Traffic measurement biases induced by partial sampling

Under equilibrium conditions, the sample average of the delays encountered by all the calls submitted during a given time interval is an unbiased estimate of the mean of the delay distribution. If some of the delays are not observed, the resulting sample average need no longer be an unbiased estimator of the corresponding population mean. This is the case when, for instance, only a limited number of delays can be timed simultaneously. The purpose of this paper is to investigate these biases for queuing systems when only one clock is available and thus one delay only can be measured at a time. It is shown that, regardless of the order of service, the expected value of the observed average delays is always smaller than the mean waiting time for all calls. Although the average delay on all calls is independent of the order of service, the measurement biases resulting when only one delay can be measured at once depend on the queue discipline. In particular, we shall show that the average delay for all calls is always larger than the average delay of the observed calls even if these calls are always served last (observed-call served-last).