Showing papers on "Retrial queue published in 2017"

Analysis of Priority Retrial Queue With Many Types of Customers and Servers Reservation as a Model of Cognitive Radio System

Abstract: Cognitive radio is emerging as one of the key information transmission technologies to enhance spectrum efficiency for dramatically increased wireless network capacity requested by end users. Dynamic spectrum access allows effective use of radio frequency and prevents its underutilization in many real-world networks. It enables unlicensed users to temporarily “borrow” unused spectrum while ensuring that the rights of the incumbent license holders are respected. Problems of optimization of joint access of primary and secondary users can be effectively solved by means of queueing theory. In this paper, the analysis of a novel queueing model suitable for the optimization of access is implemented under quite general assumptions about the system parameters. There are several types of primary customers having different requirements for the service time and preemptive priority over secondary customers. Secondary customers can share a server, while primary customers occupy the whole server. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of phase-type. Effect of retrials of secondary customers is taken into account. An effective way for the analysis of multi-server queues with many types of customers and heterogeneous requirements to the service process is provided and applied.

A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy

Abstract: In this paper, we consider a single server feedback retrial queueing system with multiple working vacations and vacation interruption. An arriving customer may balk the system at some particular times. As soon as orbit becomes empty at regular service completion instant, the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. After completion of regular service, the unsatisfied customer may rejoin into the orbit to get another service as feedback customer. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimization analysis are presented.

Two-Way Communication M/M/1//N Retrial Queue

Abstract: We consider in this paper retrial queue with one server that serves a finite number of customers, each one producing a Poisson flow of incoming calls. In addition, after some exponentially distributed idle time the server makes outgoing calls of two types - to the customers in orbit and to the customers outside it. The outgoing calls of both types follow the same exponential distribution, different from the exponential service time distribution of the incoming calls. We derive formulas for computing the steady state distribution of the system state as well as formulas expressing the main performance macro characteristics in terms of the server utilization. Numerical examples are presented.

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

Abstract: 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

Multiserver retrial queue with setup time and its application to data centers

Abstract: This paper considers a multiserver retrial queue with setup time which is motivated from application in data centers with the ON-OFF policy, where an idle server is immediately turned off The ON-OFF policy is designed to save energy consumption of idle servers because an idle server still consumes about 60% of its peak consumption processing jobs Upon arrival, a job is allocated to one of available off-servers and that server is started up Otherwise, if all the servers are not available upon arrival, the job is blocked and retries in a random time A server needs some setup time during which the server cannot process a job but consumes energy We formulate this model using a three-dimensional continuous-time Markov chain obtaining the stability condition via Foster-Lyapunov criteria Interestingly, the stability condition is different from that of the corresponding non-retrial queue Furthermore, exploiting the special structure of the Markov chain together with a heuristic technique, we develop an efficient algorithm for computing the stationary distribution Numerical results reveal that under the ON-OFF policy, allowing retrials is more power-saving than buffering jobs Furthermore, we obtain a new insight that if the setup time is relatively long, setting an appropriate retrial time could reduce both power consumption and the mean response time of jobs

Optimal design for a retrial queueing system with state-dependent service rate

Abstract: This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

Performance Modeling of Finite-Source Retrial Queueing Systems with Collisions and Non-reliable Server Using MOSEL

Abstract: In this paper we investigate a single-server retrial queueing system with collision of the customer and an unreliable server. If a customer finds the server idle, he enters into service immediately. The service times are independent exponentially distributed random variables. During the service time the source cannot generate a new primary call. Otherwise, if the server is busy, an arriving (primary or repeated) customer involves into collision with customer under service and they both moves into the orbit. The retrial time of requests are exponentially distributed. We assume that the server is unreliable and could be break down. When the server is interrupted, the call being served just before server interruption goes to the orbit. Our interest is to give the main steady-state performance measures of the system computed by the help of the MOSEL tool. Several Figures illustrate the effect of input parameters on the mean response time.

Some Features of a Finite-Source M/GI/1 Retrial Queuing System with Collisions of Customers

Abstract: In this paper a finite-source M/GI/1 retrial queuing system with collisions of customers is considered. The definition of throughput of the system as average number of customers, which are successfully served per unit time is introduced. It is shown that at some combinations of system parameter values and probability distribution of service time of customers the throughput can be arbitrarily small, and at another values of parameters throughput can be greater than the service intensity. Applying method of asymptotic analysis under the condition of unlimited growing number of sources it is proofed that limiting distribution of the number of retrials/transitions of the customer into the orbit is geometric and the sojourn/waiting time of the customer in the orbit follows a generalized exponential distribution. In addition, the mean sojourn time of the customer under service is obtained.

Comparative Analysis of Methods of Residual and Elapsed Service Time in the Study of the Closed Retrial Queuing System M/GI/1//N with Collision of the Customers and Unreliable Server

Abstract: The aim of the present paper is to investigate a finite-source M/GI/1 retrial queuing system with collision of the customers where the server is subject to random breakdowns and repairs depending on whether it is idle or busy The method of elapsed service time and the method of residual service time are considered using asymptotic approach under the condition of unlimited growing number of sources It is proved, as it was expected, that basic characteristics of the system, such as the stationary probability distribution of the server states and the asymptotic average of the normalized number of customers in the system are the same and do not depend on the applied method

Performance Modeling and ANFIS Computing for Finite Buffer Retrial Queue Under F-Policy

Abstract: This investigation is concerned with the performance prediction and admission control F-policy for the machine repair problem with retrial. To develop a Markov model, the steady state Chapman-Kolmogorov equations are constructed. The system state probabilities are obtained by using recursive method. Various performance measures are established explicitly in terms of steady state probabilities. To examine the effects of system parameters, the numerical simulation is performed by choosing a suitable illustration. The cost function is also framed to evaluate the optimal service rate and corresponding optimal cost. ANFIS soft computing technique is used to compare the numerical results obtained analytically and also by implementing ANFIS.

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

Abstract: 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.

Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision

Abstract: In this paper, a single server retrial queue with general retrial time and collisions of customers with modified M-vacations is studied. The primary calls arrive according to Poisson process with rate λ. If the server is free, the arriving customer/the customer from orbit gets served completely and leaves the system. If the server is busy, arriving customer collides with the customer in service resulting in both being shifted to the orbit. After the collision the server becomes idle. If the orbit is empty the server takes at most M vacations until at least one customer is recorded in the orbit when the server returns from a vacation. Whenever the orbit is empty the server leaves for a vacation of random length V. If no customers appear in the orbit when the server returns from vacation he again leaves for another vacation with the same length. This pattern continues until he returns from a vacation to find at least one customer recorded in the orbit or he has already taken M vacations. If the orbit is empty by the end of the Mth vacation, the server remains idle for customers in the system. The time between two successive retrials from the orbit is assumed to be general with arbitrary distribution R(t). By applying the supplementary variables method, the probability generating function of number of customers in the orbit is derived. Some special cases are also discussed. A numerical illustration is also presented.

Analysis of a Retrial Queue with Limited Processor Sharing Operating in the Random Environment

Abstract: Queueing system with limited processor sharing, which operates in the Markovian random environment, is considered. Parameters of the system (pattern of the arrival rate, capacity of the server, i.e., the number of customers than can share the server simultaneously, the service intensity, the impatience rate, etc.) depend on the state of the random environment. Customers arriving when the server capacity is exhausted join orbit and retry for service later. The stationary distribution of the system states (including the number of customers in orbit and in service) is computed and expressions for the key performance measures of the system are derived. Numerical example illustrates possibility of optimal adjustment of the server capacity to the state of the random environment.

Simulation of Finite-Source Retrial Queueing Systems with Collisions and Non-reliable Server

Abstract: The aim of the present paper is to build a simulation program to investigate finite-source retrial queuing system with collision of the customers where the server is subject to random breakdowns and repairs depending on whether it is idle or busy. All the random variables involved in the model construction are assumed to be independent and generally distributed. The novelty of the investigation is to carry sensitivity analysis of the performance measures using various distributions. Several figures show the effect of different distributions on the performance measures such as mean and variance of number of customers in the system, mean and variance of response time, mean and variance of time a customer spent in service, mean and variance of sojourn time in the orbit.

Finite source retrial queues with two phase service

Abstract: This paper is concerned with a single server retrial queue with a finite number of sources with second optional service. This queueing system assumes that each source can generate a primary call to request service when it is free. After the first phase service, the customer will choose the second phase service with probability p. Our analysis extends previous work on this topic and we can use method of discrete transformation to get some more general results on the length of the queue, busy period and waiting time. Numerical examples show the influence of key parameters on the performance of the system.

Single server retrial queues with speed scaling: Analysis and performance evaluation

Abstract: Recently, queues with speed scaling have received considerable attention due to their applicability to data centers, enabling a better balance between performance and energy consumption. This paper proposes a new model where blocked customers must leave the service area and retry after a random time, with retrial rate either varying proportionally to the number of retrying customers (linear retrial rate) or non-varying (constant retrial rate). For both, we first study a basic case and then subsequently incorporate the concepts of a setup time and a deactivation time in extended versions of the model. In all cases, we obtain a full characterization of the stationary queue length distribution. This allows us to evaluate the performance in terms of the mentioned balance between performance and energy, using an existing cost function as well as a newly proposed variant thereof. This paper presents the derivation of the stationary distribution as well as several numerical examples of the cost-based performance evaluation.

A stochastic version analysis of an M/G/1 retrial queue with Bernoulli schedule

Abstract: ‎In this work‎, ‎we derive insensitive bounds for various performance measures of a single-server‎ ‎retrial queue with generally distributed inter-retrial times and Bernoulli schedule‎, ‎under the special‎ ‎assumption that only the customer at the head of the orbit queue (ie‎, ‎a‎ ‎FCFS discipline governing the flow from the orbit to the server) is allowed‎ ‎to occupy the server‎ ‎The methodology is strongly based on stochastic comparison techniques‎ ‎Instead of studying a performance measure in a quantitative fashion‎, ‎this approach attempts to reveal the relationship between the performance measures and the parameters of the system‎ ‎We prove the monotonicity of the transition operator of the embedded Markov chain relative to strong stochastic ordering and increasing convex ordering‎ ‎We obtain comparability conditions for the distribution of the number of customers in the system‎ ‎Bounds are derived for the stationary distribution and‎ ‎some simple bounds for the mean characteristics of the system‎ ‎The proofs of these results are based on the validation of some inequalities for some cumulative probabilities associated with every state $(m‎, ‎n)$ of the system‎ ‎Finally‎, ‎the effects of various parameters on the performance of the system have been examined numerically

Waiting time distributions in an M/G/1 retrial queue with two classes of customers

Abstract: We consider an $$\textit{M/G/1}$$ retrial queueing system with two classes of customers, in which the service time distributions are different for both classes of customers. When the server is unavailable, an arriving class-1 customer is queued in the queue with infinite capacity, whereas class-2 customer enters the retrial group. In this paper, we are concerned with the analysis of the waiting time distribution. We obtain the joint transform of the waiting time of a class-2 customer and the number of class-2 customers as well as the Laplace–Stieltjes transform of the waiting time of a class-1 customer. We also obtain all the moments of the waiting time distributions of class-1 and class-2 customers.

On a tandem queue with retrials and losses and state dependent arrival, service and retrial rates

Abstract: We consider the M/M/1 → •, M/M/1/0 type retrial queueing system with state dependent parameters. The intensities of input, service (at both stations) and retrials depend in an arbitrary way on the current number of customers in the orbit. Because operation of the first station of this tandem does not depend on the state of the second station, analysis of the system is implemented in two steps. At the first step, we compute the marginal distribution of the states of the first station in a simple form. At the second step, analytical formulas are presented for computation of the steady state-distribution of the tandem. Formulas for computing the most important performance measures of the system are derived. Results can be used for solving various problems of optimal control by the system operation.

Retrial Queue M/G/1 with Impatient Calls Under Heavy Load Condition

Abstract: In the paper, the retrial queueing system of M / GI / 1 type with impatient calls is considered. The delay of calls in the orbit has exponential distribution and the impatience time of calls in the system is dynamical exponential. Asymptotic analysis method is proposed for the system studying under a heavy load condition. The theorem about the gamma form of the asymptotic probability distribution of the number of calls in the orbit is formulated and proved. During the study, the expression for the system throughput is obtained. Numerical examples compare asymptotic, exact and simulation based distributions.

Performance analysis of a discrete-time \begin{document}$ Geo/G/1$\end{document} retrial queue with non-preemptive priority, working vacations and vacation interruption

Abstract: This paper is concerned with a discrete-time \begin{document}$ Geo/G/1$\end{document} retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability \begin{document}$ α$\end{document} , or leaves the service area and enters the retrial group (orbit) with probability \begin{document}$ \mathit{\bar{\alpha }}\left( = 1-\alpha \right)$\end{document} . Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.

Analysis of a Constant Retrial Queue with Joining Strategy and Impatient Retrial Customers

Abstract: This paper treats an M/M/1 retrial queue with constant retrial times. If the server is busy at the arrival epoch, the arriving customer decides to join the retrial orbit with probability or balk with probability . Only the customer at the head of the orbit is permitted to access the server. Upon retrial, the customer immediately receives his service if the server is idle; otherwise, he may enter the orbit again or leave the system because of impatience. First, we give the performance analysis for this retrial queue and give some important performance indices. Second, based on a natural reward-cost structure, we analyze the Nash equilibrium customers’ joining strategies and give some numerical examples.

Transient Solution of M [x 1 ] ,M [x 2 ] /G 1 ,G 2 /1 Retrial queueing system with Priority services, Modified Bernoulli Vacation, Bernoulli Feedback, Negative arrival, Breakdown, Delaying repair, Setup time and Balking

Abstract: The retrial queueing systems are characterized as a customer, arriving when all servers are busy, leaves the system, but after some time makes a demand to the service facility again. These models play a vital role in computer and telecommunication networks. For example, in a telephone system, a customer might receive a busy signal due to a lack of capacity. Such a customer is not allowed to queue, but will try their luck again after some random time. Between trials, the blocked customers join a pool of unsatisfied customers called ’orbit’.