Daryl J. Daley

Comparison methods for queues and other stochastic models

Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

Population tracking of fluctuating environments and natural selection for tracking ability

The mathematics of stochastic nonlinear population models is notoriously intractable and has led some investigators to study linear approximations to make computations easier. However, nonlinear models may often better describe biological systems, and linearization may obscure dynamics of biological significance. We clarify various aspects of the dynamics of the logistic population model with a fluctuating carrying capacity. Average population size decreases with an increasing magnitude of variation in K, but N̄ is always less than or equal to K̄. This effect is mediated by the intrinsic rate of increase, r. In general, N̄ increases as r increases. This pattern should be general for models where N is a concave function of N. In environments where the magnitude of variation in K is not large, natural selection will favor genotypes which are best able to track fluctuations in K. However, when the fluctuations in K are large, natural selection may favor forms which are not highly responsive to fluctuations in K.

Long range dependence of point processes, with queueing examples

Possible definitions of the long range dependence (LRD) of a stationary point process are discussed. Examples from the standard queueing literature are considered and shown to be amenable to yielding processes with long range count dependence. In particular the effect of the single-server queueing operator, whereby one point process is transformed into another via the mechanism of a simple queue, is examined for possible long range dependence of both the counting and interval properties of the output process. For an infinite server queue, the output is long range count dependent if and only if the input is long range count dependent.

Descending chains, the lilypond model, and mutual-nearest-neighbour matching

We consider a hard-sphere model in ℝ d generated by a stationary point process N and the lilypond growth protocol: at time 0, every point of N starts growing with unit speed in all directions to form a system of balls in which any particular ball ceases its growth at the instant that it collides with another ball. Some quite general conditions are given, under which it is shown that the model is well defined and exhibits no percolation. The absence of percolation is attributable to the fact that, under our assumptions, there can be no descending chains in N. The proof of this fact forms a significant part of the paper. It is also shown that, in the absence of descending chains, mutual-nearest-neighbour matching can be used to construct a bijective point map as defined by Thorisson.

Radial basis functions with compact support for multivariate geostatistics

Matrix-valued radially symmetric covariance functions (also called radial basis functions in the numerical analysis literature) are crucial for the analysis, inference and prediction of Gaussian vector-valued random fields. This paper provides different methodologies for the construction of matrix-valued mappings that are positive definite and compactly supported over the sphere of a d-dimensional space, of a given radius. In particular, we offer a representation based on scaled mixtures of Askey functions; we also suggest a method of construction based on B-splines. Finally, we show that the very appealing convolution arguments are indeed effective when working in one dimension, prohibitive in two and feasible, but substantially useless, when working in three dimensions. We exhibit the statistical performance of the proposed models through simulation study and then discuss the computational gains that come from our constructions when the parameters are estimated via maximum likelihood. We finally apply our constructions to a North American Pacific Northwest temperatures dataset.

Classes of compactly supported covariance functions for multivariate random fields

The paper combines simple general methodologies to obtain new classes of matrix-valued covariance functions that have two important properties: (i) the domains of the compact support of the several components of the matrix-valued functions can vary between components; and (ii) the overall differentiability at the origin can also vary. These models exploit a class of functions called here the Wendland–Gneiting class; their use is illustrated via both a simulation study and an application to a North American bivariate dataset of precipitation and temperature. Because for this dataset, as for others, the empirical covariances exhibit a hole effect, the turning bands operator is extended to matrix-valued covariance functions so as to obtain matrix-valued covariance models with negative covariances.

Dimension walks and Schoenberg spectral measures

Abstract. Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ : Rd → R, φ(0) = 1, as having a representation φ(x) = ∫ R+ Ωd(tu)Gd(du), t = ‖x‖, for some uniquely identified probability measure Gd on R+ and Ωd(t) = E(e it〈e1,η〉), where η is a vector uniformly distributed on the unit spherical shell Sd−1 ⊂ Rd and e1 is a fixed unit vector. Call such Gd a d-Schoenberg measure, and let Φd denote the class of all functions f : R+ → R for which such a d-dimensional radial function φ exists with f(t) = φ(x) for t = ‖x‖. Mathéron (1965) introduced operators Ĩ and D̃, called Montée and Descente, that map suitable f ∈ Φd into Φd′ for some different dimension d′: Wendland described such mappings as dimension walks. This paper characterizes Mathéron’s operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither Ĩf nor D̃f is well defined. Because f ∈ Φd implies f ∈ Φd′ for d′ < d, any f ∈ Φd has a d′-Schoenberg measure Gd′ for 1 ≤ d′ < d and finite d ≥ 2. This paper identifies Gd′ in terms of Gd via another ‘dimension walk’ relating the Fourier transforms Ωd′ and Ωd that reflect projections on Rd ′ within Rd. A study of the Euclid hat function shows the indecomposability of Ωd.

Light traffic approximations in many-server queues

This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under light traffic conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussens (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given

Notes on Queueing Output Processes

The following notes, of which a more detailed account will appear elsewhere, survey certain features of the output process of stable queueing systems. Specifically, attention is given to distributional properties, second moment properties, and characterization problems. Kendall (1964), Reich (1965), and Cherry and Disney (1973) may be consulted for motivation and some review of the literature also.

A Light Traffic Approximation for a Single-Server Queue

It is shown that in the GI/G/1 queue, with W(S, T) denoting a stationary waiting time random variable that is determined by the generic service and interarrival times S and T,