Showing papers on "Retrial queue published in 2014"
TL;DR: This paper treats an M/G/1 retrial queue with non-persistent customers, where the server is subject to failure due to the negative arrivals and uses embedded Markov chain technique and the supplementary variable method to present the necessary and sufficient condition for the system to be stable and the joint queue length distribution in steady state.
48 citations
TL;DR: The probability generating function of joint distributions of state of the server and queue size, which is one of chief objectives of the paper, is derived and some important performance measures and reliability indices of this model are obtained.
36 citations
TL;DR: A promising technique is presented, a smoothed rate truncation method, to overcome the limitations of standard techniques and allow for the derivation of structural properties for unbounded jump Markov processes.
Abstract: The derivation of structural properties for unbounded jump Markov processes cannot be done using standard mathematical tools, since the analysis is hindered due to the fact that the system is not uniformizable. We present a promising technique, a smoothed rate truncation method, to overcome the limitations of standard techniques and allow for the derivation of structural properties. We introduce this technique by application to a processor sharing queue with impatient customers that can retry if they renege. We are interested in structural properties of the value function of the system as a function of the arrival rate.
34 citations
TL;DR: The general decomposition law for this retrial queueing system is established and the effects of various parameters on the system performance are analyzed numerically.
Abstract: This paper deals with a single server retrial queueing system with working vacation in which the server works with different service rates rather than completely terminating the service during its vacation period. We assume that both service times in a regular service period and in a working vacation period are generally distributed. The steady state distributions of the server state and the number of jobs in the orbit were obtained along with other performance measures. The general decomposition law for this retrial queueing system is established. The effects of various parameters on the system performance are also analyzed numerically.
32 citations
TL;DR: A multi-server retrial queue with two types of calls (handover and new calls) is analyzed and a constructive ergodicity condition for this chain is derived and the effective algorithm for computing the stationary distribution is presented.
30 citations
TL;DR: An M/G/1 ret trial queue with general retrial times is considered, and working vacations and vacation interruption policy is introduced into the retrial queue to prove the conditional stochastic decomposition for the queue length in orbit.
Abstract: We consider an M/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. During the working vacation period, customers can be served at a lower rate. If there are customers in the system at a service completion instant, the vacation will be interrupted and the server will come back to the normal working level. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we carry out the waiting time distribution and prove the conditional stochastic decomposition for the queue length in orbit. Finally, some numerical examples are presented.
27 citations
TL;DR: The object of this paper is to continue investigation of a single server retrial queue with finite number of sources in which the server is subjected to breakdowns and repairs, and investigates the distribution of the number of retrials, made by a customer before he reaches the server free.
Abstract: The object of this paper is to continue investigation of a single server retrial queue with finite number of sources in which the server is subjected to breakdowns and repairs. The server life time as well as the intervals between repetitions are exponentially distributed, while the repair and the service times are generally distributed. Using the formulas for the stationary system state distributions, obtained by Wang et al. [in Wang, J, L Zhao and F Zhang (2011). Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial and Management Optimization, 7, 655–676.] we investigate the distribution of the number of retrials, made by a customer before he reaches the server free. Recurrent schemes for computing this distribution in steady state as well as any arbitrary of its moments are established. Numerical results for five different distributions of the service and repair times are also presented.
24 citations
20 Nov 2014
TL;DR: This work considers a closed retrial queuing system M/M/1//N with collision of the customers with asymptotic method, and establishes formulas for computing the prelimit distribution of the number of sources in “waiting” state.
Abstract: We consider a closed retrial queuing system M/M/1//N with collision of the customers We assume that sources can be in two states: generating a primary customers and waiting for the end of successful service Source which sends the customer for service, moves into the waiting state and stays in this state till the end of the service of this customer This system is solved using the asymptotic method under conditions of infinitely increasing number of sources We establish formulas for computing the prelimit distribution of the number of sources in “waiting” state Also, we determine the range of applicability of the asymptotic results in prelimiting situation
21 citations
TL;DR: In this article, the authors considered a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate and established sufficient stability conditions for this system.
Abstract: We consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has \(c\) identical servers and can accommodate up to \(K\) jobs (including \(c\) jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. We establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that we obtained have clear probabilistic interpretation.
21 citations
TL;DR: An extensive analysis for the multiserver model is presented in which, a new formulation by a level-dependent quasi birth-and-death (QBD) process, whose block matrices have some special block structure is proposed, and an efficient algorithm utilizing the special structure for computing the stationary distribution is developed.
Abstract: This paper considers a multiserver retrial queue with two-way communication for blended call centers. Primary incoming calls arrive at the servers according to a Poisson process and request an exponentially distributed service time. Incoming calls that find all the servers fully occupied join the orbit and retry to occupy a server again after some random time. A retrial incoming call behaves the same as a primary incoming call. A server not only serves as an incoming call but also makes an outgoing call after some random idle time. We assume that the distribution of the duration of outgoing calls is different from that of incoming calls. Artalejo and Phung-Duc (2012) have extensively studied the single server case and have obtained some preliminary results for the multiserver model. In this paper, we present an extensive analysis for the multiserver model in which, we propose a new formulation by a level-dependent quasi birth-and-death (QBD) process, whose block matrices have some special block structure. Based on a matrix continued fraction approach and a censoring technique, we develop an efficient algorithm utilizing the special structure for computing the stationary distribution. Furthermore, we derive explicit formulae for the mean number of incoming calls and that of outgoing calls in the servers. Through various numerical results, we study the characteristics of the queueing system and find some insights into the optimal outgoing call rate.
20 citations
TL;DR: The present investigation deals with the bulk arrival M/G/1 retrial queue with impatient customers and modified vacation policy, where the incoming customers are impatient and may renege on seeing a long queue of the customers for the service.
Abstract: The present investigation deals with the bulk arrival M/G/1 retrial queue with impatient customers and modified vacation policy. The incoming customers join the virtual pool of customers called orbit if they find the server being busy, on vacation or in broken down state otherwise the service of the customer at the head of the batch is started by the server. The service is provided in k essential phases to all the customers by the single server which may breakdown while rendering service to the customers. The broken down server is sent to a repair facility wherein the repair is performed in d compulsory phases. As soon as the orbit becomes empty, the server goes for vacation and takes at most J vacations until at least one customer is noticed. The incoming customers are impatient and may renege on seeing a long queue of the customers for the service. The probability generating functions and queue length for the number of customers in the orbit and queue have been obtained using supplementary variable technique. Various system characteristics viz. average number of customers in the queue and the orbit, long run probabilities of the system states, etc. are obtained. Furthermore, numerical simulation has been carried out to study the sensitivity of various parameters on the system performance measures by taking an illustration.
18 Sep 2014
TL;DR: The authors illustrate numerical results of the delay, throughput and spectrum occupancy with respect to retrial rate, spectrum sensing time and primary user's busy rate.
Abstract: The authors apply the preemptive priority queueing analysis to cognitive radio networks. Using the proposed analytical model, the authors can quantify the delay, throughput and spectrum occupancy of cognitive radio networks. Cognitive users’ packet arrivals form a retrial queue with a proper choice of the retrial rate. The authors illustrate numerical results of the delay, throughput and spectrum occupancy with respect to retrial rate, spectrum sensing time and primary user's busy rate.
TL;DR: Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated and the effect of the information levels and several parameters on the customers�’ equilibrium and optimal strategies is presented.
Abstract: We consider a single-server constant retrial queueing system with a Poisson arrival process and exponential service and retrial times, in which the server may break down when it is working. The lifetime of the server is assumed to be exponentially distributed and once the server breaks down, it will be sent for repair immediately and the repair time is also exponentially distributed. There is no waiting space in front of the server and arriving customers decide whether to enter the retrial orbit or to balk depending on the available information they get upon arrival. In the paper, Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated. We consider separately the partially observable case where an arriving customer knows the state of the server but does not observe the exact number of customers waiting for service and the fully observable case where customer gets informed not only about the state of the server but also about the exact number of customers in the orbit. Some numerical examples are presented to illustrate the effect of the information levels and several parameters on the customers’ equilibrium and optimal strategies.
TL;DR: The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study.
Abstract: The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results.
TL;DR: The M/M/1 retrial queue with working vacations and negative customers is introduced and efficient methodology to compute the stationary distribution for this new queue is developed and presented.
Abstract: The M/M/1 retrial queue with working vacations and negative customers is introduced. The arrival processes of positive customers and negative customers are Poisson. Upon the arrival of a positive customer, if the server is busy the customer would enter an orbit of infinite size and the orbital customers send their requests for service with a constant retrial rate. The single server takes an exponential working vacation once customers being served depart from the system and no customers are in the orbit. Arriving negative customers kill a batch of the positive customers waiting in the orbit randomly. Efficient methodology to compute the stationary distribution for this new queue is developed and presented.
TL;DR: In this article, the authors considered the M/M/c retrial queues with multiclass of customers and showed that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/m/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity.
Abstract: We consider the M/M/c retrial queues with multiclass of customers. We show that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/M/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity. Approximation formulae for the distributions of the number of customers in service facility, the mean number of customers in orbit and the sojourn time distribution of a customer are presented. The approximations are compared with exact and simulation results.
Journal Article•
TL;DR: The main goal of this paper is to investigate various monotonicity properties of a single server retrial queue with rst-come-rst-ser ved (FCFS) orbit and general retrial times using stochastic comparisons of Markov chains.
Abstract: We propose to use a mathematical method based on stochastic comparisons of Markov chains in order to derive perfor- mance indice bounds. The main goal of this paper is to investigate various monotonicity properties of a single server retrial queue with rst-come-rst-ser ved (FCFS) orbit and general retrial times using
TL;DR: This paper considers a Poisson arrival queueing system with a single server and two essential phases of heterogeneous service, and obtains a stochastic decomposition law for the system.
Abstract: This paper has been motivated by the interactive voice response system (IVRS). This system has now become a common phenomenon in our everyday life. In this paper, we consider a Poisson arrival queueing system with a single server and two essential phases of heterogeneous service. The customer who completes the first phase has a choice of k-options to choose for the second phase of service. The customer, who finds the server busy upon arrival, can either join the orbit or he/she can leave the system. After completion of both phases, the customer can decide to try again for service by joining the orbit or he/she can leave the system. The server is subject to sudden breakdowns and repairs. We assume that the breakdowns of the system can occur even when the system is idle. By using the supplementary variables technique, we have obtained the steady state probability generating functions of the orbit size and the system size. We obtain a stochastic decomposition law for the system. We present numerical results ...
TL;DR: In this article, an analytical treatment of an GI/M/1 retrial queue with constant retrial rate was studied, where a customer who finds the server busy joins the queue in the orbit in accordance with the first-come-first-out (FCFS) discipline and only the oldest customer in the queue is allowed to make repeated attempts to reach the server.
Abstract: In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.
TL;DR: An M/M/1 retrial queue with working vacations, vacation interruption, Bernoulli feedback, and N-policy simultaneously is considered and the conditional stochastic decomposition for the queue length in the orbit is proved.
Abstract: We consider an M/M/1 retrial queue with working vacations, vacation interruption, Bernoulli feedback, and N-policy simultaneously. During the working vacation period, customers can be served at a lower rate. Using the matrix-analytic method, we get the necessary and sufficient condition for the system to be stable. Furthermore, the stationary probability distribution and some performance measures are also derived. Moreover, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, we present some numerical examples and use the parabolic method to search the optimum value of service rate in working vacation period.
TL;DR: An approximation is provided for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution and some numerical results are presented.
Abstract: We consider the PH/PH/c retrial queues with PH-retrial time. Approximation formulae for the distribution of the number of customers in service facility, sojourn time distribution and the mean number of customers in orbit are presented. We provide an approximation for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution. Some numerical results are presented.
TL;DR: In this paper, a discrete-time batch arrival retrial queue with the server subject to starting failures is considered, where each customer after service either immediately returns to the orbit for another service with probability θ or leaves the system forever with probability 1 − θ (0 ≤ θ < 1).
Abstract: This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures. Different from standard batch arrival retrial queues with starting failures, we assume that each customer after service either immediately returns to the orbit for another service with probability θ or leaves the system forever with probability 1 − θ (0 ≤ θ < 1). On the other hand, if the server is started unsuccessfully by a customer (external or repeated), the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 − q (0 ≤ q < 1). Firstly, we introduce an embedded Markov chain and obtain the necessary and sufficient condition for ergodicity of this embedded Markov chain. Secondly, we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time. We also derive a stochastic decomposition law. In the special case of individual arrivals, we develop recursive formulae for calculating the steady-state distribution of the orbit size. Besides, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on the mean orbit size.
TL;DR: A simple condition on the service and retrial rates is presented for the matrix exponential solution to be explicit or algorithmically tractable and an approximating solution is derived that allows us to obtain global error control.
TL;DR: In this article, the authors study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiiserver M/M/∞ Markov modulated queue.
Abstract: In this paper, we study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiserver M/M/∞ Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a"minimal/nondominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (a) and (b), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest. The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: (i) the irregular singularity at 0; (ii) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and (iii) the point 1, whose Taylor series coefficients are the factorial moments.
01 Jan 2014
TL;DR: This paper obtains the joint probability distribution of the server state and the number of orbiting customers in the system in terms of Laplace and z- transforms and shows how mean performance measures can be obtained.
Abstract: In this paper, we show through the example of the M/G/1 queue with working vacations, how queueing theory can help to the performance evaluation of some modern systems. We obtain the joint probability distribution of the server state and the number of orbiting customers in the system.This distribution is obtained in terms of Laplace and z- transforms. We show how mean performance measures can be obtained.
TL;DR: This paper analyses an M/G/1 retrial queue with working vacation and constant retrial policy, and derives the steady-state queue distribution of number of customer in the retrial group.
Abstract: This paper analyses an M/G/1 retrial queue with working vacation and constant retrial policy. As soon as the system becomes empty, the server begins a working vacation. The server works with different service rates rather than completely stopping service during a vacation. We construct the mathematical model and derive the steady-state queue distribution of number of customer in the retrial group. The effects of various performance measures are derived.
TL;DR: The derivation of the explicit form of the stationary system size distributions is presented visually exhibiting the effect of various parameters on the stationary distributions.
Abstract: We reconsider the M/M/∞ queue with two-state Markov modulated arrival and service processes and the single-server retrial queue analyzed in Keilson and Servi [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471]. Fuhrmann and Cooper type stochastic decomposition holds for the stationary occupancy distributions in both queues [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471; Baykal-Gursoy, M and W Xiao (2004). Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates. Queueing Systems, 48, 75–88]. The main contribution of the present paper is the derivation of the explicit form of the stationary system size distributions. Numerical examples are presented visually exhibiting the effect of various parameters on the stationary distributions.
TL;DR: The global balance equations of the Markov chain describing the system evolution are established and it is proved that this queueing system is stable as long as the customers are strict impatient and the mean retrial time is finite.
Posted Content•
TL;DR: This paper studies two families of QBD processes with linear rates, and provides a differential system for the generating function that unifies problems (a) and (b), and includes some additional features and observes that in at least one particular case the authors get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest.
Abstract: We study two families of QBD processes with linear rates: (A) the multiserver retrial queue and its easier relative; and (B) the multiserver M/M/infinity Markov modulated queue.
The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a ``minimal/non-dominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods.
Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems.
We provide a differential system for our generating function that unifies problems (A) and (B), and we also include some additional features and observe that in at least one particular case we get a special ``Okubo-type hypergeometric system", a family that recently spurred considerable interest.
The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: 1) the irregular singularity at 0; 2) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and 3) the point 1, whose Taylor series coefficients are the factorial moments.
Journal Article•
TL;DR: A single server batch arrival retrial G-queue and an unreliable server with delayed repair with expected system size, orbit size, availability and failure frequency of the server are derived.
Abstract: Single server batch arrival retrial G-queue and an unreliable server with delayed repair are analyzed. Positive customers arrive in batches according to Poisson processes. If the server is idle, one of the positive customers in the batch enters for service while the rest join the orbit. Otherwise all the customers enter the orbit. Arrival of negative customer removes the positive customer being in service from the system and causes the server breakdown. The repair of the failed server starts after a random amount of time known as delay time. Expected system size, orbit size, availability and failure frequency of the server are derived. Stochastic decomposition law is verified. Numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.