(A) busy period analysis for the M/M/c/K queueing system

Abstract

A busy period analysis plays a seminal role in efficient planning and management of the system. When analysing busy periods, the length of busy period and the number of customers served is a matter of concern. As for a single server queueing system, a busy period is easily defined as the duration that a server is in service. In contrast, for a multiple server queueing system, the definition of a busy period can be varied depend upon authors. Busy periods for a multiple server queueing system is categorised Iargely into two concepts by researchers. Those are the full busy period and the partial busy period. It is widely known that the full busy period analysis is relatively easier than the partial busy period analysis. And results which deal with finite queueing system are very few. Moreover, results that present the moments of the partial busy period of finite queueing system do not exist as far as we know. Thus we analyse the partial busy period of finite M/M/c/K queueing system. In this paper, we derived the joint transform of the length of a busy period and the number of customers served during it. Using the joint transform, we present formulae which give the first and second moments of the partial busy period. As for simple cases, explicit formulae are presented. Presented results are original and this fact makes our paper significant. Finally, verification is conducted and a relation between the length of busy period and the number of customers served is discussed briefly.