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The M/M/1 Queue with Randomly Varying Arrival and Service Rates: A Phase Substitution Solution

Author

Listed:
  • Edward P. C. Kao

    (Department of Decision and Information Sciences, University of Houston, Houston, Texas 77004)

  • Chiunsin Lin

    (Institute of Management Science, National Chiao Tung University, Taipei, Taiwan, Republic of China)

Abstract

This paper presents an alternative procedure for computing the steady state probability vector of an M/M/1 queue with randomly varying arrival and service rates. By exploiting the structure of the infinitesimal generator of the underlying continuous-time Markov chain, the approach represents an efficient adaptation of the state reduction method introduced by Grassmann for solving problems involving M/M/1 queues under a random environment. We compare computational requirements of the proposed approach with the method of Neuts and block elimination under different rush-hour congestion patterns while keeping the overall traffic intensity constant as well as under different traffic intensities. We demonstrate that the proposed method requires minimal computing time to reach convergence and moreover the time requirement does not change much when traffic intensity increases.

Suggested Citation

  • Edward P. C. Kao & Chiunsin Lin, 1989. "The M/M/1 Queue with Randomly Varying Arrival and Service Rates: A Phase Substitution Solution," Management Science, INFORMS, vol. 35(5), pages 561-570, May.
    • Handle: RePEc:inm:ormnsc:v:35:y:1989:i:5:p:561-570
    DOI: 10.1287/mnsc.35.5.561
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