is that the patience times have remained the same and unscaled for all the sys-tems, and consequently it leads to a diffusion approximation that depends only on thebehavior of the patience time distribution F ( · ) at 0. Such diffusion approximations, witha linear drift F ′ (0) x accounting for the customer abandonment, have at least two draw-backs: (Applicability) There is a wide family of distributions where F ′ (0) = 0 or ∞ , e.g.,Gamma distributions with the shape parameter α = 1; (Performance) If the underlyingpatience time distribution is of increasing (or decreasing) hazard rate (i.e., the longer onehas waited for service, the more likely to abandon), then the diffusion approximation witha linear drift F ′ (0) x invariably over-estimates (or under-estimates) the steady-state offeredwaiting time.Our objective in this paper is to establish the stationary distribution convergence ofoffered waiting time in the heavy traffic limit whilst the underlying patience times areallowed to vary and are suitably scaled, ensuring the entire distributional informationis contained in the heavy traffic diffusion limit. This relaxes the assumption of uniformabandonment time distribution in our previous work and greatly enhances the applicabilityof our results. As a result, we derive the approximation for the abandonment probabilityfor the GI/GI/ GI queue in the stationary state. It is also interesting to note that weare able to bypass the hydrodynamic scaling approach in Lee et al. [17] and simplify thearguments based on more direct and simpler pathwise stability estimates.Our result draws on past work that has developed heavy traffic approximations forthe GI/GI/ GI queue using the offered waiting time process. The offered waitingtime process, introduced in [2], tracks the amount of time an infinitely patient customermust wait for service. Its heavy traffic limit when the patience time distribution is leftunscaled is a reflected Ornstein-Uhlenbeck process (see Ward and Glynn [21]), and itsheavy traffic limit when the patience time distribution is scaled through its hazard rate isa reflected nonlinear diffusion (see Reed and Ward [20]). The work of Lee and Weerasinghe[18] incorporates both of these scaling scenarios in the general framework that can besatisfied by many other classes of patience time distributions. One key ingredient in theproof of their main result is the martingale functional central limit theorem, which helpsto accommodate the more general assumptions (e.g., the customer arrival process has astate dependent intensity). 2enerally speaking, those results (viz. the process-level convergence) are not sufficientto conclude that the stationary distribution of the offered waiting time process converges,which is the key to approximating the performance measures such as the stationary aban-donment probability and mean queue-length. Those limits were conjectured in [20], andshown through simulation to provide good approximations. However, the proof of thoselimits was left as an open question. In this paper, we analyze the case when the patiencetime distribution is scaled, following the general framework in [18].There is a growing literature on the problem of stationary distribution convergence ofservice systems in heavy traffic. Broadly speaking, there are indirect and direct methodsin approaching the problem. As for the indirect method, the papers [12] and [7] establishthe validity of the heavy traffic stationary approximation (viz. interchange of limits) for ageneralized Jackson network, without customer abandonment, using a Lyapunov functionmethod. The main difficulty in extending their methodologies to the current model is thelack of the global Lipschitz continuity property of an associated regulator (Skorokhod)mapping that helped convert the given moment bound of primitives (the inter-arrival andservice times) to the bound of the waiting time or queue-length processes. The knownregulator mapping under customer abandonment is only locally Lipschitz (that is, theLipschitz constant depends on time parameter).As for the more direct methods, there are the MGF–BAR (moment generating function–basic adjoint relationship) method and the generator comparison method (also known asStein’s method). For the latter, the authors of [13] study the Poisson arrival case (i.e.,
M/GI/ GI queue) and show the associated Brownian model is accurate uniformlyover a family of patience time distributions and universally in the heavy-traffic regime.Owing to the Poisson arrivals, it is enough to consider a one-dimensional process witha simple generator, whereas with general arrival processes, one needs to consider a two-dimensional process (tracking, e.g., the residual arrival times) and correspondingly morecomplicated generator. For the former approach, the authors of [6] consider a generalizedJackson network, without customer abandonment, and work directly with the BAR, anintegral equation that characterizes the stationary distribution of a Markov process, andestablish the convergence of the MGFs of the pre-limit stationary distributions. Whenextending their methodology to the current model, a unique challenge arises in establishinga convergence rate (in terms of the scaling parameter) of the tail estimates for the stationaryabandonment probability. Lastly, there is the so-called Drift method; see [11] and also [15]for its connection to the MGF method. The Drift method uses polynomial test functionsin an inductive manner by setting to zero the drift of the test function (i.e., equating theexpected value of the test function in two different time steps).In view of the above-mentioned technical challenges, we adopt the approach in thestudies of [23, 24], which extend the works of [12] and [7] (the interchange of limits) toa wider range of stochastic processing networks, e.g., the multiclass queueing networkand the resource-sharing network. In their studies, they relax the requirement of the3forementioned Lipschitz continuity by establishing the so-called uniform stability andthe uniform moment bound. Further extending their approach to the nonlinear dynamiccomplementarity problem, which is relevant to the GI/GI/ GI model, Lee et al. [17]establishes the stationary distribution and moment convergences when the patience timesare left unscaled. The key proof of Lee et al. [17] is to establish a kind of uniform stabilityproperty (cf. Lemmas 1-3) by applying Bramson’s hydrodynamic scaling approach and itsvariation. (By the hydrodynamic approach, the n th diffusion-scaled process breaks intomany pieces of fluid-scaled processes, with each piece covering a period of 1 / √ n in thediffusion-scaled process.)The contributions of the current paper are: (a) identifying the stability conditions ofthe pre-limit system (Theorem 1) under general scaling assumption; (b) establishing theconvergence of stationary distributions and moments of offered waiting times (Theorem 2)and offering simplified proofs based on pathwise stability estimates, replacing the moreinvolved hydrodynamic scaling approach taken in [17], and (c) establishing the convergenceof the scaled stationary abandonment probabilities (Corollary 1).The remainder of this paper is organized as follows. In Section 2, we set up the modelassumptions and recall the known process-level convergence results for the GI/GI/ GI queue under general patience time distribution scaling. In Section 3, we state our mainsresults on the convergence of the stationary distribution of the offered waiting time processand its moments. In Section 4, we present key moment bounds of the scaled state processesthat are uniform in the heavy traffic scaling parameter ( n ). Lastly, in Section 5, we providethe proofs of key lemmas. Notation and Terminology.
Use the symbol “ ≡ ” to stand for equality by definition.The set of positive integers is denoted by IN and denote IN ≡ IN ∪ { } . Let IR representthe real numbers ( −∞ , ∞ ) and IR + the non-negative real line [0 , ∞ ). For x, y ∈ IR , x ∨ y ≡ max { x, y } and x ∧ y ≡ min { x, y } . Let D ( IR ) ≡ D ( IR + , IR ) be the space ofright-continuous functions f : IR + → IR with left limits, endowed with the Skorokhod J -topology (see, for example, [3]). Lastly, the symbol “ ⇒ ” stands for the weak convergence;we make this explicit for stochastic processes in D ( IR ), otherwise, it is used for weakconvergence for a sequence of random variables. We consider a sequence of single server systems having FIFO service with abandonmentindexed by n ∈ IN , and by convention, we use superscript n for any processes or quantitiesassociated with the n -th system. For n ∈ IN , consider three independent i.i.d. sequencesof nonnegative random variables { u ni , i ≥ } , { v ni , i ≥ } , { d ni , i ≥ } , that are representinginter-arrival times, service times, and patience times, respectively, and are defined on acommon probability space (Ω , F , IP ). 4e first consider the sequences { u ni , i ≥ } , { v ni , i ≥ } which are built from independenti.i.d. sequences of random variables with unit mean { u i , i ≥ } , { v i , i ≥ } in the followingway. At time 0, the previous arrival to the system occurred at time t n <
0, so that | t n | represents the time elapsed since the last arrival in the n -th system. We let u be therandom variable representing the remaining time conditioned on | t n | time units havingpassed; that is, IP ( u > x ) = IP ( u > x | u > | t n | ) . Given positive sequences { λ n } and { µ n } , the i -th arrival to the n -th system occurs at time t ni ≡ i X j =1 u nj , u nj ≡ u j λ n , and has service time v ni ≡ v i µ n , and abandons without receiving service if processing does not begin by time t ni + d ni .We assume the following conditions.( A
1) For some p ∈ (2 , ∞ ), IE [ u p + v p ] < ∞ .The arrival and service rates in the n -th system, λ n and µ n , respectively, satisfy thefollowing heavy traffic assumption:( A λ n ≡ nλ , lim n →∞ µ n n = λ ∈ (0 , ∞ ) and lim n →∞ √ n (cid:16) λ − µ n n (cid:17) = θ ∈ IR.
Next, we consider the following assumption on the patience time distributions { d ni , i ≥ } as studied in Lee and Weerasinghe [18]. As a consequence, the drift coefficient of thelimiting diffusion is influenced by the sequence of patience time distributions in a non-linearfashion.( A
3) Let F n ( · ) be the right continuous patience time distribution function of the i.i.d.sequence ( d ni ) i ≥ . Assume that F n (0) = 0 and there is a non-negative continuouslydifferentiable function H ( · ) such that for each K > n →∞ sup x ∈ [0 ,K ] (cid:12)(cid:12)(cid:12)(cid:12) √ nF n (cid:18) x √ n (cid:19) − H ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Example 1.
We provide some examples of patience time distribution functions { F n } sat-isfying ( A ; cf. Lee and Weerasinghe [18] and Huang et al. [14]. . Take F n = F for all n , where F ( · ) is some distribution function, differentiable witha bounded derivative on [0 , δ ] for some δ > . Hence, H ( x ) = F ′ (0) x in this case.This corresponds to the result in Ward and Glynn [21].2. Take F n ( x ) = 1 − exp( − R x h ( √ nu ) du ) for x ≥ , where h ( · ) is a continuous hazardrate function. In this case, H ( x ) = R x h ( u ) du and it satisfies ( A . Indeed, forany general sequence { F n } , if F n ( x √ n ) converges to a non-negative function h ( x ) uniformly on compact sets, then { F n } satisfies ( A with the limiting function H ( x ) = R x h ( u ) du . This corresponds to the result in Reed and Ward [20].3. Take any non-negative, non-decreasing, continuously differentiable function H ( · ) whichsatisfies H (0) = 0 and H ( ∞ ) = ∞ . Let F n ( x ) = √ n min { H ( √ nx ) , √ n } for all x ≥ .Then, for each n ≥ , F n is a continuous probability distribution function and thesequence of distribution functions { F n } satisfies ( A with limiting function H ( · ) .4. More examples such as mixture of hazard-rate scaling and no scaling, and delayedhazard-rate scaling are presented in Huang et al. [14]. The Offered Waiting Time Process
The offered waiting time process { V n ( t ) : t ≥ } tracks the amount of time an incomingcustomer has to wait for service: V n ( t ) = V n (0) + A n ( t ) X j =1 v nj [ V n ( t nj − )
Consider the one-dimensional reflected diffusion process V ≡ { V ( t ) : t ≥ } : V ( t ) = V (0) + σW ( t ) + θλ t − R t H ( V ( s )) ds + L ( t ) ≥ L is non-decreasing, has L (0) = 0 and R ∞ V ( s ) dL ( s ) = 0 , (3)where { W ( t ) : t ≥ } denotes a one-dimensional standard Brownian motion, and theinfinitesimal variance parameter is σ ≡ λ − (var( u ) + var( v )) . The following weak convergence result is a simple modification of Theorem 4.10 of Leeand Weerasinghe [18] (see also Reed and Ward [20] and Lee et al. [17]).
Proposition 1.
Assuming √ nV n (0) ⇒ V (0) as n → ∞ , we have √ nV n ⇒ V in D ( IR ) as n → ∞ . (4)6 nV n ( t ) √ nV n ( ∞ ) V ( t ) V ( ∞ ) ✲✲❄ ❄ t → ∞ t → ∞ n → ∞ n → ∞ Figure 1:
A graphical representation of the limit interchange.
The weak convergence (4) motivates approximating the scaled stationary distributionsfor V n , and its moments, using the stationary distribution of V , and its moments. First ofall, we impose the following well-known stability condition for a one-dimensional diffusion V (cf. [10] and Proposition 6.1(i) in [20]):( A
4) lim x →∞ H ( x ) > θλ .We note that the above stability condition is trivially satisfied for the two examples de-scribed earlier, right below ( A H ( x ) = F ′ (0) x and H ( x ) = R x h ( u ) du converge to ∞ as x → ∞ . Identical arguments as in the proof of Proposition 6.1(i) in [20]show that, under ( A V ( t ) ⇒ V ( ∞ ) as t → ∞ (5)for V ( ∞ ) a random variable having a density function f ( x ) = M exp (cid:18) σ (cid:18) θλ x − Z x H ( s ) ds (cid:19)(cid:19) , x ≥ , (6)where M ∈ (0 , ∞ ) is such that R ∞ f ( x ) dx = 1. In summary, the convergences (4) and (5)correspond to the limit when n → ∞ is taken first and t → ∞ is taken second, depictedgraphically in Figure 1. Because of the remaining arrival time (i.e., the forward recurrence time of the arrivalprocess), the offered waiting time process { V n ( t ) : t ≥ } alone is not Markovian. Definingthe remaining arrival time τ n ( t ) ≡ t nj +1 − t for t ∈ [ t nj , t nj +1 ) , j ∈ IN , τ n (0) = u , the process X n ≡ { ( τ n ( t ) , V n ( t )) : t ≥ } having state space S ≡ IR + × IR + is strong Markov (cf. Problem 3.2 of Chapter X in [1]). For x = ( τ, v ) ∈ S ,define its norm | x | as | x | ≡ τ + v and likewise define | X n ( t ) | ≡ τ n ( t ) + V n ( t ) , t ≥ . We willassume the interarrival times are unbounded. Such an assumption has been frequently usedin the literature to verify a petite set requirement that implies positive Harris recurrenceof a Markov process, cf. Proposition 4.8 in [5].( A
5) The i.i.d. interarrival times { u i , i ≥ } are unbounded, that is, IP ( u ≥ u ) > u > Theorem 1. (Stationary Distribution Existence)
Assume ( A – ( A . For any suf-ficiently large n , there exists a unique stationary probability distribution for the Markovprocess X n . Next, we consider the diffusion-scaled process: e X n ≡ ( e τ n , e V n ) , where e τ n ( t ) ≡ √ nτ n ( t ) , e V n ( t ) ≡ √ nV n ( t ) . Notice the time is not scaled in the process e V n because the arrival and service rate pa-rameters are scaled instead (from ( A λ n and µ n are order n quantities); therefore,scaling the state by √ n produces the traditional diffusion scaling. Remark 1.
We note that Theorem 1 requires sufficiently large n , while the result in Leeet al. [17] does not. This is because the proof of Theorem 1 is based on a weaker version ofan estimate (7) in Proposition 2 below, which is lim | x |→∞ IE h | e X nx ( t | x | ) | q i / | x | q = 0 (thereis no ‘supremum over n ’), and we could establish this only for sufficiently large n . Moreprecisely, we could prove Lemma 1 below only for large enough (but fixed) n (under whichthe negative drift of netput rate is warranted; see Remark 3). Theorem 2. (Stationary Convergence)
Assume ( A – ( A . For large enough n , let π n denote the stationary distribution of e X n .(a) (Distribution) Denote by π n the marginal distribution of π n on the second coor-dinate of e X n , i.e., π n ( A ) = π n ( IR + × A ) for A ∈ B ( IR + ) . Let e V n ( ∞ ) be a randomvariable having distribution π n and also V ( ∞ ) a random variable having density (6) .We have that e V n ( ∞ ) ⇒ V ( ∞ ) as n → ∞ .(b) (Moments) For any m ∈ (0 , p − , IE [( e V n ( ∞ )) m ] → IE [( V ( ∞ )) m ] as n → ∞ .
8s a consequence, we obtain the following result on the convergence of stationaryabandonment probability. For n ≥
1, define P na ≡ IE [ F n ( V n ( ∞ ))] = IE " F n e V n ( ∞ ) √ n ! , that is, a fraction of customers who abandon the system in stationarity. Corollary 1. (Stationary Abandonment Probability Convergence)
Assume ( A – ( A . Assume further √ nF n ( x/ √ n ) ≤ C (1 + x m ) for m ∈ (0 , p − , where C ∈ (0 , ∞ ) isindependent of n . Then, as n → ∞ , √ nP na → IE [ H ( V ( ∞ ))] , where the function H ( · ) is as in ( A . Example 2.
We provide some examples of patience time distribution functions { F n } sat-isfying the assumption √ nF n ( x/ √ n ) ≤ C (1 + x m ) in Corollary 1 as well as the scalingassumption ( A .1. Take F n = F for all n (as in Ward and Glynn [21]), where F is differentiable witha derivative of polynomial growth satisfying sup y ∈ [0 ,x ] F ′ ( y ) ≤ C (1 + x m − ) .2. Take F n ( x ) = 1 − exp( − R x h ( √ nu ) du ) (as in Reed and Ward [20]), where h ( · ) issuch that sup y ∈ [0 ,x ] h ( y ) ≤ C (1 + x m − ) .3. For a general sequence { F n } , if F n ( x/ √ n ) converges to a non-negative function h ( · ) uniformly on compact sets and ≤ ( F n ) ′ ( x/ √ n ) ≤ C (1 + x m − ) for some C ∈ (0 , ∞ ) independent of n , then { F n } satisfies the stated assumption in Corollary 1 and thescaling assumption ( A . Remark 2.
Corollary 1 generalizes Propositions 2 and 3 in [16], which are establishedunder the Poisson arrivals (i.e.,
M/GI/ GI model) and essentially follow from theresults in [13]. We use the subscript x to denote the scaled Markov process e X n has an initial state( e τ n (0) , e V n (0)) = ( τ, v ) ≡ x ∈ S . Proposition 2.
Assume ( A – ( A . Let q ∈ [1 , p ) . There exist a time t ∈ (0 , ∞ ) and asufficiently large index n such that for all t ≥ t , lim | x |→∞ sup n ≥ n | x | q IE h | e X nx ( t | x | ) | q i = 0 . (7)9roposition 2 readily yields uniform (in n ) moment bounds for the scaled process e X n ,from which the Lyapunov function methods of Meyn and Tweedie [19] can yield momentbounds uniformly in time t . Ultimately, it implies the moment bounds for the stationarydistributions uniform in the scaling parameter n and hence the tightness of the collectionof the stationary distributions; see Section 5 in Lee et al. [17] for details. Proofs of Theorems 1 and 2.
Given the uniform q -th moment estimate in Proposition2, Theorems 1 and 2 follow from Section 5 in Lee et al. [17] without any modification.The crux in proving Proposition 2 lies in two versions of pathwise stability results(Lemmas 1 and 2 below), whose intuitive ideas are provided right after stating thoseresults. Proof of Proposition 2 .It is sufficient to show that there exists t ∈ IR + such that for all t ≥ t ,lim | x |→∞ sup n | x | q IE [ e τ nx ( t | x | ) q ] = 0 (8)and lim | x |→∞ sup n | x | q IE h e V nx ( t | x | ) q i = 0 . (9)First, we note that the proof of (8) follows from the proof of (11) in Lee et al. [17]without any modification. To complete the proof, we must show (9), which is more in-volved than (8), and proceeds following the approach of Ye and Yao ([23], Lemma 10 andProposition 11). First, we establish two versions of pathwise stability results (Lemmas 1and 2), one for any (fixed) n -th system and the other for the whole sequence. With themoment condition ( A
1) on the system primitives, the pathwise stability results are thenturned into the moment stability in Lemma 3, which finally leads to (9).
Lemma 1. (Stability of e V n ( · ) for large (fixed) n ) Let { r i } i ≥ be a sequence ofnumbers such that r i → ∞ as i → ∞ and assume the sequence of initial states { x i ∈ S } i ≥ satisfies | x i | ≤ r i for all i . Then, there exists some ¯ t > such that for any sufficientlylarge n , the following holds (with probability one), lim i →∞ r i e V nx i ( r i t ) = 0 , u.o.c. for t ≥ ¯ t . (10) Lemma 2. (Stability of e V n ( · ) ) Let { r n } be a sequence of numbers such that r n → ∞ as n → ∞ and assume that the sequence of initial states { x n ∈ S } n ≥ satisfies | x n | ≤ r n .Then, for some ¯ t > , the following holds (with probability one), lim n →∞ r n e V nx n ( r n t ) = 0 , u.o.c. for t ≥ ¯ t . (11)10 emma 3. (Moment stability) The following conclusions hold for some ¯ t > .(a) Letting { r i } and { x i } as in Lemma 1, and the index n be sufficiently large, lim i →∞ IE r qi e V nx i ( r i t ) q = 0 , for t ≥ ¯ t . (12) (b) Letting { r n } and { x n } as in Lemma 2, lim n →∞ IE r qn e V nx n ( r n t ) q = 0 , for t ≥ ¯ t . (13)While the proofs of the above three lemmas are provided in Section 5, we provide someintuitions here, for Lemmas 1 and 2 in particular. Consider the n -th (original) system, V n ( t ). The exogeneous arrival rate of customers is increased by n times to nλ (cf. theequation (2)). When the workload is V n ( t ) = v , the arriving customer will abandon witha probability F n ( v ), and the “effective” arrival rate of customers becomes nλ (1 − F n ( v )).Hence, assuming the server is busy with the service rate µ n , the netput rate of the systemin terms of customers , will be nλ (1 − F n ( v )) − µ n . As the service rate is µ n ( ≈ nλ ),the workload (i.e., the required service time) embodied in each customer is approximately( µ n ) − , and therefore the netput rate of the system in terms of workload will become( µ n ) − [ nλ (1 − F n ( v )) − µ n ]. Since the diffusion-scaled workload inflates the original oneby √ n (i.e., e V n ( t ) = √ nV n ( t )), the above rate immediately translates to the netput ratein terms of diffusion-scaled workload when e V n ( t ) = e v (= √ nv ): √ n ( µ n ) − [ nλ (1 − F n ( v )) − µ n ] = n ( µ n ) − [ √ n ( λ − µ n n ) − √ nF n ( e v √ n )] . (14)Observe that √ n ( λ − µ n /n ) approaches θ from ( A √ nF n ( e v/ √ n ) approaches λH ( e v ) from ( A
3) and will be (strictly) greater than θ for sufficiently large n and largeworkload ˜ v from ( A n ,if the diffusion-scaled workload starts from a large (scaled) workload state e V n (0) = r i asspecified in Lemma 1 (ignoring the residuals for simplicity), it will have a negative drift,which is indeed below a negative constant: n ( µ n ) − [ √ n ( λ − µ n n ) − √ nF n ( r i √ n )] ≤ − ¯ σ < . This workload, e V n ( t ), will reach the “normal” operating state after the initial period withan order of r i / ¯ σ . The normal operating state, scaled by 1 /r i (where r i → ∞ ), will beapproximately zero, and this is characterized by the convergence in (10) in Lemma 1.Simimlar observation applies to Lemma 2 as well. Lastly, Lemma 3 plays a pivotal role inestablishing the key moment estimate in (9). Given Lemma 3, the proof of (9) repeats theone for Proposition 11 of [23]. 11 emark 3. From a technical standpoint, the assumptions ( A and ( A can be relaxedsuch that the above drift (14) converges to a negative constant as both n and ˜ v go to + ∞ . Proof of (9) . Let ¯ t be the time given in Lemma 3, pick any time t > ¯ t , and restrictto all sufficiently large n . The rest of the proof is identical to the proof of (12) in Lee etal. [17]. Lemmas 1, 2 and 3 are almost the same as those in Lee et al. [17]. The differences are thatLemma 1 and Lemma 3(a) require the index n be sufficiently large and also the presenceof time ¯ t > c/ √ n and ǫ in Lee et al. [17]). The proof of Lemma 3 isidentical, and the proof of Lemma 1 follows a similar outline but needs modifications toaccommodate the patience time scaling assumption ( A
3) and the stability condition ( A Proof of Lemma 1.
The proof is a modification of the one for Lemma 1 of Lee et al. [17].We first show that the workload process e V n , scaled by r i , converges to a fluid limit process.Then, we analyze the fluid limit to reach our conclusion in (10). The first part of this proofbasically repeats the corresponding part of Lemma 1 of Lee et al. [17]. For the sake ofcompleteness, we include the full proof. Part 1.
Without loss of generality, assume that as i → ∞ , x i /r i → ¯ x ≡ (¯ τ , ¯ v (0)) with | ¯ x | = ¯ τ + ¯ v (0) ≤
1; otherwise, it suffices to consider any convergent subsequence. Fix theindex n throughout the proof. We also omit the index n and the subscript x i wheneverit does not cause any confusion. For the i -th copy of the n -th system, write the offeredwaiting time as:1 r i e V nx i ( r i t ) ≡ e v i ( t ) = φ i ( t ) + η i ( t ) , with (15) φ i ( t ) ≡ v i r i + n √ nµ n · r i n A n ( r i t ) X j =1 (1 − F n ( V n ( t nj − ))) − √ nt + n √ nµ n · r i (cid:18) S n ( A n ( r i t )) − S nd ( A n ( r i t )) − n M n ( A n ( r i t )) (cid:19) , (16) η i ( t ) ≡ √ nr i Z r i t { V n ( s )=0 } ds. (17)In (16), the centered processes { S n ( · ) } , { S nd ( · ) } , { M n ( · ) } are defined as follows. For k ∈ IN , S n ( k ) ≡ n k X j =1 ( v j − , S nd ( k ) ≡ n k X j =1 ( v j − { V n ( t nj − ) ≥ d nj } , n ( k ) = k X j =1 h [ V n ( t nj − ) ≥ d nj ] − F n ( V n ( t nj − )) i . First, estimate the item associated with the arrival in the above (in the first summation): A n ( r i t ) r i n = 1 r i n (cid:18) A n ( r i t ) − λn ( r i t − τ i √ n ∧ r i t ) (cid:19) + λ ( t − τ i r i √ n ∧ t ) → λ ( t − ¯ τ √ n ∧ t ) , as i → ∞ a.s. (18)Second, denote the term associated with the arrival and abandonment as ξ i ( t ) ≡ r i n A n ( r i t ) X j =1 (1 − F n ( V n ( t nj − ))) . (19)Observe that for any 0 ≤ t < t , we have0 ≤ ξ i ( t ) − ξ i ( t ) ≤ r i n ( A n ( r i t ) − A n ( r i t )) . (20)From (18), we note that the right-hand side in the above converges uniformly to λ ( t − t − ¯ τ √ n ∧ t + ¯ τ √ n ∧ t ). Therefore, any subsequence of i contains a further subsequencesuch that as i → ∞ along the further subsequence, we have the weak convergence ξ i ( · ) ⇒ ¯ ξ ( · ) , in D ( IR ) as i → ∞ , where the limit ¯ ξ ( · ) is Lipschitz continuous (recall (20)) with a Lipschitz constant λ (withprobability one). Without loss of generality, we can assume the above convergence is alongthe full sequence, and furthermore, by using the coupling technique, we can further assumethe convergence is almost surely: ξ i ( t ) → ¯ ξ ( t ) , as i → ∞ a.s . Third, for the martingale terms, we have as i → ∞ with probability one,1 r i (cid:18) S n ( A n ( r i t )) − S nd ( A n ( r i t )) − n M n ( A n ( r i t ) (cid:19) → . (21)Putting the above convergences together yields, as i → ∞ , φ i ( t ) → ¯ φ ( t ) ≡ ¯ v (0) + n √ nµ n ¯ ξ ( t ) − √ nt, u.o.c. of t ≥ . (22)13ote from (15)–(17) that the tuple ( e v i ( t ) , φ i ( t ) , η i ( t )) t ≥ satisfies the one-dimensional linearSkorokhod problem (cf. § e v i ( t ) = φ i ( t ) + η i ( t ) ≥ , dη i ( t ) ≥ η i (0) = 0 , e v i ( t ) dη i ( t ) = 0 . Hence, by invoking the Lipschitz continuity of the Skorokhod mapping (cf. Theorem 6.1of [8]), the convergence in (22) implies1 r i e V nx i ( r i t ) → ¯ v ( t ) and η i ( t ) → ¯ η ( t ) u.o.c. of t ≥ , (23)with the limit satisfying the Skorokhod problem as well:¯ v ( t ) = ¯ φ ( t ) + ¯ η ( t ) ≥ , d ¯ η ( t ) ≥ η (0) = 0 , ¯ v ( t ) d ¯ η ( t ) = 0 . (24)Next, we further examine the limit ¯ ξ ( · ) following the approach of Chen and Ye ([9],Proposition 3(b)). From (18) and (19), and noting that ξ i ( t ) ≤ A n ( r i t ) r i n , we have¯ ξ ( t ) = 0 , ≤ t ≤ ¯ τ √ n . (25)Now, consider any regular time t > ¯ τ / √ n , at which all processes concerned, i.e., ¯ v ( · ) , ¯ φ ( · ) , and ¯ η ( · ) are differentiable, and ¯ v ( t ) >
0. Note that the Lipschitz continuity of ¯ ξ i ( · ) impliesthat (¯ v ( · ) , ¯ φ ( · ) , ¯ η ( · )) are also Lipschitz continuous. Therefore, we can find (small) constants ǫ > δ > i : e v i ( t ) > ǫ i.e., e V n ( r i t ) > r i ǫ, t ∈ [ t , t + δ ) . (26)Observe that if the j -th arrival falls between A n ( r i t )+1 and A n ( r i t ), then its arrival time, t nj , shall also falls between the corresponding time epochs, i.e., r i t < t nj ≤ r i t . Given theestimate in (26), this implies the following estimate holds: e V n ( t nj ) > r i ǫ. Consequently, we have for all sufficiently large i that ξ i ( t ) − ξ i ( t ) = 1 r i n A n ( r i t ) X j = A n ( r i t )+1 (1 − F n ( V n ( t nj − ))) ≤ r i n ( A n ( r i t ) − A n ( r i t ))(1 − F n ( r i ǫ )) , t ∈ [ t , t + δ ) . Part 2.
From the assumptions in ( A
3) and ( A σ > n and x , √ nF n ( x √ n ) ≥ λ ( θ + 2¯ σ ) . t ∈ [ t , t + δ ), ξ i ( t ) − ξ i ( t ) ≤ r i n ( A n ( r i t ) − A n ( r i t ))(1 − √ nλ ( θ + 2¯ σ )) . Taking i → ∞ , this gives¯ ξ ( t ) − ¯ ξ ( t ) ≤ λ ( t − t )(1 − √ nλ ( θ + 2¯ σ )) . In summary, the above implies for any regular time t > ¯ τ / √ n with ¯ v ( t ) > d ¯ ξ ( t ) dt ≤ λ (1 − √ nλ ( θ + 2¯ σ )) . (27)Now, from the properties in (22), (24) and (27), we can see that if ¯ v ( t ) > t ≥ ¯ τ / √ n , d ¯ v ( t ) dt ≤ n √ nµ n λ (1 − √ nλ ( θ + 2¯ σ )) − √ n = nµ n (cid:18) √ n ( λ − µ n n ) − θ − λ ¯ σ ) (cid:19) According to the assumption ( A n is sufficienly large, we can ensure √ n ( λ − µ n n ) − θ ≤ λ ¯ σ . Therefore, the above implies d ¯ v ( t ) dt ≤ − ¯ σ. (28)Moreover, combined with the property (25), the above actually holds for any time t ≥ v ( t ) > v ( t ) = 0 for t ≥ ¯ v (0) / ¯ σ . This property, along with the convergence in (23) yields the desired convergencein (10) with a constant time ¯ t ≥ / ¯ σ . Proof of Lemma 2.
Without loss of generality, we assume θ ≥ A GI/GI/ GI system is dominated by a stable GI/GI/
Part 1.
Consider a special case satisfying the extra condition: there exist a sufficientlylarge K ∗ > σ > x ≥ K ∗ and all sufficiently large n , √ nF n (cid:18) x √ n (cid:19) = θλ + ¯ σ. (29)15or ease of presentation, we assume the above holds for all n . Below, we prove the lemmaunder this extra assumption in two steps. Step 1.
For the n -th system, write the offered waiting time as:1 r n e V nx n ( r n t ) ≡ e v n ( t ) = φ n ( t ) + η n ( t ) , with (30) φ n ( t ) = v n r n + φ n, ( t ) − φ n, ( t ) + φ n, ( t ) ,φ n, ( t ) = n √ nµ n · A n ( r n t ) r n n − √ nt,φ n, ( t ) = n √ nµ n · r n n A n ( r n t ) X j =1 F n ( V n ( t nj − )) , (31) φ n, ( t ) = n √ nµ n · r n (cid:18) S n ( A n ( r n t )) − S nd ( A n ( r n t )) − n M n ( A n ( r n t )) (cid:19) , (32) η n ( t ) = √ nr n Z r n t { V n ( s )=0 } ds. (33)Next, we inspect the terms in φ n ( t ). For the initial (diffusion-scaled) states, we assumewithout loss of generality that x n r n = ( ν n , τ n ) r n → (¯ ν, ¯ τ ) . (34)Otherwise, we can consider a convergent subsequent. We can also see that the last termconverges to 0, i.e., φ n, ( t ) → , u.o.c. of t ≥ . (35)For the term φ n, ( t ), similar to the display (37) of Lee et al. [17], we have A n ( r n t ) r n n = 1 r n n (cid:18) A n ( r n t ) − λn ( r n t − τ n √ n ∧ r n t ) (cid:19) + λ ( t − τ n r n √ n ∧ t ) . Using the above, we can write φ n, ( t ) as, φ n, ( t ) = nµ n √ r n · (cid:20) √ r n n (cid:18) A n ( r n t ) − λn ( r n t − τ n √ n ∧ r n t ) (cid:19)(cid:21) + nλµ n · √ n (1 − µ n nλ ) t − nλµ n ( τ n r n ∧ √ nt ) . r n → ∞ , we have1 √ r n · (cid:20) √ r n n (cid:18) A n ( r n t ) − λn ( r n t − τ n √ n ∧ r n t ) (cid:19)(cid:21) → , u.o.c. of t ≥ . From the heavy traffic condition, we have √ n (1 − µ n nλ ) → θλ . For the last term, we have for t = 0, τ n r n ∧ √ nt = 0 , and for t > τ n r n ∧ √ nt → ¯ τ . In summary, we have for t = 0, φ n, (0) = 0 , (36)and for t > φ n, ( t ) → θλ t − ¯ τ , u.o.c. (37)For the term φ n, ( t ), note that for 0 ≤ t < t , φ n, ( t ) − φ n, ( t ) = nµ n · r n n A n ( r n t ) X j = A n ( r n t )+1 √ nF n ( V n ( t nj − )) ≤ nµ n · r n n ( A n ( r n t ) − A n ( r n t )) (cid:18) θλ + ¯ σ (cid:19) → ( t − t ) (cid:18) θλ + ¯ σ (cid:19) (38)As φ n, ( t ) is increasing in t , the above implies φ n, ( t ) → ˆ φ ( t ) , u.o.c. of t ≥ . (39)where the limit ˆ φ ( t ) is increasing and Lipschitz continuous in t ≥
0, with ˆ φ (0) = 0.17utting (34, 35, 36, 37, 39) together, we have φ n ( t ) → ˆ φ ( t ) , u.o.c. of t > , (40)where the limit ˆ φ ( t ), t ≥
0, is Lipschitz continuous in (0 , ∞ ), and satisfiesˆ φ (0) = ¯ ν, ˆ φ (0+) = max(0 , ¯ ν − ¯ τ ) . (41)The above combined with the reflecting mapping given in (4), we have(˜ v n ( t ) , φ n ( t ) , η n ( t )) → (ˆ v ( t ) , ˆ φ ( t ) , ˆ η ( t )) , u.o.c. of t > , (42)where the limit forms a standard one-dimensional Skorohod mapping. Step 2.
To prove the Lemma for the special case, it suffices to show the followings forany t > ddt ˆ φ ( t ) = − ¯ σ, when ˆ v ( t ) > . (43)Indeed, the above property, along with the characterization in (40-42), imply ˆ v ( t + · ) forsome t >
0, which then justifies the conclusion in (11).Fix any t > v ( t ) >
0. Then, we can find some (small) interval [ t , t ] suchthat ˆ v ( t ′ ) > ǫ for any t ′ ∈ [ t , t ] and for some constant ǫ >
0. Hence, for sufficiently large n , we have for all t ′ ∈ [ t , t ] 1 r n e V n ( r n t ′ ) > ǫ. Note, for any t nj ∈ [ r n t , r n t ] (i.e., t nj /r n ∈ [ t , t ]), we have for sufficiently large n , V n ( t nj − ) = 1 √ n ˜ V n ( t nj − ) ≥ r n ǫ √ n ≥ K ∗ √ n . Therefore, from our assumption (29), we have for sufficiently large n , √ nF n ( V n ( t nj − )) = θλ + ¯ σ. From (31, 39), we have ddt ˆ φ ( t ) = θλ + ¯ σ, t > . The above equality and the convergence (37) together yield (43) immediately.
Part 2.
To prove the lemma for the general case, we construct dominating systemsthat satisfy the condition in the special case first. The dominating systems have thesame settings as the original
GI/GI/ { t nj } and service times { v nj } . However, the patience times in the dominating systems, denoted as d n ∗ j for the j -tharrival of the n -th queue, are generated from those in the original queues using the inversetransformation method as follows.First, under the stability condition ( A
4) and the model assumption ( A K ∗ > σ > n , √ nF n (cid:18) x √ n (cid:19) ≥ θλ + ¯ σ, for x ≥ K ∗ . (44)Now, we specify the distribution F n ∗ ( · ) of patience times for the n -th dominating queue as √ nF n ∗ (cid:18) x √ n (cid:19) = min (cid:26) √ nF n (cid:18) x √ n (cid:19) , θλ + ¯ σ (cid:27) , for x ≥ . (45)Then, the distribution F n ∗ ( · ) satisfies the condition (29), i.e., √ nF n ∗ (cid:18) x √ n (cid:19) = θλ + ¯ σ. (46)Below, for ease of presentation, we assume the above equality holds for all n .Next, we construct the patience times in the dominating queues as, for each ω ∈ Ω , n ≥ , j ≥ d n ∗ j ( ω ) = inf { x : F n ∗ ( x ) = F n ( d nj ( ω )) } . (47)Note that a patience time is allowed to be infinity, in which case the customer will neverabandon. Observe that • for each n , the sequence of patience times { d n ∗ j , j = 1 , , · · · } are i.i.d. and follow thedistribution F n ∗ ( · ), and • the dominating systems have longer patience times (for all samples): d n ∗ j ≥ d nj .Now, as the dominating systems have the patience time distribution F n ∗ ( · ) that satisfythe property (46), they fit into the special case specified in the condition (29). Hence, theconclusion in “modified Lemma 2” holds for the sequence of dominating queues, i.e., theassociated workloads, denoted as V n ∗ ( t ) (or ˜ V n ∗ ( t ) under the diffusion scaling), satisfy theconvergence property (11): lim n →∞ r n e V n ∗ x n ( r n t ) = 0 , u.o.c. for t ≥ t . (48)19oreover, since each customer in the dominating queues has a longer patience time thanthe corresponding one in the original queues, the workload of each dominating queue mustbe (roughly) as much as the original one. Indeed, we can show: V n ∗ ( t ) ≥ V n ( t ) − ν n, max ( t ) , (49)where ν n, max ( t ) is the maximum of all workloads that customers have brought in by time t ≥ ν n, max ( t ) = max { ν nj : j = 1 , · · · , A n ( t ) } . From Lemma 5.1 of [4], we have, with probability 1, the following u.o.c. convergence as n → ∞ : √ nr n ν n, max ( r n t ) (cid:20) = 1 r n ˜ ν n, max ( r n t ) (cid:21) → . (50)(Here, r n n corresponds to r of Lemma 5.1 Bramson [4]. In our definition of ν n, max , A n refers to the arrival (with ν nj being the service time), while in Bramson [4] the correspond-ing variable, v r,T, max , connects to the service times and the service (renewal) process.)Consequently, the conclusion in Lemma 2 follows from (48-50). It remains to prove (49),and this is done by induction. First, note that the inequality holds for t = t n ≡
0, asboth systems share the same initial state ( V n (0) = V n ∗ (0)). Next, we show that if for any J ≥
0, it holds for t = t nJ : V n ∗ ( t nJ ) ≥ V n ( t nJ ) − ν n, max ( t nJ ) , (51)then, it can be extended to the time t ∈ ( t nJ , t nJ +1 ].When t ∈ ( t nJ , t nJ +1 ), by using the evolution equation (1) and the relationship A n ( t nj ) = j (which hold for both the original and the dominating systems), we write V n ( t ) = V n ( t nJ ) − Z tt nJ [ V n ( s ) > ds = (cid:26) V n ( t nJ ) − ( t − t nJ ), t − t nJ < V n ( t nJ ) ,0, t − t nJ ≤ V n ( t nJ ) , (52)and similarly, V n ∗ ( t ) = (cid:26) V n ∗ ( t nJ ) − ( t − t nJ ), t − t nJ < V n ∗ ( t nJ ) ,0, t − t nJ ≤ V n ∗ ( t nJ ) . (53)By carefully examining the sample paths as characterized in (52) and (53), along withthe inductive assumption (51), we can see that the inquality (49) holds for t ∈ ( t nJ , t nJ +1 ).Particularly, this gives V n ∗ ( t nJ +1 − ) ≥ V n ( t nJ +1 − ) − ν n, max ( t nJ +1 − ) . (54)20or t = t nJ +1 , we apply the evoluation equation again to write, V n ( t nJ +1 ) = V n ( t nJ +1 − ) + ν nJ +1 [ V n ( t nJ +1 − ) The proof is identical to the one for Lemma 3 of Lee et al. [17]. Proof of Corollary 1. From the convergence in Theorem 2(a) and a generalized con-tinuous mapping theorem (see, e.g., Theorem 3.4.4 in Whitt [22]), it suffices to (i) es-tablish a uniform integrability of {√ nF n ( e V n ( ∞ ) / √ n ) } n ≥ and (ii) verify the convergence √ nF n ( x n / √ n ) → H ( x ) whenever x n → x as n → ∞ for a nonnegative sequence { x n } n ≥ .The first part (i): The uniform integrability of {√ nF n ( e V n ( ∞ ) / √ n ) } n ≥ follows immedi-ately from the assumed polynomial growth condition √ nF n ( x/ √ n ) ≤ C (1 + x m ) and theuniform integrability of { [ e V n ( ∞ )] m } n ≥ established in Theorem 2(b). The second part(ii): Since a nonnegative sequence { x n } n ≥ is convergent, there is M ∈ (0 , ∞ ) such that x n ≤ M . 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