bluuebunny/crossref_metadata_2025 huggingface dataset:
A statistical approach to identifying closed object boundaries in images
Authors: Helterbrand Jeffrey D. , Cressie Noel , Davidson Jennifer L.
Published: April 2006

In this research, we present a statistical theory, and an algorithm, to identify one-pixel-wide closed object boundaries in gray-scale images. Closed-boundary identification is an important problem because boundaries of objects are major features in images. In spite of this, most statistical approaches to image restoration and texture identification place inappropriate stationary model assumptions on the image domain. One way to characterize the structural components present in images is to identify one-pixel-wide closed boundaries that delineate objects. By defining a prior probability model on the space of one-pixel-wide closed boundary configurations and appropriately specifying transition probability functions on this space, a Markov chain Monte Carlo algorithm is constructed that theoretically converges to a statistically optimal closed boundary estimate. Moreover, this approach ensures that any approximation to the statistically optimal boundary estimate will have the necessary property of closure.
Matrix product-form solutions for Markov chains with a tree structure
Authors: Yeung Raymond W. , Sengupta Bhaskar
Published: April 2006

We have two aims in this paper. First, we generalize the well-known theory of matrix-geometric methods of Neuts to more complicated Markov chains. Second, we use the theory to solve a last-come-first-served queue with a generalized preemptive resume (LCFS-GPR) discipline. The structure of the Markov chain considered in this paper is one in which one of the variables can take values in a countable set, which is arranged in the form of a tree. The other variable takes values from a finite set. Each node of the tree can branch out into d other nodes. The steady-state solution of this Markov chain has a matrix product-form, which can be expressed as a function of d matrices Rl,· ··, Rd. We then use this theory to solve a multiclass LCFS-GPR queue, in which the service times have PH-distributions and arrivals are according to the Markov modulated Poisson process. In this discipline, when a customer's service is preempted in phase j (due to a new arrival), the resumption of service at a later time could take place in a phase which depends on j. We also obtain a closed form solution for the stationary distribution of an LCFS-GPR queue when the arrivals are Poisson. This result generalizes the known result on a LCFS preemptive resume queue, which can be obtained from Kelly's symmetric queue.
Pricing via anticipative stochastic calculus
Authors: Platen Eckhard , Rebolledo Rolando
Published: April 2006

The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.
Large deviations for discrete and continuous percolation
Authors: Penrose Mathew D. , Pisztora Agoston
Published: April 2006

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.
An operational calculus for probability distributions via Laplace transforms
Authors: Abate Joseph , Whitt Ward
Published: April 2006

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.
Homogeneous Gaussian Markov processes on general lattices
Authors: Komaki Fumiyasu
Published: April 2006

A homogeneous Gaussian Markov lattice-process model has a regression coefficient that determines the extent to which a random variable of a vertex is dependent on those of the neighbors. In many studies, the absolute value of this parameter has been assumed to be less than the reciprocal of the number of neighbors. This condition is shown to be necessary and sufficient for the existence of the Gaussian process satisfying the model equations under some assumptions on lattices using the notion of dual processes. We also give examples of models that neither satisfy the condition imposed on the region for the parameter nor the assumptions on lattices. A formula for autocovariance functions of Gaussian Markov processes on general lattices is derived, and numerical procedures to calculate the autocovariance functions are proposed.
Some asymptotic results for transient random walks
Authors: Bertoin J. , Doney R. A.
Published: April 2006

We consider a real-valued random walkSwhich drifts to –∞ and is such thatE(expθS1) < ∞ for someθ> 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.
Birth and death processes as projections of higher-dimensional Poisson processes
Authors: Garcia Nancy Lopes
Published: April 2006

Birth and death processes can be constructed as projections of higher-dimensional Poisson processes. The existence and uniqueness in the strong sense of the solutions of the time change problem are obtained. It is shown that the solution of the time change problem is equivalent to the solution of the corresponding martingale problem. Moreover, the processes obtained by the projection method are ergodic under translations.
An elementary renewal theorem for random compact convex sets
Authors: Molchanov Ilya S. , Omey Edward , Kozarovitzky Eugene
Published: April 2006

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined aswhereare Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit ofH(tK)/tasttends to infinity. ForKcontaining the origin as an interior point,wherehK(u) is the support function ofKandis the set of all unit vectorsuwithEhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.
Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics
Authors: Worsley K. J.
Published: April 2006

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.
Epidemic models featuring behaviour change
Authors: O'neill Philip
Published: April 2006

This paper considers a model for the spread of an epidemic in a closed population whose members are in either a high-risk or a low-risk activity group. Further, members of the high-risk group may change their behaviour by entering the low-risk group. Both stochastic and deterministic models are examined. A limiting model, appropriate when there is a large number of initially susceptible individuals, is used to provide a threshold analysis. The epidemic is compared to a single group epidemic, and to suitably parametrised two-group epidemics, using a coupling method. The total size distribution and effects of changing the behaviour change rate are considered.
Sample mean based index policies byO(logn) regret for the multi-armed bandit problem
Authors: Agrawal Rajeev
Published: April 2006

We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whoseregretincreases slowly with time. In their seminal work on this problem, Lai and Robbins had obtained aO(logn) lower bound on the regret with a constant that depends on the Kullback–Leibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve aO(logn) regret with a constant that is also based on the Kullback–Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a ‘contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations.
On the HNBUE property in a class of correlated cumulative shock models
Authors: Pérez-Ocón Rafael , Gámiz-Pérez M. Luz
Published: April 2006

Conditions for a correlated cumulative shock model under which the system failure time is HNBUE are given. It is shown that the proof of a theorem given by Sumita and Shanthikumar (1985) relative to this property is not correct and a correct proof of the theorem is given.
Unification of Software Reliability Models by Self-Exciting Point Processes
Authors: Chen Yiping , Singpurwalla Nozer D.
Published: April 2006

Assessing the reliability of computer software has been an active area of research in computer science for the past twenty years. To date, well over a hundred probability models for software reliability have been proposed. These models have been motivated by seemingly unrelated arguments and have been the subject of active debate and discussion. In the meantime, the search for an ideal model continues to be pursued. The purpose of this paper is to point out that practically all the proposed models for software reliability are special cases of self-exciting point processes. This perspective unifies the very diverse approaches to modeling reliability growth and provides a common structure under which problems of software reliability can be discussed.
Correlations in bivariate point processes: some biological applications
Authors: Ayala Guillermo , Simó Amelia
Published: April 2006

Let (Φ1Φ2) be a bivariate point process. Letbe the probability that (Φ1Φ2) - (0,s) (the process without the 0 andspoints) verify U when we have a point of <1>1 in the origin and a point of Φ2in s. This is the reduced cross Palm distribution. Some correlation measures for bivariate point processes based on this reduced cross Palm distribution are proposed. Their estimators and expressions under the independence and the random labelling hypothesis are considered. The differences and improvements with respect to the cross intensity function and its integrated version, the crossfunction (Stoyanet al. 1987), are studied. Some Monte Carlo tests for testing the independence and the random labelling hypothesis are proposed. They are applied to real bivariate point patterns: positions of hickories and maples in the Lansing Woods (Diggle 1983) and cases and controls of childhood leukaemia and lymphoma in North Humberside (Cuzick and Edwards 1990).
Rates of convergence for random approximations of convex sets
Authors: Dümbgen Lutz , Walther Günther
Published: April 2006

The Hausdorff distance between a compact convex setK⊂ ℝdand random setsis studied. Basic inequalities are derived for the case ofbeing a convex subset ofK.If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case ofbeing the intersection of a random family of halfspaces containingK.
Markov chain models, time series analysis and extreme value theory
Authors: Poskitt D. S. , Chung Shin-Ho
Published: April 2006

Markov chain processes are becoming increasingly popular as a means of modelling various phenomena in different disciplines. For example, a new approach to the investigation of the electrical activity of molecular structures known as ion channels is to analyse raw digitized current recordings using Markov chain models. An outstanding question which arises with the application of such models is how to determine the number of states required for the Markov chain to characterize the observed process. In this paper we derive a realization theorem showing that observations on a finite state Markov chain embedded in continuous noise can be synthesized as values obtained from an autoregressive moving-average data generating mechanism. We then use this realization result to motivate the construction of a procedure for identifying the state dimension of the hidden Markov chain. The identification technique is based on a new approach to the estimation of the order of an autoregressive moving-average process. Conditions for the method to produce strongly consistent estimates of the state dimension are given. The asymptotic distribution of the statistic underlying the identification process is also presented and shown to yield critical values commensurate with the requirements for strong consistency.
The Gaussian distribution revisited
Authors: Puente Carlos E. , López Miguel M. , Pinzón Jorge E. , Angulo José M.
Published: April 2006

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.
The M/G/1 queue with negative customers
Authors: Harrison Peter G. , Pitel Edwige
Published: April 2006

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.
Computer Tomography and Quantum Mechanics
Authors: Klebanov Lev B. , Rachev Svetlozar T.
Published: April 2006

In this paper we study some topics of interest to specialists in computer tomography. These are the following. (a) The Radon transform and its applications to computer tomography. (b) Problems of computer tomography with partially known data. Estimates of stability will be given for different types of distance in the space of probability distributions. We consider the problem with partially known tomographic data as a stability problem for appropriately chosen distances. This approach allows us to give a solution of the so-called computer tomography paradox. (c) The relation of quantum mechanics to computer tomography. An intriguing method for ‘measuring' wavefunctions by tomographic methods (CAT scans) opens a new approach to various problems in quantum mechanics. Using the method outlined for the solution of the computer tomography paradox, we derive inequalities that estimate the amount of information on the wavefunctions resulting fromrealCAT scans, i.e. CAT scans based on thefinitenumber of measured marginals (projections) of the Wigner distributions. In conclusion, we propose a new version of the mathematical justification of CAT scans.
Stochastic Analysis of ‘Simultaneous Merge–Sort'
Authors: Cramer M.
Published: April 2006

The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k–1 and Mk can be interpreted as the number of comparisons of ‘Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.
The Rao–Blackwell theorem in stereology and some counterexamples
Authors: Baddeley A. J. , Cruz-Orive L. M.
Published: April 2006

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; randoms-dimensional samples are worse than randomr-dimensional samples fors < r.Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probeincreasesefficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.
How to look at objects in a five-dimensional shape space: looking at geodesics
Authors: Kendall David G.
Published: April 2006

This paper is the third in a series with the general title ‘How to look at the five-dimensional shape space Σ4,3. Parts I and II were concerned with distributions and diffusions respectively. Here we ‘look at’ geodesics.
Random high-dimensional orders
Authors: Bollobás Béla , Brightwell Graham
Published: April 2006

The randomk-dimensional partial orderPk(n) onnpoints is defined by takingnpoints uniformly at random from [0,1]k. Previous work has concentrated on the case wherekis constant: we consider the model wherekincreases withn.

We pay particular attention to the heightHk(n) ofPk(n). We show thatk= (t/logt!) lognis a sharp threshold function for the existence of at-chain inPk(n): ifk –(t/logt!) logntends to + ∞ then the probability thatPk(n) contains at-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour ofHk(n) for the entire range ofk(n).

We also consider the maximum degree ofPk(n). We show that, for each fixedd≧ 2,is a threshold function for the appearance of an element of degreed.Thus the maximum degree undergoes very rapid growth near this value ofk.

We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension ofPk(n) for largek(n).
A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps
Authors: Kesten Harry
Published: April 2006

We consider positive matricesQ,indexed by {1,2, …}. Assume that there exists a constant 1L <∞ and sequencesu1< u2< · ··andd1d2< · ··such thatQ(i, j) = 0 wheneveri<ur< ur+L < jori > dr+ L > dr> jfor somer. IfQsatisfies some additional uniform irreducibility and aperiodicity assumptions, then fors> 0,Qhas at most one positives-harmonic function and at most ones-invariant measureµ.We use this result to show that ifQis also substochastic, then it has the strong ratio limit property, that is

for a suitableRand someR–1-harmonic functionfandR–1-invariant measureµ.Under additional conditionsµcan be taken as a probability measure on {1,2, …} andexists. An example shows that this limit may fail to exist ifQdoes not satisfy the restrictions imposed above, even thoughQmay have a minimal normalized quasi-stationary distribution (i.e. a probability measureµfor whichR–1µ = µQ).

The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by timen.
On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines
Authors: Weiss Gideon
Published: April 2006

We consider scheduling a batch of jobs with stochastic processing times on single or parallel machines, with the objective of minimizing the expected holding costs. Preemption of jobs is allowed, and the holding costs of preempted jobs may depend on the stage of completion. We provide a new proof of the optimality of a Gittins priority rule for the single machine and use the same proof to show that the Gittins priority rule is nearly optimal for parallel machines.
Some diffusion models for the mabinogion sheep problem of williams
Authors: Chan Terence
Published: April 2006

The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.
Confidence bounds for the adjustment coefficient
Authors: Pitts Susan M. , Grübel Rudolf , Embrechts Paul
Published: April 2006

Let?(u)be the probability of eventual ruin in the classical Sparre Andersen model of risk theory if the initial risk reserve isu.For a large class of such models?(u)behaves asymptotically like a multiple of exp(–Ru)whereRis the adjustment coefficient;Rdepends on the premium income rate, the claim size distribution and the distribution of the time between claim arrivals. Estimation ofRhas been considered by many authors. In the present paper we deal with confidence bounds forR. A variety of methods is used, including jackknife estimation of asymptotic variances and the bootstrap. We show that, under certain assumptions, these procedures result in interval estimates that have asymptotically the correct coverage probabilities. We also give the results of a simulation study that compares the different techniques in some particular cases.
First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)
Authors: Picard Philippe , Lefèvre Claude
Published: April 2006

The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel–Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.
Branching processes with immigration
Authors: Pakes A. G.
Published: April 2006

Consider a branching process in which each individual reproduces independently of all others and has probabilityaj(j= 0, 1, ···) of giving rise tojprogeny in the following generation, and in which there is an independent immigration component where, with probabilitybj(j= 0, 1, ···)jobjects enter the population at each generation. Then lettingXn(n= 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that {Xn} defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual,, satisfies α < 1 or α > 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.
The definition of a multi-dimensional generalization of shot noise
Authors: Daley D. J.
Published: April 2006

The paper studies the formally defined stochastic process where {tj} is a homogeneous Poisson process in Euclidean n-space En and the a.e. finite Em-valued function f(·) satisfies |f(t)| = g(t) (all |t | = t), g(t) ↓ 0 for all sufficiently large t → ∞, and with either m = 1, or m = n and f(t)/g(t) =t/t. The convergence of the sum at (*) is shown to depend on

(i)

(ii)

(iii)

. Specifically, finiteness of (i) for sufficiently large X implies absolute convergence of (*) almost surely (a.s.); finiteness of (ii) and (iii) implies a.s. convergence of the Cauchy principal value of (*) with the limit of this principal value having a probability distribution independent of t when the limit in (iii) is zero; the finiteness of (ii) alone suffices for the existence of this limiting principal value at t = 0.
A note on random walks
Authors: Phatarfod R. M. , Speed T. P. , Walker A. M.
Published: April 2006

Let {Xn} be a random walk between reflecting barriers at 0 anda> 0 with jumps {Zn}. Bywe mean the random walk between absorbing barriers at— aand 0+ with the same jumps {Zn}. It has been known for some time that when {Zn} is a sequence of mutually independent and identically distributed random variables, and 0 ≦x<a, we have for alln:
On two mathematical models of the traffic on a divided highway
Authors: Rényi A.
Published: April 2006

In the present paper we deal with two models of the traffic on a divided highway. In both models traffic is flowing in one direction only, in two lanes, of which one (the left hand lane) is used only for overtaking. We suppose that vehicles enter the highway at the same entrance so that the instants at which a vehicle enters the highway form a homogeneous Poisson process, with density λ. Thus λ is the rate at which vehicles enter the highway per unit time. We suppose that there are no junctions, or exits (i.e. the highway extends in one direction to infinity).
On the theory of resonance integrals in the statistical region
Authors: Reichel A. , Wilkins C. A.
Published: April 2006

The problem of determining infinitely dilute resonance integrals is formulated in renewal theoretical terms. The mean value of the integral for a single resonance is determined in simple closed form. On the assumption that Wigner's hypothesis holds, the resonance density is determined, and a usable approximation to it is derived. An expression for the infinitely dilute resonance integral in the statistical region is then given and its value calculated in special cases and compared with the results of a previous computation.
Diffusion approximations and models for certain congestion problems
Authors: Gaver D. P.
Published: April 2006

In a variety of the congestion or queueing problems that arise in practice, for example, in studies of the crossing and entry problems of road traffic, (see Evans, Herman, and Weiss [2]), and recently of the service afforded by large centralized and shared computer facilities, (see Scherr [12]), the understanding of system performance furnished by the present mathematical theory is inadequate. The reason is that while the consideration of simple problems typically yields elegant mathematical results, the form of these results—often expressed in terms of integral transforms—is not immediately comprehensible nor useful for simple comparisons. This fact has been remarked upon by Newell, who in [9] has suggested certain more comprehensible but approximate approaches based on diffusion theory; further promising developments and elaborations will be found in [10]. The latter approach is related to the “heavy traffic theory” of J. F. C. Kingman [8], and to some recent work of Iglehart [6]. Of course, the idea of approximating complex discrete-state processes by diffusion processes with continuous paths is not new. It has long been used in genetics, see Feller [3], and the review paper by Kimura [7]. Nonetheless, applications to congestion theory are apparently still rather rare.
Average cost semi-markov decision processes
Authors: Ross Sheldon M.
Published: April 2006

The semi-Markov decision model is considered under the criterion of long-run average cost. A new criterion, which for any policy considers the limit of the expected cost incurred during the first n transitions divided by the expected length of the first n transitions, is considered. Conditions guaranteeing that an optimal stationary (non-randomized) policy exist are then presented. It is also shown that the above criterion is equivalent to the usual one under certain conditions.
On the distribution and moments of the strength of a bundle of filaments
Authors: Suh M. W. , Bhattacharyya B. B. , Grandage A.
Published: April 2006

Small sample and large sample properties of the bundle strength of parallel filaments studied earlier by Daniels (1945) and Sen, Bhattacharyya, Suh (1969) have been developed here by probabilistic argument. The statistics belong to a family or class of statistics, each of which forms a reverse semi-martingale sequence. Certain moment properties are also discussed.
Estimation for a family of life distributions with applications to fatigue
Authors: Birnbaum Z.W. , Saunders S.C.
Published: April 2006

The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.
Recurrence times of clusters of Poisson points
Authors: Leslie R.T.
Published: April 2006

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.
On nonlinear processes involving population growth and diffusion
Authors: Montroll Elliott W.
Published: April 2006

A number of years ago, R. A. Fisher discussed the problem of the propagation of a virile mutant in a population. At about the same time, Kolmogorov, Petrovsky, and Piscounoff, whom we shall refer to as KPP, investigated a general class of partial differential equations which describe simultaneous growth and diffusion processes.
Limit theorems for random mating in infinite populations
Authors: Ellison B. E.
Published: April 2006

This paper is concerned with the distribution of “types” of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the “sexes” of the two parents, but the law is assumed to apply independently of the “sex” of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated.
An application of stationary point processes to queueing theory and textile research
Authors: Rao J. Sudarsana
Published: April 2006

Throughout this paper we shall be interested in a certain integer-valued processn(t) (t≧ 0) which is encountered in queueing theory and textile research.
On calculating extinction probabilities for branching processes in random environments
Authors: Wilkinson William E.
Published: April 2006

Consider a discrete time Markov chain {Zn} whose state space is the non-negative integers and whose transition probability matrix ║Pij║ possesses the representation

where {Pr}, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z0 = k, a finite positive integer.
The total waiting time in a busy period of a stable single-server queue, I.
Authors: Daley D. J.
Published: April 2006

A quantity of particular interest in the study of (road) traffic jams is the total waiting timeXof all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variableXis the same as the sum of waiting times of customers in a busy period of aGI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.
On the algebra of queues
Authors: Kingman J. F. C.
Published: April 2006

The more one studies the vast and ever-growing literature of the theory of queues, the more one is bewildered by the wide variety of mathematical techniques which different authors have used to analyse the same or similar problems. And yet it seems that, beneath the superficial diversity, there is a deeper unity obscured by the special devices and notations characteristic of the different approaches. It is the thesis of this paper that there is indeed such a unity, and that it is best appreciated by observing that the central results of queueing theory, despite the analytical and combinatorial accretions which, by historical accident, they have acquired, are essentially algebraic in character.
On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states
Authors: Seneta E. , Vere-Jones D.
Published: April 2006

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.
On a generalized entropy and a coding theorem
Authors: Rathie Pushpa N.
Published: April 2006

Let P= {p1,···, PN} be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as

Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).
Mean life of series and parallel systems
Authors: Marshall Albert W. , Proschan Frank
Published: April 2006

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.
Customer average and time average queue lengths and waiting times
Authors: Marshall Kneale T. , Wolff Ronald W.
Published: April 2006

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationaryGI/G/mqueue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for theGI/G/1 queue. An upper bound is also determined for thekth factorial moment of the number found in the queue by an arrival in terms of thekth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for theGI/G/1 queue.
Stochastic models for bacteriophage
Authors: Gani J.
Published: April 2006

Viruses are small particles of RNA (Ribonucleic Acid) or DNA (DeoxyRibonucleic Acid) wrapped in a protein coat, which can be crystallized into a variety of regular geometric, often polyhedral, shapes. They are much smaller than bacteria, and are capable of passage through filters designed to, arrest these. Among the numerous viruses, bacteriophages (or bacterial viruses), called phages for short, have been the subject of much concentrated study. Over the past twenty years in particular, their structure, parasitic cycle, and most recently part of their genetic mapping, have been elucidated; the isolated DNA strand used by phages as their genetic information carrier, makes them eminently suitable in investigations on the molecular basis of life.
Products of distribution functions attracted to extreme value laws
Authors: Resnick Sidney I.
Published: April 2006

When is the product of the d.f.'s H1(·), ···, Hm(·) attracted to an extreme value law φ(x)? We associate with each Hi(·) its A-function Hi(x) is attracted to φ(x) if each Hi(x) is in the domain of attraction of φ(x) and Ai(z) ~ Aj(z), 1 ≦ i, j ≦ m. Equivalence of A-functions determines an equivalence relation which partitions the domain of attraction of φ(x)into one or more convex sets. These sets fail to be closed under passages to the limit (complete convergence).
An indication of the asymptotic nature of the mendelian Markov process
Authors: Khazanie R. G.
Published: April 2006

Consider a finite Markov process {Xn} described by the one-step transition probabilities

In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.
A note on the asymptotic distribution of the traffic-time-average in a GI/G/∞ with bulk arrivals
Authors: Narasimham T.
Published: April 2006

The asymptotic distribution of the traffic average in the case of M/G/∞ with bulk arrivals has been proved in [7] to be normal with mean μCμPμF and variance where and are the variances and μP and μc are the means of the service-time and the batch-size respectively, μF being the mean inter-arrival time. We prove in this note that the asymptotic normality holds even in the more general case of a G//G/∞ with bulk arrivals.
Random selective advantages of a gene in a finite population
Authors: Jensen Louis , Pollak Edward
Published: April 2006

A problem of interest to many population geneticists is the process of change in a gene frequency. A popular model used to describe the change in a gene frequency involves the assumption that the gene frequency is Markovian. The probabilities in a Markov process can be approximated by the solution of a partial differential equation known as the Fokker-Planck equation or the forward Kolmogorov equation. Mathematically this equation iswhere subscripts indicate partial differentiation. In this equation,f(p, x; t)is the probability density that the frequency of a gene isxat timet, given that the frequency waspat timet= o. The expressions MΔXand VΔxare, respectively, the first and second moments of the change in the gene frequency during one generation. A rigorous derivation of this equation was given by Kolmogorov (1931).
The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for theM/G/1 Queueing System
Authors: Enns E.G.
Published: April 2006

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:

1.

The duration of the busy period.

2.

The number of customers served during the busy period.

3.

The maximum number of customers in the queue during the busy period.
On the Distribution of the Number of Zero Partial Sums
Authors: Percus Ora Engelberg , Percus Jerome K.
Published: April 2006

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.
The distribution of the content of finite dams
Authors: Takács Lajos
Published: April 2006

We shall consider the following model of finite dams: In the time interval (0, ∞) water is flowing into a dam (reservoir) in accordance with a random process. Denote byχ(u) the total quantity of water flowing into the dam in the time interval (0,u). The capacity of the dam is a finite positive numberm.If the dam becomes full, the excess water overflows. If the dam is not empty, there is a continuous release at a constant unit rate. Denote byη(t) the content of the dam at timet.
On quasi-stationary distributions in absorbing continuous-time finite Markov chains
Authors: Darroch J. N. , Seneta E.
Published: April 2006

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.
The initial geographical spread of host-vector and carrier-borne epidemics
Authors: Radcliffe J.
Published: April 2006

The deterministic and stochastic models developed by Bartlett (1956) to describe the initial spatial spread of an epidemic such as measles are extended to host-vector and carrier-borne epidemics such as malaria and typhoid in which more than one type of individual is involved. If the epidemic does not die out, then the behaviour predicted by the deterministic model and an examination of the mean distributions in the stochastic model is a wave of infection spreading out from the initial source of infection.
An M/G/1 model with finite waiting room in which a customer remains during part of service
Authors: Rosenlund Stig I.
Published: April 2006

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.
Results in the asymptotic and equilibrium theory of Poisson cluster processes
Authors: Westcott M.
Published: April 2006

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) ast→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.