Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers

TL;DR

This paper develops a quasi-polynomial-time approximation scheme for optimally serving customers who depart stochastically, achieving near-optimal expected rewards in a multi-stage setting.

Contribution

It introduces the first near-optimal adaptive policy algorithm with rigorous guarantees for a stochastic customer departure model.

Findings

The algorithm finds policies within 1 - epsilon of optimal in quasi-polynomial time.

The method combines stochastic analysis to understand parameter impacts on optimal policies.

Provides a framework for near-optimal adaptive decision-making in stochastic departure scenarios.

Abstract

We consider a multi-stage stochastic optimization problem originally introduced by Cygan et al. (2013), studying how a single server should prioritize stochastically departing customers. In this setting, our objective is to determine an adaptive service policy that maximizes the expected total reward collected along a discrete planning horizon, in the presence of customers who are independently departing between one stage and the next with known stationary probabilities. In spite of its deceiving structural simplicity, we are unaware of non-trivial results regarding the rigorous design of optimal or truly near-optimal policies at present time. Our main contribution resides in proposing a quasi-polynomial-time approximation scheme for adaptively serving impatient customers. Specifically, letting $n$ be the number of underlying customers, our algorithm identifies in $O( n^{ O_{ \epsilon }( \log^2 n ) } )$ time an adaptive service policy whose expected reward is within factor $1 - \epsilon$ of the optimal adaptive reward. Our method for deriving this approximation scheme synthesizes various stochastic analyses in order to investigate how the adaptive optimum is affected by alteration to several instance parameters, including the reward values, the departure probabilities, and the collection of customers itself.