Nonparametric estimation of service time distribution in the $M/G/\infty$ queue and related estimation problems

TL;DR

This paper develops a nonparametric estimator for the service time distribution in an $M/G/\infty$ queue based on incomplete queue-length data, demonstrating its near-optimal accuracy and relating it to covariance function derivative estimation.

Contribution

It introduces a new estimator for the service time distribution in $M/G/\infty$ queues and proves its rate optimality through minimax risk bounds.

Findings

The estimator achieves near-optimal accuracy.

Lower bounds confirm the estimator's rate optimality.

The problem relates to covariance derivative estimation.

Abstract

The subject of this paper is the problem of estimating service time distribution of the $M/G/\infty$ queue from incomplete data on the queue. The goal is to estimate $G$ from observations of the queue--length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. The original $M/G/\infty$ problem is closely related to the problem of estimating derivatives of the covariance function of a stationary Gaussian process. We consider the latter problem and derive lower bounds on the minimax risk. The obtained results strongly suggest that the proposed estimator of the service time distribution is rate optimal.