Service Rate Control For Jobs with Decaying Value

TL;DR

This paper studies optimal service rate control for jobs with decaying value, providing algebraic conditions for policy monotonicity and practical verification methods, applicable in healthcare, communications, and inventory management.

Contribution

It introduces algebraic conditions to determine when the optimal policy is monotonic in residual job value, simplifying analysis without extensive computation.

Findings

Optimal policy is non-decreasing in the number of jobs remaining.

Conditions for monotonicity depend on algebraic inequalities.

Simplified conditions are provided for constant reward cases.

Abstract

The task of completing jobs with decaying value arises in a number of application areas including healthcare operations, communications engineering, and perishable inventory control. We consider a system in which a single server completes a finite sequence of jobs in discrete time while a controller dynamically adjusts the service rate. During service, the value of the job decays so that a greater reward is received for having shorter service times. We incorporate a non-decreasing cost for holding jobs and a non-decreasing cost on the service rate. The controller aims to minimize the total cost of servicing the set of jobs. We show that the optimal policy is non-decreasing in the number of jobs remaining -- when there are more jobs in the system the controller should use a higher service rate. The optimal policy does not necessarily vary monotonically with the residual job value, but we give algebraic conditions which can be used to determine when it does. These conditions are then simplified in the case that the reward for completion is constant when the job has positive value and zero otherwise. These algebraic conditions are interesting because they can be verified without using algorithms like value iteration and policy iteration to explicitly compute the optimal policy. We also discuss some future modeling extensions.