On Trunks with Negative Exponential Holding Times Serving a Renewal Process

We shall study a mathematical model for the random behavior of the occupancy of trunk groups. The principal results are complete descriptions (in principle) of (a) the variations of the traffic in time, (b) the equilibrium probabilities and (c), the covariance function of the traffic found by arriving customers. These mathematical results have practical application in engineering trunk groups to have a given probability of loss, and in estimating the sampling error incurred in certain ways of measuring traffic. A "trunk group" is a set of transmission channels (trunks) between central offices. The trunks in a group are often equivalent in the sense that a call handled 011 one idle trunk could as well have been assigned another. A "holding time" of a trunk is a length of time during which it is continuously unavailable because it is being seized and used as a talk211 212 T H E DELL SYSTEM T E C H N I C A L J O U R N A L , JANUARY 1 9 5 1 ) ing path. By "interarrival times" we mean the time intervals elapsing between successive epochs at which attempts are made to place a call on the trunk group. With these definitions in mind, the theoretical model we use to describe the trunk group involves four assumptions: i. The holding times of trunks are independent random quantities having a negative exponential distribution, with mean value, (7 is the hang-up rate). This means that if a trunk is in use at time x, the chance that it is still in use at (x + dx) is 1 -- ydx -- o(dx), o(dx) denoting a quantity of order smaller than dx, irrespective of how long the trunk has been in use.