A nearly-optimal index rule for scheduling of users with abandonment

Abstract

We analyze a comprehensive model for multi-class job scheduling accounting for user abandonment, with the objective of minimizing the total discounted or time-average sum of linear holding costs and abandonment penalties. We assume geometric service times and Bernoulli abandonment probabilities. We solve analytically the case in which there are 1 or 2 users in the system to obtain an optimal index rule. For the case with more users we use recent advances from the restless bandits literature to obtain a new simple index rule, denoted by AJN, which we propose to use also in the system with arrivals. In the problem without abandonment, the proposed rule recovers the cµ-rule which is well-known to be optimal both without and with arrivals. Under certain conditions, our rule is equivalent to the cµ/θ-rule, which was recently proposed and shown to be asymptotically optimal in a multi-server system with overload conditions. We present results of an extensive computational study that suggest that our rule is almost always superior or equivalent to other rules proposed in the literature, and is often optimal.

Dynamic fluid-based scheduling in a multi-class abandonment queue

“…This rule was derived in [7] for the system SM2 (without arrivals) with the modification that the user in service also contributes to the cost.…”

“…Both models have been studied in the literature, e.g., in [20] the model SM1 is studied, while the authors of [3,4,7] consider SM2. For a given policy π, let N π k (t) denote either the number of class-k customers in the system (SM1) or the number of class-k customers in the queue (SM2).…”

“…We will see that an optimal policy can be of two possible shapes: either a switching curve emerges, i.e., we prioritize one class above the switching curve and the other class below the switching curve, or one of 1 For the model SM2, the latter was also assumed in [3,4], while [7] assumed customers in service contribute to the holding cost as well.…”

“…We also refer to [19] for a recent survey on abandonments in a many-server system. More related to our present work are the papers that deal with optimal scheduling or control aspects of multi-class queueing systems in the presence of abandonments, see for instance [23,5,25,1,3,20,4,7].…”

“…Hence, researchers have focused on obtaining approximations of the optimal control. For example, in [7] the authors study a multi-class abandonment queue without arrivals and use the Lagrangian relaxation method [34] to construct an index policy, which is optimal for a relaxed optimization problem. Another approach is to study the system in the Halfin-Whitt heavy-traffic regime.…”