Reversibility and stochastic networks

This paper considers the M/M/k queueing system where a line is being formed in front of each server. Arriving customers join the shorter line and switch to another once the latter becomes shorter. Since the expected waiting time equals that in the first come first served case (FCFS), it is of interest to compute the variance of the waiting time. We compute it explicitly for the cases where k = 1, k = 2, k = 3 and for the case of a random queue. For the case k ≥ 4, we express the variance in terms of the roots of a k-degree polynomial.