A Discrete-Time Unreliable Geo/G/1

This paper deals with the steady-state behavior of a discrete-time unreliable Geo/G/1 retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system

Single server retrial queueing models.

Most retrial queueing research assumes that each retrial customer has its own orbit, and the retrial customers retry to enter service independently of each other. A small selection of papers assume that the retrial customers themselves form a queue, and only one customer from the retrial queue can attempt to enter at any given time. Retrial queues with exponential retrial times have been extensively studied, but little attention has been paid to retrial queues with general retrial times. In this thesis, we consider four retrial queueing models of the type in which the retrial customers form their own queue. Model I is a type of M/G/1 retrial queue with general retrial times and server subject to breakdowns and repairs. In addition, we allow the customer in service to leave the service position and keep retrying for service until the server has been repaired. After repair, the server is not allowed to begin service on other customers until the current customer (in service) returns from its temporary absence. We say that the server is in reserved mode, when the current customer is absent and the server has already been repaired. We define the server to be blocked if the server is busy, under repair or in reserved mode. In Model II, we consider a single unreliable server retrial queue with general retrial times and balking customers. If an arriving primary customer finds the server blocked, the customer either enters a retrial queue with probability p or leaves the system with probability 1 - p. An unsuccessful arriving customer from the retrial queue either returns to its position at the head of the retrial queue with probability q or leaves the system with the probability 1 - q. If the server fails, the customer in service either remains in service with probability r or enters a retrial service orbit with probability 1 - r and keeps returning until the server is repaired. We give a formal description for these two retrial queueing models, with examples. The stability of the system is analyzed by using an embedded Markov chain. We get a necessary and sufficient condition for the ergodicity of the embedded Markov chain. By employing the method of supplementary variables, we describe the state of the system at each point in time. A system of partial differential equations related to the models is derived from a stochastic analysis of the model. The steady state distribution of the system is obtained by means of probability generating functions. In steady state, some performance measures of the system are reported, the distribution of some important performance characteristics in the waiting process are investigated, and the busy period is discussed. In addition, some numerical results are given. Model III consists of a single-server retrial queue with two primary sources and both a retrial queue and retrial orbits. Some results are obtained using matrix analytic methods. Also simulation results are obtained. Model IV consists of a single server system in which the retrial customers form a queue. The service times are discrete. A stability condition and performance measures are presented.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2005 .W87. Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3883. Thesis (Ph.D.)--University of Windsor (Canada), 2006

Performance evaluation of a discrete-time Geo[X]/G/1 retrial queue with general retrial times

AbstractWe consider a discrete-time Geo[X]/G/1 retrial queue with general retrial times. The system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions. These generating functions yield exact expressions for different performance measures. The present model is proved to have a stochastic decomposition law. Hence, a measure of the proximity between the distributions of the system size in the present model and the corresponding one without retrials is derived. A set of numerical results is presented with a focus on the effect of batch arrivals and general retrial times on the system performance. It appears that it is the mean batch size (and not the batch size distribution) that has the main effect on the system performance. Moreover, increasing the mean batch size is shown to have a noticeable effect on the size of the stability region. Finally, geometric retrial times are shown to have an overall better performance compared with two other distributions

A Discrete-Time G

This paper is concerned with a discrete-time Geo/G/1 retrial queueing model with J vacations and two types of breakdowns. If the orbit is empty, the server takes at most J vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system

A Discrete-Time Unreliable Geo/G/1 Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

This paper deals with the steady-state behavior of a discrete-time unreliable / /1 retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system

Optimal Pricing Strategies and Customers’ Equilibrium Behavior in an Unobservable M/M/1 Queueing System with Negative Customers and Repair

This work investigates the optimal pricing strategies of a server and the equilibrium behavior of customers in an unobservable M/M/1 queueing system with negative customers and repair. In this work, we consider two pricing schemes. The first is termed the ex-post payment scheme, where the server charges a price that is proportional to the time spent by a customer in the system. The second scheme is the ex-ante payment scheme, where the server charges a flat rate for all services. Based on the reward-cost structure, the server (or system manager) should make optimal pricing decisions in order to maximize its expected profit per time unit in each payment scheme. This study also investigates equilibrium joining/balking behavior under the server’s optimal pricing strategies in the two pricing schemes. We show, given a customer’s equilibrium, that the two pricing schemes are perfectly identical from an economic point of view. Finally, we illustrate the effect of several system parameters on the optimal joining probabilities, the optimal price, and the equilibrium behavior via numerical examples

Mathematical Analysis of Queue with Phase Service: An Overview

We discuss various aspects of phase service queueing models. A large number of models have been developed in the area of queueing theory incorporating the concept of phase service. These phase service queueing models have been investigated for resolving the congestion problems of many day-to-day as well as industrial scenarios. In this survey paper, an attempt has been made to review the work done by the prominent researchers on the phase service queues and their applications in several realistic queueing situations. The methodology used by several researchers for solving various phase service queueing models has also been described. We have classified the related literature based on modeling and methodological concepts. The main objective of present paper is to provide relevant information to the system analysts, managers, and industry people who are interested in using queueing theory to model congestion problems wherein the phase type services are prevalent

Analysis of Priority Queueing System with Working Breakdown,Vacation and Vacation Interruption under Random Environment

study priority queueing system consisting of working breakdown, repair, single vacation and vacation interruption. At a breakdown instant during the busy period, without stop the servie, the server provides service at slower rate for the current customer. After completion of priority service, depending on the environment, the server can choose type I or type II vacation. Both types of vacation are considered as a single vacation. On completion of type I vacation if the server finds thepriority queue is empty, he has the option to take type II vacation. During the type II vacation, when priority customers arrive the server has the option to interrupt the vacation. We use the established norm which is the corresponding steady state results for time dependent probability generating functions are obtained. Along with that, the expected waiting time for theexpected number of customers for both high and low priority queues are computed. Numerical results along with the graphical representations are shown elaborately

Analysis of Two Stage M[X1],M[X2]/G1,G2/1 Retrial G-queue with Discretionary Priority Services, Working Breakdown, Bernoulli Vacation, Preferred and Impatient Units

In this paper, we study M[X1] , M[X2] /G1 ,G2 /1 retrial queueing system with discretionary priority services. There are two stages of service for the ordinary units. During the first stage of service of the ordinary unit, arriving priority units can have an option to interrupt the service, but, in the second stage of service it cannot interrupt. When ordinary units enter the system, they may get the service even if the server is busy with the first stage of service of an ordinary unit or may enter into the orbit or leave the system. Also, the system may breakdown at any point of time when the server is in regular service period. During the breakdown period, the interrupted priority unit will get the fresh service at a slower rate but the ordinary unit can not get the service and the server will go for repair immediately. During the ordinary unit service period, the arrival of negative unit will interrupt the service and it may enter into an orbit or leave the system. After completion of each priority unit’s service, the server goes for a vacation with a certain probability. We allow reneging to happen during repair and vacation periods. Using the supplementary variable technique, the Laplace transforms of time-dependent probabilities of system state are derived. From this, we deduce the steady-state results. Also, the expected number of units in the respective queues and the expected waiting times, are computed. Finally, the numerical results are graphically expressed

Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints