Variance of Load Measurements in Markovian Service Systems

Abstract

The load carried by a queuing system under equilibrium conditions is the average amount of server usage per unit of time. In telephony, this parameter is often evaluated by recording the number of busy servers at regular time intervals; these readings are then cumulated and their sum, after division by the number of observations, is an unbiased estimate of the carried load. The purpose of this paper is to derive exact formulas for the computation of the variance of this measurement in systems with arbitrary input and departure rates. The results obtained here thus apply to a wide class of teletraffic models which includes, in particular, the delay-and-loss systems with finite- or infinite-source inputs, exponential service times, and arbitrary defection rates from the queue. Problems related to computations are also considered, special attention being paid to the reduction of both computer time and storage when the number of states is large.