Big jobs arrive early: From critical queues to random graphs
Gianmarco Bet, Remco van der Hofstad, Johan S. H. van Leeuwaarden
BBig jobs arrive early:From critical queues to random graphs
Gianmarco Bet, Remco van der Hofstad, and Johan S.H. van LeeuwaardenSeptember 14, 2018
Abstract
We consider a queue to which only a finite pool of n customers can arrive, at timesdepending on their service requirement. A customer with stochastic service requirement S arrives to the queue after an exponentially distributed time with mean S − α for some α ∈ [0 , α = 0 gives the so-called ∆ ( i ) /G/ α = 1 is closely related to the exploration process for inhomogeneous randomgraphs [7]. We consider the asymptotic regime in which the pool size n grows to infinityand establish that the scaled queue-length process converges to a diffusion process with anegative quadratic drift. We leverage this asymptotic result to characterize the head startthat is needed to create a long period of activity. We also describe how this first busy periodof the queue gives rise to a critically connected random forest. This paper introduces the ∆ α ( i ) /G/ n customers will join the queue. These n customers are triggered to join the queue afterindependent exponential times, but the rates of their exponential clocks depend on their servicerequirements. When a customer requires S units of service, its exponential clock rings after anexponential time with mean S − α with α ∈ [0 , α , the arrival times are i.i.d. ( α = 0) or decrease with the service requirement ( α ∈ (0 , i of the n customers have already joined the queue, waiting for service. We will take i (cid:28) n , so that withoutloss of generality we can assume that at time zero there are still n customers waiting for service.These initial customers are numbered 1 , . . . , i and the customers that arrive later are numbered i + 1 , i + 2 , . . . in order of arrival. Let A ( k ) denote the number of customers arriving duringthe service time of the k -th customer. The busy periods of this queue will then be completelycharacterized by the initial number of customers i and the random variables ( A ( k )) k ≥ . Notethat the random variables ( A ( k )) k ≥ are not i.i.d. due to the finite-pool effect and the service-dependent arrival rates. We will model and analyze this queue using the queue-length processembedded at service completions.We consider the ∆ α ( i ) /G/ n → ∞ , while imposing at thesame time a heavy-traffic regime that will stimulate the occurrence of a substantial first busyperiod. By substantial we mean that the server can work without idling for quite a while, not1 a r X i v : . [ m a t h . P R ] A p r nly serving the initial customers but also those arriving somewhat later. For this regime weshow that the embedded queue-length process converges to a Brownian motion with negativequadratic drift. For the case α = 0, referred to as the ∆ ( i ) /G/ α = 1 it is closely related to the criticalinhomogeneous random graph studied in [7, 18].While the queueing process consists of alternating busy periods and idle periods, in the∆ α ( i ) /G/ H ( i ) denote the number of customers served in the first busyperiod, starting with i initial customers. We then associate a (directed) random graph to thequeueing process as follows. Say H ( i ) = N and consider a graph with vertex set { , , . . . , N } and in which two vertices r and s are joined by an edge if and only if the r -th customer arrivesduring the service time of the s -th customer. If i = 1, then the graph is a rooted tree with N labeled vertices, the root being labeled 1. If i >
1, then the graph is a forest consisting of i distinct rooted trees whose roots are labeled 1 , . . . , i , respectively. The total number of verticesin the forest is N .This random forest is exemplary for a deep relation between queues and random graphs,perhaps best explained by interpreting the embedded ∆ α ( i ) /G/ exploration pro-cess , a generalization of a branching process that can account for dependent random variables( A ( k )) k ≥ . Exploration processes arose in the context of random graphs as a recursive algo-rithm to investigate questions concerning the size and structure of the largest components [3].For a given random graph, the exploration process declares vertices active, neutral or inactive.Initially, only one vertex is active and all others are neutral. At each time step one activevertex (e.g. the one with the smallest index) is explored, and it is declared inactive afterwards.When one vertex is explored, its neutral neighbors become active for the next time step. Astime progresses, and more vertices are already explored (inactive) or discovered (active), fewervertices are neutral. This phenomenon is known as the depletion-of-points effect and plays animportant role in the scaling limit of the random graph. Let A ( k ) denote the neutral neighborsof the k -th explored vertex. The exploration process then has increments ( A ( k )) k ≥ that eachhave a different distribution. The exploration process encodes useful information about theunderlying random graph. For example, excursions above past minima are the sizes of the con-nected components. The critical behavior of random graphs connected with the emergence ofa giant component has received tremendous attention [2, 6, 7, 8, 9, 10, 18, 14, 15]. Interpretingactive vertices as being in a queue, and vertices being explored as customers being served, wesee that the exploration process and the (embedded) ∆ α ( i ) /G/ A ( k )) k ≥ areidentical.The analysis of the ∆ α ( i ) /G/ A ( k )) k ≥ are not i.i.d. In the case of i.i.d. ( A ( k )) k ≥ , there exists an evendeeper connection between queues and random graphs, established via branching processesinstead of exploration processes [19]. To see this, declare the initial customers in the queue tobe the 0-th generation. The customers (if any) arriving during the total service time of the initial2 customers form the 1-st generation, and the customers (if any) arriving during the total servicetime of the customers in generation t form generation t +1 for t ≥
1. Note that the total progenyof this Galton-Watson branching process has the same distribution as the random variable H ( i )in the queueing process. Through this connection, properties of branching processes can becarried over to the queueing processes and associated random graphs [11, 21, 22, 24, 25, 26].Tak´acs [24, 25, 26] proved several limit theorems for the case of i.i.d. ( A ( k )) k ≥ , in whichcase the queue-length process and derivatives such as the first busy period weakly converge to(functionals of) the Brownian excursion process. In that classical line, the present paper can beviewed as an extension to exploration processes with more complicated dependency structuresin ( A ( k )) k ≥ .In Section 2 we describe the ∆ α ( i ) /G/ α ( i ) /G/ We consider a sequence of queueing systems, each with a finite (but growing) number n ofpotential customers labelled with indices i ∈ [ n ] := { , . . . , n } . Customers have i.i.d. servicerequirements with distribution F S ( · ). We denote with S i the service requirement of customer i and with S a generic random value, and S i and S all have distribution F S ( · ). In order to obtainmeaningful limits as the system grows large, we scale the service speed by n/ (1 + βn − / ) with β ∈ R so that the service time of customer i is given by˜ S i = S i (1 + βn − / ) n . (2.1)We further assume that E [ S α ] < ∞ .If the service requirement of customer i is S i , then, conditioned on S i , its arrival time T i isassumed to be exponentially distributed with mean 1 / ( λS αi ), with α ∈ [0 ,
1] and λ >
0. Hence T i d = Exp i ( λS αi ) (2.2)with d = denoting equality in distribution and Exp i ( c ) an exponential random variable with mean1 /c independent across i . Note that conditionally on the service times, the arrival times areindependent (but not identically distributed). We introduce c (1) , c (2) , . . . , c ( n ) as the indicesof the customers in order of arrival, so that T c (1) ≤ T c (2) ≤ T c (3) ≤ . . . almost surely.We will study the queueing system in heavy traffic, in a similar heavy-traffic regime as in[5, 4]. The initial traffic intensity ρ n is kept close to one by imposing the relation ρ n := λ n E [ S α ](1 + βn − / ) = 1 + βn − / + o P ( n − / ) , (2.3)where λ = λ n can depend on n and f n = o P ( n − / ) is such that lim n →∞ f n n / P →
0. Theparameter β then determines the position of the system inside the critical window: the trafficintensity is greater than one for β >
0, so that the system is initially overloaded, while thesystem is initially underloaded for β <
0. 3ur main object of study is the queue-length process embedded at service completions, givenby Q n (0) = i and Q n ( k ) = ( Q n ( k −
1) + A n ( k ) − + , (2.4)with x + = max { , x } and A n ( k ) the number of arrivals during the k -th service given by A n ( k ) = (cid:88) i/ ∈ ν k { T i ≤ ˜ S c ( k ) } (2.5)where ν k ⊆ [ n ] denotes the set of customers who have been served or are in the queue at thestart of the k -th service. Note that | ν k | = ( k −
1) + Q n ( k −
1) + 1 = k + Q n ( k − . (2.6)Given a process t (cid:55)→ X ( t ), we define its reflected version through the reflection map φ ( · ) as φ ( X )( t ) := X ( t ) − inf s ≤ t X ( s ) − . (2.7)The process Q n ( · ) can alternatively be represented as the reflected version of a certain process N n ( · ), that is Q n ( k ) = φ ( N n )( k ) , (2.8)where N n ( · ) is given by N n (0) = i and N n ( k ) = N n ( k −
1) + A n ( k ) − . (2.9)We assume that whenever the server finishes processing one customer, and the queue is empty,the customer to be placed into service is chosen according to the following size-biased distribu-tion: P (customer j is placed in service | ν i − ) = S αj (cid:80) l / ∈ ν i − S αl , j / ∈ ν i − , (2.10)where we tacitly assumed that customer j is the i -th customer to be served. With definitions(2.5) and (2.10), the process (2.4) describes the ∆ α ( i ) /G/ Remark 1 (A directed random tree) . The embedded queueing process (2.4) and (2.8) givesrise to a certain directed rooted tree. To see this, associate a vertex i to customer i and let c (1)be the root. Then, draw a directed edge to c (1) from c (2) , . . . , c ( A n (1) + 1) so to all customerswho joined during the service time of c (1). Then, draw an edge from all customers who joinedduring the service time of c (2) to c (2), and so on. This procedure draws a directed edge from c ( i ) to c ( i + (cid:80) i − j =1 A n ( j )) , . . . , c ( i + (cid:80) ij =1 A n ( j )) if A n ( i ) ≥
1. The procedure stops when thequeue is empty and there are no more customers to serve. When Q n (0) = i = 1 (resp. i ≥ c ( i ) is 1 + | A n ( i ) | and the total number of vertices in the tree (resp. forest) is given by H Q n (0) = inf { k ≥ Q n ( k ) = 0 } , (2.11)the hitting time of zero of the process Q n ( · ). 4 emark 2 (An inhomogeneous random graph) . If α = 1, the random tree constructed inRemark 1 is distributionally equivalent to the tree spanned by the exploration process of aninhomogeneous random graph. Let us elaborate on this. An inhomogeneous random graph is aset of vertices { i : i ∈ [ n ] } with (possibly random) weights ( W i ) i ∈ [ n ] and edges between them. Ina rank-1 inhomogeneous random graph , given ( W i ) i ∈ [ n ] , i and j share an edge with probability p i ↔ j := 1 − exp (cid:16) − W i W j (cid:80) i ∈ [ n ] W i (cid:17) . (2.12)The tree constructed from the ∆ i ) /G/ i ) /G/ W i ) ni =1 and λ n . Indeed, when W i = (1 + βn − / ) S i for i = 1 , . . . , n , the probabilitythat i and j are connected is given by p j ↔ i = 1 − exp (cid:16) − (1 + βn − / ) S i n S j (cid:80) l ∈ [ n ] S l /n (cid:17) = 1 − exp (cid:16) − ˜ S i S j n (cid:80) i ∈ [ n ] S i (cid:17) = P ( T j ≤ ˜ S i | ( S i ) j ∈ [ n ] ) , (2.13)where T j ∼ exp( λ n ) , (2.14)and λ n = n/ (cid:80) i ∈ [ n ] S i . The rank-1 inhomogeneous random graph with weights ( S i ) ni =1 is saidto be critical (see [7, (1.13)]) if (cid:80) i ∈ [ n ] S i (cid:80) i ∈ [ n ] S i = E [ S ] E [ S ] + o P ( n − / ) = 1 + o P ( n − / ) . (2.15)Consequently, if β = 0 and λ n = n/ (cid:80) i ∈ [ n ] S i , the heavy-traffic condition (2.3) for the ∆ i ) /G/ Remark 3 (Results for the queue-length process) . By definition, the embedded queue (2.4)neglects the idle time of the server. Via a time-change argument it is possible to prove that, inthe limit, the (cumulative) idle time is negligible and the embedded queue is arbitrarily closeto the queue-length process uniformly over compact intervals. This has been proven for the∆ ( i ) /G/ α ( i ) /G/ All the processes we consider are elements of the space D := D ([0 , ∞ )) of c`adl`ag functions thatadmit left limits and are continuous from the right. To simplify notation, for a discrete-timeprocess X ( · ) : N → R , we write X ( t ), with t ∈ [0 , ∞ ), instead of X ( (cid:98) t (cid:99) ). Note that a process5efined in this way has c`adl`ag paths. The space D is endowed with the usual Skorokhod J topology. We then say that a process converges in distribution in ( D , J ) when it converges asa random measure on the space D , when this is endowed with the J topology. We are nowable to state our main result. Recall that Q n ( · ) is the embedded queue-length process of the∆ α ( i ) /G/ Q n ( t ) := n − / Q n ( tn / ) (2.16)be the diffusion-scaled queue-length process. Theorem 1 (Scaling limit for the ∆ α ( i ) /G/ . Assume that α ∈ [0 , , E [ S α ] < ∞ andthat the heavy-traffic condition (2.3) holds. Assume further that Q n (0) = q . Then, as n → ∞ , Q n ( · ) d → φ ( W )( · ) in ( D , J ) , (2.17) where W ( · ) is the diffusion process W ( t ) = q + βt − λ E [ S α ]2 E [ S α ] t + σB ( t ) , (2.18) with σ = λ E [ S α ] E [ S α ] and B ( · ) is a standard Brownian motion. By the Continuous-Mapping Theorem and Theorem 2 we have the following:
Theorem 2 (Number of customers served in the first busy period) . Assume that α ∈ [0 , , E [ S α ] < ∞ and that the heavy-traffic condition (2.3) holds. Assume further that Q n (0) = q .Then, as n → ∞ , the number of customers served in the first busy period BP n := H Q n (0) converges to BP n d → H φ ( W ) (0) , (2.19) where W ( · ) is given in (2.18) . In particular, if |F n | denotes the number of vertices in the forest constructed from the∆ α ( i ) /G/ n → ∞ , |F n | d → H φ ( W ) (0) . (2.20)Theorem 1 implies that the typical queue length for the ∆ α ( i ) /G/ O P ( n / ), and that the typical busy period consists of O P ( n / ) services. The linear drift t → βλt describes the position of the system inside the critical window. For β > W ( · ) is more likely to cause a large initial excursion. For β < W ( · ) has a strong initial negative drift, so that φ ( W )( · ) is close to zero also forsmall t . Finally, the negative quadratic drift t → − λ E [ S α ]2 E [ S α ] t captures the depletion-of-pointseffect . Indeed, for large times, the process W ( t ) is dominated by − λ E [ S α ]2 E [ S α ] t , so that φ ( W )( t )performs only small excursions away from zero. See Figure 1.Let us now compare Theorem 1 with two known results. For α = 0, the limit diffusionsimplifies to W ( t ) = βt − t + σB ( t ) , (2.21)6 tα = 0 α = 0 . α = 1 Figure 1: Sample paths of the process Q n ( · ) for various values of α and n = 10 . The servicetimes are taken unit-mean exponential. The dashed curves represent the drift t (cid:55)→ q + βt − λ E [ S α ] / (2 E [ S α ]) t . In all plots, q = 1, β = 1, λ = 1 / E [ S α ].with σ = λ E [ S ], in agreement with [5, Theorem 5]. In [7] it is shown that, when ( W i ) i ∈ [ n ] arei.i.d. and further assuming that E [ W ] / E [ W ] = 1, the exploration process of the correspondinginhomogeneous random graph converges to W ( t ) = βt − E [ W ]2 E [ W ] t + (cid:112) E [ W ] E [ W ] E [ W ] B ( t ) . (2.22)For α = 1, (2.18) can be rewritten using (2.3) as W ( t ) = βt − E [ S ]2 E [ S ] t + (cid:112) E [ S ] E [ S ] E [ S ] B ( t ) . (2.23)Therefore the two processes coincide if W i = S i , as expected. We now use Theorem 2 to obtain numerical results for the first busy period. We shall alsouse the explicit expression of the probability density function of the first passage time of zeroof φ ( W ) obtained by Martin-L¨of [23], see also [14]. Let Ai( x ) and Bi( x ) denote the classicalAiry functions (see [1]). The first passage time of zero of W ( t ) = q + βt − / t + σB ( t ) hasprobability density [23] f ( t ; β, σ ) = e − (( t − β ) + β ) / σ − βa (cid:90) + ∞−∞ e tu Bi( cu )Ai( c ( u − a )) − Ai( cu )Bi( c ( u − a )) π (Ai( cu ) + Bi( cu ) ) d u, (2.24)where c = (2 σ ) / and a = q/σ >
0. The result (2.24) can be extended to a diffusion with ageneral quadratic drift through the scaling relation W ( τ t ) = τ ( q/τ + βτ t − τ t / σB ( t )).Figure 2 shows the empirical density of BP n , for increasing values of n and various values of α ,together with the exact limiting value (2.24).Table 1 shows the mean busy period for different choices of α and different service timedistributions. We computed the exact value for n = ∞ by numerically integrating (2.24).7 . . . . tα = 0 0 1 2 3 40 . . . tα = 0 . . . . . . . . tα = 1 n = 100 n = 1000 n = 10000 n = ∞ Figure 2: Density plot (black) and Gaussian kernel density estimates (colored) obtained byrunning 10 simulations of a ∆ α ( i ) /G/ n = 100 , , α =0 , / ,
1. In all cases, the service times are exponentially distributed and q = β = E [ S ] = 1.Deterministic Exponential Hyperexponential α n ∞ n − / E [BP n ] for different population sizes and the exact expressionfor n = ∞ computed using (2.24). The service requirements are displayed in order of increasingcoefficient of variation. In all cases q = β = E [ S ] = 1. The hyperexponential service times followa rate λ = 0 .
501 exponential distribution with probability p = 1 / λ = 250 . p = 1 − p = 1 /
2. Each value for finite n is theaverage of 10 simulations.Observe that E [BP n ] decreases with α . This might seem counterintuitive, because the larger α ,the more likely customers with larger service join the queue early, who in turn might initiate alarge busy period. Let us explain this apparent contradiction. When the arrival rate λ is fixed,assumption (2.3) does not necessarily hold and E [BP n ] increases with α , as can be seen in Table2. However, our heavy-traffic condition (2.3) implies that λ depends on α since λ = 1 / E [ S α ].The interpretation of condition (2.3) is that, on average, one customer joins the queue duringone service time. Notice that, due to the size-biasing, the average service time is not E [ S ].Exponential α n nonscaled ∆ α ( i ) /G/ λ = 0 .
01. In all cases q = 1.Each value is the average of 10 simulations. 8herefore, the number of customers that join during a (long) service is roughly equal to oneas α ↑
1. However, when customers with large services leave the system, they are not able tojoin any more. As α ↑
1, customers with large services leave the system earlier. Therefore,as α ↑
1, the resulting second order depletion-of-points effect causes shorter excursions as timeprogresses, see also Figure 1. In the limit process, this phenomenon is represented by the factthat the coefficient of the negative quadratic drift increases as α ↑
1, as shown in the followinglemma.
Lemma 1.
Let α (cid:55)→ f ( α ) := E [ S α ] E [ S α ] E [ S α ] . (2.25) Then f (cid:48) ( α ) ≥ .Proof. Since f (cid:48) ( α ) = 2 E [log( S ) S α ] E [ S α ] E [ S α ] − E [ S α ] E [log( S ) S α ] E [ S α ] E [ S α ] − E [ S α ] E [log( S ) S α ] E [ S α ] E [ S α ] , (2.26) f (cid:48) ( α ) ≥ E [log( S ) S α ] E [ S α ] E [ S α ] ≥ E [ S α ] E [ S α ] E [log( S ) S α ]+ E [ S α ] E [ S α ] E [log( S ) S α ] . (2.27)We split the left-hand side in two identical terms and show that each of them dominates oneterm on the right-hand side. That is E [log( S ) S α ] E [ S α ] E [ S α ] ≥ E [ S α ] E [ S α ] E [log( S ) S α ] , (2.28)the proof of the second bound being analogous. The inequality (2.28) is equivalent to E [(log( S ) S α ) S α ] E [ S α ] ≥ E [ S α S α ] E [ S α ] E [log( S ) S α ] E [ S α ] . (2.29)The term on the left and the two terms on the right can be rewritten as the expectation of asize-biased random variable W , so that (2.29) is equivalent to E [log( W ) W α ] ≥ E [log( W )] E [ W α ] . (2.30)Finally, the inequality (2.30) holds because W is positive with probability one and x (cid:55)→ log( x )and x (cid:55)→ x α are increasing functions. The proof of Theorem 1 extends the techniques we developed in [5]. However, the dependencystructure of the arrival times complicates the analysis considerably. Customers with larger jobsizes have a higher probability of joining the queue quickly, and this gives rise to a size-biasingreordering of the service times. In the next section we study this phenomenon in detail.9 .1 Preliminaries
Given two sequences of random variables ( X n ) n ≥ and ( Y n ) n ≥ , we say that X n converges inprobability to X , and we denote it by X n P → X , if P ( | X n − X | > ε ) → n → ε > X n = o P ( Y n ) if X n /Y n P → X n = O P ( Y n ) if ( X n /Y n ) n ≥ is tight. Given tworeal-valued random variables X , Y we say that X stochastically dominates Y and denote it by Y (cid:22) X , if P ( X ≤ x ) ≤ P ( Y ≤ x ) for all x ∈ R .For our results, we condition on the entire sequence ( S i ) i ≥ . More precisely, if the randomvariables that we consider are defined on the probability space (Ω , F , P ), then we define a newprobability space (Ω , F S , P S ), with P S ( A ) := P ( A | ( S i ) ∞ i =1 ) and F S := σ ( {F , ( S i ) ∞ i =1 } ), the σ -algebra generated by F and ( S i ) ∞ i =1 . Correspondingly, for any random variable X on Ω wedefine E S [ X ] as the expectation with respect to P S , and E [ X ] for the expectation with respectto P . We say that a sequence of events ( E n ) n ≥ holds with high probability (briefly, w.h.p.) if P ( E n ) → n → ∞ .First, we recall a well-known result that will be useful on several occasions. Lemma 2.
Assume ( X i ) ni =1 is a sequence of positive i.i.d. random variables such that E [ X i ] < ∞ . Then max i ∈ [ n ] X i = o P ( n ) .Proof. We have the inclusion of events (cid:110) max i ∈ [ n ] X i ≥ εn (cid:111) ⊆ n (cid:91) i =1 (cid:110) X i ≥ εn (cid:111) . (3.1)Therefore, P (max i ∈ [ n ] X i ≥ εn ) ≤ n (cid:88) i =1 P ( X i ≥ εn ) . (3.2)Since for any positive random variable Y , ε { Y ≥ ε } ≤ Y { Y ≥ ε } almost surely, it follows P (max i ∈ [ n ] X i ≥ εn ) ≤ (cid:80) ni =1 E [ X i { X i ≥ εn } ] εn = E [ X { X ≥ εn } ] ε . (3.3)The right-most term tends to zero as n → ∞ since E [ X ] < ∞ , and this concludes the proof.Given a vector ¯ x = ( x , x , . . . , x n ) with deterministic, real-valued entries, the size-biasedordering of ¯ x is a random vector X ( s ) = ( X ( s )1 , X ( s )2 , . . . , X ( s ) n ) such that P ( X ( s )1 = x j ) = x j (cid:80) nl =1 x l , P ( x ( s )2 = x j | X ( s )1 ) = x j (cid:80) nl =1 x l − x ( s )1 , . . . (3.4)More generally, for any α ∈ R the α -size-biased ordering of ¯ x is given by a vector ¯ X ( α ) =( X ( α )1 , X ( α )2 , . . . , X ( α ) n ) such that P ( X ( α )1 = x j ) = x αj (cid:80) nl =1 x αl , P ( X ( α )2 = x j | X ( α )1 = x i ) = x αj (cid:80) nl =1 x αl − x αi , . . . (3.5)Finally, we define S k = { c (1) , . . . , c ( k ) } (3.6)as the set of the first k customers served. The following lemma is the first step in understandingthe structure of the arrival process: 10 emma 3 (Size-biased reordering of the arrivals) . The order of appearance of customers is the α -size-biased ordering of their service times. In other words, P S ( c ( j ) = i | S j − ) = S αi (cid:80) l / ∈ S j − S αl . (3.7) Proof.
Conditioned on ( S l ) nl =1 , the arrival times are independent exponential random variables.By basic properties of exponentials, we have P S ( c ( j ) = i | S j − ) = P S (min { T l : l / ∈ S j − } = T i | S j − ) = S αi (cid:80) l / ∈ S j − S αl , (3.8)as desired.We remark that (3.8) differs from the classical size-biased reordering in that the weightsare a non-linear function of the ( S i ) ni =1 . The next lemma is crucial, establishing stochasticdomination between the service requirements of the customers in order of appearance. In ourdefinition of the queueing process (2.4)–(2.5), we do not keep track of the service requirementsof the customers that join the queue, but only of their arrival times (2.2). Therefore, at the startof service, a customer’s service requirement is a random variable that depends on the arrivaltime relative to the remaining customers. Lemma 3 then gives the precise distribution of theservice requirement of the j -th customer entering service.Recall that X stochastically dominates Y (with notation Y (cid:22) X ) if and only if there existsa probability space ( ¯Ω , ¯ F , ¯ P ) and two random variables ¯ X , ¯ Y defined on ¯Ω such that ¯ X d = X ,¯ Y d = Y and ¯ P ( ¯ Y ≤ ¯ X ) = 1. Lemma 4.
Assume that α > . Let f : R + → R be a function such that E [ f ( S ) S α ] < ∞ . Thenthere exists a constant C f, S such that almost surely, for n large enough, E S [ f ( S c ( k ) )] ≤ C f, S < ∞ , (3.9) uniformly in k ≤ cn , for a fixed c ∈ (0 , .Proof. We compute explicitly E S [ f ( S c ( k ) )] = E S (cid:104) (cid:80) j / ∈ S k − f ( S j ) S αj (cid:80) j / ∈ S k − S αj (cid:105) = E S (cid:104) (cid:80) j ∈ [ n ] f ( S j ) S αj − (cid:80) j ∈ S k f ( S j ) S αj (cid:80) j / ∈ S k − S αj (cid:105) ≤ E S (cid:104) (cid:80) j / ∈ S k − S αj (cid:105) (cid:88) j ∈ [ n ] f ( S j ) S αj . (3.10)We have the almost sure bound1 (cid:80) j / ∈ S k − S αj = 1 (cid:80) j ∈ [ n ] S αj − (cid:80) j ∈ S k − S αj ≤ (cid:80) j ∈ [ n ] S αj − (cid:80) j ∈ S k − S αj ≤ (cid:80) j ∈ [ n ] S αj − (cid:80) k − j =1 S α ( n − j +1) = 1 (cid:80) n − k +1 j =1 S α ( j ) , (3.11)11here S α (1) ≤ S α (2) ≤ . . . ≤ S α ( n ) denote the order statistics of the finite sequence ( S αi ) i ∈ [ n ] . Thereexists p ∈ (0 ,
1) such that n − k + 1 ≥ pn , for large enough n . Consequently,1 (cid:80) j / ∈ S k − S αj ≤ (cid:80) (cid:98) pn (cid:99) j =1 S α ( j ) , (3.12)so that we have E S [ f ( S c ( k ) )] ≤ (cid:80) j ∈ [ n ] f ( S j ) S αj (cid:80) (cid:98) pn (cid:99) j =1 S α ( j ) . (3.13)Let us denote by ξ p the p -th quantile of the distribution F S ( · ) and let us assume, without lossof generality, that f S ( ξ p ) > S ( (cid:98) np (cid:99) ) = F − n, S ( (cid:98) np (cid:99) /n ), where F n, S ( t ) = (cid:80) ni =1 { S i ≤ t } /n is the empirical distribu-tion function of the ( S i ) ni =1 , and ξ p = F − S ( p ). Indeed, the assumption f S ( ξ p ) > F S ( · ) is invertible in a neighborhood of ξ p . We have that, as n → ∞ , S ( (cid:98) np (cid:99) ) a . s . → ξ p . (3.14)In particular, as n → ∞ ,1 n (cid:12)(cid:12)(cid:12) (cid:88) j ∈ [ n ] S j { S j ≤ ξ p } − (cid:88) j ∈ [ n ] S j { S j ≤ S ( (cid:98) pn (cid:99) ) } (cid:12)(cid:12)(cid:12) a . s . → . (3.15)Therefore, by the strong Law of Large Numbers, as n → ∞ , (cid:80) (cid:98) pn (cid:99) j =1 S ( j ) n a . s . → E [ S { S ≤ ξ p } ] . (3.16)Then, choosing C n,f, S = E [ f ( S ) S α ] / E [ S { S ≤ ξ p } ] + ε , for an arbitrary ε >
0, gives the desiredresult.If α >
0, as is the case in our setting, the proof of Lemma 4 shows that, uniformly in k = O ( n / ), E S [ f ( S c ( k ) )] ≤ (cid:80) j ∈ [ n ] f ( S j ) S αj (cid:80) (cid:98) pn (cid:99) j =1 S α ( j ) = (cid:80) j ∈ [ n ] f ( S j ) S αj (cid:80) nj =1 S α ( j ) (cid:16) (cid:80) nj = (cid:98) pn (cid:99) S α ( j ) (cid:80) (cid:98) pn (cid:99) j =1 S α ( j ) (cid:17) , (3.17)and therefore E S [ f ( S c ( k ) )] ≤ E S [ f ( S c (1) )](1 + O P S (1)) . (3.18)If f ( · ) is an increasing function, (3.18) makes precise the intuition that, if α >
0, customerswith larger job sizes join the queue earlier. We will often make use of the expression (3.18).The following lemma will often prove useful in dealing with sums over a random index set:
Lemma 5 (Uniform convergence of random sums) . Let ( S j ) nj =1 be a sequence of positive randomvariables such that E [ S α ] < + ∞ , for α ∈ (0 , . Then, sup X ⊆ [ n ] |X | = O P ( n / ) n (cid:88) j ∈X S αj = o P (1) . (3.19)12 roof. By Lemma 2, max j ∈ [ n ] S αj = o P ( n α/ (2+ α ) ). This givessup X ⊆ [ n ] |X | = O P ( n / ) n (cid:88) j ∈X S αj ≤ max j ∈ [ n ] S αj n / O P (1) = o P ( n α − / − α/ α ) = o P ( n α − α ) . (3.20)Since α − ≤ i -th customer joining the queue (for i large) and characterize thedistribution of its service time. In particular, for α > S i . Lemma 6 (Size-biased distribution of the service times) . For every bounded, real-valued con-tinuous function f ( · ) , as n → ∞ , E S [ f ( S c ( i ) ) | F i − ] P → E [ f ( S ) S α ] E [ S α ] , (3.21) uniformly for i = O P S ( n / ) . Moreover, as n → ∞ , E S [ f ( S c ( i ) )] → E [ f ( S ) S α ] E [ S α ] , for i = O P S ( n / ) . (3.22) Proof.
First note that E S [ f ( S c ( i ) ) | F i − ] = (cid:88) j / ∈ S i − f ( S j ) P S ( c ( i ) = j | F i − ) = (cid:88) j / ∈ S i − f ( S j ) S αj (cid:80) l / ∈ S i − S αl . (3.23)This can be further decomposed as E S [ f ( S c ( i ) ) | F i − ] = (cid:80) nj =1 f ( S j ) S αj − (cid:80) j ∈ S i − f ( S j ) S αj (cid:80) nl =1 S αl − (cid:80) l ∈ S i − S αl . (3.24)Since | S i − | = i − i = O P ( n / ), by the Law of Large Numbers and Lemma 5, (cid:80) j / ∈ S i − f ( S j ) S αj n P → E [ f ( S ) S α ] , (cid:80) l / ∈ S i − S αl n P → E [ S α ] . (3.25)uniformly in i = O P ( n / ). This gives the first claim.Furthermore, we bound E S [ f ( S c ( i ) ) | F i − ] as E S [ f ( S c ( i ) ) | F i − ] = (cid:88) j / ∈ S i − f ( S j ) S αj (cid:80) l / ∈ S i − S αl ≤ sup x ≥ f ( x ) < ∞ . (3.26)Since E S [ f ( S c ( i ) )] = E S [ E S [ f ( S c ( i ) ) | F i − ]], using (3.21) and the Dominated ConvergenceTheorem the second claim follows.In Lemma 6 we have studied the distribution of the service time of the i -th customer, andwe now focus on its (conditional) moments. The following lemma should be interpreted asfollows: Because of the size-biased re-ordering of the customer arrivals, the service time of the i -th customer being served (for i large) is highly concentrated.13 emma 7. For any fixed γ ∈ [ − , , E S [ S γc ( i ) | F i − ] = E [ S γ + α ] E [ S α ] + o P (1) for i = O P S ( n / ) , (3.27) where the error term is uniform in i = O P S ( n / ) . Moreover, the convergence holds in L , i.e. E S (cid:104)(cid:12)(cid:12)(cid:12) E S [ S γc ( i ) | F i − ] − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12)(cid:105) = o P (1) , (3.28) uniformly in i = O P S ( n / ) .Proof. In order to apply Lemma 6, we first split S γc ( i ) = ( S c ( i ) ∧ K ) γ + (( S c ( i ) − K ) + ) γ , (3.29)where K > E S [ S γc ( i ) | F i − ] = E S [( S c ( i ) ∧ K ) γ | F i − ] + E S [(( S c ( i ) − K ) + ) γ | F i − ] . (3.30)The first term is bounded, and therefore converges to E [( S ∧ K ) γ S α ] / E [ S α ] by Lemma 6. Thesecond term is bounded through Markov’s inequality, as P S ( E S [(( S c ( i ) − K ) + ) γ | F i − ] ≥ ε ) ≤ E S [(( S c ( i ) − K ) + ) γ ] ε . (3.31)Next we apply Lemma 4 with f ( x ) = f K ( x ) = (( x − K ) + ) γ , E S [(( S c ( i ) − K ) + ) γ ] ≤ C f K , S . (3.32)Therefore, (cid:12)(cid:12)(cid:12) E S [ S γc ( i ) | F i − ] − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E S [( S c ( i ) ∧ K ) γ | F i − ] − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12) + C f K , S . (3.33)The proof of Lemma 4 shows that, for any ε >
0, lim K →∞ C f K , S ≤ ε , thus lim K →∞ C f K , S = 0.Therefore, by letting K → ∞ in (3.33), (3.27) follows. Next, we split E S (cid:104)(cid:12)(cid:12)(cid:12) E S [ S γc ( i ) | F i − ] − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12)(cid:105) ≤ E S (cid:104)(cid:12)(cid:12)(cid:12) ( S c ( i ) ∧ K ) γ − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12)(cid:105) + E S [(( S c ( i ) − K ) + ) γ ] . (3.34)The second term can be bounded as in (3.32). For the first term, E S (cid:104)(cid:12)(cid:12)(cid:12) ( S c ( i ) ∧ K ) γ − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12)(cid:105) ≤ (cid:12)(cid:12)(cid:12) (cid:80) nj =1 ( S j ∧ K ) γ S αj (cid:80) nj =1 S αj − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12) + E S (cid:104)(cid:12)(cid:12)(cid:12) (cid:80) nj =1 ( S j ∧ K ) γ S αj (cid:80) l ∈ S i − S αl ( (cid:80) nj =1 S αj ) (cid:12)(cid:12)(cid:12)(cid:105) + E S (cid:104)(cid:12)(cid:12)(cid:12) (cid:80) nl =1 S αl (cid:80) j ∈ S i − ( S j ∧ K ) γ S αj ( (cid:80) nj =1 S αj ) (cid:12)(cid:12)(cid:12)(cid:105) , (3.35)14here we have used that | ( a − b ) / ( c − d ) − a/c | ≤ ad/c + bc/c , for positive a , b , c , d . Thesecond and third terms converge uniformly over i = O P S ( n / ) by Lemma 5. Summarizing, E S (cid:104)(cid:12)(cid:12)(cid:12) E S [ S γc ( i ) | F i − ] − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12)(cid:105) ≤ (cid:12)(cid:12)(cid:12) (cid:80) nj =1 ( S j ∧ K ) γ S αj (cid:80) nj =1 S αj − E [ S γ + α ] E [ S α ] (cid:12)(cid:12)(cid:12) + (cid:80) nl =1 (( S l − K ) + ) γ (cid:80) nj =1 S αj + o P (1) . (3.36)Letting first n → ∞ and then K → ∞ , (3.28) follows.We will make use of Lemma 7 several times throughout the proof, with the specific choices γ ∈ { , α, } . The following lemma is of central importance in the proof of the uniform conver-gence of the quadratic part of the drift: Lemma 8. As n → ∞ , n − / sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) − E [ S α ] E [ S ] (cid:17)(cid:12)(cid:12)(cid:12) P → . (3.37) Proof.
By Lemma 7, (3.37) is equivalent to n − / sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) − E [ S αc ( i ) | F i − ] (cid:17)(cid:12)(cid:12)(cid:12) P → . (3.38)We split the event space and separately bound n − / sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) { S αc ( i ) ≤ K n } − E [ S αc ( i ) { S αc ( i ) ≤ K n } | F i − ] (cid:17)(cid:12)(cid:12)(cid:12) (3.39)and n − / sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) { S αc ( i ) >K n } − E [ S αc ( i ) { S αc ( i ) >K n } | F i − ] (cid:17)(cid:12)(cid:12)(cid:12) , (3.40)for a sequence ( K n ) n ≥ that we choose later on and is such that K n → ∞ . We start with (3.39).Since the sum inside the absolute value is a martingale as a function of j , (3.39) can be boundedthrough Doob’s L p inequality [20, Theorem 11.2] with p = 2 as P S (cid:16) sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) { S αc ( i ) ≤ K n } − E S [ S αc ( i ) { S αc ( i ) ≤ K n } | F i − ] (cid:17)(cid:12)(cid:12)(cid:12) ≥ εn / (cid:17) ≤ εn / E S (cid:104) tn / (cid:88) i =1 ( S αc ( i ) { S αc ( i ) ≤ K n } − E S [ S αc ( i ) { S αc ( i ) ≤ K n } | F i − ]) (cid:105) ≤ εn / tn / (cid:88) i =1 E S [ S αc ( i ) { S αc ( i ) ≤ K n } ] ≤ εn / tn / (cid:88) i =1 K αn E S [ S c ( i ) ] . (3.41)15emma 7 allows us to approximate E S [ S c ( i ) ] uniformly by E [ S α ] E [ S α ] . Thus, we get2 εn / tn / (cid:88) i =1 (cid:16) K αn E [ S α ] E [ S α ] + o P (1) (cid:17) = tK αn εn / O P (1) , (3.42)which converges to zero as n → ∞ if and only if K αn /n / does. We now turn to (3.40) andapply Doob’s L martingale inequality [20, Theorem 11.2] to obtain P S (cid:16) sup j ≤ tn / (cid:12)(cid:12)(cid:12) j (cid:88) i =1 (cid:16) S αc ( i ) { S αc ( i ) >K n } − E S [ S αc ( i ) { S αc ( i ) >K n } | F i − ] (cid:17)(cid:12)(cid:12)(cid:12) ≥ εn / (cid:17) ≤ εn / E S (cid:104)(cid:12)(cid:12)(cid:12) tn / (cid:88) i =1 ( S αc ( i ) { S αc ( i ) >K n } − E S [ S αc ( i ) { S αc ( i ) >K n } | F i − ]) (cid:12)(cid:12)(cid:12)(cid:105) ≤ εn / tn / (cid:88) i =1 E S [ S αc ( i ) { S αc ( i ) >K n } ] ≤ εn / tn / (cid:88) i =1 E S [ S αc (1) { S αc (1) >K n } ](1 + O P S (1))= 2 tε E S [ S αc (1) { S αc (1) >K n } ](1 + O P S (1)) = o P (1) . (3.43)We have used Lemma 7 in the second inequality, and Lemma 4 with f ( x ) = x α { x α >K n } in the third. The right-most term in (3.43) is o P (1) as n → ∞ by the strong Law of LargeNumbers. Note that this side of the bound does not impose additional conditions on K n , sothat, if we take K n = n c , it is sufficient that c < α , with the convention that = ∞ .We conclude this section with a technical lemma concerning error terms in the computationsof quadratic variations. Denote the density (resp. distribution function) of a rate λ exponentialrandom variable by f E ( · ) (resp. F E ( · )): Lemma 9.
We have E S (cid:104) (cid:88) h,q ∈ [ n ] (cid:12)(cid:12)(cid:12) F E (cid:16) S c ( i ) S αh n (cid:17) − λS c ( i ) S αh n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F E (cid:16) S c ( i ) S αq n (cid:17) − λS c ( i ) S αq n (cid:12)(cid:12)(cid:12) | F i − (cid:105) = o P (1) (3.44) uniformly in i = O ( n / ) .Proof. Since | F E ( x ) − x | = O ( x ), the bound | λS c ( i ) S αh /n − F E ( S c ( i ) S αh /n ) | ≤ C ( S c ( i ) S αh /n ) ε holds almost surely for 0 < ε < C >
0, which gives λ (cid:88) h,q ∈ [ n ] E S (cid:104)(cid:16) S c ( i ) S αh n (cid:17) ε (cid:16) S αq S c ( i ) n (cid:17) ε | F i − (cid:105) = λ n ε (cid:88) h,q ∈ [ n ] E S [ S εc ( i ) | F i − ] S α (1+ ε ) h S α (1+ ε ) q (3.45)16herefore, λ (cid:88) h,q ∈ [ n ] E S (cid:104)(cid:16) S c ( i ) S αh n (cid:17) ε (cid:16) S αq S c ( i ) n (cid:17) ε | F i − (cid:105) ≤ λ n ε max j ∈ [ n ] S εj E S [ S c ( i ) | F i − ] (cid:88) h,q ∈ [ n ] S α (1+ ε ) h S α (1+ ε ) q ≤ λ E [ S α ] E [ S α ] max j ∈ [ n ] S εj n ε n (cid:88) h,q ∈ [ n ] S α (1+ ε ) h S α (1+ ε ) q + o P (1) , (3.46)where in the last step we used Lemma 7. Note that, since E [ S α ] < ∞ , by Lemma 2max j ∈ [ n ] S εj = o P ( n ε/ (2+ α ) ). The right-most term in (3.46) then tends to zero as n tendsto infinity as long as 0 < ε < min { , /α } . We first establish some preliminary estimates on N n ( · ) that will be crucial for the proof ofconvergence. We will upper bound the process N n ( · ) by a simpler process N U n ( · ) in such a waythat the increments of N U n ( · ) almost surely dominate the increments of N n ( · ). We also showthat, after rescaling, N U n ( · ) converges in distribution to W ( · ). The process N U n ( · ) is defined as N U n (0) = N n (0), and N U n ( k ) = N U n ( k −
1) + A U n ( k ) − , (4.1)where A U n ( k ) = (cid:88) i/ ∈ S k { T i ≤ c n,β S c ( k ) /n } , (4.2)with c n,β = 1 + βn − / , (4.3)and T i d = exp i ( λS αi ) . (4.4)An interpretation of the process N U n ( · ) is that customers are not removed from the pool ofpotential customers until they have been served. Therefore, a customer could potentially jointhe queue more than once. We couple the processes N n ( · ) and N U n ( · ) as follows. Consider asequence of arrival times ( T i ) ∞ i =1 and of service times ( S i ) ∞ i =1 , then define A n ( · ) as (2.5) and A U n ( · ) as (4.2). With this coupling we have that, almost surely, A n ( k ) ≤ A U n ( k ) ∀ k ≥ . (4.5)Consequently, N n ( k ) ≤ N U n ( k ) ∀ k ≥ , (4.6)and Q n ( k ) = φ ( N n )( k ) ≤ φ ( N U n )( k ) =: Q U n ( k ) ∀ k ≥ , (4.7)almost surely.While in general only the upper bounds (4.6) and (4.7) hold, the processes N n ( · ) and N U n ( · )(resp. Q n ( · ) and Q U n ( · )) turn out to be, very close to each other. We start by proving results17or N U n ( · ) and Q U n ( · ) because they are easier to treat, and only then we are able to prove thatidentical results hold for N n ( · ) and Q n ( · ).In fact, we introduce the upper bound N Un ( · ) to deal with the complicated index set forthe summation in (2.5). The difficulty arises as follows: in order to estimate N n ( · ) one hasto estimate A n ( · ). To do this, one has to separately (uniformly) bound each element in thesum, and also estimate the number of elements in the sum. The first goal is accomplished,for example, through Lemma 7, while for the second the crude upper bound n is not strictenough. However, estimating | ν k | requires an estimate on N n ( · ) itself, as (2.6) shows. To solvethis circularity, we introduce a bootstrap argument: first, we upper bound N n ( · ) and we obtainestimates on the upper bound, from this follows an estimate on | ν k | , and this in turn allows usto estimate N n ( · ).This technique can be applied to solve a recently found technical issue in the proof of themain result of [7]. The authors in [7] prove convergence of a process which upper boundsthe exploration process of the graph. Therefore, their main result is analogous to Theorem3. However, a further step is required to complete the proof of convergence of the explorationprocess, and this is provided by our approach. Theorem 3 (Convergence of the upper bound) . n − / N U n ( tn / ) d → W ( t ) in ( D , J ) as n → ∞ , (4.8) where W ( · ) is the diffusion process in (2.18) . In particular, n − / φ ( N U n )( tn / ) d → φ ( W )( t ) in ( D , J ) as n → ∞ . (4.9)The next section is dedicated to the proof of Theorem 3. We use a classical martingale decomposition followed by a martingale FCLT. The process N U n ( · )in (4.1) can be decomposed as N U n ( k ) = M U n ( k ) + C U n ( k ), where M U n ( · ) is a martingale and C U n ( · )is a drift term, as follows: M U n ( k ) = k (cid:88) i =1 ( A U n ( i ) − E S [ A U n ( i ) | F i − ]) ,C U n ( k ) = k (cid:88) i =1 ( E S [ A U n ( i ) | F i − ] − . (4.10)Moreover, ( M U n ( k )) can be written as ( M U n ( k )) = Z U n ( k ) + B U n ( k ) with Z U n ( k ) a martingaleand B U n ( k ) the compensator, or quadratic variation, of M U n ( k ) given by B U n ( k ) = k (cid:88) i =1 ( E S [( A U n ( i )) | F i − ] − E S [ A U n ( i ) | F i − ] ) . (4.11)In order to prove convergence of N U n ( · ) we separately prove convergence of C U n ( · ) and of M U n ( · ). We prove the former directly, and the latter by applying the martingale FCLT [13,Theorem 7.1.4]. For this, we need to verify the following conditions:18i) sup t ≤ ¯ t | n − / C U n ( tn / ) − βt + λ E [ S α ]2 E [ S α ] t | P −→ , ∀ ¯ t ∈ R + ;(ii) n − / B U n ( tn / ) P −→ σ t, ∀ t ∈ R + ;(iii) lim n →∞ n − / E S [sup t ≤ ¯ t | B U n ( tn / ) − B U n ( tn / − ) | ] = 0 , ∀ ¯ t ∈ R + ;(iv) lim n →∞ n − / E S [sup t ≤ ¯ t | M U n ( tn / ) − M U n ( tn / − ) | ] = 0 , ∀ ¯ t ∈ R + . First we obtain an explicit expression for E [ A U n ( i ) | F i − ], as E S [ A U n ( i ) | F i − ] = (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) (cid:88) l / ∈ S i − ∪{ j } F E (cid:16) c n,β S j S αl n (cid:17) (4.12)= (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) n (cid:88) l =1 c n,β λS j S αl n − (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) (cid:88) l ∈ S i − ∪{ j } c n,β λS j S αl n + (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) (cid:88) l / ∈ S i − ∪{ j } (cid:16) F E (cid:16) c n,β S j S αl n (cid:17) − c n,β λS j S αl n (cid:17) The third term is an error term. Indeed, for some ζ n ∈ [0 , S c ( i ) S l /n ], E S (cid:104)(cid:12)(cid:12)(cid:12) (cid:88) l / ∈ S i − ∪{ j } F E (cid:16) S c ( i ) S αl n (cid:17) − λS c ( i ) S αl n (cid:12)(cid:12)(cid:12) | F i − (cid:105) ≤ (cid:88) l ∈ [ n ] E S (cid:104)(cid:12)(cid:12)(cid:12) F E (cid:16) S c ( i ) S αl n (cid:17) − λS c ( i ) S αl n (cid:12)(cid:12)(cid:12) | F i − (cid:105) = 12 n E S [ | F (cid:48)(cid:48) E ( ζ n ) S c ( i ) | | F i − ] (cid:88) l ∈ [ n ] S αl ≤ λ n E S [ S c ( i ) | F i − ] (cid:88) l ∈ [ n ] S αl , (4.13)since | F (cid:48)(cid:48) E ( x ) | ≤ λ for all x ≥
0. By Lemma 7 this can be bounded by λ n ( C n + o P (1)) (cid:88) l ∈ [ n ] S αl , (4.14)19here C n is bounded w.h.p. and the o P (1) term is uniform in i = O ( n / ). Therefore, the thirdterm in (4.12) is o P ( n − / ). The remaining terms in (4.12) can be simplified as E S [ A U n ( i ) | F i − ] − (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) c n,β λS j (cid:80) l ∈ [ n ] S αl n − (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) (cid:88) l ∈ S i − c n,β λS j S αl n − c n,β λ (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) S αj n − o P ( n − / )= (cid:16) c n,β λ (cid:80) l ∈ [ n ] S αl n E [ S c ( i ) | F i − ] − (cid:17) − c n,β E S [ S c ( i ) | F i − ] (cid:88) l ∈ S i − λ S αl n − c n,β λn E S [ S αc ( i ) | F i − ] + o P ( n − / ) . (4.15)For the first term of (4.15), using ca − b = ca + ca − b ba , with a = (cid:80) l ∈ [ n ] S αl and b = (cid:80) l ∈ S i − S αl , c n,β λ (cid:80) nl =1 S αl n E S [ S c ( i ) | F i − ] − c n,β λ (cid:80) l ∈ [ n ] S αl n (cid:88) j / ∈ S i − S αj (cid:80) l ∈ [ n ] S αl − c n,β λ (cid:80) l ∈ [ n ] S αl n (cid:88) j / ∈ S i − S αj (cid:80) l / ∈ S i − S αl (cid:80) s ∈ S i − S αs (cid:80) l ∈ [ n ] S αl = (cid:16) c n,β λn (cid:88) j / ∈ S i − S αj − (cid:17) + c n,β E S [ S c ( i ) | F i − ] (cid:88) s ∈ S i − λ S αs n . (4.16)Note that the right-most term in (4.16) and the second term in (4.15) cancel out. This cancel-lation is what makes the analysis of N U n ( · ) considerably easier than the analysis of N n ( · ).Moreover, Lemma 7 implies that the third term in (4.15) is also o P ( n − / ). (4.12) is thensimplified to E S [ A U n ( i ) | F i − ] − c n,β λn (cid:88) j / ∈ S i − S αj − o P ( n − / )= (cid:16) c n,β λn n (cid:88) j =1 S αj − (cid:17) − c n,β λn (cid:88) j ∈ S i − S αj + o P ( n − / )= (cid:16) c n,β λn n (cid:88) j =1 S αj − (cid:17) − c n,β λn i − (cid:88) j =1 S αc ( j ) + o P ( n − / ) , (4.17)and the o P ( n − / ) term is uniform in i = O ( n / ). We are now able to compute n − / C U n ( tn / ) = n − / tn / (cid:88) i =1 ( E S [ A U n ( i ) | F i − ] − tn / (cid:16) c n,β λn n (cid:88) j =1 S αj − (cid:17) − c n,β λn / tn / (cid:88) i =1 i − (cid:88) j =1 S αc ( j ) + o P (1) . (4.18)20ote that, since E [( S α ) α α ] < ∞ , by the Marcinkiewicz and Zygmund Theorem [12, Theorem2.5.8], if α ∈ (0 , c n,β λn n (cid:88) j =1 S αj = c n,β λ E [ S α ] + o P ( n − α ) = 1 + βn − / + o P ( n − α ) . (4.19)For α = 0, by a similar result [12, Theorem 2.5.7], for all ε > n n (cid:88) j =1 S j = E [ S ] + o P ( n − / log( n ) / ε ) . (4.20)In particular, tn / (cid:16) c n,β λn n (cid:88) j =1 S αj − (cid:17) = t ( β + o P (1)) . (4.21)By monotonicity, sup t ≤ T (cid:12)(cid:12)(cid:12) tn / (cid:16) c n,β λn n (cid:88) j =1 S αj − (cid:17) − βt (cid:12)(cid:12)(cid:12) P → , (4.22)so that, for α ∈ [0 , n − / C U n ( tn / ) = βt − c n,β λn / tn / (cid:88) i =1 i − (cid:88) j =1 S αc ( j ) + o P (1) . (4.23)Since c n,β = 1+ O ( n − / ), the second term in (4.23) converges uniformly to − t λ E [ S α ] / E [ S α ]by Lemma 8. Rewrite B U n ( k ), for k = O ( n / ), as B U n ( k ) = k (cid:88) i =1 ( E S [ A U n ( i ) | F i − ] − E S [ A U n ( i ) |F i − ] )= k (cid:88) i =1 ( E S [ A U n ( i ) | F i − ] −
1) + O P ( kn − / ) , (4.24)where we have used the asymptotics for E S [ A U n ( i ) | F i − ] in (4.17)-(4.23). Moreover, we cancompute E S [ A U n ( i ) | F i − ] as E S [ A U n ( i ) | F i − ] = E S (cid:104)(cid:16) (cid:88) h/ ∈ S i { T h ≤ c n,β S c ( i ) S h /n } (cid:17) | F i − (cid:105) (4.25)= E S [ A U n ( i ) | F i − ] + E S (cid:104) (cid:88) h,q / ∈ S i { T h ≤ c n,β S c ( i ) S h /n } { T q ≤ c n,β S c ( i ) S q /n } | F i − (cid:105) . E S [ A n ( i ) | F i − ] = 1 + o P (1), uniformly in i = O ( n / ), so that (4.24) simplifiesto B n ( k ) = k (cid:88) i =1 E S (cid:104) (cid:88) h,q / ∈ S i { T h ≤ c n,β S c ( i ) S αh /n } { T q ≤ c n,β S c ( i ) S αq /n } | F i − (cid:105) + O P ( kn − / ) . (4.26)We then focus on the second term in (4.25), which we compute as (cid:88) h,q / ∈ S i h (cid:54) = q E S [ { T h ≤ c n,β S c ( i ) S αh /n } { T q ≤ c n,β S c ( i ) S αq /n } | F i − ] (4.27)= (cid:88) j / ∈ S i − P S ( c ( i ) = j | F i − ) (cid:88) h,q / ∈ S i − ∪{ j } h (cid:54) = q E S [ { T h ≤ c n,β S j S αh /n } { T q ≤ c n,β S j S αq /n } | F i − ] . (4.28)By Lemma 9,l.h.s. (4.27) = (cid:88) j / ∈ S i − S αj (cid:80) l / ∈ S i − S αl (cid:88) h,q / ∈ S i − ∪{ j } h (cid:54) = q (cid:16) c n,β λ S j S αh S αq n + o P ( n − ) (cid:17) = ( c n,β λ ) E S [ S c ( i ) | F i − ] 1 n (cid:88) h,q / ∈ S i − ∪{ c ( i ) } h (cid:54) = q S αh S αq + o P (1)= ( c n,β λ ) n E S [ S c ( i ) | F i − ] (cid:88) ≤ h,q ≤ n S αh S αq − ( c n,β λ ) n E S (cid:104) S c ( i ) (cid:88) h,q ∈ S i − ∪{ c ( i ) }∪{ h = q } S αh S αq | F i − (cid:105) + o P (1) . The leading contribution to B U n ( k ) is given by the first term, while the second term is an errorterm by Lemma 5. We have shown that B U n ( · ) can be rewritten as B U n ( k ) = (cid:16) λn (cid:88) h ∈ [ n ] S αh (cid:17) k (cid:88) i =1 E S [ S c ( i ) | F i − ] + o P ( k ) . (4.29)Thus, n − / B U n ( n / u ) P → λ E [ S α ] E [ S α ] u, (4.30)which concludes the proof of (ii). The jumps of B U n ( k ) are given by B U n ( i ) − B U n ( i −
1) = E S [ A U n ( i ) | F i − ] − E S [ A U n ( i ) | F i − ] = E S (cid:104) (cid:88) h,q / ∈ S i h (cid:54) = q { T h ≤ c n,β S c ( i ) S αh /n } { T q ≤ c n,β S c ( i ) S αq /n } | F i − (cid:105) + (cid:0) E S [ A U n ( i ) | F i − ] − E S [ A U n ( i ) | F i − ] (cid:1) (4.31)22ince E S [ A U n ( i ) | F i − ] = 1 + O P ( n − / ) for i = O P ( n / ) by (4.17), the second term is of order O P ( n − / ), uniformly in i = O P ( n / ). The first term was computed in (4.27). Therefore, B U n ( i ) − B U n ( i − c n,β λ ) n E S [ S c ( i ) | F i − ] (cid:88) h,q ∈ [ n ] S αh S αq − ( c n,β λ ) n E S (cid:104) S c ( i ) (cid:88) h,q ∈ S i − ∪{ c ( i ) }∪{ h = q } S αh S αq | F i − (cid:105) + o P (1) ≤ ( c n,β λ ) n E S [ S c ( i ) | F i − ] (cid:88) h,q ∈ [ n ] S αh S αq . (4.32)After rescaling and taking the expectation, we obtain the bound n − / E S [ sup i ≤ ¯ tn / | B U n ( i ) − B U n ( i − | ] ≤ ( c n,β λ ) n / E S [ sup i ≤ ¯ tn / S c ( i ) ] (cid:16) n (cid:88) h,q ∈ [ n ] S αh (cid:17) . (4.33) Lemma 10. If E [ S α ] < ∞ , E S [ sup k ≤ tn / S c ( k ) ] = o P ( n / ) . (4.34) Proof.
For ε > E S [( sup k ≤ tn / S c ( k ) ) ] ≤ E S [ sup k ≤ tn / S c ( k ) { S c ( k ) >εn / } ] + ε n / . (4.35)We bound the expected value in the first term as E S [ sup k ≤ tn / S c ( k ) { S c ( k ) >εn / } ] ≤ (cid:88) k ≤ tn / n / E S [ S c ( k ) { S c ( k ) >εn / } ] ≤ n / t E S [ S c (1) { S c (1) >εn / } ](1 + O P S (1)) , (4.36)where we used Lemma 4 with f ( x ) = x { x>εn / } . Computing the expectation explicitly weget t E S [ S c (1) { S c (1) >εn / } ] = t (cid:88) i ∈ [ n ] S i { S i >εn / } P ( c (1) = i )= t (cid:88) i ∈ [ n ] S i { S i >εn / } S αi (cid:80) j ∈ [ n ] S αj , (4.37)so that the left-hand side of (4.35) is bounded by t (cid:80) j ∈ [ n ] S αj (cid:88) i ∈ [ n ] S αi { S i >εn / } + (cid:16) (cid:88) i ∈ [ n ] S αi n (cid:17) ε , (4.38)which tends to zero as n → ∞ since E [ S α ] < ∞ and ε > .1.4 Proof of (iv) for the upper bound First we split E S [ sup k ≤ tn / ( M U n ( k ) − M U n ( k − ] = E S [ sup k ≤ tn / ( A U n ( k ) − E S [ A U n ( k ) | F k − ]) ] ≤ E S [ sup k ≤ tn / | A U n ( k ) | ] + E S [ sup k ≤ tn / E [ A U n ( k ) | F k − ] ] ≤ E S [ sup k ≤ tn / | A U n ( k ) | ] . (4.39)We then stochastically dominate ( A U n ( k )) k ≤ tn / by a sequence of Poisson processes (Π k ) k ≤ tn / ,according to A U n ( k ) (cid:22) Π k (cid:16) c n,β S c ( k ) (cid:88) i ∈ [ n ] S αi n (cid:17) =: A (cid:48) n ( k ) . (4.40)Indeed, if E , E , . . . , E n are exponential random variables with parameters λ , λ , . . . , λ n , thereexists a coupling with a Poisson process Π( · ) such that (cid:80) i ≤ n { E i ≤ t } ≤ Π( (cid:80) i ≤ n λ i t ). Thecoupling is constructed as follows. Each random variable E i is coupled with a Poisson processΠ ( i ) with intensity λ i in such a way that { E i ≤ t } ≤ Π ( i ) ( λ i t ). Moreover, by basic properties ofthe Poisson process (cid:80) i ≤ n Π ( i ) ( λ i t ) d = Π( (cid:80) i ≤ n λ i t ).We bound (4.40) via martingale techniques. First, we decompose it as n − / E S [ sup k ≤ tn / | A U n ( k ) | ] ≤ n − / E S (cid:104)(cid:16) sup k ≤ tn / (cid:12)(cid:12)(cid:12) A (cid:48) n ( k ) − c n,β S c ( k ) (cid:88) i ∈ [ n ] S αi n (cid:12)(cid:12)(cid:12)(cid:17) (cid:105) + 2 n − / E S (cid:104)(cid:16) c n,β sup k ≤ tn / S c ( k ) (cid:88) i ∈ [ n ] S αi n (cid:17) (cid:105) (4.41)Applying Doob’s L martingale inequality [20, Theorem 11.2] to the first term we see that itconverges to zero, since n − / E S (cid:104)(cid:16) sup k ≤ tn / | A (cid:48) n ( k ) − S c ( k ) (cid:88) i ∈ [ n ] S αi n (cid:12)(cid:12)(cid:12)(cid:17) (cid:105) ≤ n − / E S (cid:104)(cid:12)(cid:12)(cid:12) A (cid:48) n ( tn / ) − S c ( tn / ) (cid:88) i ∈ [ n ] S αi n (cid:12)(cid:12)(cid:12) (cid:105) = 4 n − / E S (cid:104) S c ( tn / ) (cid:88) i ∈ [ n ] S αi n (cid:105) . (4.42)The last equality follows from the expression for the variance of a Poisson random variable. Theright-most term converges to zero by Lemma 7. We now bound the second term in (4.41), as n − / E S (cid:104)(cid:16) sup k ≤ tn / S c ( k ) (cid:88) i ∈ [ n ] S αi n (cid:17) (cid:105) = n − / (cid:16) (cid:88) i ∈ [ n ] S αi n (cid:17) E S [( sup k ≤ tn / S c ( k ) ) ] (4.43)By Lemma 10 the right-hand side of (4.43) converges to zero, concluding the proof of (iv). As a consequence of (4.7) and Theorem 3 we have that Q n ( k ) = O P ( n / ) for k = O ( n / ). Infact, n − / Q n ( k ) is tight when k = O ( n / ), as the following lemma shows:24 emma 11. Fix ¯ t > . The sequence n − / sup t ≤ ¯ t Q n ( tn / ) is tight.Proof. The supremum function f ( · ) (cid:55)→ sup t ≤ ¯ t f ( t ) is continuous in ( D , J ) by [27, Theorem13.4.1]. In particular, n − / sup t ≤ ¯ t Q U n ( tn / ) d → sup t ≤ ¯ t W ( t ) , in ( D , J ) . (4.44)Since Q n ( k ) ≤ Q U n ( k ), the conclusion follows.As an immediate consequence of (2.6) and Lemma 11, we have the following importantcorollary. Recall that ν i is the set of customers who have left the system or are in the queue atthe beginning of the i -th service, so that | ν i | = i + Q n ( i ). Recall also that 0 ≤ Q n ( t ) ≤ Q U n ( t ). Corollary 1. As n → ∞ , | ν i | = i + o P ( i ) , uniformly in i = O P ( n / ) . (4.45)Intuitively, this implies that the main contribution to the downwards drift in the queue-length process comes from the customers that have left the system, and not from the customersin the queue. Alternatively, the order of magnitude of the queue length, that is n / , is negligiblewith respect to the order of magnitude of the customers who have left the system, which is n / .In order to prove Theorem 1 we proceed as in the proof of Theorem 3, but we now need todeal with the more complicated drift term. As before, we decompose N n ( k ) = M n ( k ) + C n ( k ),where M n ( k ) = k (cid:88) i =1 ( A n ( i ) − E S [ A n ( i ) | F k − ]) ,C n ( k ) = k (cid:88) i =1 ( E S [ A n ( i ) | F k − ] − ,B n ( k ) = k (cid:88) i =1 ( E S [ A n ( i ) | F i − ] − E S [ A n ( i ) | F i − ] ) . (4.46)As before, we separately prove the convergence of the drift C n ( k ) and of the martingale M n ( k ),by verifying the conditions (i)-(iv) in Section 4.1. Verifying (i) proves to be the most challengingtask, while the estimates for (ii)-(iv) in Section 4.1 carry over without further complications. By expanding E S [ A n ( i ) | F i − ] − E S [ A n ( i ) | F i − ] − (cid:16) c n,β λ (cid:80) nl =1 S αl n E S [ S c ( i ) | F i − ] − (cid:17) − c n,β E S [ S c ( i ) | F i − ] (cid:88) l ∈ ν i \{ c ( i ) } λ S αl n − c n,β λn E S [ S αc ( i ) | F i − ] + o P ( n − / ) . (4.47)25y further expanding the first term in (4.47) as in (4.16), we get E S [ A n ( i ) | F i − ] − (cid:16) c n,β λn (cid:88) j / ∈ S i − S αj − (cid:17) − c n,β E S [ S c ( i ) | F i − ] i +1+ Q n ( i − (cid:88) l = i +1 λ S αc ( l ) n − c n,β λn E S [ S αc ( i ) | F i − ] + o P ( n − / ) , (4.48)where in the first equality we have used (2.6). Comparing equation (4.48) with equation (4.17),we rewrite the drift as C n ( k ) = C U n ( k ) − c n,β λ k (cid:88) i =1 E S [ S c ( i ) | F i − ] i +1+ Q n ( i − (cid:88) l = i +1 S αc ( l ) n . (4.49)Therefore, to conclude the proof of (i) it is enough to show that the second term vanishes, afterrescaling. We do this in the following lemma: Lemma 12. As n → ∞ , n − / c n,β λ ¯ tn / (cid:88) i =1 E S [ S c ( i ) | F i − ] i +1+ Q n ( i − (cid:88) l = i +1 S αc ( l ) n P → . (4.50) Proof.
By Lemma 11, sup i ≤ ¯ tn / Q n ( i ) ≤ C n / w.h.p. for a large constant C , and by Lemma7, sup i ≤ ¯ tn / E S [ S c ( i ) | F i − ] ≤ C w.h.p. for another large constant C . This implies that,w.h.p., n − / c n,β λ ¯ tn / (cid:88) i =1 E S [ S c ( i ) | F i − ] i +1+ Q n ( i − (cid:88) l = i +1 S αc ( l ) n ≤ c n,β λC tn / (cid:88) i =1 i +1+ C n / (cid:88) l = i +1 S αc ( l ) n / . (4.51)The double sum can be rewritten as c n,β λC tn / (cid:88) i =1 i +1+ C n / (cid:88) l = i +1 S αc ( l ) n / ≤ c n,β λC tn / + C n / (cid:88) j =1 min { j, C n / } S αc ( j ) n / ≤ c n,β λC C t + C ) n / (cid:88) j =1 S αc ( j ) n . (4.52)The right-most term converges to zero in probability as n → ∞ by Lemma 8. This concludesthe proof.Since n − / C n ( tn / ) = n − / C U n ( tn / ) − n − / c n,β λ tn / (cid:88) i =1 E S [ S c ( i ) | F i − ] i +1+ Q n ( i − (cid:88) l = i +1 S αc ( l ) n , (4.53)Lemma 12 and the convergence result (4.23) for n − / C U n ( tn / ) conclude the proof of (i).26 .2.2 Proof of (ii), (iii) and (iv) for the embedded queue Proceeding as before, we find that B n ( k ) = k (cid:88) i =1 ( E S [ A n ( i ) | F i − ] − E S [ A n ( i ) | F i − ] )= k (cid:88) i =1 ( E S [ A n ( i ) | F i − ] −
1) + O P ( kn − / ) , (4.54)where E S [ A n ( i ) | F i − ] = E S [ A n ( i ) | F i − ] + E S (cid:104) (cid:88) h,q / ∈ ν i − h (cid:54) = q { T h ≤ S c ( i ) S h /n } { T q ≤ S c ( i ) S q /n } | F i − (cid:105) . (4.55)Similarly as in Section 4.1.2, we get (cid:88) h,q / ∈ ν i − h (cid:54) = q E S [ { T h ≤ S c ( i ) S αh /n } { T q ≤ S c ( i ) S αq /n } | F i − ] (4.56)= E S [ S c ( i ) | F i − ] λ n (cid:16) n (cid:88) h =1 S αh (cid:17) − E S (cid:104) S c ( i ) λ n (cid:88) h,q ∈ ν i − ∪{ c ( i ) }∪{ h = q } S αh S αq | F i − (cid:105) + o P (1) . The second term is an error term by Lemma 5 and Corollary 1. This implies that B n ( · ) can berewritten as B n ( k ) = (cid:16) λn n (cid:88) h =1 S αh (cid:17) k (cid:88) i =1 E S [ S c ( i ) | F i − ] + o P ( k ) , (4.57)so that n − / B n ( n / u ) P → λ E [ S α ] E [ S α ] u, (4.58)which concludes the proof of (ii).To conclude the proof of Theorem 1, we are left to verify (iii) and (iv). However, theestimates in Sections 4.1.3 and 4.1.4 also hold for B n ( · ) and M n ( · ), since they rely respectivelyon (4.33) and (4.40) to bound the lower-order contributions to the drift. This concludes theproof of Theorem 1. In this paper we have considered a generalization of the ∆ ( i ) /G/ α ( i ) /G/ α ∈ [0 , W ( · ). A distinctive characteristic of our results is the so-called depletion-of-points effect , represented by a quadratic drift in W ( · ). A (directed) tree isassociated to the ∆ α ( i ) /G/ criticality of the associated random tree. Our result interpolates between two already knownresults. For α = 0 the arrival clocks are i.i.d. and the analysis simplifies significantly. In thiscontext, [5] proves an analogous heavy-traffic diffusion approximation result. Theorem 1 canthen be seen as a generalization of [5, Theorem 5]. If α = 1, the ∆ α ( i ) /G/ directed components of directed inhomogeneous random graphs.Lemma 6 implies that the distribution of the service time of the first O ( n / ) customersto join the queue converges to the α - size-biased distribution of S , irrespectively of the precisetime at which the customers arrive. This suggests that it is possible to prove Theorem 1 byapproximating the ∆ α ( i ) /G/ ( i ) /G/ S ∗ suchthat P ( S ∗ ∈ A ) = E [ S α { S ∈A} ] / E [ S α ] , (5.1)and i.i.d. arrival times distributed as T i ∼ exp( λ E [ S α ]). This conjecture is supported by twoobservations. First, the heavy-traffic conditions for the two queues coincide. Second, the stan-dard deviation of the Brownian motion is the same in the two limiting diffusions. However,this approximation fails to capture the higher-order contributions to the queue-length process.As a result, the coefficients of the negative quadratic drift in the two queues are different, andthus the approximation of the ∆ α ( i ) /G/ ( i ) /G/ α lies in the interval [0 ,
1] plays no role in our proof. Onthe other hand, we see from (2.18) thatmax { E [ S α ] , E [ S α ] , E [ S α ] } < ∞ (5.2)is a necessary condition for Theorem 1 to hold. From this we conclude that Theorem 1 remainstrue as long as α ∈ R is such that (5.2) is satisfied. From the modelling point of view, α > α < form of the limiting diffusion is the same for all α ∈ R , but different values of α yield differentfluctuations (standard deviation of the Brownian motion), and a different quadratic drift. Acknowledgments
This work is supported by the NWO Gravitation
Networks grant024.002.003. The work of RvdH is further supported by the NWO VICI grant 639.033.806.The work of GB and JvL is further supported by an NWO TOP-GO grant and by an ERCStarting Grant.
References [1] I. A. Abramowitz, M. and Stegun.
Handbook of Mathematical Functions: with Formulas,Graphs, and Mathematical Tables . Courier Corporation, 1964.[2] L. Addario-Berry, N. Broutin, and C. Goldschmidt. The continuum limit of critical randomgraphs.
Probability Theory and Related Fields , 152(3-4):367–406, 2012.283] D. Aldous. Brownian excursions, critical random graphs and the multiplicative coalescent.
Annals of Probability , 25(2):812–854, 1997.[4] G. Bet, R. van der Hofstad, and J. S. H. van Leeuwaarden. Finite-pool queues with heavy-tailed services. arXiv preprint arXiv preprint
Probability Theory and RelatedFields , 160(3-4):733–796, 2014.[7] S. Bhamidi, R. van der Hofstad, and J. S. H. van Leeuwaarden. Scaling limits for criticalinhomogeneous random graphs with finite third moments.
Electronic Journal of Probability ,15:1682–1702, 2010.[8] S. Bhamidi, R. van der Hofstad, and J. S. H. van Leeuwaarden. Novel scaling limits forcritical inhomogeneous random graphs.
Annals of Probability , 40(6):2299–2361, 2012.[9] S. Dhara, R. van der Hofstad, J. S. H. van Leeuwaarden, and S. Sen. Critical window forthe configuration model: finite third moment degrees. arXiv preprint arXiv preprint
Random Trees, L´evy Processes and Spatial BranchingProcesses . Societe mathematique de France, 2002.[12] R. Durrett.
Probability - Theory and Examples . Cambridge University Press, 2010.[13] S. N. Ethier and T. G. Kurtz.
Markov Processes: Characterization and Convergence . Wiley,New York, 1986.[14] R. van der Hofstad, A. Janssen, and J. S. H. van Leeuwaarden. Critical epidemics, randomgraphs, and Brownian motion with a parabolic drift.
Advances in Applied Probability ,42(4):1187–1206, 2010.[15] R. van der Hofstad, J. S. H. van Leeuwaarden, and C. Stegehuis. Mesoscopic scales inhierarchical configuration models. arXiv preprint
Queueing Systems , 80(1), 2015.[17] H. Honnappa and A. R. Ward. On transitory queueing. arXiv preprint
Annals of Applied Probability , 24(6):2560–2594, 2014.[19] D. G. Kendall. Some problems in the theory of queues.
Journal of the Royal StatisticalSociety. Series B (Methodological) , pages 151–185, 1951.2920] A. Klenke.
Probability Theory: A Comprehensive Course . Springer, London, 2008.[21] J.-F. Le Gall. Random trees and applications.
Probability Surveys , 2:245–311, 2005.[22] V. Limic. A LIFO queue in heavy traffic.
The Annals of Applied Probability , 11(2):301–331,2001.[23] A. Martin-L¨of. The final size of a nearly critical epidemic, and the first passage time of aWiener process to a parabolic barrier.
Journal of Applied Probability , 35(3):671–682, 1998.[24] L. Tak´acs. Queues, random graphs and branching processes.
Journal of Applied Mathe-matics and Simulation , 1(3):223–243, 1988.[25] L. Tak´acs. Limit distributions for queues and random rooted trees.
Journal of AppliedMathematics and Stochastic Analysis , 6(3):189–216, 1993.[26] L. Tak´acs. Queueing methods in the theory of random graphs. In
Advances in QueueingTheory, Methods, and Open Problems , chapter 2, pages 45–78. 1995.[27] W. Whitt.