## Estimating the input of a queue by Poisson sampling of the workload process

Liron Ravner, University of Amsterdam

#### Abstract:

Inferring the statistical properties of queueing systems by observing the workload is a challenging task due to the elaborate dependence between transient samples. In particular, likelihood functions are typically intractable. We consider the problem of estimating the exponent function of the Laplace-Stieltjes Transform (LST) of the input process to a Lévy driven queue. The workload of the queue is observed at random times according to an independent Poisson process. We suggest a non-parametric estimation method that relies on the empirical Laplace transform of the workload. This is achieved by using a generalized method of moments approach on the conditional LST of the workload sampled after an exponential time. The consistency of the method requires an intermediate step of estimating a constant that is related to both the input distribution and the sampling rate. To this end, for the case of an M/G/1 queue, we construct a partial maximum likelihood estimator and show that it is consistent and asymptotically normal. For spectrally positive Lévy input we construct a biased estimator for the intermediate step by considering only high workload observations above some threshold. A bound on the bias is provided and we discuss the tradeoff between the bias and variance of the estimator with respect to the chosen threshold. We further discuss how this framework can be useful in applications such as stability detection and dynamic pricing.

Joint work with Onno Boxma and Michel Mandjes