Showing papers on "Retrial queue published in 2016"

On a queueing-inventory with reservation, cancellation, common life time and retrial

Abstract: In this paper we model a queueing-inventory system that has applications in railway and airline reservation systems. Maximum items in the inventory is $$S$$ which have a random common life time; this includes those that are sold in particular cycle. A customer, on arrival to an idle server with at least one item in inventory, is immediately taken for service; or else he joins the buffer of maximum size $$S$$ depending on number of items in the inventory (the buffer capacity varies and is, at any time, equal to the number of items in the inventory). The arrival of customers constitutes a Poisson process, demanding exactly one item each from the inventory. If there is no item in the inventory, the arriving customer first queue up in a finite waiting space of capacity $$K$$ . When it overflows an arrival goes to an orbit of infinite capacity with probability $$p$$ or is lost forever with probability $$1-p$$ . From the orbit he retries for service according to an exponentially distributed inter-occurrence time. The service time follows an exponential distribution. Cancellation of sold items before its expiry is permitted. Inventory gets added through cancellation of purchased items, until the expiry time. Cancellation time is assumed to be negligible. We analyze this system. Several performance characteristics are computed; expected sojourn time of the system in a cycle with “no inventory” and also “maximum inventory” are computed. Some illustrative numerical examples are presented. An optimization problem is numerically analyzed.

Retrial queues with balanced call blending: analysis of single-server and multiserver model

Abstract: In call centers, call blending consists in the mixing of incoming and outgoing call activity, according to some call blending balance. Recently, Artalejo and Phung-Duc have developed an apt model for such a setting, with a two way communication retrial queue. However, by assuming a classical (proportional) retrial rate for the incoming calls, the short- term blending balance is heavily impacted by the number of incoming calls, which may be undesired, especially when the balance between incoming and outgoing calls is vital to the service offered. In this contribution, we consider an alternative to classical call blending, through a retrial queue with constant retrial rate for incoming calls. For the single-server case (one operator), a generating functions approach enables to derive explicit formulas for the joint stationary distribution of the number of incoming calls and the system state, and also for the factorial moments. This is complemented with a stability analysis, expressions for performance measures, and also recursive formulas, allowing reliable numerical calculation. A correlation study enables to study the system's short-term blending balance, allowing to compare it to that of the system with classical retrial rate. For the multiserver case (multiple operators), we provide a quasi-birth-and-death process formulation, enabling to derive a sufficient and necessary condition for stability in this case (in a simple form), a numerical recipe to obtain the stationary distribution, and a cost model.

A queueing model with two classes of retrial customers and paired services

Abstract: We mathematically investigate a single server system accepting two types of retrial customers and paired service. If upon arrival a customer finds the server busy, it is routed to an infinite capacity orbit queue according to each type. Upon a service completion epoch, if at least one orbit queue is non-empty, the server seeks to find customers from the orbits. If both orbit queues are non-empty, the seeking process will bring to the service area a pair of customers, one from each orbit. If there is only one non-empty, then a single customer from this orbit queue will be brought to the service area. However, if a primary customer arrives during the seeking process it will occupy the server immediately. It is shown that the joint stationary orbit queue length distribution at service completion epochs is determined by solving a Riemann boundary value problem. Stability condition is investigated, while generalizations of the main model are also discussed. A simple numerical example is obtained and yields insight into the behavior of the system.

Scaling limits for single server retrial queues with two-way communication

Abstract: This paper studies M/G/1 retrial queues in which there are two arrival flows, i.e., incoming calls made by regular customers and outgoing calls made by the server in idle time. The stationary analysis of this system has been carried out in a recent paper by Artalejo and Phung-Duc (Appl Math Model 37(4):1811–1822, 2013). In this paper, we obtain a decomposition property where we prove that the queue length is decomposed into the sum of three independent random variables with clear physical meaning. We then derive scaling limits for the queue length distribution under some extreme conditions (i) heavy traffic, (ii) slow retrials and (iii) instantaneous connection to outgoing calls. Furthermore, we also investigate the convergence of our model to that without outgoing calls.

Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers

Abstract: In this paper we study a new version of an unreliable retrial queue with persistent and impatient customers. The considered model takes into account corrective and preventive maintenances. If a preventive action occurs in a busy period, then it is postponed to an ulterior date. We give the necessary and sufficient condition for the system to be stable and obtain the joint distribution of the server state and the number of orbiting customers in the system in terms of generating functions. Some performance measures are derived. From the reliability view point, we analyze the time to the first failure of the server. The effect of the system parameters on the performance measures is shown through an illustrative numerical example.

A Retrial Queue to Model a Two-Relay Cooperative Wireless System with Simultaneous Packet Reception

Abstract: In this work we analyze a novel queueing system to model cooperative wireless networks with two relay nodes and simultaneous packet reception. We consider a network of three saturated source users, say a central and two background source users, two relay nodes and a common destination node. Source users transmit packets to the destination node with the cooperation of relays, which assist them by re-transmitting their blocked packets. We assume that the central source user forwards its blocked packets to both relay nodes in order to exploit both the spatial diversity they provide, and the broadcast nature of wireless communication. Moreover, each relay node receives also blocked packets from a dedicated background source user. We study a three-dimensional Markov process, investigate its stability condition and show that its steady-state performance is expressed in terms of the solution of a Riemann-Hilbert boundary value problem. Performance metrics are obtained, and numerical results show insights into the system behavior. Some computational issues are also discussed.

Analysis of an M [X] /G/1 unreliable retrial G -queue with orbital search and feedback under Bernoulli vacation schedule

Abstract: In this paper, we consider a batch arrival retrial queue with feedback under Bernoulli vacation schedule, where the busy server is subjected to breakdown due to the arrival of negative customers. Any arriving batch of positive customers finds the server free, one of the customers from the batch enters into the service area and the rest of them join into the orbit. Arriving positive customers may balk (or renege) the system at particular times. After completion of service the unsatisfied positive customer may rejoin into the orbit to get another regular service as feedback customer. The server takes Bernoulli vacation after service completion of positive customers. After completion of service (if the server is not taking vacation), repair or vacation the server searches for the customers in the orbit or remains idle. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Some system performance measures, reliability measures and stochastic decomposition law are discussed. Finally, some numerical examples and cost optimization analysis are presented.

Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits

Abstract: We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type i customer $$(i=1,2)$$ finds a busy server, it will join the type i orbit. After an exponential time with a constant (retrial) rate $$\mu _i$$ , a type i customer attempts to get service. This model has been recently studied by Avrachenkov et al. (Queueing Syst 77(1):1–31, 2014) by solving a Riemann–Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.

Game-theoretic analysis of opportunistic spectrum sharing with imperfect sensing

Abstract: We consider the strategic behavior of secondary users (SUs) in a cognitive radio system where SUs opportunistically share a single primary user (PU) band over a coverage area. The service of an SU can be interrupted by a PU in a preemptive manner, and the interrupted SU may abandon the system or wait until the PU band is sensed available. In the latter case, if spectrum sensing errors occur, they will cause misdetections and false alarms which impact the system’s performance heavily. In this paper, we model this problem as a retrial queueing system with server breakdowns and recoveries in which the interrupted SUs are treated as retrial customers. They will retry for using the PU band after some period of time due to interruptions or misdetections. The arrival of a PU during service of an SU is modeled as a server breakdown, and the recovery time is equivalent to the service time of this PU. We focus on the behavior of arriving SUs who can make decisions on whether to join the system or to balk based on a natural cost structure and the delays caused by PUs’ interruptions, which can be studied as a non-cooperative game. The equilibrium and optimal strategies of SUs are both derived. Furthermore, to bridge the gap between the individually and socially optimal strategies, a novel strategy of imposing an admission fee on SUs to join the retrial group is proposed. Finally, some numerical examples are presented to show the effect of several key parameters on the system performance.

Two-Way Communication M/M/1 Retrial Queue with Server-Orbit Interaction

Abstract: This paper considers an M/M/1 retrial queue where the server not only receives incoming calls but in idle time makes outgoing calls of two types whose durations follow the same exponential distribution. The outgoing calls of type 1 are directed to the customers in orbit while the outgoing calls of type 2 are directed to the customers outside the orbit. Using the generating function approach, we derive explicit expressions and recursive formulas for the stationary joint distribution of the number of customers in the orbit and the server state as well as for the partial factorial moments. A closed form formula for the mean number of customers in the orbit is also obtained.

A functional approximation for retrial queues with two way communication

Abstract: In this paper we consider retrial queues with two way communication and finite orbit. We will develop a functional approximation of the stationary performances of this queue where the parameter of interest is the outgoing rate. Specifically, we will apply the Taylor series expansion method to propagate parametric uncertainty to performance measures. We provide an analysis of the $$\hbox {M}_{1},\, \hbox {M}_{2}/\hbox {G}_{1},\, \hbox {G}_{2}/1$$ retrial queue with two way communication and finite orbit. The sensitivity analysis is carried out by considering several numerical examples. More specifically, this paper proposes a numerical approach based on Taylor series expansion with a statistical aspect for analyzing the stationary performances of the considered queueing model, where we assume that the outgoing rate is not assessed in a perfect manner. Additionally, approximate expressions of the probability density functions, the expectation and the variance of the performance measures are obtained and compared to the corresponding Monte Carlo simulations results.

Mx/G/1 retrial vacation queue for multi-optional services, phase repair and reneging

Abstract: The present paper deals with a batch arrival general retrial queue with multi-optional services, vacation and impatient customers undergoing reneging. The incoming customer (or a batch of customers) on finding the server in busy or in non-working condition forms a virtual pool of customers called orbit otherwise undergoes service if the server is free. The service is provided at two levels; first essential service or second optional service depending upon customer’s choice to opt for it or not. Moreover, the server may go for vacation if he finds no customer waiting for service and returns back to the service center again as soon as the customer approaches for the service. The server is unreliable as such subject to breakdown during the service; as soon as the server fails, it is immediately sent for repair so as to restore its functionality as before failure. The repairman assigned for the repair of the server also takes some setup time before commencing the repair process. By using supplementary...

Control F-policy for Markovian retrial queue with server breakdowns

Abstract: The present study is concerned with the control policy for the retrial queue with server breakdowns. Customers are not allowed to join if the system becomes full but still service is being provided with slower rate than that of the normal busy period. To deal with the more practical scenario, the startup time taken into account to develop Markov model. Chapman-Kolmogorov equations governing the system states have been framed to analyze the performance indices of the queueing system. To explore the effect of system descriptors on various performance indices, numerical simulation and sensitivity analysis have been conducted by taking numerical example. A cost function is also constructed to determine the optimal service rate and minimum cost. Adaptive neuro fuzzy interface system (ANFIS) has also considered to compare the numerical results obtained with neuro-fuzzy results.

Performance of an M/M/1 Retrial Queue with Working Vacation Interruption and Classical Retrial Policy

Abstract: An M/M/1 retrial queue with working vacation interruption is considered. Upon the arrival of a customer, if the server is busy, it would join the orbit of infinite size. The customers in the orbit will try for service one by one when the server is idle under the classical retrial policy with retrial rate , where is the size of the orbit. During a working vacation period, if there are customers in the system at a service completion instant, the vacation will be interrupted. Under the stable condition, the probability generating functions of the number of customers in the orbit are obtained. Various system performance measures are also developed. Finally, some numerical examples and cost optimization analysis are presented.

A repairable retrial queue under Bernoulli schedule and general retrial policy

Abstract: This paper considers a repairable M/G/1 retrial queue with Bernoulli schedule and a general retrial policy, which is motivated by a contention problem in the downlink direction of wireless base stations in cognitive radio networks. Arriving Customers (called primary arrivals) who cannot receive service upon arrival either join the infinite waiting space in front of the server (called as the normal queue) with probability $$q$$ , or enter the orbit with probability $$1-q$$ according to the FCFS discipline. If the server breaks down in the process of the service of a customer, the customer in service either joins the orbit queue or leaves the system forever. First, we study the ergodicity of two related embedded Markov chains and derive stationary distributions. Second, we find the steady-state joint generating function of the number of customers in both queues. Some important performance measures of the system are obtained. Third, the reliability analysis of the system is also given. Finally, numerical examples are given to illustrate the impact of system parameters on the system performance measures.

A discrete-time Geom/G/1 retrial queue with balking customers and second optional service

Abstract: In this paper, we discuss a discrete-time Geom/G/1 retrial queue with balking customers and second optional service where the retrial time follows a geometrical distribution. If an arriving customer finds the server is busy, he will leave the service area and go to the orbit with probability θ or leave the system with probability 1−θ; otherwise, he will begin his service immediately. In this model, after a customer finishes his first essential service, he may leave the system with probability 1−α or asks for a second optional service with probability α. Through studying the Markov chain underlying the model, we establish the probability generating functions of the orbit size and system size. Finally, some performance measures and numerical examples are presented.

N-policy for Mx/G/1 Unreliable Retrial G-Queue with Preemptive Resume and Multi-services

Abstract: Bulk arrival retrial G-queue with impatient customers and multi-services subject to server breakdowns has been analyzed. The system allows the arrival of two types of customers: positive customers and negative customers in the system. The negative customers make the server fail if they find the server in busy state, whereas positive customers are served if the server is idle otherwise they join the virtual pool of customers called orbit. The customers from the retrial orbit try their chance again for the service. The customers have the option of obtaining more than one service. Moreover, the customers are impatient and may renege from the system with probability \((1-r)\). The server is sent for repair as soon as it breakdowns; after repair, the service process starts again. Also, the server has the provision to initiate the service when there are N customers accumulated in the system. Using supplementary variables technique and generating functions, various performance measures like reliability indices and long run probabilities have been obtained.

Stochastic Decomposition in Retrial Queueing Inventory System

Abstract: The purpose of this paper is to obtain product form solution for retrial - queueing - inventory system. We study an M/M/1 retrial queue with a storage system driven by an (s, S) policy. When server is idle, external arrivals enter directly to an orbit. Inventory replenishment lead time is exponentially distributed. The interval between two successive repeat attempts is exponentially distributed and only the customer at the head of the orbit is permitted to access the server. No customer is allowed to join the orbit when the storage system is empty and also when the serer is busy. We first derive the stationary joint distribution of the queue length and the on-hand inventory in explicit product form. Using the joint distribution, we investigate long-run performance measures and a cost function. The optimal pair (s, S) is numerically investigated.

A closed-form solution for a two-server heterogeneous retrial queue with threshold policy

Abstract: In this paper, we reconsider a two-server heterogeneous retrial queue with threshold policy. However, the computation time with the existing method is prohibitively large for certain values of the threshold parameter. Applying the spectral expansion method, we derive a closed-form expression for the eigenvalues and eigenvectors matrix that are needed to determine the steady-state distribution of a quasi-birth-death process describing the queue. As a result, the computation time for the performance measures does not depend on the threshold parameter.

Retrial queueing system with retention of reneging customers

Abstract: Managing customer impatience plays a vital role in improving the efficiency of queueing systems. Reneging of impatient customers leads to loss of potential customers, which results in the loss of business. A reneged customer can be retained in many cases by employing various convincing strategies to continue in the queue until completion of service. In this paper we present an analysis of a single server retrial queue with retention of reneging customers from the orbit. The generating function technique has been used to derive the steady state probabilities of the system. Performance measures have been derived and system efficiency discussed by numerical results and graphical illustrations to demonstrate how the various parameters influence the behavior of the system. Some of the existing results have been deduced.

Analysis of an M/G/1 feedback retrial queue with unreliable server, non-persistent customers, single working vacation and vacation interruption

Abstract: In this paper, we consider a single server feedback retrial queueing system with single working vacation and vacation interruption. An arriving customer may balk (or renege) the system at some particular times. The single server provides two essential phases of regular service to each customer. When the orbit becomes empty at service completion instant; the server goes for a single working vacation. In working vacation period, the server works in lower service rate. The regular busy server may breakdown at any instance and the service channel will fail for a short interval of time. Using the method of supplementary variable technique, the steady state probability generating function for the system/orbit size is obtained. Some important system performance measures and the mean busy period are obtained. The conditional decomposition law is shown for this retrial queueing system. Finally, the effects of various parameters on the system performance are analysed numerically.

Impacts of Retrials on Power-Saving Policy in 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. The server needs some setup time during which the server cannot process a job but consumes energy. An arriving job that sees all the servers occupied (active or setup) joins the orbit and retries to enter an unoccupied server after some random time. We formulate this model using a level-dependent quasi birth- and-death process. Using Foster--Lyapunov criteria, we obtain the stability condition. We also propose a heuristic technique to determine the truncation point for the level-dependent quasi birth-and-death process.

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

Abstract: The paper deals with a research of bivariate Markov process $$\{X(t), t\ge 0\}$$ whose state space is a lattice semistrip $$S(X)=\{0,1,{\ldots },c\} \times Z_{+}$$ . The process $$\{X(t), t\ge 0\}$$ describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.