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Analysis of a Markovian feedback queue with multi-class customers and its application to the weighted round-robin queue

Author

Listed:
  • Jerim Kim

    (University of Seoul)

  • Bara Kim

    (Korea University)

  • Hsing Luh

    (National Chengchi University)

Abstract

We consider an M/G/1 Markovian feedback queue with multi-class customers. We derive functional equations for the stationary distribution of the queue size and the total response time. A system of linear equations is also derived for the moments of the queue size and the total response time distributions. The mean and the variance of the queue size and the total response time can be computed by solving the system of linear equations. By using the Markovian feedback queue with multi-class customers, we also investigate the M/PH/1 weighted round-robin queue. Numerical examples are given to show that moments of the queue sizes and of the total response times can be easily computed for the weighted round-robin queue. As the service quantum shrinks to zero, the moments of the queue sizes and the total response times converge to some limits which should be the moments of the corresponding discriminatory processor sharing queue.

Suggested Citation

  • Jerim Kim & Bara Kim & Hsing Luh, 2019. "Analysis of a Markovian feedback queue with multi-class customers and its application to the weighted round-robin queue," Annals of Operations Research, Springer, vol. 277(2), pages 137-159, June.
    • Handle: RePEc:spr:annopr:v:277:y:2019:i:2:d:10.1007_s10479-018-2917-9
    DOI: 10.1007/s10479-018-2917-9
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    References listed on IDEAS

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    1. Vikram S. Adve & Randolph Nelson, 1994. "The Relationship Between Bernoulli and Fixed Feedback Policies for the M/G/1 Queue," Operations Research, INFORMS, vol. 42(2), pages 380-385, April.
    2. Tetsuji Hirayama, 2003. "Mean sojourn times in multiclass feedback queues with gated disciplines," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 719-741, October.
    3. Kiran M. Rege & Bhaskar Sengupta, 1996. "Queue-Length Distribution for the Discriminatory Processor-Sharing Queue," Operations Research, INFORMS, vol. 44(4), pages 653-657, August.
    4. Tetsuji Hirayama, 2012. "Analysis of multiclass Markovian polling systems with feedback and composite scheduling algorithms," Annals of Operations Research, Springer, vol. 198(1), pages 83-123, September.
    5. Ralph L. Disney, 1981. "A note on sojourn times in M/G/1 queues with instantaneous, bernoulli feedback," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 28(4), pages 679-684, December.
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