Analysis of queueing systems requiring resequencing of customers.

Abstract

This dissertation describes queueing-theoretic analysis of shared service systems that require that customers leave the system in the sequence in which they arrived. This requirement makes it necessary to resequence customers before they leave the system. Resequencing adds new complications to the analysis of queueing systems. While waiting time is still important, resequencing results in a new type of "non-working" delay of a customer called the resequencing delay. This dissertation presents primarily analytical and numerical methods to determine the distribution and mean value of resequencing delay, and of total delay. In the simplest models closed form analytical expressions have been obtained, but in more complex models numerical methods have been developed to compute the distribution and mean of resequencing delay, and of total delay. This enables us to study the behavior of resequencing and total delay as system parameters are changed. For several composite server models we present expressions for the distribution and mean of resequencing delay, and of total delay. In particular we consider the M/M/∞ composite server model, the M/H(K)/∞ composite server model, the G/M/∞ composite server model, the M/M/m composite server model, and the G/M/m composite server model. The formulas are interpreted using asymptotic approximation or bounding techniques. For more general composite server models, it is difficult to obtain closed form expressions for resequencing and total delay. We develop numerical methods based on matrix-geometric methods to compute resequencing and total delay. In particular, we develop numerical methods for the computation of the mean resequencing delay, and mean total delay for the M/H₂/m composite server model, and the M/Hypo₂/m composite server model.