) as zero net probability flow between each pair of adjacent nodes. XP;-1(t) = ju.P;(t) (23.4) This gives us the infinite set of equations (Eq. 23.5) for the fixed-probability sequence (23.5) P = 1 Figure 23.1 State diagram for counting calls. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Traffic Engineering for Voice and Data Capacity and Quality Principles This sequence of probabilities comes from the single value a = Nu, and it is called the Poisson distribution. The solution to Eq. (23.5) is Eq. (23.6)3: P,=e-a (23.6) The Poisson distribution of number of channels used depends only on a = Nu, the total likelihood of the entire user population to be using the telephone. As long as the popula- tion is large and no individual makes up a significant fraction of the total telephone us- age, the distribution on channels used will follow this Poisson distribution with mean a. 23.1.3 Blocking probabilities Real telephone systems have a limited number of available channels k. When all k channels are in use, new call attempts are refused, and we say the telephone system blocked. The upper bound of k channels can be modeled in two reasonable ways. The blocked callers can continue to try and retry, still in the system but not using any chan- nels, or the blocked callers can go away and not retry their calls. In the retry case, the distribution for j > k depends on the behavior of blocked callers. If the callers keep trying and retrying and never give up, then our system becomes an M/M/c queue, as described in Sec. 23.3.1. If they coincidentally just happen to give up at the same rate as they hang up when they are talking on the telephone, then we can represent the system as a full Poisson distribution with every state j > k as a blocked state. In this case, Eq. (23.7) says that the probability of a call being blocked is the probability that the Po