TELEPHONE TRAFFIC TIME AVERAGES 1131 



and /(/) the probability that a call lasts at least /, that is, the distribution 

 function of holding times. The specializations of this, for constant holding 

 time and exponential holding time, appear in section 4. The results for 

 finite source input have a similar character. 



The procedure in obtaining these is as follows. The cumulants are de- 

 termined from the ordinary moments (about the origin) and the latter 

 are determined by the integration of expectations. Thus the first moment, 

 the m^an is determined from 



E\M{T)] = 1; jf '^ mH)] dt = E[Nit)] (2) 



where E(x) is written for the expectation or mean of x. 

 Similarly the second moment is given by 



E[M\T)] = ^[ [ E{N{t)mu)] dt du (3) 



and so on for higher moments. Correlation effects appear in (3) in 

 E[N(t)N(u)] and are included in the development by formulation of transi- 

 tion probabilities, that is, those probabilities determining the traffic flow 

 in time. The transition probability Pjk{t) is defined as the probability of 

 transition in t from j calls in progress (busy trunks) to k calls in progress, 

 and fixes the inter-relatedness of call probabilities at different time epochs. 

 Only for large values of / are these probabiHties independent. 



Hence, the first task is to determine these simple transition probabilities, 

 then those of double and triple transitions, then the expected values of 

 pairs, triples and quadruples of numbers of busy trunks, and finally the 

 moments. 



2. Transition Probabilities 



For exponential holding time, and infinite sources, infinite trunks, these 

 probabilities have already been determined by Conny Palm [5]. Palm's 

 work has been summarized both by Feller [1] and by Jensen [3], and de- 

 scribes the whole process, not merely the equilibrium condition. For the 

 equilibrium condition, a different procedure,* similar to that used by 

 Newland [4] for another purpose, allows the assumption of a more general 

 holding time characteristic. 



* Thanks are due S. O. Rice for suggesting this, as well as for many corrections and 

 improvements. I also have had the advantage of a careful reading of the mss. by E. L. 

 Kaplan. 



