bluuebunny/crossref_metadata_2025 huggingface dataset:
A solution to the generalized birthday problem with application to allozyme screening for cell culture contamination
Authors: Gail M. H. , Weiss G. H. , Mantel N. , O'Brien S. J.
Published: April 2006

This paper gives the probability that no common allozyme signature is found amongnrandomly selected cell culture lines when thesmutually exclusive and exhaustive signatures have arbitrary known probabilities. This result is useful in detecting cell culture contamination by a single cell line. This problem is a generalization of the classic problem of finding the probability of no common birthday amongnindividuals when it is assumed that each individual has a chance 1/s= 1/365 of a birthday on a given day. Both an exact recursive solution and an approximate solution are given for the general problem.
On the characterization of point processes with the order statistic property
Authors: Feigin Paul D.
Published: April 2006

We provide a probabilistic proof of the characterization of point processes (on the real line) with the order statistic property. The characterization is used to investigate the homogeneity of such processes and is also related to the martingale theory associated with point processes.
Heavy traffic theory for queues with several servers. II
Authors: Köllerström Julian
Published: April 2006

The queues being studied here are of the typeGI/G/kin statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.
A remark on the central limit question for dependent random variables
Authors: Bradley Richard C.
Published: April 2006

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.
Strategy stability in complex populations
Authors: Hines W. G. S.
Published: April 2006

In biological populations, several strategies for engaging in intraspecific contests may be present. Investigations of stability show that the mean strategy in use in the population can prove stable but that neutrally stable and gradually changing polymorphisms in strategy are to be expected when the contest affects reproductive success.
Invariant imbedding and dams with Markovian input rate
Authors: Brockwell P. J. , Pacheco-Santiago N.
Published: April 2006

The method of invariant imbedding, used extensively in astrophysics and in neutron transport theory, is applied to the analysis of dams whose input rate is a finite state-space continuous-parameter Markov chain. Passage-time distributions are considered as functions of the dam capacity. When the release rate is constant the method yields first-order non-linear differential equations with initial conditions obtained from the easily derived behaviour of a dam with zero capacity. Solutions of these equations are given. Some possible correlation functions for the increments of the cumulative input process are examined when the underlying Markov chain is stationary. Finally the analysis is extended to deal with dams having content-dependent release-rate of the form
A positive recurrence criterion associated with multidimensional queueing processes
Authors: Rosberg Zvi
Published: April 2006

A criterion is given for positive recurrence of a multidimensional, aperiodic, irreducible Markov chain with a denumerable state space. This criterion extends to the multidimensional case Foster's one-dimensional criterion. The multidimensional criterion consists of several conditions, one for each coordinate of the process. The usefulness of this criterion is shown through a queueing network example.
Note an a death process
Authors: De Vries S. A. , Stam A. J. , Reddingius J.
Published: April 2006

The forward equations for the death process in ℕ2withQ-matrix determined byq(i, j;i− 1,j)= αij, i≧ 1,q(i, j;i, j− 1) =μj, j≧ 1 are solved by giving a special interpretation to thisQ-matrix.
The stereology of location
Authors: Davy Pamela J.
Published: April 2006

It is shown how to estimate centroids and other measures of location of an unknown embedded subset of a bounded specimen by random sectioning.
On two marked point processes
Authors: Rüschendorf Ludger
Published: April 2006

Two examples for marked point processes are discussed and some characteristic parameters of these models are calculated. Both examples are in some way modifications of the counter models which are well known and treated in several textbooks.
A controlled birth and death process model of optimal product pricing under stochastically changing demand
Authors: Deshmukh S. D. , Winston Wayne
Published: April 2006

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.
Asymptotic behaviour of the Ek/G/1 queue with finite waiting room
Authors: Hokstad Per
Published: April 2006

The asymptotic behaviour of the Ek/G/1 queue with finite waiting room is studied. Using a combination of the supplementary variable and phase techniques, queue length and waiting time distributions are obtained. Also the idle period distribution and mean length of the idle and busy periods are found.
A note on the non-homogeneous poisson cluster process
Authors: Westcott Mark
Published: April 2006

It is proved that, in a non-homogeneous Poisson cluster process whose cluster centre rate is λ(t), the long-term behaviour of several associated quantities is equivalent to their behaviour in a homogeneous process with centre rate Λ and the same cluster structure if λ(t)→ λ ast→∞.
Ergodicity of diffusion and temporal uniformity of diffusion approximation
Authors: Norman M. Frank
Published: April 2006

Let {XN(t), t ≧ 0}, N = 1, 2, … be a sequence of continuous-parameter Markov processes, and let TN(t)f(x) = Ex[f(XN(t))]. Suppose that limN→∞TN(t)f(x)= T(t)f(x), and that convergence is uniform over x and over t ∈ [0, K] for all K < ∞. When is convergence uniform over t ∈ [0, ∞)? Questions of this type are considered under the auxiliary condition that T(t)f(x) converges uniformly over x as t → ∞. A criterion for such ergodicity is given for semigroups T(t) associated with one-dimensional diffusions. The theory is illustrated by applications to genetic models.
Computation of the stationary distribution of the queue size in anM/G/1 queueing system with variable service rate
Authors: Federgruen A. , Tijms H. C.
Published: April 2006

This paper presents a simple and computationally tractable method which recursively computes the stationary probabilities of the queue size in anM/G/1 queueing system with variable service rate. For each service two possible service types are available and the service rule is characterized by two switch-over levels. The computational approach discussed in this paper can be applied to a variety of queueing problems.
Further results on single-lane traffic flow
Authors: Cowan Richard
Published: April 2006

A number of stochastic properties for a model of traffic flowing in a single lane are derived. The arrival process used is one which models (a) the bunching phenomenon of traffic and (b) the fact that vehicles have a spatial dimension and travel at minimum temporal spacing. Weak assumptions about the desired speeds of vehicles are made.
A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions
Authors: Talwalker Sheela
Published: April 2006

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition

0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….
Stationary waiting-time distributions in the GI/PH/1 queue
Authors: Neuts Marcel F.
Published: April 2006

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.
On the variance of the sample correlation between two independent lattice processes
Authors: Richardson Sylvia , Hemon Denis
Published: April 2006

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u),u(Z2} and {Y(u),u(Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains inZ2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.
Weak limit for the stationary distribution of the infinite-alleles model
Authors: Avni Haim
Published: April 2006

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.

In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.
Notes on information in population genetics
Authors: Holgate P.
Published: April 2006

The change with respect to time of the information in a population is studied, when the population is classified according to genetic type. The cases considered are (i) an infinite population classified at two loci, (ii) a large finite population classified at one locus, subject to sampling fluctuations and (iii) the same, subject to the effect of mutation.
On Cohen's stochastic generalization of the strong ergodic theorem of demography
Authors: Lange Kenneth
Published: April 2006

Cohen has generalized the classical strong ergodic theorem of demography to a stochastic setting. In this setting population projection matrices are chosen according to some homogeneous Markov chain. If this Markov chain converges to the same long-run distribution regardless of its starting point, then one can define an induced Markov chain on the product space of projection matrices and age structure vectors that also has a long-run distribution independent of its starting point. The present paper gives more natural conditions under which Cohen's result holds.
Some limit theorems for the general semi-Markov storage model
Authors: Balagopal K.
Published: April 2006

In this paper we treat the general version of the semi-Markov storage model, introduced first by Senturia and Puri: transitions in the state of the system occur at a discrete sequence of time points, described by a two-state semi-Markov process. An input occurs at an instant of transition to state 1 and a demand for release occurs at an instant of transition to state 2.

Assuming general distributions for all the variables involved, we show that the dam contents just after the nth input converges properly in distribution as n →∞ under conditions of stability; likewise that after the nth demand. We also show that the demand lost due to shortage of stock, accumulated over instants of demand as well as over time, obeys a strong law and a central limit theorem.
A note on exchangeable events
Authors: Hsu Yu-Sheng
Published: April 2006

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.
Bivariate distributions as saddle points of mutual information
Authors: Kim Geung Ho , David H. T.
Published: April 2006

Fix a bivariate distributionFonX×Y, considered as a pair (α, {Fx}), where α is a marginal distribution onXand {Fx} is a collection of conditional distributions onY. For essentially every (β,{Gx}) satisfying a certain pair of moment conditions determined by (α, {Fx}),J(β, {Fx}) ≦J(α, {Fx}) ≦J(α, {Gx}), whereJis mutual information. This relates to two sorts of extremizations of mutual information of relevance to communication theory and statistics.
A uniqueness problem for the envelope of an oscillatory process
Authors: Hasofer A. M.
Published: April 2006

In a previous paper, the author has described a method for obtaining envelope processes for oscillatory stochastic processes. These are processes which can be represented as the output of a time-varying linear filter whose input is a stationary process.

It is shown in this paper that the proposed definition of the envelope process may not be unique, but may depend on the particular representation of the oscillatory process chosen.

It is then shown that for a class of oscillatory processes which is of particular interest, the class of transient processes, there is a class of natural representations which all lead to a unique envelope process.
Stoppable families of alternative bandit processes
Authors: Glazebrook K. D.
Published: April 2006

Stoppable families of alternative bandit processes are decision processes with the property that at each decision epoch the choice is between allocating service to one of the constituent bandit processes or stopping and deciding in favour of one of them. The problem is considered of finding optimal (or good suboptimal) strategies for such processes. The theory for non-stoppable families leads us to study the performance of a simple strategy. This is shown to be optimal under certain conditions. These conditions are discussed and an example relating to research planning is given.
An infinite-server queue subject to an extraneous phase process and related models
Authors: Purdue P. , Linton D.
Published: April 2006

We consider an infinite-server queueing system in an extraneous environment. Initially it is shown that the systems of interest can be decomposed into a two-stage system. The primary system is an infinite-server queue with many customer types subject to a clearing mechanism. The secondary system is a special type of bulk-arrival, infinite-server queue. We derive results for the primary and secondary systems separately and combine the results to find the mean steady-state behavior of the original system.
On occupation times for quasi-Markov processes
Authors: Bondesson Lennart
Published: April 2006

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.
The spatial distribution of Tribolium confusum
Authors: Renshaw Eric
Published: April 2006

Neyman, Park and Scott (1956) describe an experiment which they performed to determine the spatial distribution ofTribolium confusumdeveloping within a closed container. To explain the concentration of beetles at the boundary a birth–death–migration model is developed in which the beetles may migrate over a set of lattice points, and this is shown to produce a distribution of the required shape. Not only is this distribution independent of the number of lattice points, but it is also indistinguishable from the associated diffusion process.
Convergence of age and type distributions in multitype critical Bellman-Harris processes
Authors: Murthy K. Rama
Published: April 2006

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.
Optimal list order under partial memory constraints
Authors: Kan Y. C. , Ross S. M.
Published: April 2006

Suppose that we are given a set of n elements which are to be arranged in some order. At each unit of time a request is made to retrieve one of these elements — the ith being requested with probability Pi. We show that the rule which always moves the requested element one closer to the front of the line minimizes the average position of the element requested among a wide class of rules for all probability vectors of the form P1 = p, P2= · ·· = Pn = (1 – p)/(n − 1). We also consider the above problem when the decision-maker is allowed to utilize such rules as ‘only make a change if the same element has been requested k times in a row', and show that as k approaches infinity we can do as well as if we knew the values of the Pi.
On a probabilistic analogue of the Fibonacci sequence
Authors: Heyde C. C.
Published: April 2006

One of the earliest population models to be studied gives rise to the Fibonacci sequence and has a history dating back more than 750 years. A stochastic version of the model is discussed in this paper, its basic defining property beingE(Xn| Xn−1, · ··,X0) =Xn−1+Xn−2a.s. The process {Xn} mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions,a.s. asn→ where α is the ‘golden ratio'
A simple proof of Whittle's bridging condition in dynamic programming
Authors: Hartley Roger
Published: April 2006

We offer a short proof that the bridging condition introduced by Whittle is sufficient for regularity in negative dynamic programming. We exploit concavity of the optimal value operator and do not need a special treatment of the case when optimal policies do not exist.
A simple model for the balance between selection and mutation
Authors: Kingman J. F. C.
Published: April 2006

A model for the variation in time of the fitness distribution in a large haploid population is shown to have simple limiting properties which can be elucidated in fairly explicit terms. The novel feature is that mutation is not assumed to cause a small perturbation in fitness but to bring down the evolutionary ‘house of cards’. A threshold phenomenon appears: if a certain inequality holds the limiting distribution is a skewed version of the mutant fitness distribution, but otherwise an atom of probability builds up at the upper limit of fitness.
On the age distribution of a Markov chain
Authors: Pakes Anthony G.
Published: April 2006

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).

A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.
Limiting diffusions for population-size dependent branching processes
Authors: Lipow Carla
Published: April 2006

A classical theorem on the convergence to a diffusion of a sequence of branching processes will be proved in the case where the branching processes are modified to allow dependence on population size of the generating function governing reproduction. Some recent semigroup convergence theorems due to Kurtz (1969), (1975) provide the main tools.
On sequential occupancy problems
Authors: Holst Lars
Published: April 2006

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.
Multi-species population models and evolutionarily stable strategies
Authors: Hines W. G. S.
Published: April 2006

The possibility that interspecific contest tactics might be able to stabilize two species predator-prey populations is investigated using a matrix model of the population. It is found that if the interspecific tactics have no influence on intraspecific encounters, the non-trivial equilibrium population composition is necessarily unstable. The result provides some insight into a reported instability of a predator-prey population with ‘stable' strategies.
An invariance principle for solutions to stochastic difference equations
Authors: Guess Harry A.
Published: April 2006

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.
Queues with paired customers
Authors: Latouche Guy
Published: April 2006

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.
Another look at the two-lift problem
Authors: Kaplan Norman L.
Published: April 2006

How often will a store with two elevators be without service assuming that repairs are quick? In this note it is shown that, under very mild assumptions on the operating- and repair-time distributions, the times when the building is without service are asymptotically Poisson.
On the comparison of waiting times in tandem queues
Authors: Niu Shun-Chen
Published: April 2006

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.
On the classical Bonferroni inequalities and the corresponding Galambos inequalities
Authors: Walker A. M.
Published: April 2006

Let (A1A2, · ··, An) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.

Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.
Some inequalities on the distribution of ladder epochs
Authors: Downham D.Y. , Fotopoulos S.B.
Published: April 2006

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.
Representation of disease etiologies by certain stochastic models
Authors: Katz Leo
Published: April 2006

This paper considers the appropriateness of certain stochastic models for the representation of disease etiologies. Models for disease as the result of multiple events, and as the endpoint of a network of events, are discussed.
A curious binary lattice process
Authors: Pickard D. K.
Published: April 2006

A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field onZ×Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.
Some multivariate generalizations of results in univariate stationary point processes
Authors: Berman Mark
Published: April 2006

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.
Waveforms and velocities for non-nearest-neighbour contact distributions
Authors: Renshaw Eric
Published: April 2006

This paper examines a model for ecological and epidemiological spread. Expressions are derived for mean waveforms and expectation velocities for two specific contact distributions. Whilst one distribution may be bounded above by a negative exponential function the other may not, and these two situations respectively give rise to finite and infinite asymptotic expectation velocities.
Return probabilities for certain three-dimensional random walks
Authors: Daley D. J.
Published: April 2006

For some three-dimensional random walks on the cubic lattice, the probability of the walk returning to its starting point is given numerically.
Replacement problem of a parallel system in random environment
Authors: Nakagawa T.
Published: April 2006

Råde considered a parallel system with n components in a random environment. The shocks cause the components to fail with certain probabilities. This paper considers the replacement policy for the above system in which it is exchanged before failure if the total number of failed components is more than k, and it is replaced if all of n components have failed. An optimum number of k which will minimize the expected cost is obtained. A numerical example is finally presented.
Sequential rank and the Pólya urn
Authors: Robbins Herbert , Whitehead John
Published: April 2006

A sequence of independent, identically distributed random variables is observed. After a sample ofmhas been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.

The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.
Stochastically perturbed limit cycles
Authors: Holland Charles J.
Published: April 2006

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.
The expected ultimate size of a carrier-borne epidemic
Authors: Lefèvre Claude
Published: April 2006

This paper is concerned with the expected ultimate size of the Downton carrier-borne epidemic (1968). The solution obtained is slightly recursive and it may be expressed as a power series in π, the proportion of the infected susceptibles becoming carriers. It generalizes the results of Dietz (1966) and Abakuks (1973) for some special cases of this model.
A limit theorem for random coverings of a circle which do not quite cover
Authors: Kaplan Norman
Published: April 2006

We study the asymptotic behavior as α → 0 of the number of independent random arcs of length a needed to cover at least 1 − Kα (K ≧ 1) of a circle of unit circumference.
Some aspects of the Smoluchowski process
Authors: McDunnough Philip
Published: April 2006

This article deals with the generalized Smoluchowski process, {n(t),t≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained inm< ∞ disjoint regions of space. The relationship of the Smoluchowski process {n(t),t≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.
Mixing of cluster point processes
Authors: Laslett G. M.
Published: April 2006

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong,ϕ-, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated thatϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.
Epidemic process on a random graph: some preliminary results
Authors: Gertsbakh I. B.
Published: April 2006

A random graph is a collection of n points and n directed arcs: a directed arc goes equiprobably from each point to one of (n – 1) other points. m points are initially ‘infected'. We consider several schemes of epidemic process, e.g. when the infection is delivered according to arc direction. We prove that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)). Possible applications of the models to real world phenomena are briefly discussed.
Conditional limit theorems for general branching processes
Authors: Green P. J.
Published: April 2006

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.
Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations
Authors: Reh W.
Published: April 2006

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.
Almost sure asymptotic likelihood theory for diffusion processes
Authors: Lee T. S. , Kozin F.
Published: April 2006

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.
Remarks on the spatial distribution of a reproducing population
Authors: Kingman J. F. C.
Published: April 2006

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.
A conditional expectation formula for diffusion processes
Authors: Aase Knut K.
Published: April 2006

In some applications of diffusion theory it is of interest to study the expected time of the first exit from an interval given that a specified boundary will be the first exitpoint. This problem is considered here for a regular Feller process, and applications are given.
Approximate variances associated with random configurations of hard spheres
Authors: Girling A. J.
Published: April 2006

The Percus–Yevick approximation of classical liquid theory is employed to obtain variances associated with random configurations of equal spheres in ℝ3. The idea is illustrated by considering the number of spheres, and also the amount of hard material, contained within a fixed cylinder — an application of some practical value.
On a constrained optimal stopping problem
Authors: Kennedy D. P.
Published: April 2006

For a sequence of random variables {Xn, n ≧ 0}, optimal stopping is considered over stopping times T constrained so that ET ≦ α, for some fixed α > 0. It is shown that under certain circumstances a Lagrangian approach may be used to reduce the problem to an unconstrained optimal stopping problem of a conventional type. The optimal value of the natural dual problem is shown to be equal to the optimal value of the original (primal) problem when certain randomised stopping times are permitted. Two examples are considered in detail.
The structure of elementary pure birth processes
Authors: Pfeifer Dietmar
Published: April 2006

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.
Substitution times for mutant nucleotides
Authors: Watterson G. A.
Published: April 2006

It is shown that, for one model of genetic evolution, the times at which mutations at nucleotide sites become fixed in a population tend to cluster together rather than behave as a Poisson process.
On the approximation of convex, closed plane curves by multifocal ellipses
Authors: Vincze P. Erdös I.
Published: April 2006

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.
The spread of infection in a heterogeneous population
Authors: Cane Violet , McNamee Rosanne
Published: April 2006

A simple interaction model is proposed for describing variable meeting-rates in a population exposed to a contagious disease. In comparison with the usual assumption of a uniform meeting-rate it predicts, on average, a more rapid spread of infection initially and a slower spread finally; it also predicts that, for a given removal rate, an epidemic could develop among a smaller number of susceptibles.
Distributions on the circle and sphere
Authors: Watson Geoffrey S.
Published: April 2006

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.
Stationary point processes by Markov-dependent intervals and infinite intensity
Authors: Daley D. J.
Published: April 2006

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.
An M/G/c queue in which the number of servers required is random
Authors: Federgruen Awi , Green Linda
Published: April 2006

Many queueing situations such as computer, communications and emergency systems have the feature that customers may require service from several servers at the same time. They may thus be delayed until the required number of servers is available and servers may be idle when customers are waiting. We consider general server-completion-time distributions and derive approximation methods for the computation of the steady-state distribution of the number of customers in queue as well as the moments of the waiting-time distribution. Extensive computational results are reported.
On the local stability of an evolutionarily stable strategy in a diploid population
Authors: Hines W. G. S. , Bishop D. T.
Published: April 2006

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.
A threshold AR(1) model
Authors: Petruccelli Joseph D. , Woolford Samuel W.
Published: April 2006

We consider the modelwhereφ1,φ2are real coefficients, not necessarily equal, and theat,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on theφ's are given for stationarity of the process. Least squares estimators of theφ's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the caseφ1=φ2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.
The central limit theorem for the Poisson shot-noise process
Authors: Lane John A.
Published: April 2006

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.
A service system with two stages of waiting and feedback of customers
Authors: Ali Osman M. E. , Neuts Marcel F.
Published: April 2006

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.
Conditions for extinction in certain bisexual Galton–Watson branching processes
Authors: Hull David M.
Published: April 2006

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.
Optimal replacement for self-repairing shock models by general failure rate
Authors: Gottlieb Gary , Levikson Benny
Published: April 2006

A device is subject to a series of shocks which cause damage and eventually failure will occur at the time of arrival of one of the shocks. In between the shocks, the device is partially repaired as the cumulative damage decreases as some Markov process. The device must be replaced upon failure at some cost but it can also be replaced before failure at a lower cost. We consider the general case where the failure rate need not be increasing and replacement can be made at any time. The form of the optimal replacement policy is found and fairly general conditions are given for which a control limit policy is optimal.
Applications of martingale theory to some epidemic models, II
Authors: Picard Philippe
Published: April 2006

We consider Weiss's and Downton's models with parametersπ, αand β depending on i number of susceptibles and j number of carriers. A martingale argument is performed when πand α /β only depend on i or, in Weiss's case, when α /β is the product of a function of i by a function of j. In these cases the martingale approach proves very valuable and gives explicit results quite easily. In particular it shows that well-known relations between moments and integrals along a trajectory are still true for any stopping time and for more general models than the classic ones.
Shock processes by aftereffects and multivariate lack of memory
Authors: Ghurye S. G. , Marshall Albert W.
Published: April 2006

If the survival function satisfies the functional equation

where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.
A note on extinction criteria for bisexual Galton-Watson processes
Authors: Bruss F. Thomas
Published: April 2006

This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).
Isolation by distance in a hierarchically clustered population
Authors: Sawyer Stanley , Felsenstein Joseph
Published: April 2006

A biological population with local random mating, migration, and mutation is studied which exhibits clustering at several different levels. The migration is determined by the clustering rather than actual geographic or physical distance. Darwinian selection is assumed to be absent, and population densities are such that nearby individuals have a probability of being related. An expression is found for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance. Asymptotics for a small mutation rateuare discussed for both a finite number of clustering levels (and of total population size), and for an infinite number of levels. A natural example is discussed in which the probability of heterozygosity varies asuto a power times a periodic function of log(l/u).
A note on many-server queueing systems with ordered entry, with an application to conveyor theory
Authors: Nawijn W. M.
Published: April 2006

Consider a many-server queueing system in which the servers are numbered. If a customer arrives when two or more servers are idle he selects the server with lowest index (this is called the ordered entry selection rule). An explicit expression for the traffic handled by the various servers in a GI/M/s queueing system with ordered entry is derived. For the M/M/s queueing system the probability distribution of the number of busy servers among the first k(k = 1, 2,…, s) servers will be given. Finally, a formula for the traffic handled by the first server in an M/D/s system will be derived. All results are derived under steady-state conditions. As an application some numerical data for the server utilizations will be given and compared to data obtained from simulation studies of a closed-loop continuous belt-conveyor.
Evolutionarily stable strategies with two types of player I. Two-species haploid or randomly mating diploid
Authors: Cressman R. , Dash A. T.
Published: April 2006

The evolution of strategies in animal contests is examined where the dynamical equation takes into account population growth rates. This leads to a different definition of evolutionary stable strategy (ESS) from the usual one. Consequences for independent haploid species are then contrasted with the previous theory. Inheritance patterns for male–female contests with sex-dependent payoffs are considered. In particular, if males and females evolve independently to the same ESS, then so does the diploid species under random mating. Finally, the evolution of diploid populations where strategies are determined at a diallelic locus is investigated.
Selection of order of observation in optimal stopping problems
Authors: Hill Theodore P. , Hordijk Arie
Published: April 2006

In optimal stopping problems in which the player is free to choose the order of observation of the random variables as well as the stopping rule, it is shown that in general there is no function of all the moments of individual integrable random variables, nor any function of the first n moments of uniformly bounded random variables, which can determine the optimal ordering. On the other hand, several fairly general rules for identification of the optimal ordering based on individual distributions are given, and applications are made to several special classes of distributions.