Solutions of M / G /1// N -type Loops with Extensions to M / G /1 and GI / M /1 Queues

J. L. Carroll, A. van de Liefvoort and Lester Lipsky
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J. L. Carroll: San Diego State University, San Diego, California
A. van de Liefvoort: University of Kansas, Lawrence, Kansas
Lester Lipsky: University of Nebraska, Lincoln, Nebraska

Operations Research, 1982, vol. 30, issue 3, 490-514

Abstract: Closed form solutions of the joint equilibrium distribution of queue sizes are derived for a large class of M / G /1// N queues, i.e., any closed loop of two servers in which one is exponential (but possibly load dependent), and the other has a probability density function which has a rational Laplace transform and a queueing discipline that is FCFS. The class of G / M / k // N queues are included as special cases of load-dependent servers. The solutions are presented in terms of vectors and matrices whose size depends only on the distribution of the general server and not on the number of customers in the system. Efficient algorithms are outlined, and expressions for various system measurements are presented. Depending on the relative service rates of the two servers, solutions for both the M / G /1 and GI / M /1 open queues are derived as limiting cases of the M / G /1// N system. All results are contrasted with existing formulas.

Keywords: 683 algorithm for approximation of FCFS nonexponential networks; 689 explicit solution of M/G/1//N-type loops with FCFS; 692 M/G//1 and GI/M/1 queues as limits of M/G/1//N-loops (search for similar items in EconPapers)
Date: 1982
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