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Stability Analysis of a Multi-Class Retrial Queue with General Retrials and Classical Retrial Policy

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Stability Analysis of a Multi-Class Retrial Queue with General Retrials and Classical Retrial Policy ______________________________________________________PROCEEDING OF THE 28TH CONFERENCE OF FRUCT ASSOCIATION Stability Analysis of a Multi-class Retrial Queue with General Retrials and Classical Retrial Policy Ruslana Nekrasova IAMR Karelian Research Centre RAS, Petrozavodsk State University, Petrozavodsk, Russia [email protected] Abstract—We deal with a single server multi-class retrial condition under the assumption, that input stream is Poisson. model, feed by Poisson input. The system is considered under clas- The research is based on the regenerative method, which is sical retrial policy, while inter-retrial times are class dependent a strong instrument in stochastic analysis. The basic steps and generally distributed. Such systems have various applications of the proof are relayed on preliminary results from papers like multi-access protocols or cellular mobile networks, where [16] and [17], where authors considered multi-class retrial blocked messages are sent again after some waiting period. models with exponential and New Better than Used retrials, We rely on regenerative approach and results from renewal respectively. One of the key moments of the research in present theory to obtain the stability criterion of the system under paper is based on Lorden’s inequality for embedded renewal consideration and present some simulation results, to illustrate process, associated with orbit stream. We obtain that in case that obtained condition could be extended to the case with general the load coefficient is less than 1, the system is stable. The input. assumption that obtained condition guarantees the stability in case of general input (at least, bounded inter-arrival times) is I. INTRODUCTION confirmed by simulations. In this paper we consider a multi-class retrial system with The structure of the paper is the following. Section II con- the only server. The total input stream is assumed to be tains a detailed description of the model under consideration Poisson, while marginal input rates and service times are class- and introduces it’s regenerative structure. Then in Section III dependent. The model belongs to the classical retrial policy. some previous stability results, related to particular cases of Thus, blocked customer stays in the corresponding class orbit multi-class retrial systems, are presented. Next in Section IV for a generally distributed retrial time independently of other the stability condition for a model with general retrials and customers, and summary retrial rate is proportional to the unbounded inter-arrival times was established. The analysis number of customers in all orbits. was based on regenerative approach and results from renewal Retrial queues were naturally arisen from various appli- theory, namely we relied on Lorden’s inequality for residual cations with multiple access like telephone networks [1] or retrial time. Section V containes simulation results for the call centers [2] and now are successfully used in modeling system with bounded input stream. The experiments illustrate of a large number of a modern objects like wireless telecom- that stability conditions, obtained in previous section, hold for munication systems, cellular mobile networks or multi-access the model with a general renewal input. Section VI concludes protocols, where blocked packets are sent again after some the paper. waiting period. II. MODEL DESCRIPTION Retrial queuing models are widely studied in the literature. Regarding to the overview of retrial theory it is worth mention In this section we construct a multi-class and single-server the basic books [3], [4] and, for instance, quite recent survey retrial model as follows. Consider a Poisson input with a papers [5], [6]. total rate λ. Note, that arrivals join the system at instants {tn,n≥ 1}, where inter-arrival times τn := tn − tn−1 are in- Obviously, that most of research in this field is related dependent and exponentially distributed with a mean 1/λ. For to more widespread single-class Marcovian models with ex- simplicity, we assume t0 =0and define the generic element of ponential retrials. In that case authors may establish explicit iid (independent identically distributed) sequence {τn,n≥ 1} statements for stability conditions or stationary characteristics by τ. Consider K ≥ 1 different classes of customers with (see, for instance [7], [8]). The paper [9], which contains marginal input rates λi,i=1,...,K. Obviously stability results for a single-class model with the only server, was one of the first works, where authors considered non- λ = λ1 + ···+ λK . exponential distribution of inter-retrial times. Later, in [10] the analysis was extended to a multiserver single-class system, the Moreover, we consider, λi = piλ, where p =(p1,...,pK ) is research was based on the regenerative approach. The similar a given distribution. model was considered in [11], where the analysis relayed on the fluid limit approach. Assume that service times in the system under considera- tion are class-dependent. Namely, the server, captured by class- In this paper we construct a single server multi-class i customer, would be realised in a random time, stochastically model with general retrials. Our goal is to obtain stability equivalent to S(i), while the distributions of S(i) for different ISSN 2305-7254 ______________________________________________________PROCEEDING OF THE 28TH CONFERENCE OF FRUCT ASSOCIATION i may not coincide. If the n-th customer belongs to class- of the process {Xn}, while {Xn} is a regenerative process (in) in,in ∈{1,...,K}, we define it’s service time by Sn . with iid cycle lengths {S(in),n≥ 1} Note, that times n are independent. Denote class- αn = βn − βn−1,n≥ 1. (i) i load coefficient by ρi = λiES . Hence, the total system load {X } Eα<∞ coefficient is obtained as The process n is called positive recurrent,if , where α is a generic regeneration cycle length. The positive K recurrence of the basic process X is equivalent to the stability ρ := ρ . i of the model under consideration. By results from regeneration i=1 theory, for Eα<∞ its enough to show, that residual regeneration time We deal with a retrial system. Thus, if a new arrival finds β(tn):=min{tβ − tn > 0},n≥ 1 (1) the server busy, it does not lost, but joins to some kind of k k a virtual buffer, called orbit, and then, after a random time does not convergence to infinity in distribution, as n →∞. interval try to capture the server again. If the server is still {N ,n ≥ 1} busy, the orbital or secondary customer, immediately returns This result takes place, if the orbit process n to the orbit, waits a random time and then makes new a has negative drift. Note, that such a method was successfully attempts. (From this point of view, the arrivals from input applied for stability analysis of a multi-class and multi-server stream are called primary.) As a customer is described by retrial system with exponential retrials in [16] and developed K for the single-server system with Poisson input, where retrial it’s class characteristics, we consider, that the system has ξ(i) different orbits. Class-i arrival joins the corresponding orbit, waiting time belongs to the special subclass of New Better stays there for an interval distributed as ξ(i) and then try to than Used (NBU) distributions (see [17]). attacks the server again. The i-th orbit rate is defined as follows III. PRELIMINARY RESULTS γ := 1/Eξ(i),i=1,...,K. i In this section we present some stability results, obtained The model under consideration obeys to classical retrial for less general retrial systems. In section IV we rely on such policy. Thus, all the secondary customers make independent an analysis to establish stability conditions for the system attempts and the total (actual) orbit rate grows proportionally under consideration. to sum orbit size. Note, that in case γi =0we obtain the The paper [16] deals with a multi-class m ≥ 1 server loss system, while if γi = ∞, the orbit customer imme- retrial system with renewal input, general service time and diately captures the server, as in becomes empty, and the exponential retrials. The authors obtained, that the condition model is equivalent to the infinite-buffer queuing system. One ρ<m more significant feature of a model under consideration is i non-exponential retrials: class- random orbit waiting times is indeed the stability criterion. The proof of necessity is based ξ(i),i=1,...,K are generally distributed, which makes the on the balance equation in terms of workload and does not analysis more complicated. rely on inter-retrial time distribution. Thus, in [17] the analysis Define the number of customers on orbit i just before time was easily extended for the system with general retrials and instant t by N (i)(t), thus for the summary orbit we have in single-server case we have the following result. K N(t):=N (1)(t)+···+ N (K)(t),t≥ 0. Theorem 1. [17] Assume that the –class retrial system with generally distributed retrial times is stable (positive Obviously, that the only reason of instability is the infinite recurrent) and condition growth of orbit size N(t),ast →∞. max P(τ>S(i)) > 0 (2) i=1,...,K Our goal is to find the conditions, which guarantee the holds. Then stable summary orbit. The basic stability analysis is relied ρ<1. on results from regeneration theory [12], [13], [14]. First we (3) define by ν(t) ∈{0, 1} the server state (number of customers t Note, that condition (2) automatically holds in case of on service) just before time instant and assume zero initial τ state: Poisson input, as is unbound random variable. N(0) = 0,ν(0) = 0. To establish (3), in [17] we presented the total workload (the summary service times) V (t), arrived to the system X(t)=ν(t)+N(t),t ≥ 0 Then denote by the process in [0,t], as a sum of two components: D(t) and R(t) – associated with the total number of customers in the system departed and residual (on service and in the orbits) workload, and it’s discrete analogue at arrival instants by respectively.
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