Queues with Ordered Servers that Work at Different Rates: An Exact Analysis of a Model Solved Approximately by Others

In a recent paper, Ibe and Maruyama [3, p. 16] consider the following model of a heterogeneous multiserver queueing system, which they describe in Section 1 of their paper: "There are n communication links numbered 1, 2 . . . . . n, which are accessed by arriving messages in a fixed order. Specifically, an arriving message is processed by the lowest-numbered idle link, if such a link exists. When all links are busy, a single queue is formed and messages enter for service on a first-come first-served basis. We assume that messages arrive in a Poisson manner with rate ~, that message lengths are exponentially distributed with mean l / p , and that C~_ 1 >/C,, where C, is the capacity of link i, i = 2 . . . . . n. We want to compute the utilization factor of each link and the expected message delay." The authors assert that an exact analysis requires the solution of 2 n linear equations, which is impractical except when n is small. Therefore, they propose an approximation, which they compare with the exact results for n = 2 and n = 3, and with simulation results for n = 4, 5, and 10. The purpose of this short note is to point out that the model of Ibe and Maruyama is a special case of the one considered by the present author [1], who gives an exact and easily computable solution for the case of arbitrary n, and which remains valid under less restrictive assumptions about the arrival process and the queue discipline. More specifically, Cooper's model and its solution [1, pp. 72-73] are stated as follows.