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On the Gittins index in the M/G/1 queue

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Abstract

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground–Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground–Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.

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References

  1. Aalto, S., Ayesta, U.: On the nonoptimality of the foreground-background discipline for IMRL service times. J. Appl. Probab. 43, 523–534 (2006)

    Article  Google Scholar 

  2. Aalto, S., Ayesta, U.: Optimal scheduling of jobs with a DHR tail in the M/G/1 queue. In: Proceedings of ValueTools 2008, Athens, Greece (2008)

  3. Bertsimas, D., Nino-Mora, J.: Restless bandits, linear programming relaxations, and a primal-dual index heuristic. Oper. Res. 48, 80–90 (2000)

    Article  Google Scholar 

  4. Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980)

    Google Scholar 

  5. Gittins, J.C.: Multi-armed Bandit Allocation Indices. Wiley, Chichester (1989)

    Google Scholar 

  6. Kleinrock, L.: Queueing Systems, vol. 2. Wiley, New York (1976)

    Google Scholar 

  7. Klimov, G.P.: Time-sharing service systems. I. Theory Probab. Appl. 19, 532–551 (1974)

    Article  Google Scholar 

  8. Klimov, G.P.: Time-sharing service systems. II. Theory Probab. Appl. 23, 314–321 (1978)

    Article  Google Scholar 

  9. Lai, C.-D., Xie, M.: Stochastic Ageing and Dependence for Reliability. Springer, New York (2006)

    Google Scholar 

  10. Meilijson, I., Weiss, G.: Multiple feedback at a single server station. Stoch. Process. Appl. 5, 195–205 (1977)

    Article  Google Scholar 

  11. Meilijson, I., Yechiali, U.: On optimal right-of-way policies at a single server station when insertion of idle times is permitted. Stoch. Process. Appl. 6, 25–32 (1977)

    Article  Google Scholar 

  12. Osipova, N., Ayesta, U., Avrachenkov, K.E.: Optimal policy for multi-class scheduling in a single server queue. In: Proceedings of ITC-21 (2009)

  13. Righter, R., Shanthikumar, J.G.: Scheduling multiclass single server queueing systems to stochastically maximize the number of successful departures. Probab. Eng. Inf. Sci. 3, 323–334 (1989)

    Article  Google Scholar 

  14. Righter, R., Shanthikumar, J.G.: Extremal properties of the FIFO discipline in queueing networks. J. Appl. Probab. 29, 967–978 (1992)

    Article  Google Scholar 

  15. Righter, R., Shanthikumar, J.G., Yamazaki, G.: On extremal service disciplines in single-stage queueing systems. J. Appl. Probab. 27, 409–416 (1990)

    Article  Google Scholar 

  16. Schrage, L.E.: A proof of the optimality of the shortest remaining processing time discipline. Oper. Res. 16(3), 687–690 (1968)

    Article  Google Scholar 

  17. Sevcik, K.: Scheduling for minimum total loss using service time distributions. J. ACM 21, 66–75 (1974)

    Article  Google Scholar 

  18. Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)

    Book  Google Scholar 

  19. Smith, D.R.: A new proof of the optimality of the shortest remaining processing time discipline. Oper. Res. 26, 197–199 (1978)

    Article  Google Scholar 

  20. Tsitsiklis, J.N.: A short proof of the Gittins index theorem. In: IEEE CDC, pp. 389–390 (1993)

  21. Varaiya, P., Walrand, J., Buyukkoc, C.: Extensions of the multiarmed bandit problem: the discounted case. IEEE Trans. Autom. Control 30, 426–439 (1985)

    Article  Google Scholar 

  22. Weber, R.: On the Gittins index for multiarmed bandits. Ann. Appl. Probab. 2(4), 1024–1033 (1992)

    Article  Google Scholar 

  23. Whittle, P.: Restless bandits: Activity allocation in a changing world. J. Appl. Probab. 25, 287–298 (1988)

    Article  Google Scholar 

  24. Yashkov, S.F.: On feedback sharing a processor among jobs with minimal serviced length (in Russian). Tech. Sreds. Svyazi, Ser. ASU 2, 51–62 (1978)

    Google Scholar 

  25. Yashkov, S.F.: Processor sharing queues: Some progress in analysis. Queueing Syst. 2, 1–17 (1987)

    Article  Google Scholar 

  26. Yashkov, S.F.: Mathematical problems in the theory of processor sharing queueing systems. J. Sov. Math. 58, 101–147 (1992)

    Article  Google Scholar 

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Authors and Affiliations

  1. TKK Helsinki University of Technology, Espoo, Finland

    Samuli Aalto

  2. LAAS-CNRS, Toulouse, France

    Urtzi Ayesta

  3. Ikerbasque, BCAM—Basque Center for Applied Mathematics, Derio, Spain

    Urtzi Ayesta

  4. University of California at Berkeley, Berkeley, CA, USA

    Rhonda Righter

Authors
  1. Samuli Aalto
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  2. Urtzi Ayesta
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  3. Rhonda Righter
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Correspondence to Urtzi Ayesta.

Additional information

The paper is intended for the Special Issue of Queueing Systems related to the Erlang Centennial Conference.

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Aalto, S., Ayesta, U. & Righter, R. On the Gittins index in the M/G/1 queue. Queueing Syst 63, 437 (2009). https://doi.org/10.1007/s11134-009-9141-x

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  • DOI: https://doi.org/10.1007/s11134-009-9141-x

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