Asymptotic Analysis Of Markovian Retrial Queue With Two-Way Communication Under Low Rate Of Retrials Condition

In this paper we are reviewing the retrial queue with two-way communication and Poisson arrival process. If the server free, incoming call occupies it. The call that finds the server being busy joins an orbit and retries to enter the server after some exponentially distributed time. If the server is idle, it causes the outgoing call from the outside. The outgoing call can find server free, then it starts making an outgoing call in an exponentially distributed time. If the outgoing call finds the server occupied, then it is lost. To research the system in question we have derived first and second order asymptotics of a number of calls in the orbit in an asymptotic condition of a low rate of retrials. Based on found asymptotics we have built the Gaussian approximation of a number of calls in the orbit. INTRODUCTION Recently a lot of attention is being paid to the research of the retrial queues such as mathematical models of real call center systems, telecommunication networks, computer networks, economical systems (Artalejo and Gomez-Corral 2008). These systems are characterized by the fact that if the clients (calls, phone calls, messages etc. immediately they have to enter the virtual orbit where they wait out some delay before they could access the server for service again (Flajolet and Sedgewick 2009). As a rule, the ones that are considered are the retrial queues in which arriving calls are either served immediately or join the orbit where they are wait out a random delay before accessing the server again. Recently, however, server is more likely to have the ability to make an outgoing call. The example of that could be the common cellphone that has function of both incoming and outgoing calls. In different call centers operators could receive arriving calls but as soon as they have free time and are in standby mode they could make outgoing calls to advertise, promote and sell packages and services of the centre. Falin (Falin 1979) derives integral formulas for partial generating functions and some explicit expressions for characteristics of the M|G|1|1 retrial queues with outgoing calls. Choi et al. (Choi et al. 1995) extends and Resing (Artalejo and Resing 2010) have derived first moments for characteristics of the M/G/1/1 retrial queues, in which the times of serving arriving and outgoing calls are different. Martin and Artalejo (Martin and Artalejo 1995) are considering M|G|1|1 retrial queues with outgoing calls in which calls from an orbit access the server after an exponentially distributed delay in the order of arrival. Artalejo and Phung-Duc (Artalejo and Tuan 2012) are considering M|M|1|1 retrial queues with outgoing calls and a different service time for incoming and outgoing calls. In their paper the authors have found an explicit solution for two-dimensional probability distribution of a server state and a number of calls in an orbit. Likewise, the factorial moments are found, based on which the proposed numerical and recurrent algorithms may be applied. In this paper the main method of research is the asymptotic analysis method which allows to find in M|M|1|1 retrial queue with two-way communication type of limit distribution of a number of calls in the orbit in an asymptotic condition of a low rate of retrials and to show that limit distribution is Gaussian. Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD) This result is achieved by using the original asymptotic analysis method without needing to find the nonlimiting distribution. Furthermore, the discrete distribution is constructed which approximates discrete distribution of a number of calls in an orbit. This distribution will be addressed as Gaussian approximation. Research of retrial queueing system under the asymptotic condition that the retrial rate is extremely low is stated in the following papers (Nazarov and Chernikova 2014) (Nazarov and Izmailova 2016). Furthermore, we have defined conditions of applicability of obtained approximation according to system defining parameters. The remainder of the paper is presented as follows. In , we describe the model in detail and preliminaries for later asymptotic analysis. , we present our main contribution for the model with Poisson input. In Section Approximation accuracy P(i) and its application area we have defined the conditions of applicability of the obtained approximation depending on values of system-defining parameters. Section is devoted to concluding remark and future work. MATHEMATICAL MODEL Let s consider retrial queue (Figure 1) with Poisson arrival process of incoming calls with rate . Figure 1: Retrial queue with two-way communication The incoming call finds the server and goes into service for an exponentially distributed time with rate 1. If upon entering the system the call finds the server being busy the call immediately joins the orbit, where it stays during a random time distributed exponentially with rate If the server is idle (empty) it starts making outgoing calls from the outside with rate . If the outgoing call finds the server free the call goes into service for an exponentially distributed time with rate 2. If upon entering the system the outgoing call finds the server being busy the call is lost and is not considered in the future. i(t) number of calls in the orbit at the time t, n(t) server state: 0 server is free, 1 server is busy serving an incoming call, 2 server is busy serving an outgoing call. -dimensional Markovian process {i(t), n(t)} for probability distribution P{i(t) = i, n(t) = n}= Pn(i, t) setting up system of Kolmogorov equations , 0 ) ( ) ( ) ( ) ( 2 2 1 1 0 i P i P i P i ) ( ) 1 ( ) ( ) ( 0 1 1 1 i P i P i P 0 ) 1 ( ) 1 ( 0 i P i , 0 ) 1 ( ) ( ) ( ) ( 2 0 2 2 i P i P i P . (1) Introducing partial characteristic functions (Nazarov and Paul 2016), denoting 1 j , 0 ) ( ) ( i n jui n i P e u H . Rewriting system (1) in the following form du u dH j u H ) ( ) ( ) ( 0 0 , 0 ) ( ) ( 2 2 1 1 u H u H , 0 ) ( ) ( ) ( 1 0 0 1 1 du u dH e j u H u H e ju ju 0 ) ( ) ( 1 0 2 2 u H u H e ju . (2) Characteristic function H(u) of a number of incoming calls in an orbit and server states probability distribution rn are relatively easy expressed through partial characteristic functions Hn(u) by the following equations ) ( ) ( ) ( ) ( 2 1 0 ) ( u H u H u H Me u H t jui , rn = Hn(0), n = 0, 1, 2. The task is put to find these characteristics of retrial queue with two-way communication. The main content of this paper is the solution of system (2) by using asymptotic analysis method in limit condition of a low rate of retrials . This is due to the fact that for the more complicated queues with an incoming MMPP, the equation system similar to (2) is analytically unsolvable, but a solution by using asymptotic analysis method is allowed. Application of asymptotic results in prelimit situation is causing the necessity of specifying the area of its applicability, which is obtainable only through comparison of asymptotic and prelimit characteristics and that is relatively easy implemented for the retrial queue in question. For more complex systems prelimit characteristics are usually defined by results of imitational modeling or by using pretty complicated numerical algorithms. The asymptotic analysis method suggested below is implemented by sequential determination of first and second order asymptotics. FIRST ORDER ASYMPTOTIC We introduce the following notations = , u = w, Hn(u) = Fn(w, ), then we will get this system w w F j w F ) , ( ) , ( ) ( 0 0 0 ) , ( ) , ( 2 2 1 1 w F w F , ) , ( 1 1 1 w F e jw 0 ) , ( ) , ( 0 0 w w F e j w F jw , 0 ) , ( ) , ( 1 0 2 1 w F w F e jw . (3) Theorem 1. (First order asymptotic) Suppose i(t) is a number of calls in an orbit of stationary M|M|1 retrial queue with two-way communication, then the following equation is true 1 ) ( 0 lim jw t i jw e Me , where the parameter 1 is defined by the following