Traffic Measurement Biases Induced by Partial Sampling

The following remarks due to J. F. C. Kingman appear in the Proceedings of the Symposium on Congestion Theory held at the University of North Carolina in 1964 (Ref. 1, pp. 314-315): "To illustrate the pitfalls of inference from congestion systems, let me tell a (more or less true) story. It was desired to estimate the mean waiting time in a particular queuing system, and for technical reasons, only one customer could be timed at once. Thus the waiting time a>i of a customer was measured. When he entered service, the next customer to arrive was observed and his waiting time o>2 was noted. This procedure continued, the waiting times w3, w4) · · · being measured, and, for 1375 1376 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1973 large n, W-1(cd 1 + C 2 + ' * · + Oin) 0 was used as an estimate of the waiting time. It is, however, strongly biased and inconsistent, because of the selection of the customers to be observed. The mean waiting time is overestimated by a factor which becomes arbitrarily large as the traffic intensity approaches one." We stress that, according to this sampling procedure, a customer is observed if and only if it arrives when the clock is free. Another instance of biases induced by the measurement procedure is reported by Oberer and Riesz.2 These authors have investigated the possibility of estimating blocking probabilities in telephone networks by means of test calls generated by a single source repeatedly calling a dedicated number. Their study shows that the proportion of blocked test-calls does not yield a suitable estimate of the grade of service as it is markedly biased downwards.