95G 



THE BELL SYSTEM TECHNICAL JOUKXAL, JULY 1957 



If A''(0 = n, then the probabiHty is [1 — ytiAt — aAt — o{At)] that 

 there is neither a request for service nor a hang-up during A^ following 

 t, and that Z{t + AO = Z{t) + 7iAt. Therefore the conditional proba- 

 bility that N{t + AO = fi and Z{t + At) ^ z, given that no changes 

 occurred in At, is 



Fn{z - nAt, t). 



For N(t) = (n -f 1), the probability is 7(/i + 1)A^ + o(At) that one 

 conversation will end during At following /. The increment to Z{t) 

 during A^ will depend on the length x of the interval from t to the point 

 within A^ at which the conversation ended. The increment has magni- 

 tude (n -\- l)x -f n{At — x) = x + nAt, as can be verified from Fig. 3, 

 in which the shaded area is the increment. Since x is distributed uniformly 

 between and A^ the increment x + nAt is distributed uniformly be- 

 tween 71 At and (n + 1)A^ Therefore the conditional probability that 

 A^(^ + At) = n and Z{t + At) ^ z, given that one conversation ended 

 in At, is 



— / Fn+i(z — u, t) du. 



At J,iAt 



By a similar argument it can be shown that the probability that one 

 reciuest for service arrives in At is a At -f o(At), and that the conditional 

 probability that A^(^ -f At) = n and Z(t -f At) ^ z, given that one 

 request arrived during At, is 



1 It nAt 

 At J{ n-l)At 



Fn-i{z - u,t) du. 



Define Fniz, t) to be identically for negative n. Adding up the probabil- 



n + 2 - 



t + At 



Fig. 3 — Increment to Z in At. 



