Delay Distributions for One Line with Poisson Input, General Holding Times, and Various Orders of Service

Let us suppose t h a t in the time interval ( 0 , » ) calls appear before a single t r u n k line at times n , t 2 , · · · , r,, , · · · where the interarrival times Tn -- T n -i (w = 1, 2, · · · ; ro = 0) are identically distributed, mutually independent random variables with the distribution function (1) t h a t is, the input is a Poisson process of density X. If an incoming call finds the line free, a connection is realized instantaneously. If the line is busy, the call is delayed and waits for service as long as necessary (no defection). T h e holding times are identically distributed, mutually independent, positive random variables with distribution function H(x) and independent of the input process. Such a service system can be characterized by the symbol [/ ,, (.r),7 : /(.r),l] provided t h a t the order of service is specified. In this paper three orders of service are considered: (i) order of arrival (first come-first served), (ii) random order (every waiting call, independently of the others, and of its past delay, has the same probability of being chosen for service), and (Hi) reverse order of arrival (last come-first served). 487