STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 941 



The demand for telephone traffic is usually made precise by describing 

 a stochastic process which represents the way in which requests for tele- 

 phone service occur in time. A realistic description will take account of 

 the facts that, the demand is not constant, but has daily extremes, and 

 that in small systems, the demand may be materially lessened when 

 many conversations are in progress. Since taking account of the first fact 

 leads to a more complicated model in which our investigations are more 

 difficult, we ignore it, with the proviso that the results we derive are 

 only applicable to systems and observations for which the demand is 

 nearly constant. The second kind of variation in demand becomes insig- 

 nificant as the number of subscribers increases and the traffic remains 

 constant. Hence, we further confine the applicability of our results to 

 systems with large numbers of subscribers, and we assume that the de- 

 mand does not depend on the number of conversations in existence. 



With these assumptions, a mathematically convenient description of 

 the demand is specified by the condition that the time-intervals between 

 requests for service have lengths which are mutually independent posi- 

 tive random variables, with a negative exponential distribution. 



A telephone central office contains two kinds of equipment: control 

 circuits which establish a desired connection, and talking paths over 

 which a conversation takes place. The time that a reciuest for service 

 occupies a unit of equipment, be the unit a control circuit or a talking 

 path, is called the holding-time of the unit. A request for service affects 

 the availability of both kinds of equipment but, except for special cases, 

 the holding-times of talking paths are usually much longer than the 

 holding-times of control units such as markers, connectors, or registers. 

 In view of this disparity, we assume that the only holding-times of con- 

 sequence are the lengths of conversations; i.e., the holding-times of 

 talking paths. We assume also that these lengths are mutually inde- 

 pendent positive random variables, with a negative exponential distribu- 

 tion. 



For the simplest mathematical model of telephone traffic, we may 

 consider the arrangement of switches and transmission lines which con- 

 stitutes a talking path in the physical office to be replaced by an abstract 

 unit called a "trunk". A trunk is then an abstraction of the equipment 

 made unavailable by one conversation, and we may measure the supply 

 of talking paths in the office by the number of trunks in a model. The 

 word "trunk" is also used to mean a transmission line linking two central 

 offices, but as long as we have explained our use of the word there need 

 be no confusion. Often the number of transmission lines leading out of 

 an office is a major limitation on its capacity to carry conversations, 

 and in this case the two uses of the word "trunk" are verv similar. Un- 



