emand in er- langs is the sum over a large subscriber base of the small probabilities of making a call times the durations of those calls. If each of S subscribers j = 1,2, S has a proba- bility Yj of making a call of average duration dj during a time period t, then the num- ber of erlangs is in Eq. (23.1): (23.1) For our discussion on telephone system capacity, we can ignore the fact that some sub- scribers are more inclined to use the telephone than others1 and give every one of the S users the same probability Y of making a call of duration d during time period t. This gives us Eq. (23.2): $ = svd (23.2) The deep and magnificent mathematical insight is that once s, the number of subscribers, is large and Y, the probability of a subscriber making a call, is small, the behavior of the telephone system is completely determined by the erlang measure . The individual component values of s, Y, and t lose their importance once is known. 23.1.2 The Poisson distribution Let us examine a state-space diagram of the telephone system. Let the state of the tele- phone system at time t be the number of calls j in the system at time t. From any given state, two things could happen: 1. Somebody starts a new call. 2. Somebody ends a call.² Let l be the rate of users placing calls, the average number of new calls per unit time. And let be the rate of calls ending once they are started. The state of the telephone system is the number of calls j. We turn the system on with the state j set to 0 and wait for a call to start and to change the state j from 0 to 1. Then another call could come along and change the state j from 1 to 2 or the first call could end and set the state j back to 0. Never mind that the loquacious telephone subscribers are the loud talkers in the room next to you. In this discussion, we can ignore them too. 2This is not possible, of course, from the zero-calls state. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All righ