Variance of Load Measurements in Markovian Service Systems

Analysis of the stochastic behavior of traffic measurements is of considerable practical relevance, as it provides means for appraising field data as well as guidelines for selecting performance standards. Load measurements play a central role in this effort, and determination of their accuracy is therefore of particular interest. The present investigation yields an answer to this problem for a broad class of teletraffic models. Whenever statistical equilibrium prevails (and it is assumed to throughout this paper), the load carried by a service system is the average amount of server usage per unit of time or, equivalently, the average number of busy servers at an arbitrary instant. In telephony, an estimate of this parameter is often obtained by "switch-counting." 1 This statistic, which is determined by recording the number of busy servers at regular intervals and then by taking the arithmetic mean of these discrete observations, is an unbiased estimate of the carried load. 1277 The variance of this measurement, called hereafter the switch-count load to distinguish it from the estimate obtained by continuous observation, was first determined approximately by Palm 2 and Hayward 1 in the case of an infinite server group with Poisson input and exponential holding times. This result was later extended by BeneS,3 who obtained the exact variance of the switch-count load for groups of finite sizes without waiting positions (loss systems). A further generalization to loss systems with recurrent input and exponential service is due to Neal and Ivuczura.