Showing papers on "Retrial queue published in 2012"

The Unreliable M/M/1 Retrial Queue in a Random Environment

Abstract: We examine an M=M=1 retrial queue with an unreliable server whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. Provided are conditions for stability, the (approximate) orbit size distribution, and mean queueing performance measures which are obtained via matrix-analytic methods. Additionally, we consider the problem of choosing arrival and service rates for each environment state with the objective of minimizing the steady state mean time spent in orbit by an arbitrary customer, subject to cost and revenue constraints. Two numerical examples illustrate the main results.

On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations

Abstract: A single-server retrial queue with two types of customers in which the server is subject to vacations along with breakdowns and repairs is studied. Two types of customers arrive to the system in accordance with two different independent Poisson flows. The service times of the two types of customers have two different independent general distributions. We assume that when a service is completed, the server will take vacations after an exponentially distributed reserved time. It is assumed that the server has an exponentially distributed lifetime, a generally distributed vacation time and a generally distributed repair time. There is no waiting space in front of the server, therefore, if the server is found busy, or on vacation, or down, the blocked two types of customers form two sources of repeated customers. Explicit expressions are derived for the expected number of retrial customers of each type. Additionally, by assuming both types of customers face linear costs for waiting and retrial, we discuss and compare the optimal and equilibrium retrial rates regarding the situations in which the customers are cooperative or noncooperative, respectively.

Tail asymptotics for M/M/c retrial queues with non-persistent customers

Abstract: In this paper, we consider a variant of the classical M/M/c retrial queue, in which we allow non-persistent customers. When c > 1, this system does not have an explicit closed form solution for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers. Our main focus is on the tail asymptotics for the joint probabilities. We first present a matrix-product solution for the joint stationary probability vectors, which is further simplified to a scalar-product form, according to matrix-analytic theory. We then apply the censoring technique, which has been proven an efficient approach for analyzing queueing systems including retrial queues, to obtain the censored equations and the Key Lemma. In terms of these results, we finally prove an exact tail asymptotic result for the stationary probabilities.

An explicit solution for a tandem queue with retrials and losses

Abstract: We consider a retrial tandem queueing system with two servers whose service times follow two exponential distributions. There are two types of customers: type one and type two. Customers of type one arrive at the first server according to a Poisson process. An arriving customer of type one that finds the first server busy joins an orbit and retries to enter the server after some time. We assume that the arrival rate of customers from the orbit is a linear function of the number of retrial customers. After being served at the first server, a customer of type one moves to the second server. Customers of type two directly arrive at the second server according to another Poisson process. Customers of both types one and two are lost if the second server is busy upon arrival. For this model, we derive explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers. We prove that the stationary distribution is computed by a numerically stable algorithm. Numerical examples are provided to show the influence of parameters on the performance of the system.

Geo/Geo/1 retrial queue with working vacations and vacation interruption

Abstract: Consider a Geo/Geo/1 retrial queue with working vacations and vacation interruption, and assume requests in the orbit try to get service from the server with a constant retrial rate. During the working vacation period, customers can be served at a lower rate. If there are customers in the system after a service completion instant, the vacation will be interrupted and the server comes back to the normal working level. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic 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.

A Batch Arrival Retrial Queue with Two Phases of Service, Feedback and K Optional Vacations

Abstract: We consider a batch arrival queueing system with two phases of service, feedback and K optional vacations under a classical retrial policy. At the arrival epoch, if the server is busy the whole batch joins the orbit. Whereas if the server is free, then one of the arriving customer starts its service immediately and the rest joins the orbit. For each customer, the server provides two phases of service. After the completion of two phases of service, the customer may rejoin the orbit as a feedback customer for receiving another regular service with probability p. If the system is empty, then the server become inactive and begins the first essential vacation. After the completion of first essential vacation, the server may either wait idle for a customer or may take one of K additional vacations. The steady state distribution of the server state and the number of customers in the orbit are obtained. Also the effects of various parameters on the system performance are analyzed numerically.

M/M/1 retrial queue with collisions and working vacation interruption under N-policy

Abstract: Consider an M/M/1 retrial queue with collisions and working vacation interruption under N-policy. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic 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.

Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

Abstract: This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for an retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

Retrial Queuing System with Starting Failure, Single Vacation and Orbital Search

Abstract: This paper is concerned with the analysis of a single server retrial queue with vacation and orbital search. The server is subject to starting failure and repair. At the completion epoch of each service, the server may take a single vacation. After vacation completion, the server searches for the customers in the orbit or remains idle. Retrial times, service times and vacation times are assumed to be arbitrarily distributed. Various performance measures are derived and numerical results are presented.

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

Abstract: This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy. It is assumed that the server, after each service completion, begins a process of search in order to find the following customer to be served with a certain probability, or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or on vacation. Finally, the author gives two stochastic decomposition laws, and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.

Two way communication retrial queues with balanced call blending

Abstract: In call centers, call blending consists in the mixing of incoming and outgoing call activity. Artalejo and Phung-Duc recently provided 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 outgoing call activity is largely blocked when many incoming calls are in orbit, which may be unwanted, especially when outgoing calls are vital to the service offered. In this paper, we assume a balanced way of 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 deriving 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. For the multiserver case (multiple operators), we provide a quasi-birth-and-death process formulation, enabling deriving a sufficient and necessary condition for stability in this case, as well as a numerical recipe to obtain the stationary distribution.

Tool supported modeling of sensor communication networks by using finite-source priority retrial queues

Abstract: The main aim of the present paper is to draw the attention of the readers of this special issue to the modeling issues of sensor networks. The novelty of this investigation is the introduction of servers vacation combined with priority customers for finite-source retrial queues and its application to wireless sensor networks. In this paper we analyze a priority finite-source retrial queue with repeated vacations. Two types of priority customers are defined, customers with priority 1 (P1) go directly to an ordinary FIFO queue. However, if customers with priority 2 (P2) find the server in busy or unavailable state go to the orbit. These customers stay in the orbit and retry their request until find the server in idle and available state. We assume that P1 customers have non-preemptive priority over P2 customers. The server starts with a listening period and if no customer arrive during this period it will enter in the vacation mode. When the vacation period is terminated, then the node wakes up. If there is a P1 customer in the queue the server begin to serve it, and when there is no any P1 customer, the node will remain awake for exponentially distributed time period. If that period expires without arrivals the node will enter in the next sleeping period. All random variables involved in model construction are supposed to be independent and exponentially distributed ones. Our main interest is to give the main steady-state performance measures of the system computed by the help of the MOSEL tool. Sev∗Acknowledgment: This research is partially supported by the Hungarian Science and Technology Foundation, HungarianFrench Bilateral Cooperation under grant TeT 10-1-2011-0741, FR25/2010. eral Figures illustrate the effect of input parameters on the mean response time.

A batch arrival retrial queuing system for essential and optional services with server breakdown and Bernoulli vacation

Abstract: This paper studies a general retrial M X /G/1 queue with an additional phase of second optional service and Bernoulli vacation where breakdowns occur randomly at any instant while servicing the customers. If an arriving batch finds that the server is busy in providing either first essential service (FES)/second optional service (SOS) or on vacation then arriving batch enters an orbit called retrial queue. Otherwise, one customer from arriving batch starts to be served by the server while the rest join the orbit. The vacation times and service times of both first essential and second optional services are assumed to be general distributed while the retrial times are exponential distributed. Introducing supplementary variables and by employing embedded Markov chain technique, we derive some important performance measures of the system such as average orbit size, average queue size, mean waiting time, expected lengths of busy period, etc. Numerical results have been facilitated to illustrate the effect of different parameters on several performance measures.

A retrial queue with server interruptions, resumption and restart of service

Abstract: This paper analyses a single server retrial queuing model with service interruptions, resumption/restart of interrupted service. The service is assumed to get interrupted according to a Poisson process. The interrupted service is either resumed or restarted according to the realization of two competing independent, non-identically distributed random variables, the realization times of which follow exponential distributions. Arrival of primary customers constitutes a Poisson process. On arrival if a customer finds an idle server, he is immediately taken for service. If the server is busy when a customer arrives, this customer goes to an orbit of infinite capacity from where he makes repeated attempts for service according to a Poisson process with parameter β. After an unsuccessful retrial he rejoins the orbit with probability p and leaves the system without waiting for service with probability q = 1 − p. The service time durations follow PH distribution with representation (α, S) of order m, when there is no interruption. The system is found to be always stable if q > 0. The case q = 0 is also analyzed. Using Matrix analytic method, expressions for important system characteristics such as expected service time, expected number of interruptions are obtained. System performance measures are numerically explored and the effect of service interruptions in a retrial set up is studied.

A fuzzy based threshold policy for a single server retrial queue with vacations

Abstract: In this paper we deal with a single server retrial queue with vacations. The server serves the customers until the system becomes empty, then it takes a vacation. The system consists of two types of costs. The blocking cost is considered whenever a customer is blocked either because of the server is busy or off. There is also a cost each time the server is turned on. The problem is to find an effective policy for turning on the dormant server. We propose a Fuzzy Based Threshold Policy (FBTP) to control the server, substitute for conventional threshold policies. The FBTP is based on four input parameters, an inference stage and it is tuned up using a stochastic List Based Threshold Accepting (LBTA) algorithm. Simulation models are developed to validate the fuzzy controller. Numerical experiments are provided to show that the proposed method is superior to crisp threshold policies.

A repairable discrete-time retrial queue with recurrent customers, Bernoulli feedback and general retrial times

Abstract: This paper relates to a repairable Geo/G/1 retrial queue with general retrial times, Bernoulli feedback, the server subjected to starting failures and two types of customers: transit and a fixed number of recurrent customers. After service completion, recurrent customers always return to the orbit and transit customers either immediately return to the orbit for another service with probability $$\theta (0\leq\theta<1)$$ or leave the system forever with probability 1 − θ. We construct the mathematical model and present some performance measures of the model in steady-state. We provide a stochastic decomposition law and analyze the relationship between our discrete-time system and its continuous-time counterpart. Finally, some numerical examples show the influence of the parameters on the performance characteristics of the system.

Stochastic analysis of a finite source retrial queue with spares and orbit search

Abstract: This paper aims at presenting an analytic approach for investigating a single server finite-source retrial queue with spares and constant retrial rate. We assume that there is a single repair facility (server) and K independent parts (customers) in the system. The customers' life times are assumed to be exponentially distributed random variables. Once a customer breaks down, it is sent for repair immediately. If the server is idle upon the failed customer's arrival, the customer receives repair immediately. The failed customer that finds the server busy upon arrival enters into the retrial orbit. Upon completion of a repair, the server searches for a customer from orbit if any. However, when a new primary customer arrives during the seeking process, the server interrupts the seeking process and serves the new customer. There are some spares for substitution of failed machines and the system is maintained by replacing failed part by spares and by repairing failed parts so that they may become spares when they are repaired. We carry out the steady-state analysis of the model and obtain various steady-state performance measures.

Analysis of an MX/G/1 Retrial Queue with Two Phases of Service, Balking, Feedback and K Optional Vacations

Abstract: A batch arrival queueing system with two phases of service, balking, feedback, and K optional vacations under a classical retrial policy is discussed in the paper. At the arrival epoch, if the server is busy, the whole batch joins the orbit or balks the system. Whereas, if the server is free, then one of the arriving jobs starts its service immediately and the rest join the retrial group or balk the system. For each job, the server provides two phases of service. After the completion of the two phases of service, the job may rejoin the orbit as feedback for another regular service or leave the system forever. If the system is empty, then the server becomes inactive and begins its first vacation. After completion of the first essential vacation, the server may either wait for a job or may take one of `K' additional vacations.

The single server retrial queue with finite population: A BSDE approach

Abstract: This paper uses the block-structured state-dependent event (BSDE) approach to generalize the scalar version of the single server retrial queue with finite population. The simple scalar version only involves exponential random variables, which make the underlying Markov chain tractable. However, this is a drawback in applications where the exponentiality is not a realistic assumption and the flows are correlated. The BSDE approach provides a versatile tool to deal with a non-exponential model with correlated flows, but keeping tractable the dimensionality of the block-structured Markov chain. We focus on the investigation of the limiting distribution of the system state and the waiting time. The theory is illustrated by numerical experiments, which demonstrate that the proposed BSDE approach can be applied efficiently.

Cost analysis of a bulk service retrial queue

Abstract: This paper studies a batch arrival general bulk service retrial queueing model with constant retrial rate. The primary customers arrive in bulk according to Poisson process and they get service under general bulk service rule with minimum of one customer and maximum of ‘b’ customers. If the arriving batch of customers, of size ‘ζ ’, 1≤ζ ≤ b , finds the server free, then all of them get service immediately; while, if the size of the arriving batch is more than ‘b’, then, ‘b’ customers enter the service station and the remaining ζ - b customers join the orbit. However, if an arriving batch of customers finds the server busy, then the entire batch joins the orbit in order to seek service again. The customers in the orbit will try for service one by one with a constant retrial rate ‘v’ when the server is idle. For the proposed model, the probability generating function of the steady-state queue size distribution at an arbitrary time, expected number of customers in the orbit, expected waiting time, expected length of busy period and expected length of busy cycle are obtained. The cost analysis of the queueing system is discussed. The effects of several parameters on the system are analysed numerically.

Approximate Analysis of an Unreliable M/M/2 Retrial Queue

Abstract: : This thesis considers the performance evaluation of an M/M/2 retrial queue for which both servers are subject to active and idle breakdowns. Customers may abandon service requests if they are blocked from service upon arrival, or if their service is interrupted by a server failure. Customers choosing to remain in the system enter a retrial orbit for a random amount of time before attempting to re-access an available server. We assume that each server has its own dedicated repair person, and repairs begin immediately following a failure. Interfailure times, repair times and times between retrials are exponentially distributed, and all processes are assumed to be mutually independent. Modeling the number of customers in the orbit and status of the servers as a continuous-time Markov chain, we employ a phase-merging algorithm to approximately analyze the limiting behavior. Subsequently, we derive approximate expressions for several congestion and delay measures. Using a benchmark simulation model, we assess the accuracy of the approximations and show that, when the algorithm assumptions are met, the approximation procedure yields favorable results. However, as the rate of abandonment for blocked arrivals decreases, the performance declines while the results are insensitive to the rate of abandonment of customers preempted by a server failure.

A Single Server Retrial Queue with Impatient Customers

Abstract: In the present paper, a single server retrial queue with impatient customers is studied. The primary arrivals and repeating calls follow the Poisson distribution. The service time is exponentially distributed. Explicit time-dependent probabilities of an exact number of arrivals and departures from the orbit are obtained by solving the differential-difference equations recursively. Steady state solution of the number of busy servers is obtained. The numerical results are graphically displayed to illustrate the effect of arrival rate, retrial rate and service rate on different probabilities against time. Some special cases of interest are also deduced.

Retrial Queue with Threshold Recovery, Geometric Arrivals and Finite Capacity

Abstract: The present investigation deals with the performance prediction of finite Markovian retrial queues with unreliable server. The arrivals of the customers follow geometric distribution while the service pattern follows exponential distribution. The customers are served in two stages i.e. first essential service (FES) which is compulsory for all the arriving customers and second optional service (SOS) which depends on the customer’s demand. The customer occupies the server if it is idle, otherwise he is forced to join the orbit and retry for the service later with the retrial rateθ. The server is unreliable and can breakdown during any stage of service. The broken down server is sent for repair and after repair it becomes as good as before failure. The repair process follows threshold recovery according to which the repair starts when a minimum number of customers say L (≥1) has been accumulated in the system. Various performance measures like expected queue length, availability, throughput etc. have been obtained in terms of transient probabilities. Furthermore, sensitivity analysis has been done to examine the effect of different parameters on various performance indices

• performance analysis of repairable m/g/1 retrial queue with bernoulli vacation and orbital search

Abstract: In this paper we consider a repairable M/G/1 retrial queue with Bernoulli vacation and orbital search. By supplementary variable technique, the probability generating functions of the system size distribution and the orbit size distribution under steady state are obtained. Queueing as well as reliability indices to predict the behavior of the system are derived. Various models studied earlier are deduced as special cases by appropriate choice of parameter values.

Stability of the multiserver queue with addressed retrials

Abstract: In the recent paper (Mushko et al. in Ann. Oper. Res. 141:283—301, 2006) Mushko, Jacob, et al. considered an M/M/c type queueing system with retrials. Given that returning customers have access to any server they obtained a sufficient condition for the stability of the system. We suggest an alternative approach to the problem and get the necessary and sufficient condition for the stability in more general situation, when some servers are reserved for processing of primary requests and do not serve returning customers.

Unreliable M^X/(G1,G2)/1 Retrial Queue with Bernoulli Feedback Under Modified Vacation Policy

Abstract: In this paper, a batch arrival retrial queueing system with Bernoulli feedback and balking under modified vacation policy is considered wherein the server provides two phases of essential service to all arriving customers. In addition, the server may breakdown during the busy state at any time. After the completion of two phases of heterogeneous service, the server may go for 'j^th' (1 ≤ j ≤ J) vacation on finding the orbit empty until at least one customer is accumulated in the system. The inter-retrial times, service times, repair times and vacation times are assumed to be governed by the general distribution. By introducing the supplementary variables and using generating function, the expressions for the expected number of the customers in the orbit as well as in the system are derived. Finally, the numerical results are provided.