Asymptotic expansions of the sojourn time distribution functions of jobs in closed, product-form queuing networks

Abstract

Striking progress has been made recently in obtaining expressions for the sojourn time distribution function (STDF) of a job at a c -server, first-come, first-serve (FCFS) center in a closed, product-form queuing network. These results have more recently been extended, and expressions have been obtained for the joint distribution function (DF) of the sojourn times of a job at a sequence of single-server, FCFS centers lying on an “overtake-free” path. However, these formulas present considerable computational problems in the case of large, closed queuing networks. In this paper, asymptotic techniques developed by Mitra and McKenna for the calculation of the partition function of large, product-form closed queuing networks are applied to the sojourn time problem. Asymptotic expansions are obtained for the STDF of a job at c -server, FCFS center in closed, product-form queuing networks. Similar expansions are obtained for the joint DF of the sojourn times of a job at a sequence of single server, FCFS centers lying on an “overtake-free” path. In addition, integral expressions are obtained for the STDF of a job at a single server, FCPS center in a closed, product-form queuing network in which all the centers are load independent. These integral expressions also yield useful asymptotic expansions. Finally, integral expressions are also obtained for the joint DF of the sojourn times of a job at the centers of an “overtake-free” path in such a network.