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A Fluid Limit for the Engset Model an Application to Retrial Queues

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A Fluid Limit for the Engset Model an Application to Retrial Queues 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<n iµ, Y ∗(t)= + (x )e− , (4) − { } − − which leads to the following semi-martingale decom- setting λK position : α = . t λ + µ n n n X (t)= X (0) µ X (s)ds Let the hitting times − Z0 t τn := inf t 0; X¯ n(t)= 1 λ n n + (nK X (s))1 Xn(s)<n ds + M (t), { ≥ n } Z0 − { } = inf t 0; X (t)= n { ≥ } where Mn is a martingale and X n(0) [0,n]. = inf t 0; Y n(t)= n (5) ∈ { ≥ } 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− .
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