bluuebunny/crossref_metadata_2025 huggingface dataset:
A multiple-threshold AR(1) model
Authors: Chan K. S. , Petruccelli Joseph D. , Tong H. , Woolford Samuel W.
Published: April 2006

We consider the modelZt=φ(0,k)+ φ(1,k)Zt–1+at(k) wheneverrk−1<Zt−1≦rk, 1≦k≦l, withr0= –∞ andrl=∞. Here {φ(i, k);i= 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k),t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k),t≧1} independent of {at(j),t≧1} forj≠k.Necessary and sufficient conditions on the constants {φ(i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.
The threshold behaviour of epidemic models
Authors: Ball Frank
Published: April 2006

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptiblesN, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0,t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.
A connective constant for loop-erased self-avoiding random walk
Authors: Lawler Gregory F.
Published: April 2006

A ‘connective constant' is defined for self-avoiding random walk derived by erasing loops from simple random walk. Ford≧ 5, it is shown that this distribution onn-step self-avoiding paths approaches a uniform distribution in a weak sense.
On the relation between the type–token and species-area problems
Authors: Brainerd Barron
Published: April 2006

The species-area problem in biology and the type-token problem in literary studies are analogues of one another but have nearly disjoint literatures. Here their relationship is treated, a critique of models used in each is made and Markovian models are provided for two well-known empirical models from the species-area literature. Finally, some of the assumptions underlying each of these models are considered in the light of literary data.
An alternate proof for the representation of discrete distributions by equiprobable mixtures
Authors: Dieter U.
Published: April 2006

The paper contains a simple proof that every finite discrete distribution is the equiprobable mixture ofrdistributions, each of which has at mostamass points.
A queueing system with Markov-dependent arrivals
Authors: Tin Pyke
Published: April 2006

This paper considers a single-server queueing system with Markov-dependent interarrival times, with special reference to the serial correlation coefficient of the arrival process. The queue size and waiting-time processes are investigated. Both transient and limiting results are given.
On a modified counter with prolonging dead time
Authors: Dvurečenskij A. , Ososkov G. A.
Published: April 2006

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.
The busy period of the queueing system M/G/∞
Authors: Stadje Wolfgang
Published: April 2006

For the queueing system M/G/∞ some distributions connected with the associated busy periods are derived.
An order statistic characterization of the poisson renewal process
Authors: Liberman Uri
Published: April 2006

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.
An approximate solution of the integral equation of renewal theory
Authors: Deligönül Z. Șeyda
Published: April 2006

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.
Maximal, non-uniform spacings and the coverage problem
Authors: Hüsler Jürg
Published: April 2006

The spacings have been widely applied to many situations, such as for example the coverage problem of the circle or the line, under the hypothesis of uniformity. The paper examines the asymptotic behaviour of the maximal spacings under non-uniformity, assuming that the existing density f has a minimum point m with f(m) = 0. The results are applied to the coverage of the circle by randomly placed arcs.
Notes on Natvig's measure of importance of system components
Authors: Norros Ilkka
Published: April 2006

Natvig's measure of importance of system components is defined and investigated, using martingale notions and methods. Most results are proved in the case of dependent component life lengths. The sum of the ‘absolute importances' of the components is studied as a quantity expressing the ‘share of failures' in the destruction of the system, contrasted to the ‘share of time’.
Optimal allocation of components in parallel–series and series–parallel systems
Authors: El-Neweihi Emad , Proschan Frank , Sethuraman Jayaram
Published: April 2006

This paper shows how majorization and Schur-convex functions can be used to solve the problem of optimal allocation of components to parallel-series and series-parallel systems to maximize the reliability of the system. For parallel-series systems the optimal allocation is completely described and depends only on the ordering of component reliabilities. For series-parallel systems, we describe a partial ordering among allocations that can lead to the optimal allocation. Finally, we describe how these problems can be cast as integer linear programming problems and thus the results obtained in this paper show that when some linear integer programming problems are recast in a different way and the techniques of Schur functions are used, complete solutions can be obtained in some instances and better insight in others.
On the asymptotic properties of a supercritical bisexual branching process
Authors: Bagley J. H.
Published: April 2006

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.
On the almost sure convergence of controlled branching processes
Authors: Bagley J. H.
Published: April 2006

An almost sure convergence result for the normed population size of a supercritical controlled branching process is proved.
Uniformly best invariant stopping rules
Authors: Engen Steinar , Seim Eva
Published: April 2006

The class of stopping rules for a sequence of i.i.d. random variables with partially known distribution is restricted by requiring invariance with respect to certain transformations. Invariant stopping rules have an intuitive appeal when the optimal stopping problem is invariant with respect to the actual gain function. Uniformly best invariant stopping rules are derived for the gamma distribution with known shape parameter and unknown scale parameter, for the uniform distribution with both endpoints unknown, and for the normal distribution with unknown mean and variance. Some comparisons with previously published results are made.
Multivariate Lomax distribution: properties and usefulness in reliability theory
Authors: Nayak Tapan Kumar
Published: April 2006

A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.
Antistarshapedness, dispersiveness and mixtures
Authors: Lynch James
Published: April 2006

Classes of distributions are defined in terms of antistarshapedness or dispersiveness. Necessary and sufficient conditions are given for these classes to be closed under mixtures. These conditions characterize distributions with log-concave densities.
Uniformization for semi-Markov decision processes under stationary policies
Authors: Beutler Frederick J. , Ross Keith W.
Published: April 2006

Uniformization permits the replacement of a semi-Markov decision process (SMDP) by a Markov chain exhibiting the same average rewards for simple (non-randomized) policies. It is shown that various anomalies may occur, especially for stationary (randomized) policies; uniformization introduces virtual jumps with concomitant action changes not present in the original process. Since these lead to discrepancies in the average rewards for stationary processes, uniformization can be accepted as valid only for simple policies.

We generalize uniformization to yield consistent results for stationary policies also. These results are applied to constrained optimization of SMDP, in which stationary (randomized) policies appear naturally. The structure of optimal constrained SMDP policies can then be elucidated by studying the corresponding controlled Markov chains. Moreover, constrained SMDP optimal policy computations can be more easily implemented in discrete time, the generalized uniformization being employed to relate discrete- and continuous-time optimal constrained policies.
The convexity of a general performance measure for multiserver queues
Authors: Harel Arie , Zipkin Paul
Published: April 2006

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications.

We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.
On the ordering of tandem queues with exponential servers
Authors: Lehtonen Tapani
Published: April 2006

We consider tandem queues which have a general arrival process. The queueing system consists ofs(s≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.
Expected waiting times in a multiclass batch arrival retrial queue
Authors: Kulkarni V. G.
Published: April 2006

Expressions are derived for the expected waiting times for the customers of two types who arrive in batches (in a compound Poisson fashion) at a single-server queueing station with no waiting room. Those who cannot get served immediately keep returning to the system after random exponential amounts of time until they get served. The result is shown to agree with similar results for three special cases studied in the literature.
The origin and growth of the Applied Probability Trust
Authors: Hitchcock Mavis
Published: April 2006

An account is given of the foundation of the Applied Probability Trust and the development of its office at the University of Sheffield. The publishing achievements of its first quarter-century are reviewed.
Theory of semiregenerative phenomena
Authors: Prabhu N. U.
Published: April 2006

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.
Some thoughts on stationary processes and linear time series analysis
Authors: Heyde C. C.
Published: April 2006

In this paper we trace the development of the asymptotic analysis of autocorrelations for stationary purely non-deterministic time series. We emphasize the interplay between mathematical requirements and modelling philosophy. We then proceed to extend the theory to the case where only a certain weak form of asymptotic independence of the linear prediction errors is needed rather than the earlier martingale difference or independence requirements.
Mesoadditive processes and the specific conductivity of lattices
Authors: Hammersley J. M.
Published: April 2006

The hypercubic lattice with bonds of random electrical resistance affords a model for the specific conductivity of microscopically irregular material. The resulting stochastic process lies between a subadditive and a superadditive process: mesoadditive processes of this type provide several unsolved problems for pure mathematicians.
Symmetric sampling procedures, general epidemic processes and their threshold limit theorems
Authors: Martin-Löf Anders
Published: April 2006

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.
A genealogical approach to variable-population-size models in population genetics
Authors: Donnelly Peter
Published: April 2006

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.
Some exact distributions of a last one-sided exit time in the simple random walk
Authors: Chao Chern-Ching , Slivka John
Published: April 2006

For each positive integer n, let Sn be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).
Benes's formula for M/G/1–FIFO ‘explained' by preemptive-resume LIFO
Authors: Cooper Robert B. , Niu Shun-Chen
Published: April 2006

We provide a term-by-term interpretation of Beneš's well-known but mysterious inversion of the Pollaczek–Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/1–FIFO with remaining work in M/G/1–LIFO–preemptive resume. In the process, we give a new and simple derivation of some known results for M/G/1–LIFO–preemptive resume.
Transition probabilities for some ‘special' diffusions
Authors: Beneš Václav E. , Karatzas Ioannis
Published: April 2006

We study the transition probabilities of the diffusionsdXt= (1 – exp(Xt))dt+dWtanddXt= – tanhXtdt+dWt, in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).
On an optimal selection problem of Cowan and Zabczyk
Authors: Bruss F. Thomas
Published: April 2006

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.
Quasi-stationary distributions and one-dimensional circuit-switched networks
Authors: Ziedins Ilze
Published: April 2006

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceedingK.An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.
Inequalities for the M/G/∞ queue and related shot noise processes
Authors: Huffer Fred W.
Published: April 2006

Suppose that pulses arrive according to a Poisson process of rate λ with the duration of each pulse independently chosen from a distribution F having finite mean. Let X(t) be the shot noise process formed by the superposition of these pulses. We consider functionals H(X) of the sample path of X(t). H is said to be L-superadditive if for all functions f and g. For any distribution F for the pulse durations, we define H(F) = EH(X). We prove that if H is L-superadditive and for all convex functions ϕ, then . Various consequences of this result are explored.
Insensitive bounds for the stationary distribution of non-reversible Markov chains
Authors: Hordijk Arie , Ridder AD
Published: April 2006

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.
The transient behaviour of the simple random walk in the plane
Authors: Downham D. Y. , Fotopoulos S. B.
Published: April 2006

For the simple two-dimensional random walk on the vertices of a rectangular lattice, the asymptotic forms of several properties are well known, but their forms can be insufficiently accurate to describe the transient process. Inequalities with the correct asymptotic form are derived for six such properties. The rates of approach to the asymptotic form are derived. The accuracy of the bounds and some practical implications of the results are discussed.
Scale renormalization and random solutions of the Burgers equation
Authors: Rosenblatt M.
Published: April 2006

Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.
The range of the mean-value quantities of planar tessellations
Authors: Kendall Wilfrid S. , Mecke Joseph
Published: April 2006

Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.
Sojourn times in queuing networks with multiserver modes
Authors: Schassberger R. , Daduna H.
Published: April 2006

This paper generalizes previous results for sojourn-time distributions along so-called overtake-free routes in product-form networks of queues.
First passage times and lumpability of semi-Markov processes
Authors: Sumita Ushio , Rieders Maria
Published: April 2006

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.
Diffusion-reaction in one dimension
Authors: Balding David
Published: April 2006

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.
On the reliability function of a system of components sharing a common environment
Authors: Currit Allen , Singpurwalla Nozer D.
Published: April 2006

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.
Asymptotic normality of some Graph-Related statistics
Authors: Baldi Pierre , Rinott Yosef
Published: April 2006

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.
Order estimation by accumulated prediction errors
Authors: Rissanen Jorma
Published: April 2006

This paper presents a new criterion based on prediction error which allows the estimation of the number of parameters as well as structures in statistical models. The criterion is valid for short and long samples alike. Unlike Akaike's earlier criterion, also based on prediction error, the criterion proposed here appears to produce consistent error estimates in ARMA processes.
Regression for randomly sampled spatial series: the trigonometric case
Authors: Brillinger David R.
Published: April 2006

The model Y(t) = s(t | θ) + ε(t) is studied in the case that observations are made at scattered points τ j in a subset of Rp and θ is a finite-dimensional parameter. The particular cases of

0 = (α, β) and (α, β, ω) are considered in detail. Consistency and asymptotic normality results are developed assuming that the spatial series ε(·) and the point process {τ j} are independent, stationary and mixing. The estimates considered are equivalent to least squares asymptotically and are not generally asymptotically efficient.

Contributions of the paper include: study of the Rp case, management of irregularly placed observations, allowance for abnormal domains of observation and the discovery that aliasing complications do not arise when the point process {τ j} is mixing. There is a brief discussion of the construction and properties of maximum likelihood estimates for the spatial-temporal case.
On the equivalence of two stochastic approaches to spline smoothing
Authors: Ansley Craig F. , Kohn Robert
Published: April 2006

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.
Some explicit results for correlated random walks
Authors: Lal Ram , Bhat U. Narayan
Published: April 2006

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i)n-step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).
Some new results on the subexponential class
Authors: Murphree Emily S.
Published: April 2006

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F(2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.
Analysis of planar anisotropy by means of the Steiner compact
Authors: Rataj Jan , Saxl Ivan
Published: April 2006

A graphical method for the estimation of the anisotropy of planar fibre systems based on the Steiner compact set is proposed and discussed. Upper bounds for the deviation in probability of the graphical estimate of the Steiner compact are given and a consistency theorem is proved.
Collision probabilities for convex sets
Authors: Weil Wolfgang
Published: April 2006

Let K, L ⊂ En be non-empty, closed, convex sets, K bounded, and suppose boundary sets αof K and ß of L are painted. If K undergoes a random motion such that K and L touch, the probability for a paint-to-paint contact is expressed by curvature measures of K and L. This generalizes and simplifies previous work of Molter (1986) on infinite cylinders L touching a convex body K.
Some results on two-armed bandits when both projects vary
Authors: O'Flaherty Brendan
Published: April 2006

In the multi-armed bandit problem, the decision-maker must choose each period a single project to work on. From the chosen project she receives an immediate reward that depends on the current state of the project. Next period the chosen project makes a stochastic transition to a new state, but projects that are not chosen remain in the same state. What happens in a two-armed bandit context if projects not chosen do not remain in the same state? We derive two sufficient conditions for the optimal policy to be myopic: either the transition function for chosen projects has in a certain sense uniformly stronger stochastic dominance than the transition function for unchosen projects, or both transition processes are normal martingales, the variance of which is independent of the history of process choices.
On higher moments of the population size in a subcritical branching process
Authors: Willekens Eric
Published: April 2006

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.
A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials
Authors: Chryssaphinou O. , Papastavridis S.
Published: April 2006

A sequence of independent experiments is performed, each producing a letter from a given alphabet. Using a result by Barbour and Eagleson (1984) we prove that under general conditions the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution.
Optimality of myopic stopping times for geometric discounting
Authors: Fristedt Bert , Berry Donald A.
Published: April 2006

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.
The extinction time of a general birth and death process with catastrophes
Authors: Brockwell P. J.
Published: April 2006

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy λ0 = 0, λ j > 0 for each j > 0, and . Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λ j = jλ, µj = jμ) with catastrophes of several different types.
Brownian motion with quadratic killing and some implications
Authors: Wenocur Michael L.
Published: April 2006

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.
An optimal inspection and replacement policy for a deteriorating system
Authors: Ohnishi Masamitsu , Kawai Hajime , Mine Hisashi
Published: April 2006

This paper investigates a system whose deterioration is expressed as a continuous-time Markov process. It is assumed that the state of the system cannot be identified without inspection. This paper derives an optimal policy minimizing the expected total long-run average cost per unit time. It gives the optimal time interval between successive inspections and determines the states at which the system is to be replaced. Furthermore, under some reasonable assumptions reflecting the practical meaning of the deterioration, it is shown that the optimal policy has monotonic properties. A control limit rule holds for replacement, and the time interval between successive inspections decreases as the degree of deterioration increases.
Products of 2 × 2 stochastic matrices with random entries
Authors: Van Assche Walter
Published: April 2006

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.
On approximately optimal index strategies for generalised arm problems
Authors: Fay N. A. , Walrand J. C.
Published: April 2006

Nash has extended Gittins' work to describe optimal strategies for a class of generalised bandit problems. Here we use a forwards induction argument to analyseε-optimal strategies for generalised bandit problems. An evaluation procedure for such problems is described; this may be used to analyse models in research planning and stochastic scheduling.
An asymptotically exact decomposition of coupled Brownian systems
Authors: Fendick Kerry W.
Published: April 2006

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.
The density of interfaces: a new first-passage problem
Authors: Chayes L. , Winfield C.
Published: April 2006

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.
A note on exceedances and rare events of non-stationary sequences
Authors: Hüsler J.
Published: April 2006

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.
Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains
Authors: Kalpazidou S.
Published: April 2006

The asymptotic behaviour of the sequence (𝒞n(ω), wc,n(ω)/n), is studied where 𝒞n(ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n) until time n and wc,n(ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ωc) which represents uniquely the chain (ξ n) as a circuit chain, and ω c is given a probabilistic interpretation.
Saddlepoint approximations for bivariate distributions
Authors: Wang Suojin
Published: April 2006

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean ofnindependent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.
On the maximum and its uniqueness for geometric random samples
Authors: Bruss F. Thomas , O'cinneide Colm Art
Published: April 2006

Givennindependent, identically distributed random variables, letρndenote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of theρn's in the case of geometric random variables. We find a function Φsuch that (ρn–Φ(n)) → 0 asn→∞. In particular, we show thatρndoes not convergeasn→∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.
Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period
Authors: Sericola Bruno
Published: April 2006

Markov process are widely used to model computer systems. De Souza e Silva and Gail [3] calculated numerically the distribution of the cumulative operational time of repairable computer systems modelled by Markovian processes, that is, the distribution of the total time during which the system was in operation over a finite observation period. An extension of their approach is presented here. A closed-form solution is obtained for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period, which is theoretically and numerically interesting. We also give an application of this result to a fault-tolerant system.
Transition probability density of a certain diffusion process concentrated on a finite spatial interval
Authors: Milian A.
Published: April 2006

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.
The Mk/M/∞ queue with heterogeneous customers in a batch
Authors: Choi Bong Dae , Park Kwang Kyu
Published: April 2006

We consider the Mk/M/∞ queue with k heterogeneous customers in a batch where the customer of type i in a batch requires an exponential service time with parameter µi. In steady state, the joint generating function of the number of customers of type i being served in the system is derived explicitly by solving a partial differential equation.
Spatial patterns arising from plant dispersal as modelled by a correlated random walk
Authors: Routledge R. D.
Published: April 2006

A correlated random walk in the plane is studied for which the direction of a step depends on the direction of the previous step. Both step direction and step length are continuous random variables. Such a random walk has been used to model the vegetative dispersal of certain plant populations. The analysis provides general conclusions about dependencies on parameters, an efficient scheme for generating numerical results, and testable predictions for plant populations.
A functional central limit theorem for the Ewens sampling formula
Authors: Hansen Jennie C.
Published: April 2006

For eachn> 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integern. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures onD[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.
On the two-boundary first-crossing-time problem for diffusion processes
Authors: Buonocore A. , Giorno V. , Nobile A. G. , Ricciardi L. M.
Published: April 2006

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.
On a relationship between the inverse of a stationary covariance matrix and the linear interpolator
Authors: Bhansali R. J.
Published: April 2006

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices ofxB(k) = [x′–1,x′–2, · ··,x′–k]′,xF(k) = [x′1,x′2· ··x′k] andx= [·· ·x′–2,x′–1,x′0,x′1,x′2· ··]′ respectively, wherek≧ 1, is finite or infinite. Also letφm,n(j) and δm,n(u) be the coefficients ofxt+jandxt–urespectively in the linear least-squares interpolator ofxtfromxt+ 1, · ··,xt+m;xt− 1, · ··,xt–n, wherem, n≧ 0, 0 ≦j≦m, 0 ≦u≦nare integers,zt(m, n) denote the interpolation error and τ2(m, n) =E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1and Γ–1is given by relating these components to theφm,n(j)δm,n(u) andτ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–land the interpolation parameters when these have been obtained from a realization of lengthTof {xt}. Some of the applications of the results are considered.
Expected size distributions in models of group dynamics
Authors: Morgan Byron J. T.
Published: April 2006

This paper considers several models for biological processes in which animate individuals live and die as members of groups which can split to form smaller groups. Resulting distributions of individuals over groups are compared and contrasted. In particular, two qualitatively different types of distributions are identified. It is clear that distinguishing between models giving rise to the same distribution types is difficult. Implications for more complex models are discussed and avenues for further research are outlined.
On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case
Authors: Kalpazidou S.
Published: April 2006

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.
The stationary work in system of a G/G/1 gradual input queue
Authors: Kino Issei , Miyazawa Masakiyo
Published: April 2006

This paper is devoted to the study of a stationary G/G/1 queue in which work input is gradually injected into the system and the work load is processed at unit rate. First, assuming that the input process is stationary, the key formula for the stationary distribution of work in system is derived by appeal to the waiting time of the associated regular G/G/1 queue. The main contribution of this paper is the derivation of Laplace-Stieltjes transforms (LSTs) for the stationary distributions of work in system and related random variables in terms of integrations with respect to a waiting time distribution of the associated regular queue. The results are exemplified by giving explicit formulas for the LST of total work for M/Ek/1 and M/H2/1. The results generalize the results of Pan et al. (1991) for the M/M/1 gradual input queue.
A constraint on the random packing of disks
Authors: Cowan Richard
Published: April 2006

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.
A competitive best-choice problem with Poisson arrivals
Authors: Enns E. G. , Ferenstein E. Z.
Published: April 2006

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0, T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.
Optimal R&D programs in a random environment
Authors: Posner M. J. M. , Zuckerman D.
Published: April 2006

Our study examines a stochastic R&D model with flexible termination time and without rivalry. Specifically, we assume a stochastic relationship between expenditures rate and the project's status. Furthermore, the termination time of the project is incorporated into the R&D model as a decision variable by allowing the controller to ‘sell' the obtained technology from the project at any point of time. The proposed framework extends the classical approach in the R&D literature.

The main purpose of our study is to determine the optimal stopping time of the project and to characterize qualitatively the firm's expenditure strategy. We show that under certain realistic conditions, the optimal stopping strategy is a control limit policy. Furthermore, the research effort increases monotonically over the development time of the project.
On arrivals that see time averages: a martingale approach
Authors: Melamed Benjamin , Whitt Ward
Published: April 2006

This paper is a sequel to our previous paper investigating when arrivals see time averages (ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.