Advertisement

Queueing Systems

, 63:437 | Cite as

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

  • Samuli Aalto
  • Urtzi Ayesta
  • Rhonda Righter
Article

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.

Keywords

M/G/1 Gittins index policy Optimal scheduling Bathtub hazard rate 

Mathematics Subject Classification (2000)

60K25 90B36 68M20 

References

  1. 1.
    Aalto, S., Ayesta, U.: On the nonoptimality of the foreground-background discipline for IMRL service times. J. Appl. Probab. 43, 523–534 (2006) CrossRefGoogle Scholar
  2. 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) Google Scholar
  3. 3.
    Bertsimas, D., Nino-Mora, J.: Restless bandits, linear programming relaxations, and a primal-dual index heuristic. Oper. Res. 48, 80–90 (2000) CrossRefGoogle Scholar
  4. 4.
    Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980) Google Scholar
  5. 5.
    Gittins, J.C.: Multi-armed Bandit Allocation Indices. Wiley, Chichester (1989) Google Scholar
  6. 6.
    Kleinrock, L.: Queueing Systems, vol. 2. Wiley, New York (1976) Google Scholar
  7. 7.
    Klimov, G.P.: Time-sharing service systems. I. Theory Probab. Appl. 19, 532–551 (1974) CrossRefGoogle Scholar
  8. 8.
    Klimov, G.P.: Time-sharing service systems. II. Theory Probab. Appl. 23, 314–321 (1978) CrossRefGoogle Scholar
  9. 9.
    Lai, C.-D., Xie, M.: Stochastic Ageing and Dependence for Reliability. Springer, New York (2006) Google Scholar
  10. 10.
    Meilijson, I., Weiss, G.: Multiple feedback at a single server station. Stoch. Process. Appl. 5, 195–205 (1977) CrossRefGoogle Scholar
  11. 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) CrossRefGoogle Scholar
  12. 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) Google Scholar
  13. 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) CrossRefGoogle Scholar
  14. 14.
    Righter, R., Shanthikumar, J.G.: Extremal properties of the FIFO discipline in queueing networks. J. Appl. Probab. 29, 967–978 (1992) CrossRefGoogle Scholar
  15. 15.
    Righter, R., Shanthikumar, J.G., Yamazaki, G.: On extremal service disciplines in single-stage queueing systems. J. Appl. Probab. 27, 409–416 (1990) CrossRefGoogle Scholar
  16. 16.
    Schrage, L.E.: A proof of the optimality of the shortest remaining processing time discipline. Oper. Res. 16(3), 687–690 (1968) CrossRefGoogle Scholar
  17. 17.
    Sevcik, K.: Scheduling for minimum total loss using service time distributions. J. ACM 21, 66–75 (1974) CrossRefGoogle Scholar
  18. 18.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007) CrossRefGoogle Scholar
  19. 19.
    Smith, D.R.: A new proof of the optimality of the shortest remaining processing time discipline. Oper. Res. 26, 197–199 (1978) CrossRefGoogle Scholar
  20. 20.
    Tsitsiklis, J.N.: A short proof of the Gittins index theorem. In: IEEE CDC, pp. 389–390 (1993) Google Scholar
  21. 21.
    Varaiya, P., Walrand, J., Buyukkoc, C.: Extensions of the multiarmed bandit problem: the discounted case. IEEE Trans. Autom. Control 30, 426–439 (1985) CrossRefGoogle Scholar
  22. 22.
    Weber, R.: On the Gittins index for multiarmed bandits. Ann. Appl. Probab. 2(4), 1024–1033 (1992) CrossRefGoogle Scholar
  23. 23.
    Whittle, P.: Restless bandits: Activity allocation in a changing world. J. Appl. Probab. 25, 287–298 (1988) CrossRefGoogle Scholar
  24. 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. 25.
    Yashkov, S.F.: Processor sharing queues: Some progress in analysis. Queueing Syst. 2, 1–17 (1987) CrossRefGoogle Scholar
  26. 26.
    Yashkov, S.F.: Mathematical problems in the theory of processor sharing queueing systems. J. Sov. Math. 58, 101–147 (1992) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.TKK Helsinki University of TechnologyEspooFinland
  2. 2.LAAS-CNRSToulouseFrance
  3. 3.IkerbasqueBCAM—Basque Center for Applied MathematicsDerioSpain
  4. 4.University of California at BerkeleyBerkeleyUSA

Personalised recommendations