Mean value analysis is an elegant tool for determining mean performance measures in queueing models. In this paper we show how mean value analysis can be applied to retrial queues. First, we illustrate the technique for the standard M/G/1 retrial queue with exponential retrial times. After that we show how the relations can be adapted to obtain mean performance measures in more advanced M/G/1-type retrial queues.

Mean value analysis of single server retrial queues

Cloud computing is a new paradigm where a company makes money by selling computing resources including both software and hardware. The core part or infrastructure of cloud computing is the data center where a large number of servers are available for processing incoming data traffic. These servers not only consume a large amount of energy to process data, but also need a large amount of energy to keep cool. Therefore, a reduction of a few percent of the power consumption means saving a substantial amount of money for the company as well as reduce our impact on the environment. As it currently stands, an idle server still consumes about 60 % of its peak energy usage. Thus, a natural suggestion to reduce energy consumption is to turn off servers which are not processing data. However, turning off servers can affect the customer experience. Customers trying to access computing power will experience delays if their data cannot be processed quickly enough. Moreover, servers require setup times in order to move from the off state to the on state. In the setup phase, servers consume energy, but cannot process data. Therefore, there exists a trade-off between power consumption and delay performance. In [7, 9], the authors analyze this tradeoff using an M/M/c queue with setup time for which they present a decomposition property by solving difference equations. In this paper, we complement recent stationary analysis of these types of models by studying the sample path behavior of the queueing model. In this regard, we prove a weak law of large numbers or fluid limit theorem for the queue length and server processes as the number of arrivals and number of servers tends to infinity. This methodology allows us to consider the impact of nonstationary arrivals and abandonment, which have not been considered in the literature so far.

A law of large numbers for m/m/c/delayoff-setup queues with nonstationary arrivals

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

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

The goal of this paper is to investigate the two-way communication by the help of finite source retrial queuing systems. Incoming calls from sources (primary calls) arrive to the server according to a Poisson process. If an incoming call finds the server idle, its service starts. Otherwise, if the server is busy, an arriving (primary or secondary - from the orbit) call moves into the orbit and after some exponentially distributed time it retries to enter to the server. When the server is idle it generates an outgoing call after an exponentially distributed time with different parameters to the calls in the orbit and in the sources, respectively. The service time of the incoming and outgoing calls are exponentially distributed with different rates. Results on two-way communication assume, that after the service an outgoing call (primary or secondary) is sent back to the source. The novelty of this paper is investigating two cases. In Case 1 the secondary outgoing call is sent back to the orbit, thus the pending incoming call will not be lost. In Case 2 after service of the secondary outgoing call its incoming service request will be started immediately. This means a two-phase service. The balance equations are solved by the help of MOSEL-2 tool. Graphical results and comparisons of the cases are presented.

A Contribution to Modeling Two-Way Communication with Retrial Queueing Systems

Analysis of the influence of the current-voltage characteristics of the voltage rectifiers on the static characteristics of hydrogen electrolyzer load

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.

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Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

In this paper, we consider an MMPP/M/1/1 retrial queue where incoming fresh calls arrive at the server according to a Markov modulated Poisson process. Upon arrival, an incoming call either occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. The server makes an outgoing call in its idle time. Our contribution is to derive the asymptotics of the number of calls in retrial queue under the conditions of high rate of making outgoing calls and low rate of service time of outgoing calls.

Heavy Outgoing Call Asymptotics for $${MMPP{\slash }}M{\slash }1{\slash }1$$ Retrial Queue with Two-Way Communication

Heavy outgoing call asymptotics for MMPP/M/1/1 retrial queue with two-way communication

In this paper, we consider a retrial queue with batch Poisson input process, arbitrarily distributed service times and arbitrarily distributed number of calls in the batch. Upon arrival, a call from the batch occupies the server if it is idle. The other calls from the batch join the orbit. If the server is busy, all calls from the batch go to the orbit. In the orbit, incoming calls stay for an exponentially distributed random delay and repeat their request for service. Besides incoming calls, the server can also make outgoing calls when idle. We assume that there are several types of outgoing calls in the system. The aim of the current research is to obtain asymptotic probability distribution of the number of incoming calls in the system using an asymptotic-diffusion analysis method.

Svetlana Paul

Papers

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

Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

Heavy Outgoing Call Asymptotics for $${MMPP{\slash }}M{\slash }1{\slash }1$$ Retrial Queue with Two-Way Communication

Heavy outgoing call asymptotics for MMPP/M/1/1 retrial queue with two-way communication

Asymptotic-Diffusion Analysis for Retrial Queue with Batch Poisson Input and Multiple Types of Outgoing Calls