Queueing system with retrials and vacations

Abstract

The main purpose of the thesis is to find the distributions of queue size and waiting time of the retrial queue and queues with vacations. These queueing systems have applications to applied communication networks, computer and production systems. In chapter 2, we consider an M/G/1 retrial queueing systems with two types of calls. In the case that arriving calls are blocked due to the channel being busy, the outgoing calls are queued in priority group whereas the incoming calls enter the retrial group in order to seek service again after a random amount of time. In this chapter we derive the Laplace-Stieltjes transform of the virtual waiting time and the generating function for the number of retrials of an incoming call. When the arrival rate of outgoing calls is zero, it is shown that our result is consistent with the known result for a retrial queueing system with only one type of calls. In chapter 3, we consider a $G/M^{a,b}/1$ queue with multiple vacation discipline. Calls are served in batches according to the following bulk service rule in which at least ``a`` calls are needed to start a service and maximum capacity of the server is ``b`` at a time. When the server either finishes a service or returns from a vacation, if he finds less than ``a`` calls in the system, he takes a vacation with exponential distribution. When the server either finishes a service or returns from a vacation, if he finds more than ``a`` calls in the system, he serves a bulk of maximum of ``b`` calls at a time. With the supplementary variable method, we explicitly obtain the queue length probabilities at arrival time points and arbitrary time points simultaneously. The results for our model in the special case of a = b = 1 coincide with known results for G/M/1 queue with multiple vacation obtained by imbedded Markov chain method. As the rate of exponential vacation time tends to infinite, the queue size distribution for queueing system with vacation will approach to that for the q...