A Fluid Limit for the Engset Model An Application to Retrial Queues
Stylianos Georgiadis1, Pascal Moyal1, Tam´as B´erczes2 and J´anos Sztrik2 1Laboratoire de Math´ematiques Appliqu´ees de Compi`egne, Universit´ede Technologie de Compi`egne, Centre de Recherches de Royallieu, BP 20529, 60205, Compi`egne, France 2Faculty of Informatics, University of Debrecen, Egyetem tr 1, P.O. Box 12, 4010 Debrecen, Hungary
Keywords: Engset Model, Fluid Limit, Semi-martingale Decomposition, Retrial Queues.
Abstract: We represent the classical Engset-loss model by the stochastic process counting the number of customers in the system. A fluid limit for this process is established for all the possible values of the various parameters of the system, as the number of servers tends to infinity along with the number of sources. Our results are derived through a semi-martingale decomposition method. A numerical application is provided to illustrate these results. Then, we represent a finite-source retrial queue considering in addition the number of sources in orbit. Finally, we extend the fluid limit results to a retrial queueing system, discussing different cases.
1 INTRODUCTION tend to infinity. In our case, the number of servers tends to infinity along with the number of sources. In many real-life queueing systems of finite capacity, Such techniques have been applied fruitfully to many a customer may find a fullsystem upon arrival. In sev- queueing systems (Robert, 2000; Asmussen, 2003; eral finite-source models, this request can return to the Anisimov, 2007; Decreusefond and Moyal, 2012). source and stay there for a randomly distributed time Recently, (Feuillet and Robert, 2012) constructed ex- until it tries again to reach a server. The Engset model ponential martingales for the Engset model, allow- represents a loss queueing system having this input ing to derive asymptotic estimates for several hitting mechanism for several finite sources producing Pois- times of interest. We build on these results to de- son processes of the same intensity (see, e.g., (Engset, rive the fluid limit of an Engset model having a single 1918)). We suppose that the system has no buffer, server (Section 3), and then several servers (Section hence a request is either immediately served or im- 4). Simulations are presented in Section 5. mediately lost, whenever no server is available upon In a finite source retrial queue, the messages arrival. which couldnot reach a serverare sent to the so-called Such a model has been applied to a variety of re- orbit, from which they are re-emitted on and on, at a alistic computer and telecommunication systems and rate that is possibly higher than the original one. It is networks. For exemple, an Engset system is adequate then easily seen that the Engset model in nothing but a to representa radio-mobilenetwork in which the radio particular case of a retrial queueing system for which sources emit messages only if no message of the same the two emission rates are equal. Based on this obser- source is currently in service. One could think that the vation, in Section 6 we investigate some applications radio sources re-emit the same message as long as the of our initial result to derive the fluid approximation latter is refused due to the fact that all channels are of a retrial queue, under various conditions on the sys- busy, and wait to re-issue a new message whenever tem parameters. the previous message is in treatment. This model has a wide field of applications, so it has been studied extensively through analytical and algorithmic methods as well. However, when the sys- 2 THE ENGSET MODEL tem becomes very large, several complexity problems may appear. The fluid limit techniqueoffers the possi- We consider an Engset system with S (S 1) servers. bility to approximate the exact values of some charac- There are K (K > S) independent Poisson≥ sources teristics of the system, when one or more parameters emitting requests with intensity λ. The service times
Georgiadis S., Moyal P., Bérczes T. and Sztrik J.. 315 A Fluid Limit for the Engset Model - An Application to Retrial Queues. DOI: 10.5220/0004203101170122 In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 117-122 ISBN: 978-989-8565-40-2 Copyright c 2013 SCITEPRESS (Science and Technology Publications, Lda.) ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems
of the requests are exponentially distributed of param- 2.2 The Free Process eter µ. Whenever a request finds all servers busy, it is immediately lost. If not, the request enters service The free process describes an Engset model without and the corresponding source remain inactive during limitation in the number of servers, hence an infinite its treatment. As soon as the latter service has been server queues M/M/∞/∞/nK, having the same input completed, the source becomes active again, and re- mechanism. As above, the process Y n counting the emit jobs according to a Poisson process of intensity λ number of customers in the system, satisfies the semi- that is independent from the past. In particular, there martingale decomposition t is no dependence between the holding times and idle n n n periods of the sources. Y (t)= Y (0) µ Y (s)ds − Z0 Let XS := (XS(t); t 0) denote the process count- t ing the number of customers≥ in the system (i.e., the + λ (nK Y n(s)) ds + Pn(t) Z0 − number of busy servers) at current time. XS is a t Markov process, whose stationary measure is well- = Y n(0) (λ + µ) Y n(s)ds + λnKt + Pn(t), known and easily derived. − Z0 We first examine the Engset queueing model with (2) n a single server, and denote X := X1 the corresponding where P is a martingale. process.
2.1 Semi-martingale Decompositions 3 FLUIDLIMIT
Recall (Jacod and Shiryaev, 2003; Decreusefond and We are interested in the asymptotic behavior of X n Moyal, 2012) that for a given Feller Markov process Z (properly rescaled), as the number of servers goes to of state space E and infinitesimal generator A defined infinity together with the number of sources. We ob- for all bounded f : E R by tain hereafter a fluid limit that coincides with that of a → 1 loss Erlang system M/M/n/n, and proceed as in Sec- A f (i)= lim E[ f (X(h)) X(0)= i] f (i) , i E, h 0 h | − ∈ tion 6.7 of (Robert, 2000). → the process We normalize the various processes as follows. t For all t 0, ≥ n n M f : t f (Z(t)) f (Z(0)) A f (Z(s)) ds (1) n X (t) n Y (t) → − − Z0 X¯ (t)= ; Y¯ (t)= . n n is a martingale w.r.t. the natural filtration of Z. Assume that the deterministic initial condition satis- For n 1 fixed, consider an Engset system ≥ n fies M/M/n/n/nK, and add a superscript to all the param- X¯ n(0) x, n ∞ eters involved. It is easily seen that the infinitesimal −→→ generator An of X n reads for all bounded f : R R where x [0,1] fixed. ∈ and all i 0,...,n , → We easily check that the semi-martingale equation ∈{ } (2) is similar to that of an M/M/∞ system of arrival in- An f (i)= tensity λnK and service durations of parameter λ + µ. λ(K i)( f (i + 1) f (i))+ µi( f (i 1) f (i)), Whenever Y¯ n(0) x, it then follows from Theorem − − − − n−→∞ = i 1,...,n 1 ; 6.13 of (Robert, 2000)→ that for all T 0, ∈{ − } ≥ µn( f (n 1) f (n)), i = n. − − E sup Y¯ n t Y t 0, (3) ( ) ∗( ) ∞ So, taking f as the identity function of 0,...,n in 0 t T | − | n−→ ≤ ≤ → (1), we get { } where, for all t 0, ≥ n α α (λ+µ)t A f (i)= λ(nK i)1 i 316 AFluidLimitfortheEngsetModel-AnApplicationtoRetrialQueues and Proof. Notice that τ = inf t 0;Y ∗(t)= 1 . (6) n { ≥ } P sup X¯ (t) X ∗(t) 0 t T | − | 3.1 Heavy Traffic ≤ ≤ n n P(τ T)+ P sup Y¯ (t) Y ∗(t) > ε , First, we examine the case α > 1. Then, we get ≤ ≤ 0 t T | − | ≤ ≤ 1 α x and apply (3) together with Markov inequality. The τ = log − . (7) τn λ + µ α 1 first term vanishes as is asymptotically of the or- − der of nαn 1, as can be shown along the lines of The following lemma follows from Proposition 6 of ( )− (Feuillet and Robert, 2012). (Feuillet and Robert, 2012). Lemma 1. In heavy traffic, the following convergence 3.3 Critical Case in probability holds for the hitting time (5) : P α τn τ, (8) Suppose that = 1. We reason as above : n ∞ n −→→ Theorem 3. In the critical case, τ is of the order of τ where is defined by (7). log√n, hence the same convergence as in Theorem 2 Theorem 1. For all ε > 0 and all T 0, we have holds true. ≥ P sup X¯ n t X t > ε 0, ( ) ∗( ) ∞ 0 t T | − | n−→ ≤ ≤ → where, for all t 0, 4 MULTISERVER ENGSET ≥ (λ+µ)t MODEL X ∗(t)= 1 α + (x α)e− . ∧ − Proof. We have We now check that the results of Section 3 still hold true for an Engset queue with S identical servers and n P sup X¯ (t) X ∗(t) > ε the process XS counting the busy servers. We consider 0 t T | − | ≤ ≤ the Engset model M/M/nS/nS/nK. Easily, we obtain n n n the infinitesimal generator AS of XS P sup Y¯ (t) X ∗(t) > ε τn n λ ≤ 0 t T | − | AS f (i)= (nK i)1 i 317 ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems Theorem 4. For all ε > 0 and all T 0, we have ≥ 500 P sup X¯ n t X t > ε 0, 450 S ( ) S∗( ) ∞ 0 t T | − | n−→ 400 ≤ ≤ → 350 where, for all t 0, we consider the following cases : ≥ 300 if α > S, 250 Busy Servers 200 α α (λ+µ)t X ∗(t)= S + (xS )e− ; (11) 150 S ∧ − 100 if α S, ≤ 50 Trajectory λ Fluid Limit α α ( +µ)t 0 XS∗(t)= + (xS )e− . (12) 0 5 10 15 20 25 30 35 40 45 50 − Time As a consequence, the fluid limit for the Engset Figure 1: Heavy traffic (K > 3000). system can be derived for any arbitrary, but fixed, val- ues of the number of sources and servers. 500 450 400 5 NUMERICAL RESULTS 350 300 In this section, we present a numerical example for 250 the multiserver Engset model concerning the different Busy Servers 200 cases discussed in Section 4. Consider a M/M/S/S/K 150 λ queueing system with parameters S = 500, = 0.1 100 and µ = 0.5. The critical value for the number of 50 Trajectory Fluid Limit sources is, therefore, K = 3000. Setting various val- 0 0 5 10 15 20 25 30 35 40 45 50 ues for the number of sources, we may obtain the Time heavy traffic (α > 500), the light traffic (α < 500) or Figure 2: Light traffic (K < 3000). the critical case (α = 500). We set the initial value xS = 100 for the busy servers. 500 In the following figures, we present a realization 450 of the process X 1 of the model described above in the S 400 time interval [0,50], along with the fluid limit, for the 350 three cases as given in (11) and (12). 300 Notice that, in Figure 1, the hitting time of S 250 servers for the realization (4.6381) is fairly close to Busy Servers 200 the theoretical value of τS (5.3648). 150 100 50 Trajectory Fluid Limit 0 6 APPLICATION TO SINGLE 0 5 10 15 20 25 30 35 40 45 50 SERVER RETRIAL QUEUES Time Figure 3: Critical case (K = 3000). Retrial queues follow the following scenario : when a customer arrives with all servers and waiting positions and leaves it after completion of service. The rate of (if any) being busy, he leaves the service area but after service time is denoted by µ. If all servers are busy some randomly distributed time repeats his demand. at arrival time, then the source goes into the orbit (a For a review of the main results on the topic see (Falin secondary queue of infinite size) and starts the gener- and Templeton, 1997) and the references therein. ation of requests with rate ν until it finds a free server. The general queueing system with retrials is de- After completion of service, the source returns to the scribed more precisely as follows. There are finitely initial state and it can generate a new request, while many identical independent fully available servers at the server may serve a new request. All the times in- which requests arrive. Each source can generate a re- volved in the model are assumed to be mutually inde- quest with rate λ. If an arriving request finds at least pendent of each other. one server free, it immediately occupies the server The presence of the orbit makes the retrial queue- 318 AFluidLimitfortheEngsetModel-AnApplicationtoRetrialQueues ing model more flexible than the Engset one, once the and source may generate a message with a different rate t n n ν λ n from the rate of the initial state. Actually, an Engset C (t)=C (0) + ( ) N (s)1 Cn(s) 319 ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems interval as n goes large. Moreover, the Aldous- 7 CONCLUSIONS Reboledo tightness criterion for semi-martingales (see, e.g. (Joffe and Mtivier, 1986)) is easily met by We have derived the fluid limit of an Engset queueing both N¯ n and C¯n. Thus, the sequence N¯ n( ), C¯n( ) is system with several servers. After discussing the con- tight, and any subsequential limit (N∗ ( ), C∗( )) reads nection between the Engset queue and retrial queue- for all t 0, the equations (13) and (14). The weak ing models, we present several fluid limit results for limit is then≥ a solution of this system. The unique- retrial queues. The generalization of the fluid limit ness of the latter is easily checked by showing that for all possibles values of the parameters λ, µ and ν the underlying mapping is locally-Lipschitz continu- of the retrial model, and the numerical confirmation ous. of their accuracy is a challenging problem that is cur- rently under investigation. 6.2 Discussion We discuss applications of the latter result in several REFERENCES cases. (i) If λ = ν, from (4) and (14), C∗ coincides with the Anisimov, V. V. (2007). Switching Processes in Queuing fluid limit Y ∗ of Section 3, in the various cases. Models. ISTE-Wiley, Washington. Setting, whenever τ is finite (i.e. in the heavy traf- Asmussen, S. (2003). Applied Probability and Queues. fic case), Spring-Verlag, New York, 2nd edition. λτ τ0 = n0e− , Decreusefond, L. and Moyal, P. (2012). Stochastic Model- ing and Analysis of Telecom Networks. ISTE-Wiley. from the equation (13), we obtain that Engset, T. (1918). Die wahrscheinlichkeitsrechnung zur λt N∗(t)=n0e− 1 t<τ bestimmung der wahleranzahl in automatischen fern- { } sprechamtern. Elektrotech. Zeit., 39(31):304–306. τ λ(t τ) + (K 1 + [ 0 (K 1)]e− − )1 t τ . Falin, G. and Templeton, J. (1997). Retrial Queues. Chap- − − − { ≥ } We check as well the intuitive result that asymp- man & Hall, London. totically, the orbit is either full (heavy traffic case) Feuillet, M. and Robert, P. (2012). On the transient behav- ior of Ehrenfest and Engset processes. Advances in or empty (light traffic/ critical case). Applied Probability, 44:562–582. (ii) Suppose now that Kλ λ + µ, and fix the initial Jacod, J. and Shiryaev, A. (2003). Limit Theorems for ≤ condition n0 = 0. Let Stochastic Processes. Springer, Berlin, 2nd edition. ρn = inf t 0; Nn(t) > 0 . Joffe, A. and Mtivier, M. (1986). Weak convergence { ≥ } of sequences of semimartingales with applications to Up to ρn, no arrival occurs from the orbit, so the multitype branching processes. Advances in Applied value of ν is irrelevant. It is easily seen that Probability, 18:20–65. ρn τn Robert, P. (2000). R´eseaux et Files d’Attente : M´ethodes = in distribution, Probabilistes. Springer, Paris. where τn corresponds to the previous hitting time for an Engset model. So, from the results of the previous section in the light traffic and critical cases, a proof similar to that of Theorem 2 shows that, for all t 0, ≥ N∗(t)= 0; C∗(t)= Y ∗(t), where Y ∗(t) is given in (10). (iii) All the same, if finally λ < ν and Kλ > ν + µ, we show by stochastic comparison of Markov pro- cesses that n n ρ st τ , ≤ where τn corresponds to a heavily loaded Engset model of arrival rate λ. So, Theorem 1 entails that for some 0 < ρ τ, where τ is defined by (6), ≤ C∗(t)= 1, for all t ρ, ≥ i.e. the server is busy all the time after ρ, at the fluid level. 320