Analysing Ek:Er:c Queues

Adan, I.J.B.F., Waarsenburg, W.A. van de, & Wessels, J. (1992). Analysing Ek:Er:c Queues. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-92-047

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Abstract

In this paper we study a system consisting of parallel identical servers and a common queue. The service times are Erlang-r distributed and the interarrival times are Erlang-k distributed. The service discipline is first-come first-served. Bertsimas has proved that the equilibrium probability for a saturated state can be written as a linear combination of geometric terms. In the present paper it is shown that the coefficients also have a geometric form. It is also shown how the factors may be found efficiently. The present paper uses a direct approach for solving the equilibrium equations rather than a generating function approach as Bertsimas does. The direct approach was inspired by previous work of two of the authors on the shortest queue problem, in particular, and on the two-dimensional random walk, more generally. Although the paper extends results of Bertsimas it is self-contained.

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