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

  • S.I.: Queueing Theory and Network Applications
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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.

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References

  • Adve, V. S., & Nelson, R. (1994). The relationship between Bernoulli and fixed feedback policies for the M/G/1 queue. Operations Research, 42(2), 380–385.

    Article  Google Scholar 

  • Altman, E., Avrachenkov, K. E., & Ayesta, U. (2006). A survey on discriminatory processor sharing. Queueing Systems, 53, 53–63.

    Article  Google Scholar 

  • Balogh, T. & Medvecký, M. (2011). Performance evaluation of WFQ, WF$^2$Q+ and WRR queue scheduling algorithms. In 2011 34th International conference on telecommunications and signal processing (TSP) (pp. 136–140), Budapest.

  • Boxma, O. J., & Yechiali, U. (1997). An M/G/1 queue with multiple types of feedback and gated vacations. Journal of Applied Probability, 34(3), 773–784.

    Article  Google Scholar 

  • Chaskar, H. M., & Madhow, U. (2003). Fair scheduling with tunable latency: A round-robin approach. IEEE/ACM Transactions on Networking, 11(4), 592–601.

    Article  Google Scholar 

  • Choi, B. D., & Kim, B. (2002). An M/G/1 queueing system with fixed feedback policy. The ANZIAM Journal, 44(2), 283–297.

    Article  Google Scholar 

  • Choi, B. D., Kim, B., & Choi, S. H. (2000). On the M/G/1 Bernoulli feedback queue with multi-class customers. Computers & Operations Research, 27(3), 269–286.

    Article  Google Scholar 

  • Choi, B. D., Kim, B., & Choi, S. H. (2003). An M/G/1 queue with multiple types of feedback, gated vacations and FCFS policy. Computers & Operations Research, 30(9), 1289–1309.

    Article  Google Scholar 

  • Disney, R. L. (1981). A note on sojourn times in M/G/1 queues with instantaneous, Bernoulli feedback. Naval Research Logist Quarterly, 28, 679–684.

    Article  Google Scholar 

  • Disney, R. L., König, D., & Schmidt, V. (1984). Stationary queue-length and waiting-time distributions in single-server feedback queues. Advances in Applied Probability, 16(2), 437–446.

    Article  Google Scholar 

  • Disney, R. L., McNickle, D. C., & Simon, B. (1980). M/G/1 queues with instantaneous, Bernoulli feedback. Naval Research Logistics Quarterly, 27, 635–644.

    Article  Google Scholar 

  • Doshi, B. T., & Kaufman, J. S. (1988). Sojourn time in an M/G/1 queue with Bernoulli feedback. In Queueing theory and its applications, CWI Monographs 7. Amsterdam: North-Holland.

  • Epema, D. H. J. (1996). Joint queue-lengths and sojourn-time distributions in a general preemptive feedback queue. Performance Evaluation, 27&28, 99–116.

    Article  Google Scholar 

  • Fayolle, G., Mitrani, I., & Iasnogorodski, R. (1980). Sharing a processor among many job classes. Journal of ACM, 27, 519–532.

    Article  Google Scholar 

  • Fontana, B., & Berzosa, C. D. (1985). M/G/l queue with N-priorities and feedback, joint queue-length distributions and response time distribution for any particular sequence. In M. Akiyama (Ed.), Teletraffic issues in an Advanced Information Society ITC-II (pp. 452–458). Amsterdam: North-Holland.

    Google Scholar 

  • Hirayama, T. (2003). Mean sojourn times in multiclass feedback queues with gated disciplines. Naval Research Logistics (NRL), 50(7), 719–741.

    Article  Google Scholar 

  • Hirayama, T. (2012). Analysis of multiclass Markovian polling systems with feedback and composite scheduling algorithms. Annals of Operations Research, 198(1), 83–123.

    Article  Google Scholar 

  • Hunter, J. J. (1989). Sojourn time problems in feedback queues. Queueing Systems, 5, 55–75.

    Article  Google Scholar 

  • Katevenis, M., Sidiropoulos, S., & Courcoubetis, C. (1991). Weighted round-robin cell multiplexing in a general-purpose ATM switch chip. IEEE Journal on Selected Areas in Communications, 9(8), 1265–1279.

    Article  Google Scholar 

  • Kim, B., & Kim, J. (2006). Comparison of DPS and PS systems according to DPS weights. IEEE Communications Letters, 10(7), 558–560.

    Article  Google Scholar 

  • Kim, J., & Kim, B. (2004). Sojourn time distribution in the M/M/1 queue with discriminatory processor-sharing. Performance Evaluation, 58(4), 341–365.

    Article  Google Scholar 

  • Lam, S. S., & Shankar, A. (1981). A derivation of response time distributions for a multi-class feedback queueing systems. Performance Evaluation, 1(1), 48–61.

    Article  Google Scholar 

  • Liu, J. W. S. (2000). Real-time systems. London: Pearson Education.

    Google Scholar 

  • Rege, K. (1993). On the M/G/1 queue with Bernoulli feedback. Operations Research Letters, 14(3), 163–170.

    Article  Google Scholar 

  • Rege, K. M., & Sengupta, B. (1996). Queue-length distribution for the discriminatory processor-sharing queue. Operations Research, 44(4), 653–657.

    Article  Google Scholar 

  • Resing, J. A. C., Hooghiemstra, G., & Keane, M. S. (1990). The M/G/1 processor sharing queue as the almost sure limit of feedback queues. Journal of Applied Probability, 27(4), 913–918.

    Article  Google Scholar 

  • Simon, B. (1984). Priority queues with feedback. Journal of ACM, 31(1), 134–149.

    Article  Google Scholar 

  • Takács, L. (1963). A single-server queue with feedback. Bell System Technical Journal, 42(2), 505–519.

    Article  Google Scholar 

  • Takagi, H. (1991). Queueing analysis, a foundation of performance evaluation, Vol. 1, vacation and priority systems, Part 1. Amsterdam: North-Holland.

  • Takine, T., Takagi, H., & Hasegawa, T. (1991). Sojourn times in vacation and polling systems with Bernoulli feedback. Journal of Applied Probability, 28(2), 422–432.

    Article  Google Scholar 

  • van den Berg, J., Boxma, O. J., & Groenendijk, W. P. (1989). Sojourn times in the M/G/1 queue with deterministic feedback. Communications in Statistics. Stochastic Models, 5(1), 115–129.

    Article  Google Scholar 

  • Wortman, M. A., Disney, R. L., & Kiessler, P. C. (1991). The M/GI/1 Bernoulli feedback queue with vacations. Queueing Systems, 9(4), 353–363.

    Article  Google Scholar 

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Acknowledgements

We are grateful to the reviewers for valuable comments and suggestions, which greatly improved this paper. J. Kim’s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2016R1C1B1016602). B. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2017R1A2B4012676).

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Authors and Affiliations

  1. Department of Mathematics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul, 02504, Korea

    Jerim Kim

  2. Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Korea

    Bara Kim

  3. Department of Mathematical Sciences, National Chengchi University, Taipei City, 11605, Taiwan

    Hsing Luh

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  1. Jerim Kim
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  2. Bara Kim
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  3. Hsing Luh
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Correspondence to Bara Kim.

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Kim, J., Kim, B. & Luh, H. Analysis of a Markovian feedback queue with multi-class customers and its application to the weighted round-robin queue. Ann Oper Res 277, 137–159 (2019). https://doi.org/10.1007/s10479-018-2917-9

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  • DOI: https://doi.org/10.1007/s10479-018-2917-9

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