Queueing output processes

Abstract

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing systemG/G/s/Nwith general arrival process, mutually independent service times,sservers (1 ≦s≦ ∞), and waiting room of sizeN(0 ≦N≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-serverM/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.