A Study Of Some Discrete-time Queueing Systems

Discrete-time queueing systems have been received increasing attention during the last decades due to their wide applications in various fields, such as computer and communication systems, telecommunication systems, production management, etc. This dissertation is devoted to studying some discrete-time queueing systems and concerns the following two parts:discrete-time retrial queues and working vacation queues. The thesis is organized as follows.Firstly, Chapter 2 treats a discrete-time single-server retrial queue with two classes of customers, where the blocked class-1 customers leave the system forever whereas the blocked class-2 customers leave the service area and enter the orbit and try their luck again some time later. Each class-2 customer after service either immediately returns to the orbit for another service with probability?(0??<1) or leaves the system forever with probability 1-?. Firstly, we study the Markov chain underlying the queueing system. The generating functions of the number of customers in the orbit and in the system are obtained along with the marginal distributions of the orbit size when the server is idle, busy with class-1 or busy with class-2. Some performance measures of the system in steady-state are also derived. Secondly, we investigate the relationship between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.Chapter 3 considers a repairable Geo/G/l 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?(0??< 1) or leave the system forever with probability 1-?. 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, numerical examples show the impact of the parameters on the performance measures of the system.Chapter 4 treats a discrete-time batch arrival queue with single working vacation. The main purpose of this chapter is to present a performance analysis of this system by using the supplementary variable technique. For this purpose, we first analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. Next, we present the stationary distributions of the system length as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method, we give the probability generating function (PGF) of the number of customers at the beginning of a busy period and give a stochastic decomposition formulae for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.Chapter 5 considers a discrete-time batch arrival queue with geometrically working vacations and vacation interruption. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive the PGF of the stationary system length at the departure epochs based on the embedded Markov chain technique. Next, we present the stationary distributions of the system lengths as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method, we give a distribution for the number of the customers at the beginning of a busy period and give a stochastic decomposition formula for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.In chapter 6, we study a discrete-time finite buffer batch arrival queue with multiple geometrically working vacations and vacation interruption. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed by using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present PGFs of different customers'actual waiting-time distributions in the system and some performance measures are carried out.