The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times

Abstract

The single-channel queuing equations considered in this paper are characterized by a Poisson-distributed arrival rate and a general class of holding-time distributions. General solutions of the equations are obtained for the case in which the traffic intensity i is a continuous function of time and possesses continuous derivatives of all orders. The following particular cases are considered in detail (a) i constant and the holding-time distribution Pearson type-III (in this case the general solution is obtained in closed form in terms of a newly introduced function I n k ( z ), many of the properties of which are derived in the Appendix), (b) i directly proportional to time and the holding time exponentially distributed.