Estimating the input of a Lévy-driven queue by Poisson sampling of the workload process

Abstract

This paper aims at semi-parametrically estimating the input process to a Lévy-driven queue by sampling the workload process at Poisson times. We construct a method-ofmoments based estimator for the Lévy process' characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimating a constant (related to both the input distribution and the sampling rate); this constant also features in the asymptotic analysis. For subordinator Lévy input, a partial MLE is constructed for the intermediate step and we show that it satisfies the consistency and asymptotic normality conditions. For general spectrally-positive Lévy input a biased estimator is proposed that only uses workload observations above some threshold; the bias can be made arbitrarily small by appropriately choosing the threshold. * To appear in Bernoulli.

Hypothesis testing for a Lévy-driven storage system by Poisson sampling

“…If Φ i , i = 1, 2, are parametric families, e.g., Φ 0 corresponds to Compound Poisson and Φ 1 corresponds to Gamma processes, then ad hoc computational methods for the GLRT computation can potentially be constructed. For the second approach, namely the conditional likelihood ratio test, this task becomes easier because the value of the suprema can be evaluated by applying the MLE method of [12]. The second challenge is the asymptotic performance analysis of the GLRT.…”

“…In the situation we are considering the systems's input is a non-decreasing Lévy process minus a deterministic drift, while the workload is sampled according at Poisson epochs. This setup has been considered before in [12], where a method was developed for consistent and asymptotically normal semi-parametric estimation of the Laplace exponent, based on workload observations at Poisson epochs. The present paper can be seen as the hypothesis testing counterpart of [12].…”

“…Evaluating the above derivatives and taking the conditional expectation with respect to V (0) immediately yields (12) and (14).…”

A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations

“…Dai [21] introduced a unified stochastic queueing network of coupled forward-backward stochastic differential equations with Lévy jumps and double completely-S skew reflections. Ravner et al [22] gave semi-parametrically estimating the input process to a Lévy-driven queue by sampling the workload process at Poisson times. Berkelmansa et al [23] dealt with the workload correlation function of a queue with general Lévy input, i.e., queue fed by a spectrally two-sided Lévy process.…”

Hypothesis testing for a Lévy-driven storage system by Poisson sampling

“…This setup has been considered before in [12], where a method was developed for consistent and asymptotically normal semi-parametric estimation of the Laplace exponent, based on workload observations at Poisson epochs. The present paper can be seen as the hypothesis testing counterpart of [12].…”