bluuebunny/crossref_metadata_2025 huggingface dataset:
Estimating Functionals of a Stochastic Process
Authors: Istas Jacques , Laredo Catherine
Published: April 2006

The problem of estimating the integral of a stochastic process from observations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied this problem for processes with mean 0 and Hölder index K + ½, K ; ℕ These results are here extended to processes with arbitrary Hölder index. The estimators built here are linear in the observations and do not require the a priori knowledge of the smoothness of the process. If the process satisfies a Hölder condition with index s in quadratic mean, we prove that the rate of convergence of the mean square error is N2s+1 as N goes to ∞, and build estimators that achieve this rate.
Size-and-shape distributions for paired landmark data
Authors: Mardia K. V. , Walder A. N.
Published: April 2006

In this paper we study the distribution associated with paired size-and-shape for two sets of correlated points having an underlying Gaussian distribution in the plane. This extends work previously done just for paired shape. Special cases are studied, and various properties described. In addition we give further results for the paired shape density.
Second-order stereology for planar fibre processes
Authors: Weiss V. , Nagel W.
Published: April 2006

Three different stereological methods for the determination of second-order quantities of planar fibre processes which have been suggested in the literature are considered. Proofs of the formulae are given (also by using a new integral geometric formula), relations between the methods are derived and the prerequisites are discussed. Furthermore, edge-corrected unbiased estimators for the second-order quantities are given.
On the theory of Markov set-chains
Authors: Hartfiel D. J. , Seneta E.
Published: April 2006

In the theory of homogeneous Markov chains, states are classified according to their connectivity to other states and this classification leads to a classification of the Markov chains themselves. In this paper we classify Markov set-chains analogously, particularly into ergodic, regular, and absorbing Markov set-chains. A weak law of large numbers is developed for regular Markov set-chains. Examples are used to illustrate analysis of behavior of Markov set-chains.
Approximations of large trunk line systems under heavy traffic
Authors: Kushner Harold J.
Published: April 2006

The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.
A sample path approach to mean busy periods for Markov-modulated queues and fluids
Authors: Asmussen Søren , Bladt Mogens
Published: April 2006

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.
A flow conservation law for surface processes
Authors: Last G. , Schassberger R.
Published: April 2006

The object studied in this paper is a pair (Φ,Y), where Φ is a random surface inandYa random vector field on. The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector fieldYis smooth outside Φ but may have discontinuities on Φ. Gauss' divergence theorem is applied to derive a flow conservation law forY.Forthis specializes to a well-known rate conservation law for point processes. As an application, relationships for the linear contact distribution of Φ are derived.
Spectral analysis ofM/G/1 andG/M/1 type Markov chains
Authors: Gail H. R. , Hantler S. L. , Taylor B. A.
Published: April 2006

When analyzing the equilibrium behavior ofM/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class ofG/M/1 type Markov chains by similar methods is also presented.
Throughput properties of a queueing network with distributed dynamic routing and flow control
Authors: Tassiulas Leandros , Ephremides Anthony
Published: April 2006

A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.
Limit theorems for the time of completion of Johnson-Mehl tessellations
Authors: Chiu S. N.
Published: April 2006

Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝd and k-dimensional sectional tessellations, where 1 ≦ k < d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.
Comparison of replacement policies via point processes
Authors: Shaked Moshe , Szekli Ryszard
Published: April 2006

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.
Asymptotic Expansions in the Exponent: a Compound Poisson Approach
Authors: Čekanavičius V.
Published: April 2006

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.
Stochastic Orders Generated by Integrals: a Unified Study
Authors: Müller Alfred
Published: April 2006

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.
Estimating the reduced moments of a random measure
Authors: Kiêu Kiên , Mora Marianne
Published: April 2006

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.
Conditional simulation of a Boolean model
Authors: Lantuéjoul Ch.
Published: April 2006

A Boolean model in ℝd is the union of independent random compact subsets of ℝd (called ‘grains’) located according to a Poisson point process. A sequential algorithm is proposed to conditionally simulate a Boolean model, i.e. so that a finite set of specified points belongs to the grains and another set to the complementary phase (referred to as the ‘pores’).
Methods of simulating random patterns of non-spherical objects and their application
Authors: Tanemura Masaharu
Published: April 2006

We consider two mechanisms for simulating spatial patterns of hard-core non-spherical particles, namely the random sequential packing (RSP) and the Markov chain Monte Carlo (MCMC) procedures. The former is described as follows: we put a particle one-by-one into a finite region by sampling its location x and direction θ uniformly at random; if it does not overlap with other particles put before, it is put successfully, otherwise, we discard it and try another uniform sampling of (x, θ); by repeating the above, we can obtain a set of non-overlapping particles. The MCMC procedure is the following: we first give a certain non-overlapping pattern of non-spherical particles prepared in a random or a regular manner; then we select a particle and sample its new trial location x and direction θ at random; if the new sample (x, θ) is accepted, i.e. it does not overlap with other particles, the selected particle is moved to the new ‘position’, otherwise the particle is retained at the old position; by repeating the above, a series of a set of non-overlapping particles is generated.
Multiscaling and random cascades
Authors: Waymire Edward C.
Published: April 2006

The study of spatial distributions which arise from independent multiplicative cascades can be traced back at least to the efforts by the Russian school led by Andrei N. Kolmogorov on the statistical theory of turbulence. However, largely owing to the intriguing statistical/geometric scaling properties so heavily emphasized by Benoit Mandelbrot, and to the rich mathematical foundations begun by Jean-Pierre Kahane and Jacques Peyriere in the middle 1970's and continuing to the present, there has been a growing interest in random cascades both in the physical sciences and in mathematics. Our objective is to review some of the principal aspects of the theory and applications of independent cascades as well as some recent extensions.
Markov properties of cluster processes
Authors: Baddeley A. J. , Van Lieshout M. N. M. , Møller J.
Published: April 2006

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.
The sectional Poisson Voronoi tessellation is not a Voronoi tessellation
Authors: Chiu S. N. , Van De Weygaert R. , Stoyan D.
Published: April 2006

Is the intersection between an arbitrary but fixed plane and the spatial Poisson Voronoi tessellation a planar Voronoi tessellation? In this paper a negative answer is given to this long-standing question in stochastic geometry. The answer remains negative for the intersection between at-dimensional linear affine space and thed-dimensional Poisson Voronoi tesssellation, where 2 ≦t≦d− 1. Moreover, it is shown that each cell on this intersection is almost surely a non-Voronoi cell.
Perpetuities with thin tails
Authors: Goldie Charles M. , Grübel Rudolf
Published: April 2006

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.
Random discrete distributions invariant under size-biased permutation
Authors: Pitman Jim
Published: April 2006

Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.
Queues with marked customers
Authors: He Qi-Ming
Published: April 2006

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.
The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d
Authors: Appel Martin J. B. , Russo Ralph P.
Published: April 2006

On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.
Whole Genome Amplification and Branching Processes
Authors: Sun Fengzhu , Waterman Michael S.
Published: April 2006

Whole genome amplification is important for multipoint mapping by sperm or oocyte typing and genetic disease diagnosis. Polymerase chain reaction is not suitable for amplifying long DNA sequences. This paper studies a new technique, designated PEP-primer-extension-preamplification, for amplifying long DNA sequences using the theory of branching processes. A mathematical model for PEP is constructed and a closed formula for the expected target yield is obtained. A central limit theorem and a strong law of large numbers for the number of kth generation target sequences are proved.
Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk
Authors: Fuh C. D.
Published: April 2006

Let (X, S) = {(Xn, Sn); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | Xo= i}, and a two-barrier probability P{ST ≧b | Xo = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.
Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times
Authors: Lee Shiowjen , Lynch J.
Published: April 2006

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.
Asymptotic Estimates for Blocking Probabilities in a Large Multi-Rate Loss Network
Authors: Simonian A. , Roberts J. W. , Théberge F. , Mazumdar R.
Published: April 2006

In this paper, asymptotic estimates for the blocking probability of a call pertaining to a given route in a large multi-rate circuit-switched network are given. Concentrating on low load and critical load conditions, these estimates are essentially derived by using probability change techniques applied to the distribution of the number of occupied links. Such estimates for blocking probabilities are also given a uniform expression applicable to both load regimes. This uniform expression is numerically validated via simple examples.
Statistics of the Boolean model: from the estimation of means to the estimation of distributions
Authors: Molchanov Ilya S.
Published: April 2006

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.
On the volume of parallel bodies: a probabilistic derivation of the Steiner formula
Authors: Vitale Richard A.
Published: April 2006

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.
The Estimation of mean shape and mean particle number in overlapping particle systems in the plane
Authors: Weil Wolfgang
Published: April 2006

A stationary (but not necessarily isotropic) Boolean modelYin the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensityyof the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union setY.The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body ofY, which in turn makes it possible to estimate the mean grain of the particle process.
Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state
Authors: Pakes Anthony G.
Published: April 2006

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose Pi(T < ∞) ≡ 1 and (*) limi→∞Pi(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff Ei (e∊T) < ∞ for some ∊ > 0.

The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.
Dependencies in Markovian networks
Authors: Daduna H. , Szekli R.
Published: April 2006

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.
Asymptotic analysis of single resource loss systems in heavy traffic, with applications to integrated networks
Authors: Bean N. G. , Gibbens R. J. , Zachary S.
Published: April 2006

In this paper we consider the analysis of call blocking at a single resource with differing capacity requirements as well as differing arrival rates and holding times. We include in our analysis trunk reservation parameters which provide an important mechanism for tuning the relative call blockings to desired levels. We base our work on an asymptotic regime where the resource is in heavy traffic. We further derive, from our asymptotic analysis. methods for the analysis of finite systems. Empirical results suggest that these methods perform well for a wide class of examples.
Line transects, covariance functions and set convergence
Authors: Cabo A. J. , Baddeley A. J.
Published: April 2006

We define the ‘linear scan transform' G of a set in ℝd using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric η for ‘regular' subsets of ℝd defined as the L1 distance between their linear scan transforms. For convex sets only, η is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.
Large-deviation expressions for the distribution of first-passage coordinates
Authors: Whittle P.
Published: April 2006

We consider the distribution of the free coordinates of a time-homogeneous Markov process at the time of its first passage into a prescribed stopping set. This calculation (for an uncontrolled process) is of interest because under some circumstances it enables one to calculate the optimal control for a related controlled process. Scaling assumptions are made which allow the application of large deviation techniques. However, the first-order evaluation obtained by these techniques is often too crude to be useful, and the second-order correction term must be calculated. An expression for this correction term as an integral over time is obtained in Equation (20). The integration can be performed in some cases to yield the conclusions of Theorems 1 and 2, expressed in Equations (7) and (9). Theorem 1 gives the probability density of the state vector (to the required degree of approximation) at a prescribed time for a class of processes we may reasonably termlinear.Theorem 2 evaluates (without any assumption of linearity) the ratio of this density to the probability density of the coordinates under general stopping rules.
Markov population replacement processes
Authors: Gerontidis Ioannis I.
Published: April 2006

We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.
Event coupling and performance sensitivity analysis of generalized semi-Markov processes
Authors: Cao Xi-Ren
Published: April 2006

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.
Analysis of an M/G/1 queue with two types of impatient units
Authors: Martin M. , Artalejo J. R.
Published: April 2006

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later.

We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.
The range of a simple random walk on ℤ
Authors: Vallois P.
Published: April 2006

Letθ(a) be the first time when the range (Rn;n≧ 0) is equal toa, Rnbeing equal to the difference of the maximum and the minimum, taken at timen, of a simple random walk on ℤ. We compute the g.f. ofθ(a); this allows us to compute the distributions ofθ(a) andRn.We also investigate the asymptotic behaviour ofθ(n),ngoing to infinity.
A central limit theorem for the parsimony length of trees
Authors: Steel Mike , Goldstein Larry , Waterman Michael S.
Published: April 2006

In phylogenetic analysis it is useful to study the distribution of the parsimony length of a tree under the null model, by which the leaves are independently assigned letters according to prescribed probabilities. Except in one special case, this distribution is difficult to describe exactly. Here we analyze this distribution by providing a recursive and readily computable description, establishing large deviation bounds for the parsimony length of a fixed tree on a single site and for the minimum length (maximum parsimony) tree over several sites. We also show that, under very general conditions, the former distribution converges asymptotically to the normal, thereby settling a recent conjecture. Furthermore, we show how the mean and variance of this distribution can be efficiently calculated. The proof of normality requires a number of new and recent results, as the parsimony length is not directly expressible as a sum of independent random variables, and so normality does not follow immediately from a standard central limit theorem.
The secretary problem: minimizing the expected rank with I.I.D. random variables
Authors: Assaf David , Samuel-Cahn Ester
Published: April 2006

n candidates, represented by n i.i.d. continuous random variables X1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X1, …, Xn) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn, and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak, let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.
The quasi-stationary distribution of the closed endemic sis model
Authors: Nåsell Ingemar
Published: April 2006

The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratioR0passes the deterministic threshold value 1. Approximations are derived that describe these changes. The quasi-stationary distribution is approximated by a geometric distribution (discrete!) forR0distinctly below 1 and by a normal distribution (continuous!) forR0distinctly above 1. Uniformity of the approximation with respect toR0allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.
On invariant measures for simple branching processes
Authors: Seneta E.
Published: April 2006

This paper was initially motivated by a problem raised earlier by the author (Seneta (1969), Section 5.3) viz. that of the existence and uniqueness of an invariant measure for a supercritical Galton-Watson process with immigration; and, indeed, in the sequel we show that such a measure always exists, but is not in general unique.
Multiplicative population processes
Authors: Moyal J. E.
Published: April 2006

The theory ofmultiplicative(orbranching)population processeswhere the states of each individual in the population range over a fixed finite set hasbeen studied by a number of authors: see for example, Everett and Ulam, Kolmogorov and Sevastyanov, Harris; the continuous parameter case is treated in Arley; further references will be found in the books by Bartlett and Bharucha-Reid. The purpose of the present paper is to study the theory of such processes in the case where the individual state space is arbitrary (i.e., an abstract space).
Delays to road traffic at an intersection
Authors: Yeo G. F. , Weesakul B.
Published: April 2006

A model for road traffic delays at intersections is considered where vehicles arriving, possibly in bunches, in a Poisson process in a one way minor road yield right of way to traffic, which forms alternate bunches and gaps, in a major road. The gap acceptance times are random variables, and depend on whether or not a minor road vehicle is immediately following another minor road vehicle into the intersection or not.

The transforms of the stationary waiting time and queue size distributions, and the mean stationary delay, for minor road vehicles are obtained by substitution of determined service time distributions into results for a generalisation of theM/G/1 queueing system. Some numerical results are given to illustrate the increase in the mean delay for variable gap acceptance times for a Borel-Tanner distribution of major road traffic, and a partial solution is given for a two way major road.
A class of distribution function processes which have derivatives
Authors: Kraft Charles H.
Published: April 2006

In the .author and van Eeden considered, as prior distributions for the cumulative, F, of the bio-assay problem, processes whose sample functions are, with probability one, distribution functions. The example we considered there had the undesirable property that its mean, E(F), was singular with respect to Lebesgue measure. In fact, Dubins and Freedman have shown that a class of such processes, which includes the example we considered, has sample functions F which are, with probability one, singular.
Incomplete discontinuous Markov processes
Authors: Moyal J. E.
Published: April 2006

The present paper generalizes the results of the author's paper [2] on “Discontinuous Markov processes” to processes that are incompletely specified. These results are then used to develop a step-by-step method of solution for processes involving several kinds of “jumps”, where at the first step one solves for the transition probabilities involving jumps of the first kind, but not of the second, third, etc., at the second step for transition probabilities involving jumps of the first and second, but not the third, fourth, etc. kinds, and so on. Some examples are given.
A renewal theoretic model for chronic disease statistics
Authors: Berman Simeon M.
Published: April 2006

The health of an individual during his lifetime may be considered as a stochastic process over time. He is usually born free of disease; at some timexafter his birth, he becomes the victim of a mortal disease or accident, and dies at timey ≧ x.Comparative studies of chronic diseases try to evaluate the relative forces of diseases in a population, and to estimate the survival timey − x.One type of study is of the “time-oriented” type: a group of individuals is followed by a periodic survey over several years. Several papers on these problems, and comprehensive bibliographies, are contained in the volume [3].
A mathematical analysis of the stepping stone model of genetic correlation
Authors: Weiss George H. , Kimura Motoo
Published: April 2006

In this paper we analyze the correlation coefficient between members of any two colonies out of an infinite ensemble of colonies. It is assumed that each colony has the same number of members, that migration takes place between the different colonies, and that there is a constant rate of mutation in each colony. An explicit formula is derived for the correlation function and the long distance form of this function is derived. It is shown that under rather weak restrictions on the pattern of migration the asymptotic form of the correlation function is characteristic of the dimension of the model.
On the solution of differential equations arising in some attachment models of virology
Authors: Morgan Byron J. T.
Published: April 2006

(1.1) In this paper, acquaintance will be assumed with the basic mechanics of phage/bacterium, antibody/virus interactions; for background reading see, for example, Adams (1959), Bouanchaud (1970), Durham and King (1969) and Fraser (1967).
Physical nearest-neighbour models and non-linear time-series
Authors: Bartlett M. S.
Published: April 2006

A general class of spatial-temporal Markov processes is defined leading to the standard spatial equilibrium distribution for nearest-neighbour models on a multi-dimensional lattice. Physical properties are obtainable from the marginal spatial spectral function. However, only the simplest one-dimensional case corresponds to a linear model with a readily derived spectrum. Non-linear models corresponding to two- and three-dimensional lattices are presented in their simplest terms, and a preliminary discussion of approximate solutions is included.
Busy periods in vehicle queues
Authors: Shaw Leonard
Published: April 2006

Queueing models have been used to describe the slowing or stopping of a stream of vehicles due to a traffic light, a merging disturbance, or an accident. The buildup of stopped vehicles, and their eventual dispersal after the delaying factor has been removed, has been likened to the busy period of a queue, e.g., Jewell (1964). Some studies have dealt with the total delay to all drivers involved in such a queue, Daley and Jacobs (1969), Gaver (1969); or the combination of this total delay and the distance of the last vehicle from the originating incident, McNeil (1969), Shaw (1970).
Equilibrium behavior of population genetic models with non-random mating. Part II: Pedigrees, Homozygosity and Stochastic Models
Authors: Karlin Samuel
Published: April 2006

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.
Extreme nth moments for distributions on [0, 1] and the inverse of a moment space map
Authors: Skibinsky Morris
Published: April 2006

Let n be a positive integer. Denote by Mn the convex, closed, bounded, and n-dimensional set of all n-tuples (c1, c2,···, cn) such that

for some probability measure a on the Borel subsets of the unit interval I = [0,1].
A two-unit standby redundant system with repair and preventive maintenance
Authors: Osaki Shunji , Asakura Tatsuyuki
Published: April 2006

A two-unit standby redundant system with repair and preventive maintenance is discussed. Assuming that the failure, repair, inspection, and preventive maintenance (preventive repair) time distributions of each unit for the system are all arbitrary, the Laplace-Stieltjes transform (L.S.T.) of the distribution of time to the first system down is derived and the mean time obtained from it. It is further shown that under suitable assumptions, the preventive maintenance policy for the system is effective in the sense that it increases the mean time to the first system down.
Selective interaction of a Poisson and renewal process: first-order stationary point results
Authors: Lawrance A. J.
Published: April 2006

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.
A note on the Ehrenfest multiurn model
Authors: Severo Norman C.
Published: April 2006

Denote bySN,rthe set ofr-tuplesnhaving non-negative integer components summing toN,and byeither-tuple having a one in theith position and zeros elsewhere. Formandnbelonging toSN r,we letP(n, t|m, s) represent the transition probability of moving from statemat timesto statenat timet. Bartlett's conservative process [1] assumes that forn–ei+ejandn∈SN,rand Δt> 0where the λij(t) are non-negative continuous functions oft.
The distribution of radioactive recovery in randomly cut and sedimented DNA
Authors: Litwin Samuel
Published: April 2006

Viral DNA is studied by the technique of radioactive labelling and subsequent ultra centrifugation which separates it from reference DNA according to length or density. One hypothesis that frequently is of service is that the strands of DNA being studied are randomly cut along their lengths. Often the average number of cuts, λ, per unit length is either known by the nature of the experiment or is desired as a statistic. If radioactive phosphorous, P32, is used to label the DNA, e.g., 6 P32 atoms per unit length of 200,000 nucleotides (each of which is connected to its neighbor by two phosphorous-containing links) then within two weeks about 3 P32 atoms will have decayed and the DNA obtained, once separated into single stranded form (denatured), will form a predictable distribution of sizes. Of course 3 decays may be obtained more quickly by starting with more than 6 P32 atoms per unit length.
Asymptotic properties and equilibrium conditions for branching Poisson processes
Authors: Lewis P.A.W.
Published: April 2006

Some previously obtained asymptotic results for branching Poisson processes are extended and sharpened. It is shown that under rather general conditions the number of events in both the transient and the equilibrium processes, suitably normalized, have a unit normal distribution. Finally, unique initial conditions are derived for the equilibrium process.
On infinite server queues with batch arrivals
Authors: Shanbhag D. N.
Published: April 2006

The queueing system studied in this paper is the one in which

(i)

there are an infinite number of servers,

(ii)

initially (att= 0) all the servers are idle,

(iii)

one server serves only one customer at a time and the service times are independent and identically distributed with distribution functionB(t) (t> 0) and mean β(< ∞),

(iv)

the arrivals are in batches such that a batch arrives during (t,t+ δt) with probability λ(t)δt+o(δt) (λ(t) > 0) and no arrival takes place during (t,t+ δt) with the probability 1 –λ(t)δt+o(δt),

(v)

the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given byar(r≧ 1),

(vi)

the batch sizes, the service times and the arrivals are independent.
Another approach to some Markov chain models in population genetics
Authors: Lillestøl Jostein
Published: April 2006

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.
A survey of some recent Czechoslovak work in automatic statistical process control
Authors: Žaludová A. H. , Režný Z. , Ullrich M.
Published: April 2006

The article first reviews briefly the field of statistical process control and its relation to automatic process control. Some recent results obtained by various workers in Czechoslovakia in the field of statistical aspects of automatic process control are then presented.
Two queues in series with a finite, intermediate waitingroom
Authors: Neuts Marcel F.
Published: April 2006

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full.

The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.
Use of erlangian distributions for single-server queueing systems
Authors: Kotiah T. C. T. , Thompson J. W. , Waugh W. A. O'N.
Published: April 2006

The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.
Branching and clustering models associated with the ‘lost-games distribution’
Authors: Kemp Adrienne W. , Kemp C. D.
Published: April 2006

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variablesXiwith probability generating functionsgi(s), i= 1, 2, 3, ifg3(s) =g1[g2(s)], we say thatX3isX1generalized byX2and writeX3~X1VX2.
Characterization of distributions by the identical distribution of linear forms
Authors: Eaton Morris L.
Published: April 2006

Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.
Solutions of ordinary differential equations as limits of pure jump markov processes
Authors: Kurtz Thomas G.
Published: April 2006

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation

used to describe a number of processes including radioactive decay and population growth.