Some Properties of the Variance of the Switch-Count Load

When statistical equilibrium prevails, the load carried by a group of servers is defined as the average amount of traffic handled per unit of time. In telephony, this parameter is often evaluated by "switchcounting."2'3 According to this method, the number of servers in use is recorded at regular time intervals; these numbers are then added together and their sum, divided by the number of observations (scans), is an unbiased estimate of the carried load. This measurement is called hereafter the switch-count load to distinguish it from estimates based on continuous observation. The latter are obtained by dividing the aggregated usage of the servers by the length of the measurement interval and can be viewed as limits of switch-count load measurements in which the number of scans tends to infinity while the length of the observation period is kept unchanged. 59 The problem of finding the variance of the switch-count load in loss systems with exponential service times has attracted a good deal of attention. Description of some earlier contributions to this subject can be found in Ref. 1. In a recent paper,1 formulas for the variance of the switch-count load were derived for delay-and-loss systems with state-dependent input rates and exponential service times. (As is customary, the state of the system at some instant, t, is defined as the number of customers who are either being served or are waiting at that time.) More precisely, the assumptions made here are as follows: (i) Calls originate at rate Xn ( > 0 ) whenever the system is in state n.