Fedorova, Ekaterina A.

Moiseev, Alexander

Nazarov, Anatoly A.

English

Tomsk State University Repository

11th International Workshop onRetrial Queuesand Related TopicsWRQ-2016, AMSTERDAMAugust 31 – September 2,2016Tinbergen Institute,Gustav Mahlerplein 117,1082 MS Amsterdam,The Netherlands.Programme Schedule & AbstractsRetrial queueing system under heavy load conditionEkaterina Fedorova, Alexander Moiseev, Anatoly Nazarov,Tomsk state university,Tomsk, Russia.e-mail: mosskos@gmail.comOur group of researchers from Tomsk State University develops the method of asymptotic anal-ysis for diﬀerent types of retrial queueing systems. Single-server and multi-server retrial queueswith collision, priority or impatient customers, various protocols are studied. We consider systemsunder various asymptotic conditions: heavy load, long delay, long time of patience, etc. In thepaper, we describe the asymptotic analysis method under a heavy load condition and presentresults obtained for some types of retrial queues.Asymptotic analysis of retrial queue M/M/N. Let us consider retrial queueing system ofM/M/N type. The arrival process is Poisson with a rate λ. There are N servers. Service times areexponentially distributed with a rate μ. If a call arrives when there is a free server, the call occupiesit for the service. If all servers are busy, the call goes to the orbit. There, call is waiting during arandom time distributed by exponential law with parameter σ. Then the call from the orbit makesan attempt to get a service again. If there is a free server, the call occupies it, otherwise the callinstantly returns to the orbit. Let i(t) be a number of calls in the orbit and k(t) be a number ofbusy servers. Denote by P (k, i, t) = P{k(t) = k, i(t) = i} the probability that there are k callsservicing and i calls waiting in the orbit at the time moment t.We write Kolmogorov diﬀerential equations for partial characteristic functionsHk(u) =∑iejuiP (k, i),where P (k, i) = limt→∞P (k, i, t) is a probability distribution for stationary regime. Denote ρ = λ/Nμas a load parameter. We solve obtaining system by the method of asymptotic analysis under aheavy load condition ρ ↑ 1 or ε ↓ 0 , where ε = 1− ρ > 0. First of all, we introduce notations:u = εw, H0(u) = εNF0(w, ε), . . . , Hk(u) = εN−kFk(w, ε), . . . , HN(u) = FN(w, ε).In the next step, using expansions Fk(w, ε) = Fk(w) + εfk(w) + O(ε2) and limits limε→0Fk(w, ε) =Fk(w), we solve the system of equations for asymptotic functions. As a result, we derive the asymp-totic characteristic function in the form h(u) =N∑k=0Hk(u) = FN(u1− ρ)+O(ε). Making inversesubstitutions, we obtain that the required function is equal to h(u) =(1− ju1− ρ)−μ+σσ. So, wecan make a conclusion that the probability distribution of number of calls in the orbit under aheavy load condition is a gamma distribution asymptotically with parameters α = (μ+ σ)/σ andβ = 1− ρ.Results for other types of retrial queues. Similar investigations were performed for retrialqueues with Markovian modulated Poisson arrivals and general distribution of service time. Itis interesting that for all considering systems, probability distributions of number of calls in theorbit are gamma distributions asymptotically under a heavy load condition. Parameters of theseAlexander Moiseev et al. 9gamma distributions are showed in the Table 1. Here we use the following notations: b and b2are a mean and a second initial moment of the service time distribution, Λ = diag{λn}, V is asolution of the equation VQ = R(Ib−1−Λ), R is a stationary distribution of states of the MMPPunderlying chain, e is a vector of 1s, I is an identity matrix.Table 1: Gamma distribution parametersType of RQ M |M |1 M |GI|1 MMPP |M |1 MMPP |GI|1αμ+ σσ1 +1bσβ 1 +μσβ 1 +1bσββ 12bb2μVΛe− μVe+ μ1b(VΛe−Ve1b+ b22b3)As a result of additional study, we have obtained that in the case of impatient calls (when wait-ing time has exponential distribution with a rate α/i) the asymptotic distribution of number of callsunder a heavy load condition is also a gamma distribution with parameters α = S(μ+σ)/σ−α/σand β = S(S − ρ), where S = 1 + α/μ is a system throughput value.Acknowledgments. This work is performed under the state order of the Ministry of Educationand Science of the Russian Federation No. 1.511.2014/K.10 Alexander Moiseev et al.

Retrial queueing system under heavy load condition

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Retrial queueing system under heavy load condition

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