Service Scheduling for Random Requests with Quadratic Waiting Costs

TL;DR

This paper analyzes service scheduling with quadratic waiting costs in a stochastic environment, characterizing optimal policies, Nash equilibria, and proposing algorithms for unknown parameters.

Contribution

It provides explicit linear optimal policies, characterizes Nash equilibria in scheduling games, and introduces parameter estimation algorithms for complex quadratic cost systems.

Findings

Optimal policy is linear in system state.

Nash equilibria are derived for scheduling games.

Algorithm for parameter estimation with bounded cost difference.

Abstract

We study service scheduling problems in a slotted system in which agents arrive with service requests according to a Bernoulli process and have to leave within two slots after arrival, service costs are quadratic in service rates, and there are also waiting costs. We consider quadratic waiting costs. We frame the problems as average cost Markov decision processes. While the studied system is a linear system with quadratic costs, it has state dependent control. Moreover, it also possesses a non-standard cost function structure in the case of fixed waiting costs, rendering the optimization problem complex. We characterize optimal policy. We provide an explicit expression showing that the optimal policy is linear in the system state. We also consider systems in which the agents make scheduling decisions for their respective service requests keeping their own cost in view. We consider quadratic waiting costs and frame these scheduling problems as stochastic games. We provide Nash equilibria of this game. To address the issue of unknown system parameters, we propose an algorithm to estimate them. We also bound the cost difference of the actual cost incurred and the cost incurred using estimated parameters.