Perfect sampling of GI/GI/c queues

Transcription

1 Perfect samping of GI/GI/c queues Jose Banchet 1 Jing Dong 2 Yanan Pei 3 Received: 7 October 215 / Revised: 12 February 218 Springer Science+Business Media, LLC, part of Springer Nature 218 Abstract We introduce the first cass of perfect samping agorithms for the steadystate distribution of muti-server queues with genera interarriva time and service time distributions. Our agorithm is buit on the cassica dominated couping from the past protoco. In particuar, we use a couped muti-server vacation system as the upper bound process and deveop an agorithm to simuate the vacation system bacward in time from stationarity at time zero. The agorithm has finite expected termination time with mid moment assumptions on the interarriva time and service time distributions. Keywords Perfect samping FCFS muti-server queue Dominated couping from the past Random was Mathematics Subject Cassification 6K25 Support from NSF through the Grants CMMI , DMS and DMS is gratefuy acnowedged. B Jose Banchet jose.banchet@stanford.edu Jing Dong jing.dong@gsb.coumbia.edu Yanan Pei yp2342@coumbia.edu 1 Management Science and Engineering, Stanford University, 475 Via Ortega, Suite 31, Stanford, CA 9435, USA 2 Graduate Schoo of Business, Coumbia University, 322 Broadway, New Yor, NY 127, USA 3 Department of IEOR, Coumbia University, 5 West 12th Street, New Yor, NY 127, USA

2 1 Introduction In this paper, we present the first cass of perfect samping agorithms for the steadystate distribution of muti-server queues with genera interarriva time and service time distributions. Our agorithm has finite expected running time under the assumption that the interarriva times and service times have finite 2 + ɛ moments for some ɛ>. The goa of perfect samping is to sampe without any bias from the steady-state distribution of a given ergodic process. The most popuar perfect samping protoco, nown as Couping From The Past CFTP, was introduced by Propp and Wison in the semina paper [23]; see aso [2] for another important eary reference on perfect simuation. Foss and Tweedie [15] proved that CFTP can be appied if and ony if the underying process is uniformy ergodic, which is not a property appicabe to muti-server queues. So, we use a variation of the CFTP protoco caed Dominated CFTP DCFTP introduced by Kenda in [19] and ater extended in [2,21]. A typica impementation of DCFTP requires at east four ingredients: a a stationary upper bound process for the target process, b a stationary ower bound process for the target process, c the abiity to simuate a and b bacward in time i.e., from time to t, for any t >, d a finite time T < at which the state of the target process is determined typicay by having the upper and ower bound processes coaesce, and the abiity to reconstruct the target process from T up to time couped with the two bounding processes. The time T is caed the coaescence time, and it is desirabe to have E [T ] <. The ingredients are typicay combined as foows. One simuates a and b bacward in time by appying c unti the processes meet. The target process is sandwiched between a and b. Therefore, if we can find a time T < when processes a and b coincide, the state of the target process is nown at T as we. Then, appying d, we reconstruct the target process from T up to time. The agorithm outputs the state of the target process at time. It is quite intuitive that the output of the above construction is stationary. Specificay, assume that the sampe path of the target process couped with a and b is given from, ]. Then, we can thin of the simuation procedure in c as simpy observing or unveiing the paths of a and b during [ t, ]. When we find a time T < at which the paths of a and b tae the same vaue, because of the sandwiching property, the target process must share this common vaue at T. Starting from that point, property d simpy unveis the path of the target process. Since this path has been coming from the infinite distant past we simpy observed it from time T, the output is stationary at time. Notice that whie T is a random time, the output is the state of the target process at the fixed time. One can often improve the performance of a DCFTP protoco if the underying target process is monotone [2], as in the muti-server queue setting. A process is monotone if there exists a certain partia order,, such that if w and w are initia states where w w, and one uses common random numbers to simuate two paths, one starting from w and the other from w, then the order is preserved when comparing

3 the states of these two paths at any point in time. Thus, instead of using the bounds a and b directy to detect coaescence, one coud appy monotonicity to detect coaescence as foows: At any time t <, one can start two paths of the target process, one from the state w obtained from the upper bound a observed at time t, and the other from the state w w obtained from the ower bound b observed at time t. Then, we run these two paths using common random numbers, which are consistent with the bacward simuation of a and b, in reverse order according to the dynamics of the target process, and chec whether these two paths meet before time zero. If they do, the coaescence occurs at such a meeting time. We aso notice that because we are using common random numbers and system dynamics, these two paths wi merge into a singe path from the coaescence time forward, and the state at time zero wi be the desired stationary draw. If coaescence does not occur, then one can simpy et t 2t and repeat the above procedure. For this iterative search procedure, we must show that the search terminates in finite time. Whie the DCFTP protoco is reativey easy to understand, its appication is not straightforward. In most appications, the most difficut part has to do with eement c. Then, there is an issue of finding good bounding processes eements a and b, in the sense of having short coaescence times which we interpret as maing sure that E [T ] <. There has been a substantia amount of research that deveops generic agorithms for Marov chains see, for exampe, [1] and [8]. These methods rey on having access to the transition ernes, which are difficut to obtain in our case. Perfect simuation for queueing systems has aso received a significant amount of attention in recent years, though most perfect simuation agorithms for queues impose Poisson assumptions on the arriva process. Sigman [25,26] appied the DCFTP and regenerative idea to deveop perfect samping agorithms for stabe M/G/c queues. The agorithm in [25] requires the system to be super-stabe i.e., the system can be dominated by a stabe M/G/ 1 queue. The agorithm in [26] wors under natura stabiity conditions, but it has infinite expected termination time. A recent wor by Connor and Kenda [9] extends Sigman s agorithm [26] to sampe stationary M/G/c queues, and the agorithm has finite expected termination time, but it sti requires the arrivas to be Poisson. The main reason for the Poisson arriva assumption is that under this assumption, one can find dominating processes which are quasi-reversibe see Chapter 3 of [18] and therefore can be simuated bacward in time using standard Marov chain constructions eement c. In genera, constructing eements a and b, a in particuar, as b can often be taen as the trivia ower bound,, in the muti-server queue setting requires proving sampe path amost sure dominance under different service/routing discipines. The sampe path method has been widey used in the contro of queues [22]. Comparison of muti-server queues, under the amost sure dominance or the stochastic dominance, has been studied in the iterature see, for exampe, [12,13,27] and references therein. For genera renewa arriva process, our wor is cose in the spirit to [4,11] and [6], but the mode treated is fundamentay different. Thus, it requires some new deveopments. We aso use a different couping construction than that introduced in [26] and refined in [9]. In particuar, we tae advantage of a vacation system which aows us to transform the probem into simuating the running infinite horizon maximums from time t to infinity of renewa processes, compensated with negative drifts so

4 that the infinite horizon maximums are we defined. Finay, we note that a significant advantage of our method, in contrast to [26], is that we do not need to wait unti the upper bound system empties to achieve coaescence. Due to the monotonicity of our process, we can appy the iterative method introduced above. This is important in many-server queues in heavy traffic for which it woud tae an exponentia amount of time in the arriva rate, or sometimes be impossibe, to observe an empty system. We demonstrate the performance of our procedure for different many-server heavy traffic regimes using simuation experiments in Sect. 5. The rest of the paper is organized as foows: In Sect. 2, we describe our simuation strategy, invoving eements a d, and we concude the section with the statement of a resut which summarizes our main contribution Theorem 1. Subsequent sections Sects. 3 and 4 provide more detais of our simuation strategy. In Sect. 5, we conduct some numerica experiments an onine companion of this paper incudes a MATLAB impementation of the agorithm. Section 6 contains the proofs of some technica resuts. Lasty, we provide a ist of seected notation in the Appendix. 2 Simuation strategy and main resut Our target process is the stationary process generated by a muti-server queue with independent and identicay distributed iid interarriva times and iid service times which are independent of the arrivas. There are c 1 identica servers, each can serve at most one customer at a time. Customers are served on a first-come-firstserved FCFS basis. Let G and Ḡ = 1 G resp. F and F = 1 F denote the cumuative distribution function, CDF, and the tai CDF of the interarriva times resp. service times. We sha use A to denote a random variabe with CDF G, and V to denote a random variabe with CDF F. Assumption 1 A1Both A and V are stricty positive with probabiity one, and there exists ɛ> such that E[A 2+ɛ ] <, E[V 2+ɛ ] <. The previous assumption wi aow us to concude that the coaescence time of our agorithm has finite expectation. The agorithm wi terminate with probabiity one if E[A 1+ɛ ]+E[V 1+ɛ ] <. We assume that G and F are nown so that the required parameters in our agorithmic deveopment can be obtained. We write λ = Ḡtdt 1 = 1/E [A] as the arriva rate, and μ = Ftdt 1 = 1/E[V ] as the service rate. In order to ensure the existence of the stationary distribution of the system, we require the foowing stabiity condition: λ/cμ < Eements of the simuation strategy: upper bound and couping We refer to the upper bound process as the vacation system, the construction that we use is based on that given in [16]. Let us first expain in words how the vacation system

5 operates. Customers arrive at the vacation system according to the renewa arriva process, and the system operates simiary to a GI/GI/c queue, except that every time aserversayserveri finishes an activity i.e., a service or a vacation, if there is no customer waiting to be served in the queue, server i taes a vacation which has the same distribution as the service times. If there is at east one customer waiting, the first customer waiting in the queue starts to be served by server i. Using a suitabe couping, the wor of [16] shows that the tota number of jobs in the vacation system is an upper bound of the tota number of jobs in the corresponding muti-server queue. In this paper, we estabish bounds for other system-reated processes, such as the Kiefer Wofowitz vectors, which are of independent interest. We next provide more detais about the vacation system. We introduce c + 1 time-stationary renewa processes, which are used to describe the vacation system. Let T := T n : n Z\ be a time-stationary renewa point process with Tn > and T n <, n 1theT n are sorted in a non-decreasing order in n. For n 1, Tn represents the arriva time of the n-th customer into the system after time zero, and T n is the arriva time of the n-th customer, counting bacward in time, from time zero. We aso define T,+ n := inf Tm : T m > T n, that is, the arriva time of the next customer after Tn,+.Ifn 1orn 2, T However, T,+ 1 = T1. Simiary, we write T, n := sup Tm : T m < T n, n = T n+1. i.e., the arriva time of the previous customer before Tn. Define A n := Tn,+ Tn for a n Z\. Note that A n is the interarriva time between the customer arriving at time Tn and the next customer. A n has CDF G for n 1 and n 2, but A 1 has a different distribution due to the inspection paradox. Figure 1a provides a pictoria iustration of the renewa process T. Simiary, for i 1, 2,...,c, we introduce iid time-stationary renewa point processes T i := T i n : n Z\. As before, we have that Tn i > and T n i < forn 1 with the T n i sorted in a nondecreasing order. We aso define Tn i,+ := inftm i : T m i > T n i i, and Tn := suptm i : Tm i < T n i. Then, we et V n i i,+ := Tn Tn i. We assume that V n i has CDF F for n 1 and n 2. The Vn i are activities services and vacations, which are executed by the i-th server in the vacation system.

6 a b T 3 T 2 T 1 T 1 T 2 T 3 T 3 t T 2 T 1 T 1 T 2 T 3 A 3 A 2 A 1 A 1 A 2 N t t = 2 Fig. 1 Renewa processes. a Definition of A i, b definition of N u t Next, we define, for each i, 1,...,c, and any u,, a counting process N i u t := [u, u + t] T i, for t, where denotes cardinaity. Note that as T 1 i < < T 1 i by stationarity, N i =. In particuar, the quantity N u t is the number of customers who arrive during the time interva [u, u + t] see Fig. 1b. The quantity Nu i t is the number of activities initiated by server i during the time interva [u, u + t] when i =. For simpicity of the notation, et us write N i t = N i t if t and N i t = Nt i t if t The upper bound process: vacation system Let Q v t denote the number of peope waiting in queue at time t in the stationary vacation system. We write Q v t := im s t Q v s and dq v t := Q v t Q v t. Aso, for any t, i,...,c and each u,, define N i u t := im h N i u h t, and et dnu i t := N u i t N u i t for a t note that as Nu i =, dnu i shoud equa Nu i. Simiary, for t, N i t = Nt i t. We aso introduce X u t := Nu t c i=1 Nu i t. For simpicity of the notation, we aso write X t = X t if t, and X t = X t t if t. Then, the dynamics of Q v t : t > satisfy dq v t = dx t + I Q v t = c dn i t, 1 given Q v. Note that here we are using the fact that arrivas do not occur at the same time as the start of activity times; this is because the processes T i are independent time-stationary renewa processes in continuous time so that T 1 i and T 1 i have a density. It foows from standard arguments of Sorohod mapping [7] that, for t, i=1 Q v t = Q v + X t inf s t X s + Q v,

7 where X s + Q v = min X s + Q v,. Moreover, using Lyons construction, we have that, for t, Q v t = sup X s X t 2 s t see, for exampe, Proposition 1 of [3]. Q v t : t, is a we-defined process by virtue of the stabiity condition λ/μc < The couping: extracting service times for each customer The vacation system and the target process the GI/GI/c queue wi be couped by using the same arriva stream of customers, T, and assuming that each customer brings his own service time. In particuar, the evoution of the underying GI/GI/c queue is described using a sequence of the form Tn, V n : n Z\, where Vn is the service time of the customer arriving at time Tn. In simuation, we start by simuating the upper bound process vacation system. Thus, the V n must be extracted from the evoution of Q v so that the same service times are matched to the common arriva stream both in the vacation system and in the target process. In order to match the service times to each of the arriving customers in the vacation system, we define the foowing auxiiary processes: For every i 1,...,c, any t >, and any u,, etσu i t denote the number of service initiations by server i during the time interva [u, u + t]. Observe that σ i u t = [u,u+t] I Q v s > dnu i s u. That is, we count activity initiations at time T i [u, u +t] as service initiations if and ony if Q v T i >. Once again, here we use the fact that arriva times and activity initiation times do not occur simutaneousy. We now expain how to match service time for the customer arriving at Tn, n Z\. First, such a customer occupies position Q v T n 1 when he enters the queue. Let Dn be the deay or waiting time inside the queue of the customer arriving at Tn. Then we have that and therefore, Dn = inf t : Q v T n = V n = i=1 c σ i Tn t, i=1 c V i N i Tn +D n dn i Tn + D n. 3 Observe that the previous equation is vaid, because for each n Z\, there is a unique i n 1,...,c for which dn in T n + D n = 1 and dn j T n + D n =

8 if j = i n ties are not possibe because of the time stationarity of the T i, so we obtain that 3 is equivaent to V n = V in N in T n +D n. We sha expain in Sect. 6.1 that V n : n Z\ and T n : n Z\ are two independent sequences and the V n are iid copies of V, i.e., the extraction procedure here does not create any bias. 2.2 Monotonicity properties and the stationary GI/GI/c queue A famiy of GI/GI/c queues and the target GI/GI/c stationary system We now describe the evoution of a famiy of standard GI/GI/c queues. Once we have the sequence T n, V n : n Z\, we can proceed to construct a famiy of continuous-time Marov processes Z u t; z : t for each u,, given the initia condition Z u ; z = z. We write z = q, r, eu, and set Z u t; z := Q u t; z, R u t; z, E u t; z, for t, where Q u t; z is the number of peope in the queue at time u+t Q u ; z = q, R u t; z is the vector of ordered ascending remaining service times of the c servers at time u + t R u ; z = r, and E u t; z is the time eapsed since the previous arriva at time u + t E u ; z = eu. We sha aways use E u ; z = eu = u suptn : T n u, and we sha seect q and r appropriatey based on the upper bound. The evoution of the process Z u s; z : < s t is obtained by feeding the traffic Tn, V n : u < T n u + s for s, t] into a FCFS GI/GI/c queue with initia conditions given by z. Constructing Z u s; z : < s t using the traffic trace Tn, V n : u < T n u + s for s, t] is standard see, for exampe, Chapter 3 of [24]. One can further describe the evoution of the underying GI/GI/c queue at arriva epochs, using the Kiefer Wofowitz vector [1]. In particuar, for every non-negative vector w R c such that w i w i+1 where w i is the i-th entry of wfor1 i c 1, and each Z\, the famiy of processes W T n ; w : n, n Z\ satisfies W Tn,+ ; w = S + W Tn ; w + V n e 1 A n 1, 4 with initia condition W T ; w = w, where e 1 = 1,,..., T R c, 1 = 1,...,1 T R c, and S is a sorting operator which arranges the entries in a vector in ascending order. In simpe words, W T n ; w for Z\ describes the Kiefer Wofowitz vector as observed by the customer arriving at Tn, assuming that customer who arrived at T, n, experienced the Kiefer Wofowitz state w. Reca that the first entry of W T n ; w, namey W 1 T n ; w, is the waiting time of the customer arriving at Tn given the initia condition w at T. More generay,

9 the i-th entry of W T n ; w, namey W i T n ; w, is the virtua waiting time of the customer arriving at Tn if he decides to enter service immediatey after there are at east i servers free once he reaches the head of the ine. In other words, one can aso interpret W T n ; w as the vector of remaining woroads sorted in ascending order that woud be processed by each of the c servers at Tn, if there are no more arrivas after time Tn. We are now ready to construct the stationary version of the GI/GI/c queue. Namey, for each n Z\ and every t,, we define W n and Z t via W n := im W T n ;, 5 Z t := Q t, R t, E t = im u Z u t u, z u, where z u =,, eu. We sha show in Proposition 1 that these imits are we defined The anaogue of the Kiefer Wofowitz process for the upper bound system In order to compete the couping strategy, we aso describe the evoution of the anaogous Kiefer Wofowitz vector induced by the vacation system, which we denote by W v n : n Z\, where v stands for vacation. As with the i-th entry of the Kiefer Wofowitz vector of a GI/GI/c queue, the i-th entry of W v n, namey W v i n, is the virtua waiting time of the customer arriving at time Tn if he decides to enter service immediatey after there are at east i servers free once he reaches the head of the ine assuming that servers become ide once they see, after the competion of current activity, the customer in queue waiting in the head of the ine. To describe the Kiefer Wofowitz vector induced by the vacation system precisey, et U i t be the time unti the next renewa after time t in T i, that is U i t = inftn i : T n i > t t. So, for exampe, U Tn = An for n Z\. LetU t = U 1 t,...,u c t T. We then have that W v n = Dn U 1+S Tn + D n. 6 In particuar, note that W v 1 n = Dn, i.e., the deay the customer arriving at T n woud experience. We next introduce a recursive way of constructing/defining the Kiefer Wofowitz vector induced by the vacation system. We define W v n = W v n + V n e 1 A n 1, and et W v i n be the i-th entry of W v n.letw v n+ denote the Kiefer Wofowitz vector seen by the customer arriving at Tn,+. From the definition of W v n,wehave W v n+ = S W v n + + Ξn,

10 where Ξ i n = I W v i n < U j i n Tn,+ i.e., j i n is the server whose remaining activity time immediatey before Tn i-th smaest in order. So, 6 actuay satisfies is the where Ξ n = Ξ 1 n W v n+ = S W v n + V n e 1 A n 1 + +Ξ n, 7,...,Ξ c T Monotonicity properties n In this section, we wi present severa emmas which contain usefu monotonicity properties. The proofs of the emmas are given in Sect. 6.2 in order to quicy arrive at the main point of this section, which is the construction of a stationary version of the GI/GI/c queue. First, we reca that the Kiefer Wofowitz vector of a GI/GI/c queue is monotone in the initia condition 8 and invoe a property 9 which wi aow us to construct a stationary version of the Kiefer Wofowitz vector of our underying GI/GI/c queue, using Lyons construction. Lemma 1 For n,, n Z\, w + >w, W T n ; w+ W T n ; w. 8 Moreover, if n, W Tn ; W T n ;. 9 The second resut aows us to mae precise how the vacation system dominates a suitabe famiy of GI/GI/c systems, in terms of the underying Kiefer Wofowitz vectors. Lemma 2 For n,, n Z\, W v n W T n ; W v. The next resut shows that in terms of queue ength processes, the vacation system aso dominates a famiy of GI/GI/c queues, which we sha use to construct the upper bounds. Lemma 3 Let q = Q v u, r = S U u, and e = u suptn : T n z + = q, r, e and z =,, e. Then, for t u, u, so that

11 Q u t u; z Q u t u; z + Q v t. Using Lemmas 1 3, we can estabish the foowing resut. Proposition 1 The imits defining W n and Z t in 5 exist amost surey. Moreover, we have W Tn ; W n W T n ; W v. 1 Proof Using Lemmas 1 and 2, we have that W v n W T n ; W v W T n ;. Then, by property 9 in Lemma 1, we concude that the imit defining W n exists amost surey and that W n W v n. 11 Simiary, using Lemma 3, we can obtain the existence of the imit Qt and we have that Q t Q v t. Moreover, by convergence of the Kiefer Wofowitz vectors, we obtain the i-th entry of R Tn + W 1 n, namey + R i Tn + W 1 n = W i n W n 1, where i 1,...,c. Lasty, since the age process has been taen from the underying renewa process T, we have that E t = t suptn : T n t. The fact that the imits are stationary foows directy from the imiting procedure and is standard in Lyons-type constructions. For 1, we use the identity W n = W T n ; W, combined with Lemma 1, to obtain W T n ; W T n ; W = W n, and then we appy Lemma 2, together with 11, to obtain W n = W Tn ; W W T n ; W v. 2.3 Description of simuation strategy and main resut We now describe how the variation of DCFTP that we mentioned in the Introduction, using monotonicity of the muti-server queue, and eements a d, appy to our setting. Define a fixed inspection sequence κ j : j 1 with κ j <κ j 1 <, and define κ =. We start from the first inspection time Tκ 1 j = 1. The upper bound is initiaized using the Kiefer Wofowitz process associated with the vacation system at Tκ j. The ower bound is initiaized with a nu vector. We run the two bounding

12 GI/GI/c queues forward in time using Tn, V n : κ j n κ j 1. If the two processes meet before time zero, then we can unvei the state of the stationary GI/GI/c queue; otherwise, we go bacward in time to the next inspection time Tκ j+1 j j + 1 and construct two new bounding GI/GI/c queues accordingy. We repeat the procedure unti the coaescence is detected. The strategy combines the foowing facts which we sha discuss in the seque. Fact I We can simuate sup s t X s and N i t : t i= c jointy for any given t. This part, which corresponds to item c is executed by appying an agorithm from [6] designed to sampe the infinite horizon running time maximum of a random wa with negative drift. We sha provide more detais about this in Sect. 4. Fact II For a 1and every n 1, by Proposition 1, we have that W Tn ; W n W T n ; W v. This portion expoits the upper bound a i.e., W v and the ower bound b i.e.,. Fact III We can detect that coaescence occurs at some time T [T, ] for some 1byfinding some n Z, n, such that Tn + W 1 T n ; W v and W T n ; W v = W T n ;. This is precisey the coaescence detection strategy which uses monotonicity of the Kiefer Wofowitz vector and the coaescence time T = Tn + W 1 T n ; W v. Fact IV We can combine Facts I III to concude that ZT T ; Q T, S U T, = Z 12 is stationary. We aso have that W T 1 ; = W 1, which foows the stationary distribution of the Kiefer Wofowitz vector of a GI/GI/c queue. The main resut of this paper is stated in the foowing theorem. Theorem 1 Assume A1 is in force, with λ/cμ, 1. Then, Facts I IV hod true. We can detect coaescence at a time T < such that E T <. The rest of the paper is dedicated to the proof of Theorem 1. We have verified a number of monotonicity properties in Sect , which in particuar aow us to concude that the construction of W n and Z t is egitimate i.e., the imits exist amost surey. The monotonicity properties aso yied Fact II and pave the way to verify Fact III. Section 3 proves the finite expectation of the coaescence time. In Sect. 4, we provide more agorithmic detais about our perfect samping construction.

13 3 Coaescence detection in finite time In this section, we give more detais about the coaescence detection scheme. The next resut corresponds to Fact III and Fact IV. Proposition 2 Suppose that w + = W v for some 1, and w =. If W T n ; w + = W T n ; w for some n 1, then W T m ; w + = W m = W T m ; w for a m n. Moreover, for a t Tn + W 1 T n ; w +, Z T t T ; Q v T, S U T, = Z T t T ;,, = Z t. 13 Proof The fact that W T m ; w+ = W m = W T m ; w for m n foows immediatey from the recursion defining the Kiefer Wofowitz vector. Now, to show the first equaity in 13, it suffices to consider t = Tn + W 1 T n ; w +, since from t Tn the input is exacty the same and everyone coming after Tn wi depart the queue and enter service after time T n + W 1 T n ; w +.The arriva processes i.e., E u ceary agree, so we just need to verify that the queue engths and the residua service times agree. First, note that R T T n + W 1 Tn ; w+ T ; Q v T, S U T, = W Tn ; w+ W 1 Tn ; w+ 1 = W Tn ; w W 1 Tn ; w 1 = R T T n + W 1 Tn ; w T ;,,. 14 So, the residua service times of both upper and ower bound processes agree. The agreement of the queue engths foows from Lemma 3. Finay, the second equaity in 13 foows from Proposition 1. Next, we anayze properties of the coaescence time. Define T = sup T : inf T t Z T Z T t T ;,, = t T ; Q v. Notice that if at time T we start an upper bound queue, T Z T ; Q v T, SUT,,, S U T,

14 and a ower bound queue, Z T ;,,, they wi coaesce before time. Thus, if we simuate the system up to T, we wi be abe to detect a coaescence. We next estabish that E[ T ] <. By stationarity, we have that T is equa in distribution to T = inf T : inf t T Z t; Q v, SU, Z t;,, =. Proposition 3 If E[V n ] < ce[a n ] for n 1 and Assumption A1 hods, E[T ] <. Proof Define τ = inf n 1 : W 1 Tn ; W v 1 = W 1 T n ;. By Wad s identity, EA n <, for any n 1; it suffices to show that E[τ] <. We start with an outine of the proof, which invoves two main components. I We first construct a sequence of events which ead to the occurrence of τ. The events that we construct put constraints on the interarriva times and service times so that we see a decreasing trend on the Kiefer Wofowitz vectors. When putting a number of these events together consecutivey, the waiting time of the upper bound system wi drop to zero. We further impose the events for c more arrivas after the waiting time drops to zero. Notice that these c arrivas do not have to wait in both the upper bound and the ower bound systems. Thus, by the time of c-th such arriva, the two systems wi have the same set of customers with the same remaining service times. II Based on events constructed in I, we then spit the process W 1 T n ; W v1 : n 1 into cyces where: IIa the probabiity that the desired event, which eads to coaescence, happens during each cyce is bounded from beow by a positive constant, and IIb the expected cyce ength is bounded from above by a constant. IIa aows us to bound the number of cyces we need to chec before finding τ by a geometric random variabe. Then, we appy Wad s identity using IIb to estabish an upper bound for E[τ]. We next provide more detais of the proof, which are divided into part I and II as outined above. Part I We first construct the sequence of events, Ω : 2, which enjoys the property that if Ω happens, the two bounding systems wi have coaesced by time of the + ck/ɛ 1-th arriva. As E[V n ] < ce[a n ],forn 2, we can find m,ɛ > such that for every n 2, the event H n =V n < cm ɛ, A n > m is nontrivia in the sense that P H n >δ for some δ>. Now, pic K > cm arge enough, and define, for 2, Ω = W c 1 T ; W v1 K + ck/ɛ 1 n= H n.

15 To see the coaescence of the two bounding systems, et W = K, K,...,K T be a c-dimensiona vector with each eement equa to K. We notice that, under Ω, W W 1 T ; W v 1. For n, define Ṽ n = cm ɛ, Ã n = m, and the auxiiary Kiefer Wofowitz sequence + W n+1 = S W n + Ṽ n e 1 Ã n 1. Then, Ω impies V n < Ṽ n and A n > Ã n for n, which in turn impies W 1 T n ; W v 1 W n. Moreover, under Ω,wehave W 1 n = and W n c T < cm for n = + ck/ɛ c + 1,..., + ck/ɛ. Then, W 1 1 n ; W v 1 = and W c 1 T n ; W v 1 < cm for n = + ck/ɛ c + 1,..., + ck/ɛ. This indicates that under Ω, 1 a the arrivas between the + ck/ɛ c + 1-th arriva and the + ck/ɛ -th arriva incuded enter service immediatey upon arriva have zero waiting time, and 2 the customers initiay seen by the + ck/ɛ c + 1-th arriva woud have eft the system by the time of the + ck/ɛ -th arriva. The same anaysis hods assuming that we repace W 1 T ; W v 1 by W 1 T ;. Therefore, by the time of the + ck/ɛ 1- th arriva, the two bounding systems woud have exacty the same set of customers with exacty the same remaining service times, which is equa to their service times minus the time eapsed since their arriva times since a of them start service immediatey upon arriva. We aso notice that since there is no customer waiting, the sorted remaining service time at T+ ck/ɛ 1 coincides with the Kiefer Wofowitz vector W + ck/ɛ 1. Part II We first introduce how to spit the process into cyces, which are denoted as κ i, κ i+1, i 1.LetU K := w : w c K. We define and for i 2, define κ i := κ 1 := inf n 1 : W 1 Tn ; W v 1 U K, n > κ i 1 + ck/ɛ 1 : W 1 T n ; W v1 U K. We denote Θ i = κ i + ck/ɛ 1 n= κ i H n for i 1. We next show that the event Θ i happens during the i-th cyce with positive probabiity. Since PH n >δ, PΘ i δ ck/ɛ >

16 . Let N denote the first i for which Θ i occurs. Then, N is stochasticay bounded by a geometric random variabe with probabiity of success δ ck/ɛ. In particuar, E[N] δ ck/ɛ <. We next show that E[ κ i+1 κ i ] is bounded using the standard Lyapunov argument. Under Assumption A1 and λ<cμ, W 1 T n ; w 1 : n 1 for any fixed initia condition w1 is a positive recurrent Harris chain [1]. Under Assumption A1, we aso have that Q v t : t, is a we-defined process with E[Q v t] < see the random-wa bound in 18. Thus, [ c ] E W v i 1 <. i=1 Consider the Lyapunov function gw = W c, i.e., gw and gw as W. Then, for K arge enough, as λ<cμ, we can find δ, c/λ 1/μ such that [ ] E g W 1 Tc+1,w1 gw1 δ for w1 / U K. 15 We aso have [ ] E g W 1 Tc+1,w1 K + c/μ for w1 U K. Then, by Theorem 2 in [14], E[ κ 1 ] < and we can find a constant M > such that E[ κ i κ i 1 ] < M for i 2. We comment that here we need to oo c steps ahead to identify the downward drift in 15, Thus, we use a genera version of the Lyapunov argument deveoped in [14]. Lasty, by Wad s identity, we have setting κ = that E[τ] E[ κ N ]+ ck/ɛ 1 N = E κ i κ i 1 + ck/ɛ 1 i=1 E[N] M + E[ κ 1 ]+ ck/ɛ 1 <. Remar 1 Foowing the proof, we can aso concude that the number of activities either vacations or services to simuate in the vacation system, denoted by N V,is aso finite in expectation. Since coaescence is detected by the τ-th arriva, we ony need to simuate the vacation system forward in time from time unti we are abe to extract the first Q v + τ service time requirements to match the customers waiting in queue at time and the arrivas from time to coaescence time T. For any m < such that E[V m ] >, we et N i t, i = 1,...,c, denote the counting process corresponding to the i-th truncated vacation process with independent activity times capped by m, i.e., V m. Foowing a standard argument as in the proof of Ward s identity in [1], a oose upper bound for E[N v ] is given by

17 E [N V ] c E i=1 c E i=1 c 4 Simuation procedure [ ] N i T E [Q v + τ] [ N i T + 1] + E [Q v ] + E [τ] E [T ] + m E [V m ] + E [Q v] + E [τ] <. In this section, we first address the vaidity of Fact I, namey, that we can simuate the vacation system bacward in time, jointy with Tn i : m n 1 for i c, for any m Z. Let G e = λ Ḡxdx and F e = μ Fxdx denote equiibrium CDFs of the interarriva time and service time distributions, respectivey. We first notice that simuating the stationary arriva process Tn : n 1 and stationary service/vacation competion process Tn i : n 1 for each 1 i c is straightforward by the reversibiity of Tn i for i c. Specificay, we can simuate the renewa arriva process forward in time from time with the first interarriva time foowing G e and subsequent interarriva times foowing G. We then set T = T for a 1. Liewise, we can aso simuate the service/vacation process of server i, fori = 1,...,c, forward in time from time with the first service/vacation initiation time foowing F e and subsequent service/vacation time requirements distributed as F. LetT i, 1, denote the -th service/vacation initiation time of server i counting forward in time. Then, we set T i = T i. Simiary, we have the equaity in distribution, for a t jointy, X t d = X t ; therefore, we have from 2 that the foowing equaity in distribution hods for a t jointy: Q v t = d sup X s X t. s t The chaenge in simuating Q v t invoves samping Mt = max s t X s jointy with X t during any time interva of the form [, T ] for T >. The rest of the section is devoted to sove this chaenge. The idea is to identify a sequence of random times such that max T t T + X t max X t. t T + Then, MT = max t T X t = max T t T + X t. In particuar, to cacuate MT, we ony need to oo at the maximum of X t over a finite time

18 interva, [T, T + ]. To find such, we appy two trics here. The first tric is to decompose X t into c + 1 random was with negative drift associated with N i for i =, 1,...,c. This is based on the fact that for λ < cμ, we can pic a λ, cμ such that N t at and a/ct N i t are drifting downward to negative infinity. We can then bound Mt by the corresponding running time maximum of the random was with negative drift. The second tric is a miestone event construction, which aows us to identify random times beyond which a random wa with negative drift wi never go above a previousy achieved eve. The miestone events are simiar to the adder height decomposition of a random wa, but we cannot directy use adder height theory because the corresponding expressions for the probabiities of interest for exampe the probabiity of an infinite stricty increasing adder epoch are rarey computabe in cosed form. The miestone construction introduces a parameter m which, together with change of measure ideas, aows us to simuate without bias the occurrence of object such as the time the random wa reaches a certain barrier, for exampe. Putting these miestone events of the random was together and using the fact that Mt can be bounded by the appropriate running time maximums of the random was, we can find the desired. We next provide the detais of the construction. Decomposition Choose a λ, cμ. Then, for t >, X t = N t c N i t = N t at + i=1 c i=1 a c t N i t. We define c+1 random was with negative drift associated with N i t s as foows: S =, S 1 = at1 + 1, S n = S n 1 + aa n for n In particuar, S n = N Tn at n.fori = 1,...,c, S i = a c T 1 i, Si n = S i n a c V n i for n Here, S n i = N i Tn i ati n. Figure 2 pots the reationship between N t at : t and S n : n, and the reationship between a c t N i t : t and : n for i = 1,...,c. In particuar, we notice from Fig. 2 that S i n max N s as =max N t at, s t and for i = 1,...,c, a max s s t c s N i = max n N i t max n N t+1 S n max S n N n, t S n i. max n N i t 1 + S n i.

19 a S N t S 1 at S 2 t b a c t N i t S i S 3 S 4 S 1 i S 2 i S 3 i S 4 i t Fig. 2 The reationship between the renewa processes and the random was a N t at and S n a/ct N i t and S n i b We then notice that, for any given T, c MT = maxx t =max N t at + t T t T i=1 c max N t at + t T max S n N n + T c i=1 a c t N i t max t T max n N i=1 i T 1 a c t N i t S n i. 18 Miestone construction We use the miestone events construction to generate the c+1 random was with negative drift, S i, together with their running time maxima, M i := max n S n i,, i =, 1,...c. This construction is introduced in [5,6], and we sha provide a brief overview here. Fix m > and L 1 such that Pm < M i L+1m >fori =,...,c.the vaues of m and L do not seem to have significant impact on agorithm performance, as ong as they are chosen to be sma. In our numerica impementation, we choose m = 1 and L = 3. For each random wa S n i : n, i =, 1,...,c, we sha define a sequence of downward and upward miestone events, which we denoted as Φ i j and Υ j i, respectivey, for j as foows: Φ i :=, Υi :=, and for j 1, Φ i j := inf n Υ j 1 i I Υ j 1 i < Φi j 1 : Si n Υ j i := inf n Φ i j : Si n > S i + m. Φ i j < S i Lm, Φ i j 1

20 Notice that PΦ i j < = 1 whie PΥ j i < <1, as the random was have negative drift. In fact, under A1, Proposition 2.1 in [6] showspυ j i =, i.o. = 1. We observe that when the event Υ j i = happens, we now that the random wa wi never go above S i + m beyond Φ i Φ i j. This important observation aows j us to find the running time maximum M i. In particuar, et Φi denote the first downward miestone at or after step, and et Φ i be the first downward miestone after Φ with Υ i =. Then, after step Φi, the random wa Si wi never go above the eve S i + m, and S i + m < S i. Therefore, Φ Φ Φ M i = max n S i n Lm + m S i Φ i =max n Φ i S n i, i.e., we just need to find the maximum vaue of the random wa between step and step Φ i. Figure 3 provides a pictoria expanation of the construction. We are now ready to use the miestone events across the c + 1 random was to identify associated with each T 1, such that N i T 1fori = 1,...,c. Define Λ := min Φ j > N T j 1 : S Φ j S N T j =, 19 and, for i = 1,...,c, Λ i := min j 1 Φ i j > N i T 1 : S i S i Φ i j N i T 1 m,υi j =. 2 S n Fig. 3 This figure pots a reaization of the sampe path S n : n 11. Here, we set m = 1and L = 3. Then, Φ1 = 3, Υ 1 = 4, Φ 2 = 7. If Υ 2 =, then for n 7, S n wi stay beow the eve S 7 + m, which is demonstrated by the bod dashed ine. Thus, M 2 = max n 2 S n =S 2 by ony comparing the random wa vaues between step 2 and step 7

21 In particuar, the random wa S n i for n Λ i, i =,...,c. Let : n wi never go above the eve S i + m Λ i := max T Λ, max T i 1 i c Λ i +1 T. 21 Since N T + Λ and N i T + 1 Λ i for i = 1,...,c, max n N T + Therefore, max t T + S n S Λ X t max n N T + S + m + Λ S N T + + m and max n N i T + 1 S n + c i=1 c i=1 N T at + = X T c S i + m Λ i S i N i T 1 max c i=1 T t T + n N i=1 i T + 1 S n i S i + m. Λ i max a c T N i T X t. S n i Under A1, the time it taes to find using the miestone construction has finite expectation Theorem 2.2 in [6]. We sha provide the agorithmic detais to generate the random wa with negative drift together with the miestone events for the ight-taied case in Sect. 4.1 to demonstrate the basic idea. The genera case can be found in [6]. We aso provide the agorithm to match the service time requirements to the customers in the vacation system between two consecutive inspection times in Sect Lasty, the exact simuation agorithm of the GI/GI/c queue is summarized in Sect Simuate a random wa with negative drift jointy with miestone events To demonstrate the basic idea, we wor with a generic random wa with negative drift S n := S n 1 + X n,forn, with S given. We aso impose the ight-tai assumption on X n, i.e., there exist θ> such that E[expθ X n ] <.Let Φ :=, Υ :=,

22 and, for j 1, Φ j := inf n Υ j 1 I Υ j 1 < Φ j 1 : S n < S Φ j 1 Lm, Υ j := inf n Φ j : S n > S Φ j + m. We aso denote τ m = infn : S n > m, S =. Notice that PΥ j = = Pτ m = >. Samping Φ j is straightforward. We just sampe the random wa, S n, unti S n < S Φ j 1 Lm. Samping Υ j and the path conditiona on Υ j < require more advanced simuation techniques, as PΥ j = >. In particuar, we use the exponentia titing idea discussed in [1]. Let ψ X θ = og E [ expθ X n ] be the og moment generating function of X n, then we have EX n = ψ X < and VarX n = ψ X >. By the convexity of ψ X, we can aways find η>with ψ X η = and ψ X η,. Hence, we can define a new measure P η based on exponentia titing so that dp η dp X n = expηx n. Under P η, S n is a random wa with positive drift ψ X η. Thus, P ητ m < = 1. By our choice of η,weasohavepτ m < = E η exp ηs τm. In impementation, we sha generate the path S n under P η unti τ m and chec whether U exp ηs τm, where U is a uniform random variabe independent of everything. If U exp ηs τm, we caim that τ m < and accept the path S n : n τ m as the path of the random wa conditiona on τ m <. The agorithm to sampe the random wa together with the miestone events goes as foows. Throughout this paper, sampe in the pseudocode means samping independenty from everything that has aready been samped. Agorithm RWS: Sampe a random wa with negative drift unti stopping criteria are met Input: L, m, S,...,S n, Φ,...,Φ j, Υ,...,Υ j and stopping criteria H. Note that n = Φ j if Υ j =, and n = Υ j otherwise. If there is no previous simuated partia random wa, then we initiaize n =, j =, Φ =, Υ =, and S as needed. 1. Whie the stopping criteria H are not satisfied, set j j + 1. a Downward miestone simuation Sampe S : n + 1 Φ j under the nomina measure, i.e., generate the random wa unti S n < S Φ j 1 Lm. Update n = Φ j. b Upward miestone simuation Sampe S 1,..., S τm from the tited measure P η. Sampe U Uniform[,1]. If U exp η S τm,setυ j = n + τ m, S n+ = S n + S for = 1,...,τ m and update n n + τ m ; otherwise set Υ j =. 2. Output updated S,...,S n, Φ,...,Φ j and Υ,...,Υ j.

23 4.2 Simuate the vacation system between inspection times To summarize our discussion above, in this section, we provide the pseudocode for generating the vacation system between the inspection time Tκ and Tκ +1,for, κ =, and κ +1 <κ. Agorithm VSS: sampe vacation system between Tκ and Tκ 1, and extract corresponding service times Input: m, L, κ, κ 1, S i,...,si n i, Φ i,...,φi j i, Υ i,...,υi j i for i =, 1,...,c. 1. Appy Agorithm RWS to further sampe S with the stopping criterion H being n κ. Then, find T κ. 2. Appy Agorithm RWS to further sampe S with the stopping criterion H being n = Λ κ, with Λ κ defined in Eq For i = 1,...,c, appy Agorithm RWS to further sampe S i unti the stopping criterion H being n i = Λ i κ, with Λi κ defined in Eq Compute κ as defined in Eq. 21. For i =, 1,...,c, appy Agorithm RWS to further sampe S i with the stopping criterion H being Tn i i T κ + κ. 5. Construct the bacward renewa processes N i t : Tκ κ t using S n i : n n i for i =, 1,...,c. In particuar, we sha set T n i = T n i. Then, construct X t = N t c i=1 N i t for t [Tκ κ, ]. 6. Set MTκ = max T κ κ t Tκ X t and then compute Q v T κ = MTκ X Tκ to be the number of peope waiting in the queue at time Tκ. The remaining activity times are U i Tκ,fori = 1,...,c. 7. If Q v Tκ >1, then for 1 j Q v Tκ 1, the j-th peope waiting in queue arrive at time T κ Q v Tκ + j.let D j = inft : j = c i=1 σ i t, then extract T κ V κ Q v Tκ + j = c i=1 V i N i Tκ + D dn i T j κ + D j as his service time. 8. For κ n 1, use Eq. 3 to extract the service times V κ,...,v Output a service times of the peope waiting in queue at time Tκ excuding the arriva at Tκ, i.e., nu if Q v Tκ = 1 and V κ Q v Tκ +1,...,V κ 1 in the order of arriva if Q v Tκ >1. b matched arriva times and service times Tj, V j : κ j 1 in the order of arriva. c updated random was S i,...,si n i with updated miestone events Φ i,...,φi j i, Υ i,...,υi j i for i =, 1,...,c. 4.3 Overa exact simuation procedure In this section, we provide the overa pseudocode for our exact simuation agorithm. Agorithm PS: sampe stationary GI/GI/c queue at time Input: m, L, F, G, c

24 1. For i =, 1,...,c, initiate Φ i = Υ i =, and Si as defined in Eqs. 16, Set κ =, κ 1 = 1, = a Appy Agorithm VSS to sampe vacation system between Tκ and Tκ 1, and extract corresponding service times. b Start two GI/GI/c queues, both from Tκ, one initiaized with Q v Tκ, S U Tκ, and the other initiaized with. Evove the two queues forward in time unti time and cacuate C = min Z T κ j 1 κ Tj Tκ ; Q v Tκ, S U Tκ, Z T κ Tj Tκ ;. 4. If C =, output Z = Z T κ T κ ;. Otherwise C >, set + 1, κ = 2κ 1, then go bac to Step 3. 5 Numerica experiments As a sanity chec, we have impemented our MATLAB code in the case of an Erang 1,λ/Erang 2,μ/c queue. We provide a detaied MATLAB impementation of each of the agorithms required to execute Facts I IV in the onine Appendix to this paper. Firsty, in the context of the M/M/c queue, which is a specia case of Erang 1,λ/Erang 2,μ/c when 1 = 2 = 1 and whose stationary distribution can be computed in cosed form, we have compared the theoretica distribution to the empirica distribution of the number of customers in the system at stationarity. The empirica distribution is produced from a arge number of runs using our perfect simuation agorithm. Figure 4 shows a comparison of these distributions when λ = 3, μ = 2, and c = 2. Gray bars show the empirica resut of 5 draws using our perfect simuation agorithm, and bac bars show the theoretica distribution of the number of customers in the system. The two are very cose to each other. Foowing [9], we test the goodness of fit using a Pearson s chi-squared test; under the nu hypothesis, the empirica histogram converges to theoretica distribution as the sampe size increases. The test yieds a p vaue equa to.686, indicating cose agreement i.e., we can not reject the nu hypothesis. Simiary, Fig. 5 provides another comparison with a different set of parameters, λ = 1, μ = 2, c = 1, with a p vaue being.6454 from the chi-squared test. Aso, for a genera Erang 1,λ/Erang 2,μ/c queue 1 > 1, 2 > 1 when ρ = λ 1 2 / cλ 2 1 =.9, we have compared the empirica distribution obtained from simuation with the numerica resuts with precision at east 1 4 provided in Tabe III of [17]. Figure 6 shows the comparison for an E 3 /E 2 /5 queue with ρ =.9. We observe that the two histograms are very cose to each other. A Pearson s chi-

25 .25 Number of Customers for an M/M/c queue in equiibrium with ambda = 3, mu = 2, c = 2 5 draws Perfect Simuation Theoretica Fig. 4 Number of customers for an M/M/c queue in stationarity when λ = 3, μ = 2, and c = Number of Customers for an M/M/c queue in equiibrium with ambda = 1, mu = 2, c = 1 5 draws Perfect Simuation Theoretica Fig. 5 Number of customers for an M/M/c queue in stationarity when λ = 1, μ = 2, and c = 1 squared test between the simuated distribution and the numerica one gives a p vaue of Next, we run numerica experiments in the M/M/c case to see how the running time of our agorithm, measured by mean coaescence time of two bounding systems, scaes as the number of servers grows and the traffic intensity ρ changes. Starting from time, the upper bound queue has its queue ength samped from the theoretica distribution of an M/M/c vacation system and a servers busy with remaining service times drawn from the equiibrium distribution of the service/vacation time; and the ower bound queue is empty. Then, we run both the upper bound and ower bound queues forward in time with the same stream of arriva times and service requirements unti they coaesce. Tabe 1 shows the estimated average coaescence time, E[T ], based on 5 iid sampes, for different system scaes in the quaity-driven regime QD. We observe that E[T ] does not increase much as the system scae parameter, s, grows. Tabe 2 shows simiar resuts for the quaity-and-efficiency-driven operating

26 .15 Number of Customers for an E3/E2/5 queue in equiibrium with rho=.9 5 draws Perfect Simuation Numerica Tabe Fig. 6 Number of customers for an Erang 1,λ/Erang 2,μ/c queue in stationarity when 1 = 3, λ = 4.5, 2 = 2, μ = 2/3, c = 5, and ρ =.9 Tabe 1 Simuation resut for coaescence time of M/M/c queue s Mean 95% confidence interva QD: λ s = s, c s = 1.2s,μ= [6.292, ] [6.9848, ] [7.6667, ] Tabe 2 Simuation resut for coaescence time of M/M/c queue s Mean 95% confidence interva QED: λ s = s, c s = s + 2 s,μ= [6.3771, ] [8.4361, ] [9.341, 9.645] regime QED. In this case, E[T ] increases at a faster rate with s than the QD case, but the magnitude of increment is sti not significant. Finay, we run a numerica experiment in the M/M/c case aiming to test how computationa compexity of our agorithm changes with traffic intensity, ρ = λ/cμ. Here, we define the computationa compexity as the tota number of renewas incuding arrivas and services/vacations the agorithm sampes in tota to find the coaescence time. We expect the compexity to scae ie c + 11 ρ 2 E[T ρ], where c + 1 is the number of renewa processes we need to simuate, 1 ρ 2 is on average the amount of renewas we need to sampe to find its running time maximum for each renewa process, and E[T ρ] is the mean coaescence time when the traffic intensity is ρ. Tabe 3 summarizes our numera resuts, based on 5 independent runs of the agorithm for each ρ. We run the coaescence chec at 1 2 for = 1, 2,..., unti we find the coaescence. We observe that as ρ increases, the computationa compexity increases significanty, but when mutipied by 1 ρ 2, the resuting products are of