Analyzing E k|E r|c queues

Abstract

In this paper we study a system consisting of c 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. The waiting process may be characterized by ( n −1, n 0, n 1,…, n c ) where n −1 represents the number of remaining arrival stages, n 0 the number of waiting jobs and n i , i = 1,…, c, the number of remaining service stages for server i. Bertsimas has proved that the equilibrium probability for saturated states (i.e. states with all servers busy) can be written as a linear combination of geometric terms with n 0 as exponent. In the present paper it is shown that the coefficients also have a geometric form with respect to n −1, n 1, …, n c . 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 is based on separation of variables and has been inspired by previous work of two of the authors on the shortest queue problem in particular and the two-dimensional random walk more generally. The characterization of the equilibrium probabilities leads to exact expressions for performance measures such as the moments of the queue length and the waiting time, which are useful for numerical computations. Numerical results are presented.