Analysis of Switching Networks

The application of probability theory to telephone traffic problems owes its origin to the pioneering work of A. K. Erlang, T. C. Fry, E. C. Molina and others. 1 ' 2 ' 3 Since then much has been written on these and other related problems. On the other hand, except for several recent papers on the subject, 4 , 5 , 6 , 7 the literature on the application of probability t h e o r y to large size switching s y s t e m s h a s been c o m p a r a t i v e l y meager, mainly because the sheer size and complexity of these systems tend to render exact analysis unmanageable. To fix ideas, let us peer into a telephone central office and point our attention at a single link (or crosspoint) in the system. As time progresses the link becomes busy and idle in some fashion and gives rise to a sequence of pairs of observations: (to, busy), (/i , change to idle), (h , change to busy), · · · Let xt be a function such that xt is 1 if the link is idle and is 0 if the link is busy, then the sequence of pairs of observations corresponds to the behavior of xt as t changes. A plot of the values of xt versus t would look perhaps as shown in Fig. 1.1. The function xt (or the sequence of pairs of observations) is one of a large family of possible functions for the link. Since there are in general several thousand such links in a telephone central office, a complete description would involve several thousand families of functions xt. We may add that the situation is made somewhat worse by the fact that these links are not independent of each other, for example, the establishment of a telephone conversation involves in general not one but several links in series.