On Markov-Krein Characterization of Mean Sojourn Time in Queueing Systems

We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multi-server system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 round-robin queue. We argue that rather than looking for exact expressions for the mean response time as a function of the job size distribution, a more fruitful approach is to find distributions which minimize or maximize the mean response time given the first n moments of the job size distribution. We prove that for the M/G/k system in light-traffic asymptote and given first n (= 2, 3) moments of the job size distribution, analogous to the classical Markov-Krein Theorem, these ‘extremal’ distributions are given by the principal representations of the moment sequence. Furthermore, if we restrict the distributions to lie in the class of Completely Monotone (CM) distributions, then for all the three queueing systems, for any n, the extremal distributions under the appropriate “light traffic” asymptotics are hyper-exponential distributions with finite number of phases. We conjecture that the property of extremality should be invariant to the system load, and thus our light traffic results should hold for general load as well, and propose potential strategies for a unified approach to finding moments-based bounds for queueing systems. By identifying the extremal distributions, our results allow numerically obtaining tight bounds on the performance of these queueing systems.