Nonparametric estimation of the job-size distribution for an M/G/1 queue with Poisson sampling

Abstract

This work presents a non-parametric estimator for the cumulative distribution function (CDF) of the job-size distribution for a queue with compound Poisson input. The workload process is observed according to an independent Poisson sampling process. The nonparametric estimator is constructed by first estimating the characteristic function (CF) and then applying an inversion formula. The convergence rate of the CF estimator at $s$ is shown to be of the order of $s^2/n$, where $n$ is the sample size. This convergence rate is leveraged to explore the bias-variance tradeoff of the inversion estimator. It is demonstrated that within a certain class of continuous distributions, the risk, in terms of MSE, is uniformly bounded by $C n^{-\frac{\eta}{1+\eta}}$, where $C$ is a positive constant and the parameter $\eta>0$ depends on the smoothness of the underlying class of distributions. A heuristic method is further developed to address the case of an unknown rate of the compound Poisson input process.