doi:10.1016/j.peva.2003.11.001
Copyright © 2003 Elsevier B.V. All rights reserved.
-MG1: an efficient technique for the analysis of a class of M/G/1-type processes by aggregation*1
Department of Computer Science, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA
Received 13 October 2000;
Revised 10 November 2003.
Available online 24 January 2004.
Abstract
We extend the
approach, initially proposed for the efficient numerical solution of a class of quasi-birth–death processes, to a more complex class of M/G/1-type Markov processes where arbitrary forward transitions are allowed but backward transitions must be to a single state to the previous level. The new technique reduces the exact solution of this class of M/G/1-type models to that of a finite linear system. We demonstrate the utility of our method by describing the exact computation of an extensive class of Markov reward functions that include the expected queue length or its higher moments. We also provide an algorithm that finds an appropriate state reordering satisfying our applicability conditions, if one such order exists. We illustrate the method, discuss its complexity and numerical stability, and present comparisons with other traditional techniques.
Author Keywords: M/G/1-type Markov chain; Matrix-analytic solution; Bulk arrival
Fig. 1. Aggregation of an infinite into a finite number of states.
Fig. 2. The column vectors used to prove linear independence.
Fig. 3. State repartitioning.
Fig. 4. The CTMC that models a dynamic multiprocessor scheduling policy.
Fig. 5. Computation time (in s) for a process with bulk sizes of (a) 50 and (b) 100.
Fig. 6. Ratio of the matrix-analytic and -MG1 computation times for a process with bulk sizes of 50 (a) or 100 (b).
Fig. 7. Perturbation of the solution as result of the perturbation of the input for both -MG1 and matrix-analytic method for 50 different experiments, with matrix size n=32 and bulk size of p=64, for perturbation magnitudes of (a) 10−12 and (b) 10−14.
Fig. 8. Maximum difference in the solutions obtained from -MG1 and the matrix-analytic method for one perturbation experiment as a function of different problem sizes, with perturbation magnitudes of (a) 10−12 and (b) 10−14.
Fig. 9. Perturbation of the solution as result of the perturbation of the stiff input for both -MG1 and the matrix-analytic method for 50 different experiments, with matrix size n=32 and bulk size of p=64, with perturbation magnitudes of (a) 10−12 and (b) 10−14.
Fig. 10. Maximum difference in the solutions obtained from -MG1 and the matrix-analytic method for one perturbation experiment with stiff input as a function of different problem sizes, with perturbation magnitudes of (a) 10−12 and (b) 10−14.
Table 1. Computational and storage complexity of
-MG1 and the matrix-analytic methoda
Corresponding author. Fax: +1-757-221-1717.
*1 A preliminary version of this paper was presented at the Third Meeting on the Numerical Solution of Markov Chains, Zaragoza, Spain, 1999 [2]. This work was supported by National Science Foundation under grants no. EIA-9974992, CCR-0098278 and ACR-0090221, by the National Aeronautics and Space Administration under NASA Grant NAG-1-2168, and by a William and Mary Summer Research Grant.
1 Present address: Seagate Research, 1251 Waterfront Place, Pittsburgh, PA 15222, USA.
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