Light Traffic Limits of Sojourn Time Distributions in Markovian Queueing Networks

The calculation of sojourn time distributions in queueing networks is both an important and difficult problem. We consider a class of Markovian queueing networks with Poisson input, and show how the results of Reiman and Simon [9] can be applied to calculate light traffic limits for the sojourn time distribution and its moments. Let W(x, λ) denote the probability that a stationary sojourn time is greater than x when the arrival rate is X. We provide a method for calculating W(n) (x, 0), the nth derivative of W with respect to λ at λ = 0. Light traffic limits for moments follow straightforwardly. To illustrate the method we calculate light traffic limits for some otherwise intractable queueing systems.