Nonparametric Inference About Service Time Distribution from Indirect Measurements

Abstract

In studies of properties of queues, for example in relation to Internet traffic, a subject that is of particular interest is the 'shape' of service time distribution. For example, we might wish to know whether the service time density is unimodal, suggesting that service time distribution is possibly homogeneous, or whether it is multimodal, indicating that there are two or more distinct customer populations. However, even in relatively controlled experiments we may not have access to explicit service time data. Our only information might be the durations of service time clusters, i.e. of busy periods. We wish to 'deconvolve' these concatenations, and to construct empirical approximations to the distribution and, particularly, the density function of service time. Explicit solutions of these problems will be suggested. In particular, a kernel-based 'deconvolution' estimator of service time density will be introduced, admitting conventional approaches to the choice of bandwidth. Copyright 2004 Royal Statistical Society.

Sojourn Time Estimation in an M/G/∞ Queue with Partial Information

“…Some of the literature reviewed in Section 1 estimated G via the busy-period process {I (Q t >0) } (see Bingham and Pitts (1999), Hall and Park (2004), and Park (2007)). A necessary condition for obtaining even the crudest estimate from such methods is to have more than one busy period in the sampling period; to obtain a reasonable estimate, quite a few busy periods are required.…”

“…Bingham and Pitts (1999) studied the same problem, and also considered estimation under information of type (ii). Hall and Park (2004), and later Park (2007), proposed a kernel-based deconvolution estimator for the density of the processing time, under information of type (ii). In a recent related work, Grübel and Wegener (2011) derived a method for matching arrivals and departures in an M/G/∞ queue, when the distribution G is known and the available data are the order statistics of the arrivals and departures.…”

Convergence rates of Laplace-transform based estimators

“…Hansen and Pitts [18] use the Pollaczek-Khinthcine formula to construct estimators for the servicetime distribution and its stationary excess distribution in an M/G/1 queue, and show that the estimated stationary excess distribution is asymptotically Normal. Some related papers that use Fourier instead of Laplace inversion are [15], [8], [9] and [17]. Van Es et al [15] estimate the density of B i by inverting the empirical Fourier transform associated with a sample of X, and prove that this estimator is weakly consistent and asymptotically normal.…”

Matchmaking and Testing for Exponentiality in the M/G/∞ Queue

“…In [4] and [11] related statistical questions are investigated, such as the possibility of consistent estimation of the correlation coefficient for bivariate normal distributions. In the delay situation, estimation of Q on the basis of the ordered arrival and departure data has been considered in the M/G/∞ context in [8] and [14]; see also the references given therein. The problem that originally motivated the present investigation comes from insurance mathematics: in the context of multivariate risk theory the class of models introduced in [6] incorporates dependence between the components by assuming that original events may trigger later claims in different business lines.…”