Congestion in Blocking Systems - A Simple Approximation Technique

The study of many important stochastic server systems often leads to models that are extremely difficult to treat via exact analysis. An important example in telephony is the study of a (secondary) trunk group which is offered the superposition of several overflow streams [blocked calls from other (primary) trunk groups]. While the traffic characteristics of this pooled stream are rather complex (the stream is not even renewal), a useful characterization has been via its peakedness, defined as the variance-to-mean ratio of the number of busy 805 servers on an infinite trunk group offered this traffic. This peakedness concept is the heart of the "equivalent random" method introduced by Wilkinson1 as an approximation technique for this trunking problem.f Although peakedness is generally not a complete characterization of traffic, it has been found to be quite useful in many applications besides the analysis of trunk groups offered overflow traffic. Peakedness was used by W. S. Hayward (c. 1959) as the basis for an especially simple but surprisingly accurate approximation to the blocking experienced by the overflow traffic on the secondary trunk group in this trunking problem. If we are given an overflow stream with mean arrival rate X offered to N (exponential) servers with rate /A, then Hayward's approximation for the resulting blocking probability (in telephony terminology, call congestion), Bc, is obtained in the following manner. First compute the peakedness z of the given overflow stream relative to an infinite group of (exponential, rate /i) servers, and then approximate Bc via (H)