Analysis of queueing models under the D-policy

Abstract

In this dissertation, we analyze the $D$-policy queueing models. $D$-policy queueing models are operated as follows: the server is turned off each time the system becomes empty and is turned on only when the sum of the service times of the waiting customers who are present in the system exceeds a predetermined value $D$. Under the $D$-policy, the service times of the customers who arrive during an idle period are dependent on $D$. This is the reason why the $D$-policy is difficult to analyze. Application can be found, for example, in internet telephone systems, in dam model and in continuous production systems. We consider the two queueing models of the $D$-policy. First, we consider the M/G/1 queueing models with $D$-policy. As performance measures, we obtain the distributions of unfinished work, queue length and queue waiting time. For the unfinished work, we show that the unfinished work is decomposed into two random variables: one is the unfinished work of the ordinary M/G/1 queue and the other depends on the $D$-policy. For the queue length and queue waiting time, we obtain the distributions by conditioning on the number of cumtomers who arrive during an idle period. Using these performance measures, we do the cost analysis and show that the optimal $D$ which minimizes the average cost per unit time exists. Second, we consider the M/G/1 queueing models with $D$-policy and multiple vacations. As a performance measure, we obtain the distribution of unfinished work. We show that the unfinished work is decomposed into three randm variables: one is the unfinished work of the ordinary M/G/1 queue, another depends on the $D$-policy, and the last depends on the vacations. Using this performance measure we do the cost analysis and show that the optimal $D$ which minimizes the average cost per unit time exists.