Queueing networks with a single shared server: light and heavy traffic
aa r X i v : . [ m a t h . P R ] A ug Queueing networks with a single shared server:Light and heavy traffic
M.A.A. Boon
Department of Mathematicsand Computer ScienceEindhoven University ofTechnologyP.O. Box 513, 5600MBEindhoven, The Netherlands [email protected] R.D. van der Mei
Centre for Mathematics andComputer Science (CWI)Department of Probability andStochastic Networks1098 SJ Amsterdam, TheNetherlands [email protected] E.M.M. Winands
Department of Mathematics,Section StochasticsVU UniversityDe Boelelaan 1081a, 1081HVAmsterdam, The Netherlands [email protected]
ABSTRACT
We study a queueing network with a single shared server,that serves the queues in a cyclic order according to thegated service discipline. External customers arrive at thequeues according to independent Poisson processes. Aftercompleting service, a customer either leaves the system oris routed to another queue. This model is very generic andfinds many applications in computer systems, communica-tion networks, manufacturing systems and robotics. Specialcases of the introduced network include well-known pollingmodels and tandem queues. We derive exact limits of themean delays under both heavy-traffic and light-traffic condi-tions. By interpolating between these asymptotic regimes,we develop simple closed-form approximations for the meandelays for arbitrary loads.
Categories and Subject Descriptors
G.3 [
Mathematics of Computing ]: Probability and Statis-tics—
Queueing Theory
General Terms
Theory, Performance
Keywords
Queueing network, heavy traffic, light traffic, approximation
1. INTRODUCTION
In this paper we study a queueing network served by asingle shared server, that visits the queues in a cyclic order.Each queue receives gated service. Customers from the out-side arrive at the queues according to independent Poissonprocesses, and the service time and switch-over time dis-tributions are general. After receiving service at queue i , acustomer is either routed to queue j with probability p i,j , orleaves the system with probability p i, . This model can beseen as an extension of the standard polling model by cus-tomer routing, introduced and analyzed by Sidi et al. [7, 8].The possibility of re-routing of customers further enhancesthe already-extensive modeling capabilities of polling mod-els, which find applications in diverse areas such as computersystems, communication networks, logistics, flexible manu-facturing systems, robotics systems, production systems andmaintenance systems (see, for example, [1] for overviews). Applications of this type of customer routing can be found,for example, in manufacturing when products undergo ser-vice in a number of stages or in the context of rework. Thepresent research can be seen as a unifying analysis for a vari-ety of special cases (besides polling systems), such as tandemqueues [5], multi-stage queueing models with parallel servers[4], feedback vacation queues [2] and many others.The main contribution of the present paper is twofold.Firstly, we study exact heavy-traffic (HT) asymptotics of thesystem under consideration. Under HT scalings we derive aclosed-form expression for the joint queue-length vector atpolling instants under HT scalings. This result, in turn, isshown to lead to a closed-form expression for the expecteddelay at a queue at an arbitrary moment. This expression isstrikingly simple and shows explicitly how the expected de-lays depend on the system parameters, and in particular, onthe routing probabilities p i,j . Secondly, we derive a closed-form approximation for the mean delay for arbitrary loads,based on an interpolation between the light-traffic (LT) andHT limits. Numerical results are presented to assess the ac-curacy of this approximation. We would like to note thatthe analysis of the present paper can be extended in sev-eral directions, for example different server routing policiesor service disciplines. For reasons of compactness, these re-sults are however not discussed.The remainder of this paper is organized as follows. InSection 2 we describe the model in further detail. Thismodel is analyzed under HT scalings in Section 3. In Sec-tion 4 we develop an approximation based on the LT andHT limits for the mean waiting times and give a numericalexample.
2. MODEL DESCRIPTION
In this paper we consider a queueing network consistingof N ≥ Q , . . . , Q N . External cus-tomers arrive at Q i according to a Poisson arrival processwith rate λ i , and have a generally distributed service re-quirement B i at Q i , with mean value b i := E [ B i ], and sec-ond moment b (2) i := E [ B i ]. The queues are served by a sin-gle server in cyclic order. Whenever the server switches from Q i to Q i +1 , a switch-over time R i is incurred, with mean r i .The cycle time C i is the time between successive momentswhen the server arrives at Q i . The total switch-over timein a cycle is denoted by R = P Ni =1 R i and its first two mo-ents are r := E [ R ] and r (2) := E [ R ]. Indices throughoutthe paper are modulo N , so Q N +1 actually refers to Q . Allservice times and switch-over times are mutually indepen-dent. Each queue receives gated service, which means thatonly those customers present at the server’s arrival at Q i will be served before the server switches to the next queue.This queueing network can be modeled as a polling system with the specific feature that it allows for routing of thecustomers: upon completion of service at Q i , a customer iseither routed to Q j with probability p i,j , or leaves the sys-tem with probability p i, . Note that P Nj =0 p i,j = 1 for all i , and that the transition of a customer from Q i to Q j takesno time. The model under consideration has a branchingstructure, which is discussed in more detail by Resing [6].The total arrival rate at Q i is denoted by γ i , which is theunique solution of the following set of linear equations: γ i = λ i + N X j =1 γ j p j,i , i = 1 , . . . , N. The offered load to Q i is ρ i := γ i b i and the total utilisationis ρ := P Ni =1 ρ i . We assume that the system is stable, whichmeans that ρ should be less than one (see [8]). The totalservice time of a customer is the total amount of servicegiven during the presence of the customer in the network,denoted by e B i , and its first two moments by ˜ b i := E [ e B i ]and ˜ b (2) i := E [ e B i ]. The first two moments are uniquelydetermined by the following set of linear equations: For i =1 , . . . , N , ˜ b i = b i + N X j =1 ˜ b j p i,j , ˜ b (2) i = b (2) i + 2 b i N X j =1 ˜ b j p i,j + N X j =1 ˜ b (2) j p i,j . For each variable x (which may be a scalar, a vector or amatrix) that is a function of ρ , we denote its value evaluatedat ρ = 1 by ˆ x . Also, we will be taking HT limits, letting ρ ↑
1. To be precise, the limit is taken such that the arrivalrates λ , . . . , λ N are increased, while keeping the service andswitch-over time distributions, the routing probabilities andthe ratios between these arrival rates fixed. Also, an N -dimensional vector x has components ( x , . . . , x N ).
3. HEAVY-TRAFFIC ANALYSIS
Let X := ( X . . . , X N ) be the N -dimensional vector thatdescribes the joint queue length at a visit beginning at Q .Sidi et al. [8] show that the joint queue-length process atsuccessive visit beginnings to Q constitutes an N -dimensionalmulti-type branching process (MTBP) with immigration ineach state. In this section we focus on the limiting behaviorof X as ρ goes to 1. Theorem 1
The joint queue-length vector at polling instantsat Q has the following asymptotic behavior: (1 − ρ ) X ... X N d → ˜ b (2) b (1) δ ˆ u ... ˆ u N Γ( α,
1) ( ρ ↑ , with u i := λ i N X j = i ρ j + N X j = i γ j p j,i ( i = 1 , . . . , N ) ,δ := ˆ u ⊤ ˜ b = N X i =1 ˆ λ i ˜ b i N X j = i +1 ˆ ρ j + N X i =1 ˜ b i N X j = i ˆ γ j p j,i , (1) α := 2 rδ ˜ b (1) ˜ b (2) , ˜ b ( k ) := N X i =1 λ i E [ e B ki ] / N X j =1 λ j , and where Γ( α, is a gamma-distributed random variablewith shape parameter α and scale parameter 1. For reasons of compactness we omit the proof, noting thatthe basis for the proof is given by the general framework asproposed in [9].Next, we focus on the workload in the individual queues.To this end, we note that simple balancing arguments canbe used to show that E [ C i ] = r/ (1 − ρ ), which does not de-pend on i . To obtain HT-results for the amount of work ineach queue, we use the Heavy Traffic Averaging Principle(HTAP) for polling systems [3]. When the system becomessaturated, two limiting processes take place. The scaled to-tal workload tends to a Bessel-type diffusion whereas thework in each queue is changing at a much faster rate thanthe total workload. This implies that during the course ofa cycle, the total workload can be considered as constant,while the workloads of the individual queues fluctuate ac-cording to a fluid model. The HTAP relates these two lim-iting processes. The fluid limit of the per-queue workload isobtained by dividing by r/ (1 − ρ ) and letting ρ ↑
1. For ourmodel, the fluid model for the workload at Q i is a piecewiselinear function. More precisely, it is easy to show that thefluid limit of the mean amount of work at Q i at the begin-ning of a visit to Q j is P j + N − k = i ˆ γ k ˜ b i (ˆ λ i b k + p k,i ) for j = i and ˆ γ i b i for j = i . Moreover, in the fluid limit the proba-bility that at an arbitrary moment the server is visiting Q j is ˆ ρ j ( j = 1 , . . . , N ). Combining these observations, one canobtain the following expression for δ i , defined as the fluidlimit of the average amount of work at Q i . Lemma 1
For i = 1 , . . . , N , δ i = 12 ˆ ρ i ˆ γ i ˜ b i (1 + ˆ λ i b i + p i,i ) (2)+ i + N − X j = i +1 ˆ ρ j (cid:0)
12 ˆ γ j ˜ b i (ˆ λ i ˜ b j + p j,i ) + j − X k = i ˆ γ k ˜ b i (ˆ λ i b k + p k,i ) (cid:1) . Note that in the classical case where p i,j = 0 for all i, j wehave δ i = ˆ ρ i (1 + ˆ ρ i ) /
2. Moreover, it is easily verified that P Ni =1 δ i = δ , where δ is defined in (1).Subsequently, the diffusion limit of the total workload pro-cess and the workload in the individual queues can be relatedusing the HTAP. To this end, we start with the cycle-timedistribution under HT scalings. emma 2 For i = 1 , . . . , N , (1 − ρ ) C i d → Γ( α, µ ) , ( ρ ↑ , with α = 2 rδ ˜ b (1) / ˜ b (2) and µ = 2 δ ˜ b (1) / ˜ b (2) . Note that the distribution of C i no longer depends on i inthe HT limit. The proof can be found along the same linesas in [9]. Lemma 2 implies that the HT limit of the mean (scaled) amount of work found by an arbitrary customer is δ i E [ C ], where C is the length-biased cycle time, with limitingdistribution Γ( α + 1 , µ ). Again, see [9] for more details.The (HT limit of the) mean waiting time of an arbitrarycustomer in Q i can be found by application of Little’s Lawto the mean queue length at Q i , which is simply the meanamount of work in Q i divided by the mean total servicetime. Theorem 2
For i = 1 , . . . , N , (1 − ρ ) E [ W i ] → (cid:18) r + ˜ b (2) δ ˜ b (1) (cid:19) δ i ˜ b i ˆ γ i , ( ρ ↑ . (3) Remark 1 (Insensitivity)
Equation (3) reveals a varietyof properties about the dependence of the limiting mean delaywith respect to the system parameters. The mean waitingtimes E [ W i ] are independent of the visit order of the server,depend on the switch-over time distributions only through r , and depend on the service-time distributions only through ˜ b (1) and ˜ b (2) .
4. APPROXIMATION
The LT limit of E [ W i ] can be found by conditioning onthe customer type (external or internally routed). Theorem 3
For i = 1 , . . . , N , E [ W i ] → λ i γ i r (2) r + i − X j = i − N γ j p j,i γ i i − X k = j r k , ( ρ ↓ . (4)In light traffic we ignore all O ( ρ ) terms, which implies thatwe can consider a customer as being alone in the system.Equation (4) can be interpreted as follows. An arbitrarycustomer in Q i has arrived from outside the network withprobability λ i γ i . In this case he has to wait for a residualtotal switchover time with mean r (2) / r . If a customer in Q i arrives after being served in another queue, say Q j (withprobability γ j p j,i /γ i ), he has to wait for the mean switch-over times r j , . . . , r i − .Subsequently, we construct an interpolation between theLT and HT limits that can be used as an approximation forthe mean waiting times. For i = 1 , . . . , N , W approx i = w LT i + ( w HT i − w LT i ) ρ − ρ , (5)where w LT i and w HT i are the LT and HT limits respectively,as given in (4) and (3). Because of the way W approx i is con-structed, it has the nice properties that it is exact as ρ ↓ ρ ↑
1. Furthermore, it satisfies a so-called pseudo-conservation law for the mean waiting times, which is de-rived in [8]. This implies that the W approx i yields exact re-sults for symmetric (and, hence, single-queue) systems. We do not aim at giving an extensive numerical studyto assess the accuracy of the approximation. Instead, wegive one numerical example that indicates the versatility ofthe model that we have discussed, and shows the practicalusage of the approximation (5). To this end, we use anexample that was introduced by Katayama [4], who studiesa network consisting of three queues. Customers arrive at Q and Q , and are routed to Q after being served. Thismodel, which is referred to as a tandem queueing modelwith parallel queues in the first stage, is a special case of themodel discussed in the present paper. We simply put p , = p , = p , = 1 and all other p i,j are zero. We use the samevalues as in [4]: λ = λ /
10, service times are deterministicwith b = b = 1, and b = 5. The server visits the queuesin cyclic order: 1, 2, 3, 1, . . . . The only difference with themodel discussed in [4] is that we introduce (deterministic)switch-over times r = r = 2. We assume that no time isrequired to switch between the two queues in the first stage,so r = 0. In Table 1 we show the mean waiting timesof customers at the three queues and their approximatedvalues. From this table we can see that the accuracy is bestfor values of ρ close to 0 or 1, but the overall accuracy isvery good in general. ρ .
01 0 . . . . . . E [ W ] 2 .
05 2 .
53 3 .
87 6 .
07 10 .
95 34 .
87 356 . W approx .
04 2 .
40 3 .
53 5 .
57 10 .
34 34 .
17 355 . E [ W ] 2 .
05 2 .
56 4 .
01 6 .
45 11 .
95 39 .
05 403 . W approx .
04 2 .
45 3 .
74 6 .
05 11 .
46 38 .
49 403 . E [ W ] 2 .
02 2 .
26 3 .
18 5 .
04 9 .
62 33 .
00 349 . W approx .
04 2 .
39 3 .
51 5 .
52 10 .
22 33 .
69 350 . Table 1: Results for the numerical example.
5. REFERENCES [1] M. A. A. Boon, R. D. van der Mei, and E. M. M.Winands. Applications of polling systems.
Surveys inOperations Research and Management Science ,16:67–82, 2011.[2] O. J. Boxma and U. Yechiali. An
M/G/
Journalof Applied Probability , 34:773–784, 1997.[3] E. G. Coffman, Jr., A. A. Puhalskii, and M. I. Reiman.Polling systems in heavy-traffic: A Bessel process limit.
Mathematics of Operations Research , 23:257–304, 1998.[4] T. Katayama. A cyclic service tandem queueing modelwith parallel queues in the first stage.
StochasticModels , 4:421–443, 1988.[5] S. S. Nair. A single server tandem queue.
Journal ofApplied Probability , 8(1):95–109, 1971.[6] J. A. C. Resing. Polling systems and multitypebranching processes.
Queueing Systems , 13:409–426,1993.[7] M. Sidi and H. Levy. Customer routing in pollingsystems. In P. King, I. Mitrani, and R. Pooley, editors,
Proceedings Performance ’90 , pages 319–331.North-Holland, Amsterdam, 1990.[8] M. Sidi, H. Levy, and S. W. Fuhrmann. A queueingnetwork with a single cyclically roving server.
QueueingSystems , 11:121–144, 1992.9] R. D. van der Mei. Towards a unifying theory onbranching-type polling models in heavy traffic.