The $M/G/\infty$ estimation problem revisited

Abstract

The subject of this paper is the M/G/∞ estimation problem: the goal is to estimate the service time distribution G of the M/G/∞ queue from the arrival-departure observations without identification of customers. We develop estimators of G and derive exact non-asymptotic expressions for their mean squared errors. The problem of estimating the service time expectation is addressed as well. We present some numerical results on comparison of different estimators of the service time distribution.

Dispersal density estimation across scales

“…We first analyse the interaction between parents X and children Y via the correlation structure between the measures M (dx) and N (dy). In Proposition 1 in Section 2.1 below, building on the approach of Goldenshluger (2018), we establish the formula…”

“…Example 1: Service time estimation in M/G/∞ queuing models Dating back to Brown (1970), certain M/G/∞ queuing models are embedded in our approach, see in particular the recents results of Goldenshluger (2016Goldenshluger ( , 2018; Goldenshluger and Koops (2019).…”

“…The starting point of our estimation approach lies in the analysis of the correlation structure between M and N , that builds upon the approach by Goldenshluger (2018). In our study, the fact that the parent data X are distributed on a bounded interval has a considerable impact on the correlation structure.…”

“…Note that the convolution term in ( 7) is uninformative if M is a homogeneous Poisson point process on whole real line as in Goldenshluger (2018) or on the torus as in Hunt et al (2018).…”

“…We discuss this approach in Section 3.3 in our context. An interaction estimator was also applied by Goldenshluger (2018) in a setting where the intensity measure of N is the Lebesgue measure on whole real line and which corresponds to nσ = 1. As we will see below the interaction estimator is still applicable if σ > 1/n as long as σ is not too large.…”