Time-Dependent Approximations For Oueues With High Traffic Intensities

Abstract

AbstractFor a certain class of queues, the time-dependeiit analysis with a Laplace transform-generating function technique involves the complex root u&> inside the unit circle of a polynomial. The exact value of u&> gives for the queue size distribution Pn(t) and the mean queue size E[X(t)] transforms that are, in general, difficult to invert. We discuss some iterative procedures that give successive rational approximations for u&>. The resulting transforms for Pn(t) and E[X(t)] are rational and more easily inverted to give approximations for the queue size distribution and the mean queue size respectively. Applications to the M/M/c queue and the M/Ep/1 queue suggest that a first approximation for u&> by Newton’s method is sufficient to yield a good approximation for the mean queue size for a wide range of values of t, particularly for large t, provided the traffic intensity exceeds, but is not too close to, one. A fairly accurate approximation for Pn(t) is also obtained but only for rather small values o...