Stationary analysis of the shortest queue problem

Abstract

Abstract. A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity K, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distributions with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression for the blocking probability is derived, which as far as we know is original. Using coupling arguments and explicit formulas, comparisons with related loss systems are then provided. Bounds are similarly obtained for the average total number of customers, with the stationary distribution explicitly determined on {K, . . . , 2K}, and elsewhere upper bounded. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states (0, k) for 0 ≤ k ≤ K. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initial symmetric finite capacity model, the stationary probabilities of states (0, k) can be obtained recursively from the blocking probability. In the other cases, they are implicitly determined through a functional equation that characterizes their generating function. The whole approach shows that the stationary distribution of the infinite capacity symmetric process is the limit of the corresponding finite capacity distributions. For the infinite capacity symmetric model, we provide an elementary proof of a result by Cohen which gives the solution of the functional equation in terms of an infinite product with explicit zeroes and poles.

Martingales and buffer overflow for the symmetric shortest queue model

“…The latter result is the main motivation for the present work, which initially aimed at understanding the strikingly simple expression of the blocking probability obtained in [17]. The idea is that enlarging the model might help interpret the result, by examining how the different parameters contribute to the formula.…”

“…Before proceeding to the main result, let us describe how the martingale M(t), hereafter defined, emerges through revisiting the chain of relations that determine the blocking probability in [17]. Such 'kernel relations,' as introduced by Kingman in [31] for the infinite capacity JSQ model, have, since then, flourished in the literature on reflected random walks (see the book by Fayolle, Iasnogorodski and Malyshev [21]).…”

“…For the finite capacity K case, the single point (K , K ) constitutes an additional face, so that relations similar to (8) hold, with an additional term (i = 4) involving the stationary blocking probability π(K , K ). In [17], the latter is calculated by combining, on the one hand, the extended relations (8) at three particular pairs (x, y) satisfying c 0 (x, y) = 0, and, on the other hand, the derivatives w.r.t. x, at x = 1, of relations (8) at (x, 1) and (x, x).…”

“…The present brief overview of the literature, though far from exhaustive, testifies to the long history of the subject. We refer to [17] for a more detailed review. The issues so far addressed go from stability criteria, as for example in [9,16,25,34], and steady-state analysis [1,2,[13][14][15]20,23,31], including bounds and comparisons with other policies [5,29,[48][49][50], to asymptotic approaches, such as steady-state tail asymptotics [35,36], mean-field and other scaling limits [10,18,32,37,45,47] or large deviations and rare events [24,39,41,46].…”

“…The state space is shown to be partitioned into different regions on which the invariant distribution exhibits different behaviors. In [17], this distribution, for fixed finite K , is expressed as a function of the stationary probabilities of states (0, j) for 0 ≤ j ≤ K , which are recursively characterized. The stationary blocking probability is moreover given an explicit, simple formula.…”