ON THE TIME-DEPENDENT OCCUPANCY DISTRIBUTION OF THE G/G/1 QUEUING SYSTEM

This article offers an approach for studying the time-dependent occupancy distribution for a modest generalization of the GI/G/1 queuing system in which interarrival times and service times, although mutually independent, are not necessarily identically distributed.We develop and explore an analytical model leading to a computational approach that gives tight bounds on the occupancy distribution. Although there is no general closed-form characterization of probability law dynamics for occupancy in the GI/G/1 queue, our results offer what might be termed “near-closed-form” in that accurate plots of the transient occupancy distribution can be constructed with an insignificant computational burden.We believe that our results are unique; we are unaware of any alternative analytical approach leading to a numerical characterization of the time-dependent occupancy distribution for the G/G/1 queuing systems considered here. Our analyses employ a marked point process that converges to the occupancy process at any fixed time t; it is shown that this process forms a Markov chain from which the transient occupancy law is available. We verify our analytical approach via comparison with the well-known closed-form expressions for time-dependent occupancy distribution of theM/M/1 queue. Additionally,we suggest the viability of our approach, as a computational means of obtaining the time-dependent occupancy distribution, through straightforward application to a Gamma[x] /Weibull/1 queuing system having batch arrivals and batch job services.