The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - α ||x||, ||x|| < 1, to observed correlations at (appropriately scaled) spatial lag x in d-dimensional space. We show that such an approach is permissible if and only if

the upper bound depending on the spatial dimension d. The proof relies on Matheron's turning bands operator and an extension theorem for positive definite functions due to Rudin. Side results and examples include a general discussion of isotropic correlation functions defined on d-dimensional balls.

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probe increases efficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.

The paper investigates some properties of the first-order bilinear time series model: stationarity and invertibility. Estimates of the parameters are obtained by a modified least squares method and shown to be strongly consistent.

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.

A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.