1132 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



For infinite sources, and calls arriving individually and collectively at 

 random with average density a, the well-known formula for the probability 

 that exactly k calls arrive in time interval / is the Poisson 



TT.W = e-'^Xatf/kl (4) 



Then, if P,y(/; k) is the conditional probabiUty of transition from i to j 

 when k calls arrive in time /, 



Piiit) = IlPij(t;kUk{t) ' (5) 



fc=0 



Consider Pij{t; 0), that is the (conditional) transition probabilities when 

 no calls arrive. Let the probabiUty that a call lasts at least t be/(/), so that 

 the average holding time h is given by 



h = f u[-f(u)] du = f f{u) du (6) 



Jo Jq 



The i calls initially in process are independent of each other. Select one of 

 them and suppose the time from its arrival (its age) is h . Then the proba- 

 biUty that it will also exist / units later is the conditional proba- 

 biUty /(/ + ti)/f{ti). Since in equilibrium conditions all moments of arrival 

 have equal probability, the corresponding probability for an arbitrary call is 



git) = j[ f{t + h) dh ^ jf m dt=^-j^ fiu) du (7) 



Hence the transitional probability Pij(t', 0) is the binomial expression 



i>,,(/;0) = (j)g'(l-g)-' (8) 



and its generating function is 



Piit, x; 0) = ZPiiii; 0)x^ = [1 + (,: - l)gY (9) 



In (8) and (9), for brevity, the argument of g is omitted. 



Now, suppose one call arrives in interval /. The moment of arrival is 

 uniformly distributed in /; that is, if Ui is the moment of arrival, 



Pr(u < Ui < u -\- du) = du/i 



and the probability that a caU arriving at an arbitrary moment wiU be in 

 existence at time t is, say, 



QU) = ('f{t-u)^ = \ /"/(«) d« = 7 (1 - ««)) 



Jo t I Jo t 



(10) 



