bluuebunny/crossref_metadata_2025 huggingface dataset:
Connecting internally balanced quasi-reversible Markov processes
Authors: Henderson W. , Pearce C. E. M. , Pollett P. K. , Taylor P. G.
Published: April 2006

We provide a general framework for interconnecting a collection of quasi-reversible nodes in such a way that the resulting process exhibits a product-form invariant measure. The individual nodes can be quite general, although some degree of internal balance will be assumed. Any of the nodes may possess a feedback mechanism. Indeed, we pay particular attention to a class of feedback queues, characterized by the fact that their state description allows one to maintain a record of the order in which events occur. We also examine in some detail the problem of determining for which values of the arrival rates a node does exhibit quasi-reversibility.
A finite form for the wrapped Poisson distribution
Authors: Ball Frank , Blackwell Paul
Published: April 2006

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.
Preservation of phase-type distributions under Poisson shock models
Authors: Manoharan M. , Singh Harshinder , Misra Neeraj
Published: April 2006

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.
Discrete-time queueing networks with geometric release probabilities
Authors: Henderson W. , Taylor P. G.
Published: April 2006

This note is concerned with the continuing misconception that a discrete-time network of queues, with independent customer routing and the number of arrivals and services in a time interval following geometric and truncated geometric distributions respectively, has a product-form equilibrium distribution.
Consistent ordered sampling distributions: characterization and convergence
Authors: Donnelly Peter , Joyce Paul
Published: April 2006

This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.
Basic properties and prediction of max-ARMA processes
Authors: Davis Richard A. , Resnick Sidney I.
Published: April 2006

A max-autoregressive moving average (MARMA(p, q)) process {Xt} satisfies the recursion

for all t where φ i, , and {Zt} is i.i.d. with common distribution function Φ1,σ (X): = exp {–σ x–1} for . Such processes have finite-dimensional distributions which are max-stable and hence are examples of max-stable processes. We provide necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion and we examine the reducibility of the process to a MARMA(p′, q′) with p′ <p or q′ < q. After introducing a natural metric between two jointly max-stable random variables, we consider the prediction problem for MARMA processes. Assuming that X1, …, Xn have been observed, we restrict our class of predictors to be max-linear, i.e. of the form , and find b1, …, bn to minimize the distance between this predictor and Xn+k for k 1. The optimality criterion is designed to minimize the probability of large errors and is similar in spirit to the dispersion criterion adopted in Cline and Brockwell (Stoch. Proc. Appl. 19 (1985), 281-296) for the prediction of ARMA processes with stable noise. Most of our results remain valid for the case when the distribution of Z1 is only in the domain of attraction of Φ1,σ. In addition, we give a naive estimation procedure for the φ 's and the θ 's which, with probability 1, identifies the true parameter values exactly for n sufficiently large.
On the stability condition of a precedence-based queueing discipline
Authors: Baccelli François , Liu Zhen
Published: April 2006

The queueing model known asprecedence-based queueing discipline, proposed in [8] by Tsitsiklis et al. is addressed. In this queueing model, there are infinitely many servers and for any pair of customersiandjsuch thatiarrived later thanj,there is a fixed probabilitypthatiwill have to wait for the end of the execution ofjbefore starting its execution. In the case where the customer service times are deterministic and the arrival process is Poisson, these authors have derived the stability condition which determines the maximum arrival rate that keeps the system stable. For more general statistics, they conjectured that the stability condition would possibly depend on the complete service time distribution functions but only on the first moment of the inter-arrival times.

In this paper, we consider this queueing model and relax the restrictive statistical assumptions mentioned above by only assuming that the service times and the inter-arrival times are stationary and ergodic sequences, so that these variables can receive general distribution functions and be correlated.

We derive a general expression for the stability condition which in turn proves the above conjectures. The results are obtained by establishing pathwise evolution equations for these queueing systems, and then a schema which, in certain sense, generalizes the schema of Loynes for theG/G/1 queues.
On the first hitting place of the integrated Wiener process
Authors: Lefebvre Mario , Léonard Éric
Published: April 2006

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).
Large deviations in the supercritical branching process
Authors: Biggins J. D. , Bingham N. H.
Published: April 2006

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.
Compound poisson approximations for the numbers of extreme spacings
Authors: Roos Małgorzata
Published: April 2006

The accuracy of the Poisson approximation to the distribution of the numbers of large and smallm-spacings, whennpoints are placed at random on the circle, was analysed using the Stein–Chen method in Barbour et al. (1992b). The Poisson approximation for m≧2 was found not to be as good as for 1-spacings. In this paper, rates of approximation of these distributions to suitable compound Poisson distributions are worked out, using the CP–Stein–Chen method and an appropriate coupling argument. The rates are better than for Poisson approximation for m≧2, and are of orderO((logn)2/n) for largem-spacings and of orderO(1/n) for smallm-spacings, for any fixed m≧2, if the expected number of spacings is held constant asn → ∞.
The Representation and Decomposition of Integrated Stationary Time Series
Authors: Chen Zhao-Guo , Anderson Oliver D.
Published: April 2006

Learning from Matheron's representation (1973), and using the increment vector (PIV) methodology introduced by Cressie (1988) and developed by Chen and Anderson (1994), this paper presents a theory for the representation and decomposition of integrated stationary time series and gives some applications.
Another Look at the Ehrenfest Urn Via Electric Networks
Authors: Palacios José Luis
Published: April 2006

Using the electric network approach, we give closed-form formulas for the expected hitting times in the Ehrenfest urn model.
Bivariate random closed sets and nerve fibre degeneration
Authors: Ayala Guillermo , Simó Amelia
Published: April 2006

A biphase image, representing the normal and degenerated fibres in a vertical cross-section of a nerve, is considered. A random set model based on a Gibbs point process is proposed for the union of the two phases. A kind of independence between the degeneration process and the original fibres is defined and tested.
On the mean shape of particle processes
Authors: Weil Wolfgang
Published: April 2006

For a stationary point processXof sets in the convex ring in ℝd, a relation is given between the mean particles of the section processX∩E(whereEvaries through the set ofk-dimensional subspaces in ℝd) and a mean particle ofX.In particular, it is shown that the mean bodies of all planar sections ofXdetermine the Blaschke body ofXand hence the mean normal distribution ofX.
Empirical clustering of bursts of openings in markov and semi-markov models of single channel gating incorporating time interval omission
Authors: Ball Frank
Published: April 2006

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than t0. Let N(t) be the number of bursts commencing in (0, t]. Then

are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.

The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.
Explicit asymptotic results for open times in ion channel models
Authors: Stefanov Valeri T. , Yeo Geoffrey F.
Published: April 2006

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.
Shapes of Rectangular Prisms After Repeated Random Division
Authors: Cowan Richard
Published: April 2006

The shape of a rectangular prism in (d + 1)-dimensions is defined as Y = (Y1, Y2, · ··, Yd), Yn = Ln/Ln+1 where the Ln are the prism's edge lengths, in ascending order. We investigate shape distributions that are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, 1]d arise.
The Multivariate Ginar(p) Process
Authors: Latour Alain
Published: April 2006

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.
Queueing in space
Authors: Altman Eitan , Levy Hanoch
Published: April 2006

We consider a problem in which a single server must serve a stream of customers whose arrivals are distributed over a finite-size convex space. Under the assumption that the server has full information on the customer location, obvious service policies are the FCFS and the greedy (serve-the-closest-customer) approaches. These algorithms are, however, either inefficient (FCFS) or ‘unfair' (greedy).

We propose and study two alternative algorithms, the gated-greedy policy and the gated-scan policy, which are more ‘fair' than the pure greedy method. We show that the stability conditions of the gated-greedy are p < 1 (where p is the expected rate at which work arrives at the system), implying that the method is at least as efficient (in terms of system stability) as any other discipline, in particular the greedy one. For the gated-scan policy we show that for any p < 1 one can design a stable gated-scan policy; however, for any fixed gated-scan policy there exists p < 1 for which the policy is unstable. We evaluate the performance of the gated-scan policy, and present bounds for the performance of the gated-greedy policy.

These results are derived for systems in which the arrivals occur on a two-dimensional space (a square) but they are not limited to this configuration; rather they hold for more complex N-dimensional spaces, in particular for serving customers in (three-dimensional) convex space and serving customers on a line.
Conditional cyclic Markov random fields
Authors: Kent John T. , Mardia Kanti V. , Walder Alistair N.
Published: April 2006

Grenanderet al.(1991) proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes' (Kasset al.1987) is established. Applications are given in Grenanderet al.(1991), Mardiaet al.(1991) and Kentet al.(1992).
Scanning Brownian Processes
Authors: Adler Robert J. , Pyke Ron
Published: April 2006

The ‘scanning process'Z(t),t ∈ ℝkof the title is a Gaussian random field obtained by associating withZ(t)the value of a set-indexed Brownian motion on the translatet+A0of some ‘scanning set'A0. We study the basic properties of the random fieldZrelating, for example, its continuity and other sample path properties to the geometrical properties ofA0. We ask if the setA0determines the scanning process, and investigate when, and how, it is possible to recover the structure ofA0from realisations of the sample paths of the random fieldZ.
Bounds for the Joint Survival and Incidence Functions Through Coherent System Data
Authors: Deshpande J. V. , Karia S. R.
Published: April 2006

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.
A Geometric Product-Form Distribution for a Queueing Network by Non-Standard Batch Arrivals and Batch Transfers
Authors: Miyazawa Masakiyo , Taylor Peter G.
Published: April 2006

We introduce a batch service discipline, calledassemble-transfer batch service,for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed.

Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.
The random division of faces in a planar graph
Authors: Cowan Richard , Chen Simone
Published: April 2006

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.
An induction proof of the generalized Blaschke-Petkantschin formula
Authors: Jensen E. B. Vedel
Published: April 2006

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.
A point-process approach to almost-sure behaviour of record values and order statistics
Authors: Goldie Charles M. , Maller Ross A.
Published: April 2006

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima (the rth largest among a sample of n i.i.d. random variables), partial record values and differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties as a.s. for some finite constant c, or a.s. for some constant c in [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail of F, which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.
Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments
Authors: Seleznjev Oleg
Published: April 2006

We consider the piecewise linear interpolation of Gaussian processes with continuous sample paths and stationary increments. The interrelation between the smoothness of the incremental variance function,d(t – s) =E[(X(t)– X(s))2], and the interpolation errors in mean square and uniform metrics is studied. The method of investigation can also be applied to the analysis of different methods of interpolation. It is based on some limit results for large deviations of a sequence of Gaussian non-stationary processes and related point processes. Non-stationarity in our case means mainly the local stationary condition for the sequence of correlation functionsrn(t,s),n= 1, 2, ···, which has to hold uniformly inn.Finally, we discuss some examples and an application to the calculation of the distribution function of the maximum of a continuous Gaussian process with a given precision.
On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input
Authors: Miyazawa Masakiyo , Schassberger Rolf , Schmidt Volker
Published: April 2006

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).
On a Voronoi aggregative process related to a bivariate Poisson process
Authors: Foss S. G. , Zuyev S. A.
Published: April 2006

We consider two independent homogeneous Poisson processes Π0and Π1in the plane with intensities λ0and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.
Consistency in systematic sampling
Authors: Arnau X. Gual , Cruz-Orive L. M.
Published: April 2006

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points). Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. The purpose of this paper is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling. Relevant mechanisms to increase sample size, compatible with stereological practice, are considered.
Dynamic routing and jockeying controls in a two-station queueing system
Authors: Xu Susan H. , Zhao Y. Quennel
Published: April 2006

This paper studies optimal routing and jockeying policies in a two-station parallel queueing system. It is assumed that jobs arrive to the system in a Poisson stream with rate λand are routed to one of two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, μ1 and μ2. Jockeying between stations is permitted. The jockeying cost is cij when a job in station i jockeys to station j, i ≠ j. There is no cost when a new job joins either station. The holding cost in station j is hj, h1 ≦ h2, per job per unit time. We characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both discounted and long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically non-decreasing functions. We study the properties of these control functions, their relationships, and their asymptotic behavior. We show that some well-known queueing control models, such as optimal routing to symmetric and asymmetric queues, preemptive or non-preemptive scheduling on homogeneous or heterogeneous servers, are special cases of our system.
Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms
Authors: Wang I-Jeng , Chong Edwin K. P. , Kulkarni Sanjeev R.
Published: April 2006

We consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark's condition, Chen's condition, a decomposition condition, and Kulkarni and Horn's condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.
The application of diffusion theory to two population genetic models of Moran
Authors: Watterson G. A.
Published: April 2006

Moran (1958a) (1958b) introduced two population genetic models of a general kind to investigate the combined effects on evolution of four factors,
The Martin boundary for Polya's urn scheme, and an application to stochastic population growth
Authors: Blackwell David , Kendall David
Published: April 2006

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box containsbblack andrred balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ballof the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keepm= 2 and it will be convenient further to simplify the scheme by takingb=r= 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary numberk(≧2) of colours. Thus initially the box will containkdifferently coloured balls and successive random drawings will be followed by double replacement as before. We writesn(ak-vector withjth component) for the numerical composition of the box immediately after the nth replacement, so thatand we observe thatis a Markov process for which the state-space consists of all orderedk-ads of positive integers, the (constant) transition-probability matrix having elements determined bywhere Snis the sum of the components of snand (e(i))j= δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.
A three-player extension of the gambler's ruin problem
Authors: Barnett V. D.
Published: April 2006

The classical two-player gambler's ruin problem is well known and has attracted a great deal of attention in the literature in various forms. Extensions of these two-player problems to three or more players do not appear to have been discussed however, and this paper is concerned with one particular extension of the problem to the case of three players.
The use of a ballot theorem in order statistics
Authors: Takács Lajos
Published: April 2006

The following generalization of the classical ballot theorem has many possible applications in order statistics.
On the two sided barrier problem
Authors: Shinozuka Masanobu
Published: April 2006

Upper and lower bounds are given for the probability that a separable random processX(t) will take values outside the interval (—λ1,λ2) for 0 ≦t≦T, whereλ1andλ2are positive constants.

The random process needs to be neither stationary, Gaussian nor purely random (white noise).

In engineering applications,X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake.

Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.
On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains
Authors: Darroch J. N. , Seneta E.
Published: April 2006

The time to absorption from the setTof transient states of a Markov chain may be sufficiently long for the probability distribution overTto settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.
Probability of random paths across elementary geometrical shapes
Authors: Horowitz Maurice
Published: April 2006

There are several practical situations where one requires the statistical properties of the straight path length l across a specified geometrical shape. Typical examples are the length of the path of a gamma-ray to the wall of a nuclear reactor (Primak (1956)), the length of a sound ray in a room from one reflection to the next (Kosten (1960)), or the length of a straight path across a square (Horowitz (1964)).
General customer impatience in the queue GI/G/1
Authors: Daley D. J.
Published: April 2006

For the queueing system GI/G/1 with both waiting-line and service-line customer impatience an integral equation for the limiting waiting-time distribution function W(x) is derived and the existence of W(x) proved when the system satisfies certain conditions. Two counter examples are constructed showing that W(x) does not necessarily exist. A sufficient condition for its existence in GI/G/1 with deterministic waiting-line impatience is given. Formal solutions for M/G/l are derived for deterministic impatience and negative exponential impatience.
On the maximum of the queueGI/M/1
Authors: Heathcote C. R.
Published: April 2006

SupposeX1,X2,X3,… are independent and identically distributed positive nonlattice random variables with common distribution functionF(x).
Denumerable Markov processes with bounded generators: a routine for calculating pij(∞)
Authors: Jensen Arne , Kendall David
Published: April 2006

1. Let the (honest) Markov process with transition functions (pij(0)) have transition rates (qij) and suppose that, for some M,

so that the matrix Q = (qij) determines a bounded operator on the Banach space l1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and ΩF will be a closed restriction of Q with dense domain, so that ΩF = Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt = (pij(t)) is given by

where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P∞ = (limt→∞pij(t)).
Migratory populations in branching processes
Authors: Crump Kenny S.
Published: April 2006

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases.
Linear programming and continuous markovian decision problems
Authors: Mine Hisashi , Tabata Yoshio
Published: April 2006

This paper is concerned with a continuous time parameter Markovian sequential decision process, and presents a method which transforms a given continuous parameter problem into a discrete one. It is proved that the optimal stationary policy for the resulting discrete time parameter Markovian decision process is also the optimal stationary policy for the original continuous one, and vice versa. The resulting discrete parameter problem may be more easily solved than the continuous one by applying the linear programming method. A simple numerical example is presented.
On the expected number of crossings of a level in certain stochastic processes
Authors: Roes P. B. M.
Published: April 2006

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].
A generalisation of erlang's formulas in queueing theory
Authors: Mejzler D.
Published: April 2006

We follow the terminology of Khintchine's monograph [4].

Let us consider a random stream of calls entering a service system which consists of n lines. The service durations are assumed to be identically distributed random variables which are independent both of each other and of the course of the incoming stream.
A Mendelian Markov process with multinomial transition probabilities
Authors: Khazanie R. G. , McKean H. E.
Published: April 2006

The utility of the moments of the binomial distribution in finding the n-stage transition probabilities was indicated by Khazanie and McKean (1966). In this paper, maintaining the same breeding structure as in the two-allele case, it is proposed to extend the technique to cover the multiallelic case in general, with specific emphasis on the triallelic case. The existing literature on the multiple allelic case is at best scanty (see for example Kimura (1955)). Section 6 will briefly touch on a further extension to multiple loci which segregate independently.
Moments of coverage and coverage spaces
Authors: Ailam Gedalia
Published: April 2006

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.
A semi-markov model for clinical trials
Authors: Weiss George H. , Zelen Marvin
Published: April 2006

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.

Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.
Solutions of some two-sided boundary problems for sums of variables with alternating distributions
Authors: Chover J. , Yeo G.
Published: April 2006

In this paper we present a method for obtaining explicit results for some two-sided boundary problems involving sums of independent random variables with alternating distributions. We apply the method to finding the first passage time to either one of two finite barriers, and to some situations arising in queueing and dam theory. The results can be expressed in terms of a finite sum of simple repeated integrals (or sums) of known functions (cf. formulae (3.6)– (3.11)).
On some problems concerning a restricted random walk
Authors: Engelberg Ora
Published: April 2006

A particle that is located at the position (0, 0) in the (x, y) plane at the moment x = 0 is subjected to random impulses at the moments of time x = 1, 2, …, a + b. As the result of each impulse, the particle may be displaced either by a positive or by a negative step with respect to the y-axis; it is given that the particle is restricted by the condition that it has exactly a positive and b negative displacements. Hence in the (x, y) plane the path of the particle at each impulse is depicted by a shift of a unit to the right and by one unit upward or downward; that is the particle's path starts at (0, 0) and terminates at (a + b, a – b) after a + b steps. We assume that all possible different paths are equally probable, and assume also, without loss of generality, that a ≧ b.
An asymptotic result in traffic theory
Authors: Gani J. , Lehoczky J.
Published: April 2006

In problems of traffic theory, we are frequently concerned with the queues of vehicles which form at an intersection. A standard model is the vehicle queue whose length is increased during unit time intervals [t, t + 1) by non-negative integer inputs which form a sequence of i.i.d. or Markovian random variables; see Gani (1970) and Lehoczky ((1969), (1971)). The output from the queue at the end of each unit of time is one vehicle if the queue is non-empty, and no vehicle if there are none waiting.
Bounds for Certain Branching Processes
Authors: Pollak Edward
Published: April 2006

We consider a branching process for which the offspring distribution has the generating function f(t) and mean f'(1) = m < 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.
Some results for general cell-size-dependent branching processes
Authors: Sudbury Aidan , Clifford Peter
Published: April 2006

A general model for the growth and division of cells in which the growth rate and division probability at any instant depend only on their size at that time is introduced. Conditions under which (a) the distribution of cell-size at division converges ergodically, (b) the sizes tend to 0 or ∞, are exhibited, and bounds to the correlation between the sizes at division of sister cells are given in a wide class of cases.
First-emptiness problems in applied probability under first-order dependent input and general output conditions
Authors: Lehoczky J. P.
Published: April 2006

Results for the first-emptiness time of a semi-infinite reservoir and the integral functional of the process up to first-emptiness time are derived under Markov chain input conditions and general output conditions. The results are further extended to allow an input process which is the sum of k consecutive elements of the Markov chain, k ≧ 1.
Estimation of the mean of a stationary time series by sampling
Authors: Brillinger David R.
Published: April 2006

Let X(t), – ∞ < t < ∞, be a stationary time series with mean cx. Let 0 < τ1 < τ2 < … < τN ≦ T denote A given sampling times in the interval (0, T]. We determine the asymptotic distribution of the estimate [X(τ1) + … + X(τN)]/N of cx when the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the series X(t). The results obtained may be viewed as non-standard central limit theorems.
Extremal processes and record value times
Authors: Resnick Sidney I.
Published: April 2006

Let {Xn, n ≧ 1} be i.i.d. and Yn = max {X1,…, Xn}. Xj is a record value of {Xn} if Yj > Yj–1 The record value times are Ln, n ≧ 1 and inter-record times are Δn, n ≧ 1. The known limiting behavior of {Ln} and {Δn} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.
On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures
Authors: Sharma S. D.
Published: April 2006

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.
Busy period of a finite queue with phase type 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 two phases, during the first of which a place in the waiting room is occupied. Starting from a determinant expression for the Laplace-Stieltjes transform of the busy period given in [2] we obtain contour integral expressions for the transform and the expectation, thereby generalising a result by Cohen [1]. This is effected by developing the determinants along the first row.
The asymptotic behaviour of the minimal total expected cost for the denumerable state Markov decision model
Authors: Hordijk Arie , Schweitzer Paul J. , Tijms Henk
Published: April 2006

This paper considers the discrete time Markov decision model with a denumerable state space and finite action space. Under certain conditions it is proved that the minimal total expected cost for a planning horizon of n epochs minus n times the minimal long-run average expected cost per unit time has a finite limit as n → ∞ for each initial state.
Weak convergence and first passage times
Authors: Gut Allan
Published: April 2006

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].
A note on roots of Markov shifts
Authors: Rudolfer S. M.
Published: April 2006

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.
Interconnected population processes
Authors: Renshaw E.
Published: April 2006

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.
An approximate characterisation of optimal stopping boundaries
Authors: Whittle P.
Published: April 2006

An identity, of the type of Green's equation, is deduced for the loss function of a stopping process. This yields a set of relations determining the optimal (minimal loss) stopping boundary, which do not require simultaneous determination of the loss function in the stopping region. A no-overshoot approximation is invoked, but a bound on the magnitude of the terms neglected is obtained by appeal to a general version of Chernoff's tangency condition at an optimal boundary.
An algebraic approach to the waiting time process in GI/M/S
Authors: Loris-Teghem Jacqueline
Published: April 2006

The transient behaviour of the waiting time process in GI/M/S is studied algebraically by means of a two-dimensional Markovian process {(vn, ln} , where the variables vn denote the times of full occupation immediately after the arrival instants Tn and where ln = max {0,– 1 + number of idle servers at (Tn + 0)}.
A note on the cost of carrier-borne, right-shift, epidemic models
Authors: Kryscio Richard J. , Saunders Roy
Published: April 2006

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.