STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 945 



II DESCRIPTION OF THE MATHEMATICAL MODEL 



Throughout the rest of the paper we follow a simplified form of the 

 iiotational conventions of J. Riordan's paper" wherever possible. A sum- 

 mary of notations is given in Section IV. The model we study has the 

 following properties: 



(i) Demands for service arise individually and collectively at random 

 at the rate of a calls per second. Thus the chance of one or more demands 

 ill a small time-interval A^ is 



aAt + o{At), 



where o{At) denotes a quantity of order smaller than A^ The chance of 

 more than one demand in At is of order smaller than A^ It can be shown 

 (Feller,' p. 864 et seq.) that this description of the demand is equivalent 

 to saying that the intervals between successive demands for service are 

 all independent, with the negative exponential distribution 



1 —at 



I — e . 



This again is equivalent to saying that the call arrivals form a Poisson 

 process;" i.e., that for any time interval, t, the probability that exactly ?i 

 demands are registered in / is 



—at / ,\n 



e {at) 

 nl 



Thus the number of demands in t has a Poisson distribution with mean 

 at. 



(ii) The holding-times of distinct conversations are independent vari- 

 ates having the negative exponential distribution 



1 - r^"\ 



where y is the reciprocal of the mean holding-time h. This description of 

 the holding-time distribution is the same as saying that the probability 

 that a conversation, which is in progress, ends during a small time- 

 interval A^ is 



yAt + o{At), 



without regard to the length of time that the conversation has lasted 

 Feller, p. 375). 



(iii) The model contains an infinite number of trunks. Thus, at no time 

 will there be insufficient central office equipment to handle a demand 

 for service, and no provision need be made for dealing with demands that 

 cannot be satisfied. 



