Fluctuation of Telephone Traffic

T h e aim of this p a p e r is to s t u d y t h e average fluctuations of telephone traffic in an exchange, by m e a n s of a simple m a t h e m a t i c a l model to which we a p p l y concepts used in t h e t h e o r y of stochastic processes a n d in t h e analysis of noise. T h e m a t h e m a t i c a l model we use is based on t h e following a s s u m p t i o n s : (1) requests for telephone service arise individually a n d collectively at r a n d o m at an average r a t e of a per second; (2) t h e holding-times of calls are m u t u a l l y i n d e p e n d e n t r a n d o m variables h a v i n g t h e common probability d e n s i t y function h(u); a n d (3) t h e capacity of t h e exchange is effectively unlimited, a n d no call is blocked or delayed by lack of e q u i p m e n t . T h i s telephone exchange model h a s been described b y J . Riordan.6 As a measure of traffic, it is n a t u r a l to use t h e n u m b e r of calls in progress in the exchange. We are t h u s led to consider a r a n d o m step-function of time N(t), defined as t h e n u m b e r of calls in progress at time t. N(t) fluctuates a b o u t an average in a m a n n e r d e p e n d i n g on t h e calling-rate, a, a n d t h e holding-time density, h(u). II P R O O F OF R I O R D A N ' S FORMULA FOR T R A N S I T I O N P R O B A B I L I T I E S