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

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. © Springer Science+Business Media, LLC 2009

Restless bandit marginal productivity indices II: multiproject case and scheduling a multiclass make-to-order/-stock M/G/1 queue

This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author's companion paper "Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue", based on characterization of a project's marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)'s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter's near-optimality across the cases investigated

Optimal policy for multi-class scheduling in a single server queue

In this paper we apply the Gittins optimality result to characterize the optimal scheduling discipline in a multi-class M/G/1 queue. We apply the general result to several cases of practical interest where the service time distributions belong to the set of decreasing hazard rate distributions, like Pareto or hyper-exponential. When there is only one class it is known that in this case the Least Attained Service policy is optimal. We show that in the multi-class case the optimal policy is a priority discipline, where jobs of the various classes depending on their attained service are classified into several priority levels. Using a tagged-job approach we obtain, for every class, the mean conditional sojourn time. This allows us to compare numerically the mean sojourn time in the system between the Gittins optimal and popular policies like Processor Sharing, First Come First Serve and Least Attained Service (LAS). We implement the Gittins' optimal algorithm in NS-2 and we perform numerical experiments to evaluate the achievable performance gain. We find that the Gittins policy can outperform by nearly 10% the LAS policy

SEH: Size Estimate Hedging for Single-Server Queues

For a single server system, Shortest Remaining Processing Time (SRPT) is an optimal size-based policy. In this paper, we discuss scheduling a single-server system when exact information about the jobs' processing times is not available. When the SRPT policy uses estimated processing times, the underestimation of large jobs can significantly degrade performance. We propose a simple heuristic, Size Estimate Hedging (SEH), that only uses jobs' estimated processing times for scheduling decisions. A job's priority is increased dynamically according to an SRPT rule until it is determined that it is underestimated, at which time the priority is frozen. Numerical results suggest that SEH has desirable performance when estimation errors are not unreasonably large

Restless bandits, partial conservation laws and indexability

We show that if performance measures in a stochastic scheduling problem satisfy a set of so-called partial conservation laws (PCL), which extend previously studied generalized conservation laws (GCL), then the problem is solved optimally by a priority-index policy for an appropriate range of linear performance objectives, where the optimal indices are computed by a one-pass adaptive-greedy algorithm, based on Klimov's. We further apply this framework to investigate the indexability property of restless bandits introduced by Whittle, obtaining the following results: (1) we identify a class of restless bandits (PCL-indexable) which are indexable; membership in this class is tested through a single run of the adaptive-greedy algorithm, which also computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (2) we further indentify the class of GCL-indexable bandits, which includes classical bandits, having the property that they are indexable under any linear reward objective. The analysis is based on the so-called achievable region method, as the results follow from new linear programming formulations for the problems investigated.Stochastic scheduling, Markov decision chains, bandit problems, achievable region