Showing papers on "Retrial queue published in 2013"
TL;DR: The main aim of this paper is to study the steady state behavior of an M/G/1-type retrial queue in which there are two flows of arrivals namely ingoing calls made by regular customers and outgoing calls making by the server when it is idle.
74 citations
TL;DR: For a retrial queue accepting two types of positive customers and negative arrivals, mixed priorities, unreliable server and multiple vacations, stability conditions and the system state probabilities are investigated both in a transient and in a steady state.
47 citations
TL;DR: An M/G/1 retrial queue with negative customers and priority under Bernoulli vacation schedule subject to the server breakdowns and repairs is considered and the stochastic decomposition law is investigated.
45 citations
TL;DR: An analytic approach for investigating a single-server retrial queue with finite population of customers where the server is subject to interruptions is presented and includes the analysis of the arriving customer's distribution, the busy period, and the waiting time process.
Abstract: This paper aims at presenting an analytic approach for investigating a single-server retrial queue with finite population of customers where the server is subject to interruptions. A free source may generate a primary call to request service. If the server is free upon arrival, the call starts to be served and the service times are independent, generally distributed random variables. During the service time the source cannot generate a new primary call. After service the source moves into the free state and can generate a new primary call. There is no waiting space in front of the server, and a call who finds the server unavailable upon arrival joins an orbit of unsatisfied customers. The server is subject to interruptions during the service processes. When the server is interrupted, the call being served just before server interruption goes to the retrial orbit and will retry its luck after a random amount of time until it finds the server available. The recovery times of the interrupted server are assumed to be generally distributed. Our analysis extends previous work on this topic and includes the analysis of the arriving customer’s distribution, the busy period, and the waiting time process.
38 citations
TL;DR: The Markov chain underlying the considered queueing system and its ergodicity condition is analyzed, the system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions.
29 citations
TL;DR: In this article, a single server retrial queueing model, in which customers arrive according to a batch Markovian arrival process (BMAP), is considered, and steady state analysis of the model is performed.
Abstract: A single server retrial queueing model, in which customers arrive according to a batch Markovian arrival process (BMAP), is considered. An arriving batch, finding server busy, enters an orbit. Otherwise one customer from the arriving batch enters for service immediately while the rest join the orbit. The customers from the orbit try to reach the server subsequently and the inter-retrial times are exponentially distributed. Additionally, at each service completion epoch, two different search mechanisms are switched-on. Thus, when the server is idle, a competition takes place between primary customers, the customers coming by retrial and the two types of searches. It is assumed that if the type II search reaches the service facility ahead of the rest, all customers in the orbit are taken for service simultaneously, while in the other two cases, only a single customer is qualified to enter the service. We assume that the service times of the four types of customers namely, primary, repeated and those by the two types of searches are arbitrarily distributed with different distributions. Steady state analysis of the model is performed.
28 citations
TL;DR: The present investigation deals with Mx/G/1 retrial queue with unreliable server and general retrial times, and queueing analysis has been done to obtain the queue size and orbit size distributions, and some other queueing as well as reliability measures.
Abstract: The present investigation deals with Mx/G/1 retrial queue with unreliable server and general retrial times. The server is subject to breakdowns and takes some setup time before starting the repair. The server renders first essential phase of service (FES) to all the arriving customers whereas second optional phase services (SOS) are provided after FES to only those customers who opt for it. The impatient customers are allowed to balk depending upon server’s status; they may also renege after waiting sometime in the queue. By incorporating the supplementary variables corresponding to service time, repair time, retrial time and setup time and then using generating function method, the queueing analysis has been done to obtain the queue size and orbit size distributions, and some other queueing as well as reliability measures. The effects of several parameters on the system performance are examined numerically by taking an illustration.
23 citations
TL;DR: This paper considers an unreliable single server retrial queue accepting two types of customers, with negative arrivals, preemptive resume priorities and vacations, and proves that the model satisfies the stochastic decomposition property.
Abstract: In this paper we consider an unreliable single server retrial queue accepting two types of customers, with negative arrivals, preemptive resume priorities and vacations. A distinguishing feature of the model is that the rates of the Poisson arrival process depends on the server state. For this model we investigate the stability conditions and the joint queue length distribution in steady state. We also prove that our model satisfies the stochastic decomposition property. Transient, as well as steady state solutions for reliability measures are obtained. Finally, numerical results demonstrate the typical features of the model under consideration.
20 citations
TL;DR: In this article, a discrete-time 1 X Geo G retrial queue with general retrial times was considered, and working vacations and vacation interruption policy was introduced into the retrial queuing.
Abstract: We consider a discrete-time 1 X Geo G retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. Firstly, we analyze the stationary condition for the embedded Markov chain at the departure epochs. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.
17 citations
TL;DR: In this article, the authors considered a Geo/Geo/1 retrial queue with nonpersistent customers and working vacations, and developed the recursive formulae for the stationary distribution by using matrix-geometric solution method.
Abstract: In this paper, we consider a Geo/Geo/1 retrial queue with non-persistent customers and working vacations. The server works at a lower service rate in a working vacation period. Assume that the customers waiting in the orbit request for service with a constant retrial rate, if the arriving retrial customer finds the server busy, the customer will go back to the orbit with probability q (0≤q≤1), or depart from the system immediately with probability $\bar{q}=1-q$
. Based on the necessary and sufficient condition for the system to be stable, we develop the recursive formulae for the stationary distribution by using matrix-geometric solution method. Furthermore, some performance measures of the system are calculated and an average cost function is also given. We finally illustrate the effect of the parameters on the performance measures by some numerical examples.
17 citations
TL;DR: In this article, the authors considered the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment, and two different types of multi-dimensional Markov chains were investigated describing the behavior of the system based on state space arrangements.
Abstract: We consider the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment. Two different types of multi-dimensional Markov chains are investigated describing the behavior of the system based on state space arrangements. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated, based on which the main performance measures are obtained, and numerical examples are presented as well.
TL;DR: In this paper, the authors considered a single-server queuing system with a finite number of sources, where customers are not allowed to queue; instead of queuing, they make repeated attempts, or retrials, in order to enter service after some time.
Abstract: We consider a single-server queuing system with a finite number of sources, where customers are not allowed to queue; instead of that, they make repeated attempts, or retrials, in order to enter service after some time. This queuing system and its variants are widely used to model disk memory systems, star-like local area networks, and other communication systems. The article extends previous works on this topic and deals with the number of retrials, produced by a tagged customer, until he finds the server available. An algorithm for determination of all moments of this number is obtained, and the results of numerical experiments are presented.
TL;DR: A single server retrial queue with negative customers and two types of Bernoulli feedback is considered, and a necessary and sufficient condition for the system to be stable is investigated.
Abstract: A single server retrial queue with negative customers and two types of Bernoulli feedback is considered. A necessary and sufficient condition for the system to be stable is investigated. The system size probabilities at output epochs are obtained by using an embedded Markov chain. Further, the joint generating functions of queue length and server status are studied by using supplementary variables method. Some important system performance measures are derived. Busy period of the system is also discussed. Finally, extensive numerical illustrations are provided.
TL;DR: This paper provides the exact expression for the conditional mean number of customers based on the computation of the rate matrix R with the time complexity of O(c) and constructs an efficient algorithm for the stationary distribution with the determination of a threshold that allows the computationof performance measures with a specific accuracy.
28 Jan 2013
TL;DR: A multi-server retrial queueing model in which customers arrive according to a Markovian arrival process and the service times are assumed to be of phase type (PH-type) is studied, illustrating how one can underestimate or overestimate some key system performance measures by incorrectly assuming the retrial times to be exponential.
Abstract: In this paper we study a multi-server retrial queueing model in which customers arrive according to a Markovian arrival process (MAP) and the service times are assumed to be of phase type (PH-type). An arriving customer finding all servers busy will enter into a (retrial) orbit of infinite size. The customers in orbit will try to capture a free server after a random amount of time which is assumed to be of PH-type. Thus, every customer in the orbit has his/her own phase type distribution before attempting to get into service. Due to the complexity of the model and lack of attention to such models in the literature, we study this via simulation. After validating our simulated results against known results (both exact and approximation) for some special cases, we illustrate how one can underestimate or overestimate some key system performance measures by incorrectly assuming the retrial times to be exponential.
TL;DR: The authors discuss a discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times with steady-state analysis for the considered queueing system and obtains the generating functions of the number of customers in the orbit and in the system.
Abstract: The authors discuss a discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times. As soon as the orbit is empty, the server takes a vacation. However, the server is allowed to take a maximum number J of vacations, if the system remains empty after the end of a vacation. If there is at least one customer in the orbit at the end of a vacation, the server begins to serve the new arrivals or the arriving customers from the orbit. For this model, the authors focus on the steady-state analysis for the considered queueing system. Firstly, the authors obtain the generating functions of the number of customers in the orbit and in the system. Then, the authors obtain the closed-form expressions of some performance measures of the system and also give a stochastic decomposition result for the system size. Besides, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided.
TL;DR: A queueing model with N channels and an orbit with infinite waiting positions to model a specific cell in wireless networks and derives many performance measures such as the blocking probability of new calls and the dropping probability of handoff calls, the idle probability of the target cell, the mean queue lengths of two types of calls.
15 May 2013
TL;DR: A multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution and description of several independent Markov processes in parallel that allows to compute the stationary distribution of the system for large number of servers is considered.
Abstract: We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a quite general queueing system suits for modeling, e.g., modern wireless communication networks. We assume that arriving customers, which do not succeed to start the service immediately upon arrival due to the lack of available servers, may leave the system forever (balk) or join the orbit for further retrials. Customers in the orbit are impatient (they may leave the system forever after exponentially distributed duration of the stay in the orbit) and non-persistent (they may leave the system forever after any unsuccessful attempt to reach the service). Approach by V. Ramaswami and D. Lucantoni for description of several independent Markov processes in parallel that allows to compute the stationary distribution of the system for large number of servers is used along with the results for multi-dimensional asymptotically quasi-Toeplitz Markov chains for computation of steady state distribution of the system states and some its performance measures.
TL;DR: A priority retrial queue with repeated inhomogeneous vacations with power-saving mechanism in wireless communication systems in which the sleep duration of the device corresponds to the vacation of the server and the listening duration to the time-limited idle period.
Abstract: In this paper, we analyse a priority retrial queue with repeated inhomogeneous vacations. Two types of customers arrive at the system and if they find the server unavailable, the first type join an ordinary queue, while the second have to reattempt after a random period. The server departs for a vacation when the ordinary queue is empty upon a service completion. Under this scheme, when a vacation period expires, the server wakes up. If the priority queue is non-empty, it starts serving it exhaustively. Otherwise, it remains awake for a limited time period, waiting for a possible request. If no customers arrive during this period, it goes for another vacation with a different probability distribution from the previous one. The theoretical model has application to the power-saving mechanism in wireless communication systems in which the sleep duration of the device corresponds to the vacation of the server and the listening duration to the time-limited idle period. Various system performance and energy metrics are derived. Moreover, we compare the proposed policy with other vacation schemes. Optimization problems are formulated for finding best values for the critical parameters of the model which maximize the economy of energy achieved in the power save mode, subject to performance constraints.
TL;DR: Under the assumption that the service times are exponentially distributed with possibly different rates, the model is analyzed using matrix-analytic methods and Illustrative numerical examples to bring out the qualitative aspects of the model under study are presented.
Abstract: Generally in call centers, voice calls (say Type 1 calls) are given higher priority over e-mails (say Type 2 calls). An arriving Type 1 call has a preemptive priority over a Type 2 call in service, if any, and the preempted Type 2 call enters into a retrial buffer (of finite capacity). Any arriving call not able to get into service immediately will enter into the pool of repeated calls provided the buffer is not full; otherwise, the call is considered lost. The calls in the retrial pool are treated alike (like Type 1) and compete for service after a random amount of time, and can preempt a Type 2 call in service. We assume that the two types of calls arrive according to a Markovian arrival process (MAP) and the services are offered with preemptive priority rule. Under the assumption that the service times are exponentially distributed with possibly different rates, we analyze the model using matrix-analytic methods. Illustrative numerical examples to bring out the qualitative aspects of the model under study are presented.
TL;DR: In this paper, the performance evaluation of two retrial queueing systems is considered and the relationship between retrials, arrivals, balking rates, and the retrial queuing length is investigated.
Abstract: In this paper, we consider the performance evaluation of two retrial queueing system. Customers arrive to the system, if upon arrival, the queue is full, the new arriving customers either move into one of the orbits, from which they make a new attempts to reach the primary queue, until they find the server idle or balk and leave the system, these later, and after getting a service may comeback to the system requiring another service. So, we derive for this system, the joint distribution of the server state and retrial queue lengths. Then, we give some numerical results that clarify the relationship between the retrials, arrivals, balking rates, and the retrial queue length.
TL;DR: The non-preemptive assignment of a single server to two infinite-capacity retrial queues is analyzed and the conditions under which server assignment to either queue 1 or queue 2 is cost-optimal are presented.
Abstract: We analyze the non-preemptive assignment of a single server to two infinite-capacity retrial queues. Customers arrive at both queues according to Poisson processes. They are served on first-come-first-served basis following a cost-optimal routing policy. The customer at the head of a queue generates a Poisson stream of repeated requests for service, that is, we have a constant retrial policy. All service times are exponential, with rates depending on the queues. The costs to be minimized consist of costs per unit time that a customer spends in the system. In case of a scheduling problem that arise when no new customers arrive an explicit condition for server allocation to the first or the second queue is given. The condition presented covers all possible parameter choices. For scheduling that also considers new arrivals, we present the conditions under which server assignment to either queue 1 or queue 2 is cost-optimal.
TL;DR: In this article, the steady-state behavior of a discrete-time unreliable Geo/G/1 retrial queueing system with balking customers and second optional service is investigated.
Abstract: This paper deals with the steady-state behavior of a discrete-time unreliable Geo/G/1 retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system.
TL;DR: Analysis of the non-Markov dynamic retrial queue (RQ) system is performed and prelimit probability distributions of the number of requests are found in the retrial call source at various service time distributions.
Abstract: This article deals with the non-Markov dynamic retrial queue (RQ) system, i.e., the unilinear queue system with the retrial call source (RCS), the incoming Markov modulated (Poisson) flow (MMP flow) of requests and the arbitrary distribution of the service time of requests; the system is controlled by the dynamic access report. Analysis of the given RQ system is performed and prelimit probability distributions of the number of requests are found in the retrial call source at various service time distributions. The stabilization property of the sequence of relations p(i+ 1)/p(i) is found. For approximation of the probability distributions p(i) the quasigeometric distribution of the defect n is suggested.
TL;DR: The stochastic decomposition property and the corresponding continuous-time queueing system are investigated and some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.
Abstract: This paper is concerned with a discrete-time retrial queueing model with vacations and two types of breakdowns. If the orbit is empty, the server takes at most vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.
TL;DR: In this paper, the authors considered an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy.
Abstract: We consider an M
X
/G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system, including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.
TL;DR: For an unreliable single server retrial queue with batch arrivals, accepting two types of customers, say ordinary and transit, the stability conditions and the joint queue length distribution in steady state are investigated while operating characteristics and numerical results are presented.
TL;DR: The proposed scheme was applied to solve an optimization problem in the class of threshold strategies to solve the problem of stationary mode optimization in trial queues with variable arrival rate.
Abstract: Retrial queues with variable arrival rate are considered. For such queues, the condition of the existence of stationary mode is found. The approximation approach is used to calculate the stationary probabilities. The proposed scheme was applied to solve an optimization problem in the class of threshold strategies.
TL;DR: The ergodicity of the embedded Markov chains and their stationary distributions are studied, the steady-state solutions for both queueing measures and reliability quantities are obtained and the stochastic decomposition law is investigated.
Abstract: We consider a single server retrial queue with waiting places in service area and three classes of customers subject to the server breakdowns and repairs. When the server is unavailable, the arriving class-1 customer is queued in the priority queue with infinite capacity whereas class-2 customer enters the retrial group. The class-3 customers which are also called negative customers do not receive service. If the server is found serving a customer, the arriving class-3 customer breaks the server down and simultaneously deletes the customer under service. The failed server is sent to repair immediately and after repair it is assumed as good as new. We study the ergodicity of the embedded Markov chains and their stationary distributions. We obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law, the busy period of the system and the virtual waiting times. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analyzed numerically.
TL;DR: The waiting time distribution in the M/PH/1 retrial queue is analyzed to provide a numerical algorithm for calculating the Laplace-Stieltjes transform of the waiting time.