Nonparametric Estimation of the Service Time Distribution in the Discrete-Time $GI/G/\infty$ Queue with Partial Information

TL;DR

This paper develops a nonparametric method to estimate service time distributions in discrete-time $GI/G/$ queues using only arrival and departure data, extending existing results with a new CLT and bootstrap approach.

Contribution

It introduces a functional CLT for the sequence of differences estimator and demonstrates the applicability of the moving block bootstrap for covariance estimation.

Findings

Proves a functional central limit theorem for the estimator.

Shows the bootstrap method works under mild conditions.

Extends existing results for nonparametric queue analysis.

Abstract

Estimation of the service time distribution in the discrete-time $GI/G/\infty$-queue based solely on information on the arrival and departure processes is considered. The focus is put on the estimation approach via the so called "sequence of differences". Existing results for this approach are substantially extended by proving a functional central limit theorem for the resultant estimator. Here, the underlying function space is taken to be the space of sequences converging to zero. The moving block bootstrap technique is considered for the estimation of the resultant covariance kernel and is shown to be applicable under mild additional conditions.