Abstract
When controlled stochastic systems have performances which satisfy generalisedconservation laws (GCL), an objective which is linear in the performance is optimised by aGittins index policy. We develop measures of the extent to which a system fails to satisfyGCL and derive suboptimality bounds for suitable index policies in terms of such measures.These bounds are used, inter alia, to explore the robustness in performance of cm‐typerules for a multiclass G/G/1 queueing system to departures from an assumption of exponentialservice times. We also study Gittins index policies for parallel processor versions of theclassical undiscounted and discounted multi‐armed bandit problems. In the undiscountedcase, the cost of an index policy comes within a constant of the optimal cost ‐ thisconstant being independent of the number of projects submitted for scheduling. In thediscounted case, under fairly mild conditions, Gittins index policies come within an O(1) quantity ofoptimality and are hence average reward optimal when the discount rate is small enough.
Similar content being viewed by others
References
D. Bertsimas and J. Niño-Mora, Conservation laws, extended polymatroids and multi-armed bandit problems; a unified approach to indexable systems, Mathematics of Operations Research 21(1996) 257-306.
D. Blackwell, Discrete dynamic programming, Annals of Mathematical Statistics 33(1962)719-726.
A. Federgruen and H. Groenevelt, Characterization and optimization of achievable performance in general queueing systems, Operations Research 36(1988)733-741.
A. Federgruen and H. Groenevelt, M/G/c queueing systems with multiple customer classes. Characterization and control of achievable performance under non-preemptive priority rules, Management Science 34(1988)1121-1138.
E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems, Academic Press, New York, 1980.
J.C. Gittins, Bandit processes and dynamic allocation indices, Journal of the Royal Statistical Society B41(1979)148-177.
J.C. Gittins, Bandit Processes and Dynamic Allocation Indices, Wiley, New York, 1989.
J.C. Gittins and D.M. Jones, A dynamic allocation index for the sequential design of experiments, in: Progress in Statistics, eds. J. Gani et al., North-Holland, Amsterdam, 1974, pp. 241-266.
K.D. Glazebrook, On the undiscounted tax problem with precedence constraints, Advances in Applied Probability 28(1996)1123-1144.
K.D. Glazebrook and R. Garbe, Reflections on a new approach to Gittins indexation, Journal of the Operational Research Society 47(1996)1301-1309.
K.D. Glazebrook and R. Garbe, Almost optimal policies for stochastic systems which almost satisfy conservation laws, Technical Report, Newcastle University.
G.P. Klimov, Time sharing service systems I, Theory of Probability and its Applications 19(1974) 532-551.
J.G. Shanthikumar and D.D. Yao, Multiclass queueing systems: Polymatroidal structure and optimal scheduling control, Operations Research 40,Supplement 2 (1992)S293-299.
H.C. Tijms, Stochastic Models — An Algorithmic Approach, Wiley, Chichester, 1994.
J.C. Walrand, Introduction to Queueing Networks, Prentice-Hall, Englewood Cliffs, 1988.
R.R. Weber, On the Gittins index for multi-armed bandits, Annals of Applied Probability 2(1992) 1024-1033.
R.R. Weber, Scheduling jobs with stochastic processing requirements on parallel machines to minimize makespan or flowtime, Journal of Applied Probability 19(1982)167-182.
R.R. Weber, P. Varaiya and J. Walrand, Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime, Journal of Applied Probability 23(1986) 841-847.
G. Weiss, Approximation results in parallel machines stochastic scheduling, Annals of Operations Research 26(1990)195-242.
G. Weiss, Turnpike optimality of Smith's rule in parallel machines stochastic scheduling, Mathematics of Operations Research 17(1992)255-270.
G. Weiss, On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines, Advances in Applied Probability 27(1995)821-839.
P. Whittle, Multi-armed bandits and the Gittins index, Journal of the Royal Statistical Society B42(1980)143-149.
P. Whittle, Restless bandits, Journal of Applied Probability 25A(1988)287-298.
Rights and permissions
About this article
Cite this article
Glazebrook, K., Garbe, R. Almost optimal policies for stochastic systemswhich almost satisfy conservation laws. Annals of Operations Research 92, 19–43 (1999). https://doi.org/10.1023/A:1018934714800
Issue Date:
DOI: https://doi.org/10.1023/A:1018934714800