Showing papers on "Retrial queue published in 2015"

Retrial queueing system with balking, optional service and vacation

Abstract: In this study, we propose a single server retrial queueing system with balking, second optional service and single vacation. At the arrival epoch, if the server is busy, the arriving job join the orbit or balks the system whereas if the server is free, then the arriving job starts its service immediately. For each job, the server provides two phases of service. All the jobs demand the first essential service, whereas only some of the jobs demand for the second optional service. If the system is empty, then the server becomes inactive and begins a single vacation. If server comes back from the vacation, it does not go for another vacation even if the system is still empty at that time. The steady state distributions of the server state and the number of jobs in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analyzed numerically. A general decomposition law for this retrial queueing system is established.

Sojourn Time Analysis of Finite Source Markov Retrial Queuing System with Collision

Abstract: This paper deals with a finite source retrial queueing system of type M/M/1//N with collision of the customers. This means that the system has one server and N sources. Analysis of the sojourn time in the system is presented. The analysis is performed under an asymptotic condition of infinitely increasing number of sources. The approximation of the distribution of the total sojourn time in the system is derived.

A preemptive priority retrial queue with two classes of customers and general retrial times

Abstract: In this paper, we consider a continuous-time retrial queue with two classes of customers: priority customers and ordinary customers, where priority customers don’t queue and have an exclusive preemptive priority to receive their services over ordinary customers. If an arriving ordinary customer finds the server busy, it enters a retrial group (called orbit) according to FCFS discipline. Only the ordinary customer at the head of the retrial queue is allowed to access the server. Firstly, we obtain the necessary and sufficient condition for the system to be stable by embedded Markov chain approach. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures of interest. Thirdly, we give the analysis of the sojourn time in the system of an arbitrary ordinary customer. Lastly, numerical examples are given to show the effect of system parameters on several performance measures.

Transient Analysis of Finite F-Policy Retrial Queues with Delayed Repair and Threshold Recovery

Abstract: The transient analysis of F-policy retrial queue has been carried out using Runge–Kutta method of fourth order. The server is unreliable and may break down while providing service to the customers. The failed server is sent to the repair facility where after required setup time, the repair is done as per pre-specified rule known as ‘threshold recovery policy’ for the repair. The cost optimization of the system has been carried out and threshold parameters are computed.

Analysis of M[X]/G/1 retrial queue with two phase service under Bernoulli vacation schedule and random breakdown

Abstract: This paper investigates the steady state behaviour of an M[x]/G/1 retrial queue with two phases of service under Bernoulli vacation schedule and breakdown. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters the service immediately while the rest join the orbit. After completion of each two phases of service, the server either goes for a vacation with probability p or may wait for serving the next customer with probability (1-p). While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. We construct the mathematical model and derive the probability generating functions of number of customers in the system when it is idle, busy, on vacation and under repair. Some system performances are obtained.

An Unreliable Server Retrial Queue with Two Phases of Service and General Retrial Times Under Bernoulli Vacation Schedule

Abstract: This paper deals with the steady state behavior of an M/G/1 retrial queue with two successive phases of service and general retrial times under Bernoulli vacation schedule for an unreliable server. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. The primary customers finding the server busy, down, or on vacation are queued in the orbit in accordance with the FCFS (first come, first served) retrial policy. After the completion of the second phase of service, the server either goes for a vacation of random length with probability p or serves the next unit, if any, with probability (1 – p). For this model, we first obtain the condition under which the system is stable. Then, we derive the system size distribution at a departure epoch and the probability generating function of the joint distributions of the server state and orbit size, and prove the decomposition property. We also provide a reliabili...

Analysis of a Multiserver Queueing-Inventory System

Abstract: We attempt to derive the steady-state distribution of the queueing-inventory system with positive service time. First we analyze the case of servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair and the corresponding expected minimum cost are computed. As in the case of retrial queue with , we conjecture that for , queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutive to transitions of the inventory level (i.e., the first return time to ) is computed. We also obtain several system performance measures.

The multi-server retrial system with Bernoulli feedback and starting failures

Abstract: In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probability p or to join the retrial orbit again for another service with complementary probability 1−p. We analyse this queueing system as a quasi-birth–death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model.

Steady state analysis of batch arrival feedback retrial queue with two phases of service, negative customers, Bernoulli vacation and server breakdown

Abstract: In steady state, a batch arrival feedback retrial queue with negative customers has been discussed. Any arriving batch of positive customers finds the server is free, one of the customers from the batch enters into the service area and the rest of them join into the orbit/retrial group, where the server provides two essential phases of service to each positive customer. The negative customer, arriving during the service time of a positive customer, will remove the positive customer in-service and the interrupted positive customer either enters into the orbit with probability θ or leaves the system with probability 1-θ. The busy server may breakdown at any instant and the service channel will fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Numerical illustrations are analysed to see the effect of system parameters.

Analysis of a single server batch arrival retrial queueing system with modified vacations and N-policy

Abstract: In this paper, a batch arrival single server retrial queue with modified vacations under N -policy is considered. If an arriving batch of customers finds the server busy or on vacation, then the entire batch joins the orbit in order to seek the service again. Otherwise, one customer from the arriving batch receives the service, while the rest joins the orbit. The customers in the orbit will try for service one by one when the server is idle with a classical retrial policy with the retrial rate ‘jv ’, where ‘j ’ is the size of the orbit. At a service completion epoch, if the number of customers in the orbit is zero, then the server leaves for a secondary job (vacation) of random length. At a vacation completion epoch, if the orbit size is at least N , then the server remains in the system to render service for the primary customers or orbital customers. On the other hand, if the number of customers in the orbit is less than ‘N ’ at a vacation completion epoch, the server avails multiple vacations subject to maximum ‘M ’ repeated vacations. After availing ‘M ’ consecutive vacations, the server returns to the system to render service irrespective of the orbit size. The model is studied using supplementary variable technique. For the proposed queueing system, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained. Various performance measures are derived. A cost model for the queueing system is developed. Numerical illustration is provided.

Strategic Behavior in the Single-Server Constant Retrial Queue with Individual Removal

Abstract: We consider the system with no waiting space in which primary customers arrive according to a Poisson process. The arriving customer receives service immediately as the server is idle, otherwise, he will enter a retrial orbit asking for repeated requests. When a negative customer comes, the customer being served is deleted immediately and it also causes the server’s breakdown. Whenever the server breaks down, it is sent for repair immediately. If no customer arrives after every service completion, the server selects the first customer of the orbit to serve. The time required to find the first customer is assumed exponentially distributed with a constant rate referred as constant retrial rate. Based on a natural reward-cost structure, we study the equilibrium behavior of the customers and derive the corresponding Nash equilibrium joining strategies and social net benefit maximization problem with respect to the levels of information available to customers upon arrival.

The Second Order Asymptotic Analysis Under Heavy Load Condition for Retrial Queueing System MMPP/M/1

Abstract: In the paper, the retrial queueing system of MMPP|M|1 type is studied by means of the second order asymptotic analysis method under heavy load condition. During the investigation, the theorem about the form of the asymptotic characteristic function of the number of calls in the orbit is formulated and proved. The asymptotic distribution is compared with the exact one obtained by means of numerical algorithm. The conclusion about method application area is made.

Solving two-dimensional Markov chain model for call centers

Abstract: Purpose – The purpose of this paper is to develop a novel model of a call center that must treat calls with distinctly different service depending on whether they orginate from VIP or regular customers. VIP calls must be responded to immediately but regular calls can be routed to a retrial queue if the operators are busy. Design/methodology/approach – This study’s proposed model can easily reveal the optimal arrangement of operators while minimizing computational time and without losing any precision of the performance measure when dealing with a call center with more operators. Findings – Based on the results of the comparison between the exact method and the proposed approximation method, the approach shows that the larger the number of operators or inbound calls, the smaller the error between the two methods. Originality/value – This investigation presents a computational method and management cost function intended to identify the optimal number of operators for a call center. Because of computational...

Analysis of Customer-Induced Interruption and Retrial of Interrupted Customers

Abstract: SYNOPTIC ABSTRACTWe consider a single-server retrial queue with infinite capacity of the primary buffer and finite capacity of the orbit to which customers arrive according to a Poisson process, and the service time follows phase-type distribution. The customer-induced interruption while in service occurs according to a Poisson, process. The self-interrupted customers enter into orbit. Any interrupted customer, finding the orbit full, is considered lost. The interrupted customers retries for service after the interruption is completed. We investigate the behavior of this queueing system. Several performance measures are evaluated. Numerical illustrations of the system behavior are also provided.

Working vacation policy for a discrete-time Geo X /Geo/1 retrial queue

Abstract: Due to rapid growth of today’s technology in fast speeding digital networks and industrial organizations, there is a need to develop a model which proves to be useful in handling designing issues of computer and telecommunication systems and many other related digital systems. To accomplish this, we have made an attempt to provide a remedy for modeling some discrete-time (digital) systems of day-to-day life viz. Broadband Integrated Services Digital Network (BISDN), Asynchronous Transfer Mode (ATM) and related computer communication technologies, wherein the models for continuous-time queues fail. This is due to the fact that the discrete-time systems are more appropriate than their continuous-time equivalents to model digital systems. In these systems, time is treated as discrete random variable and is measured in fixed size data units such as machine cycle time, bits, bytes, packets, etc. In this study, we analyze a GeoX/Geo/1 retrial queue wherein the service facility may leave for more economical type of vacation schedule, called as working vacation. The inter-arrival-time, retrial time, service time and working vacation time are assumed to be geometric distributed in discrete environment. We have used matrix geometric method to compute various useful performance measures of interest. Further, we obtain joint optimal values of most sensitive parameters such as vacation returning rate (η) and service rate of the server during working vacation (μV) via direct search method based on heuristic approach. Numerical results are also facilitated to depict the performance of the developed model.

Tail asymptotics for the queue size distribution in an m x /g/1 retrial queue

Abstract: We consider an MX/G/1 retrial queue, where the batch size and service time distributions have finite exponential moments. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. Our result generalizes the result of Kim et al. (2007) to the MX/G/1 retrial queue.

A Discrete-Time Geo/G/1 Retrial Queue with Two Different Types of Vacations

Abstract: We analyze a discrete-time 1 retrial queue with two different types of vacations and general retrial times. Two different types of vacation policies are investigated in this model, one of which is nonexhaustive urgent vacation during serving and the other is normal exhaustive vacation. For this model, we give the steady-state analysis for the considered queueing system. Firstly, we obtain the generating functions of the number of customers in our model. Then, we obtain the closed-form expressions of some performance measures and also give a stochastic decomposition result for the system size. Moreover, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided to illustrate the effect of nonexhaustive urgent vacation on some performance characteristics of the system.

Analysis of bulk retrial queue using maximum entropy principle

Abstract: In the present investigation, we study the maximum entropy principle (MEP) for the Mx/G/1 retrial queue with general retrial times and unreliable server. The customers arrive in bulk in Poisson fashion and are served if they find the server idle otherwise join the virtual pool of customers called orbit. By using maximum entropy principle, approximate results for the queue length as well as waiting time are obtained. We construct the maximum entropy function by using several known constraints so as to determine the probability distributions. Moreover, a comparative study has been done between the exact analytic expressions obtained by the supplementary variable technique and approximate results obtained by MEP.

Maximum Entropy Analysis of MX/G/1 Retrial Queue with k-Phases of Heterogeneous Service and Impatient Calls Under Different Vacation Policies

Abstract: SYNOPTIC ABSTRACTThis work studies the steady state behavior of an Mx/G/1 retrial queue with k-phases of heterogeneous service under different vacation policies. If the server is busy or on vacation, an arriving batch of calls either becomes impatient and balks (leaves) the system with some probability or enters in the virtual pool of blocked calls called “orbit.” The server provides service in k-phases to all the calls. After completing the kth phase of essential service of a call, the server has an option to go for lth (l = 1,2,…,M) type of vacation with probability ηl (l = 1,2,…,M) or to continue to serve the next call (if any) with probability η0where . The explicit expressions for the expected number of calls and expected waiting time of the calls in the retrial group are obtained using supplementary variable and generating function techniques. The maximum entropy principle is also employed to derive various system performance characteristics. A comparative analysis between the approximate results an...

Simulation optimisation of total cost in M/G/C retrial queuing systems with geometric loss, feedback and linear retrial policy

Abstract: This paper presents an integrated computer simulation model for optimisation of total cost in an M/G/C retrial queue system with geometric loss, feedback and linear retrial policy. First, the computer simulation model is used to verify and validate an M/G/C retrial queue system. Second, certain important performance measures are obtained under steady–state condition. Finally, the simulation model is used to identify the preferred alternatives to optimise the current system. Significantly, it should be noted that there is no mathematical model to analyse an M/G/C retrial queue system. In this study, an integrated approach is used to develop the simulation model of the M/G/C retrial queue system. The objective of simulation model is to minimise the total cost and find the optimum solution. Also, a case study provided in which incoming calls of a clinic are modelled and analysed, because the most obvious retrial queue application lies in phone calls. This is the first study that utilises an integrated simulation approach for optimisation of total cost in an M/G/C queuing system with geometric loss and linear retrial policy.

On the performance of the m1,m2/g1,g2/1 retrial queue with pre-emptive resume policy

Abstract: Priority mechanism is an invaluable scheduling method that allows customers to receive different quality of service. Service priority is clearly today a main feature of the operation of any manufacturing system. We are interested by an M 1 ,M 2 /G 1 ,G 2 /1 priority retrial queue with pre-emptive resume policy. For model in question, we discuss the problem of ergodicity and, by using the method of supplementary variables, find the partial generating functions of the steady state system state distribution. Moreover, some pertinent performance measures are obtained and numerical study is also performed.

Performance Analysis of an Unreliable $M/G/1$ Retrial Queue with Two-way Communication

Abstract: Efficient use of call center operators through technological innovations more often come at the expense of added operation management issues. In this paper, the stationary characteristics of an $M/G/1$ retrial queue is investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. The incoming calls arriving at the server follow a Poisson arrival process, while outgoing calls are made in an exponentially distributed time. On finding the server unavailable (either busy or temporarily broken down), incoming calls intrinsically join the virtual orbit from which they re-attempt for service at exponentially distributed time intervals. The system stability condition along with probability generating functions for the joint queue length distribution of the number of calls in the orbit and the state of the server are derived and evaluated numerically in the context of mean system size, server availability, failure frequency and orbit waiting time.

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 $\mu_i$, an type $i$ customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali~\cite{ANY2014} 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 (QBD) 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 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.

Discrete-time Retrial Queue with Bernoulli Vacation, Preemptive Resume and Feedback Customers

Abstract: Purpose: We consider a discrete-time Geo/G/1 retrial queue where the retrial time follows a general distribution, the server subject to Bernoulli vacation policy and the customer has preemptive resume priority, Bernoulli feedback strategy. The main purpose of this paper is to derive the generating functions of the stationary distribution of the system state, the orbit size and some important performance measures. Design/methodology: Using probability generating function technique, some valuable and interesting performance measures of the system are obtained. We also investigate two stochastic decomposition laws and present some numerical results. Findings: We obtain the probability generating functions of the system state distribution as well as those of the orbit size and the system size distributions. We also obtain some analytical expressions for various performance measures such as idle and busy probabilities, mean orbit and system sizes. Originality/value: The analysis of discrete-time retrial queues with Bernoulli vacation, preemptive resume and feedback customers is interesting and to the best of our knowledge, no other scientific journal paper has dealt with this question. This fact gives the reason why efforts should be taken to plug this gap.

Optimal Analysis for M/G/1 Retrial Queue with Two-Way Communication

Abstract: We consider an \(M/G/1\) retrial queue with two types of calls: incoming calls (regular one’s) and outgoing calls (which are made when the server is free). A blocked incoming call joins the orbit and retries for service after some random time while an outgoing call is made by the server after some random idle time. We assume that incoming and outgoing calls have random amount of works which are processed by the server at two distinct speeds. This assumption is suitable for evaluating the power consumption that depends on the speed of the server. We obtain the joint probability distribution of the server state and the number of requests in the orbit in terms of Laplace and \(z\)- transforms. From these transforms, we obtain some performance metrics of interest such as the probability that the server is idle or busy by an incoming (outgoing) call and the mean number of requests in orbit. We propose two optimization problems to find the optimal outgoing call rate and service speeds.

On a retrial queueing model with single/batch service and search of customers from the orbit

Abstract: A single server retrial queueing system, where customers arrive according to a batch Poisson process and are served either in single or as a batch, is considered here. An arriving batch, finding the server busy, enters an orbit. Otherwise, one customer, a few customers, or all customers from the arriving batch, depending on if the batch size exceeds a threshold value or not, enter service immediately, while the rest join the orbit. Customers from the orbit try to reach the server subsequently with the inter-retrial times, exponentially distributed. Additionally, at each service completion epoch, one of the two types of search mechanisms say, type I and type II search, to bring the orbital customers to service, is switched on—type I search when the orbit size is less than the threshold value and type II search otherwise. This means that, while the server is idle, a competition takes place among primary customers, customers who come by retrial and by one of the two types of search as the case may be. A type I search selects a single customer whereas a type II search takes a batch of customers from the orbit. In the case of primary customers and those who come by type II search, maximum size of the batch taken into service is restricted to a pre-assigned value. Both single and batch service are assumed to be arbitrarily distributed with different distributions, which are independent of each other. Steady state analysis is performed. Some important system descriptors are computed algorithmically and numerical illustrations are provided.

Analysis of a retrial queue with multiple vacations and state dependent arrivals

Abstract: This paper examines an M/G/1 retrial queueing system with multiple vacations and different arrival rates. Whenever the system is empty, the server immediately takes a vacation. At a vacation completion epoch, if the number of customers in the orbit is at least one the server remains in the system to activate service, otherwise the server avails multiple vacations until at least one customer is recorded in the orbit. The primary arrival rate is λ 1 when the server in idle and the primary arrival rate is λ 2 when the server is busy or on vacation (λ 1 > λ 2 ). The steady state queue size distribution of number of customers in the retrial group, expected number of customers in the retrial group and expected number of customers in the system are obtained. Some special cases are also discussed. Numerical illustrations are also provided.