bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarks on renewal theory for percolation processes
Authors: Smythe R. T.
Published: April 2006

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) anL1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the lineX = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.
A continuous-time population model with poisson recruitment
Authors: McClean Sally I.
Published: April 2006

A continuous-time model of a multigrade system is developed, which includes Poisson arrivals, interaction between grades and a leaving process. It therefore constitutes a continuous-time analogue of Pollard's hierarchical population model with Poisson recruitment. An expression is found for the first and second moments of grade size at any time. A general formulation of the joint probability generating function of the numbers in each grade is given, and the limiting distribution of grade size is shown to be Poisson.
The mean number of maxima above high levels in Gaussian random fields
Authors: Hasofer A. M.
Published: April 2006

An asymptotic formula for the mean number of maxima above a level of an n-dimensional stationary Gaussian field has been given by Nosko without proof. In this note a short general proof of this formula is given.
Reliability systems in random environment
Authors: Råde L.
Published: April 2006

A parallel reliability system is subject to shocks which are generated by a renewal point process. The shocks cause the components to fail with certain probabilities. First we study the case when the components fail independently of each other with the same probability. For this case simple closed expressions for the Laplace-Stieltjes transform and expectation of time to system failure are derived. Then a case of dependent components is studied: the probability of failure of a component depends on the number of functioning components. Some extensions are mentioned.
On a point process with independent locations
Authors: Isham Valerie
Published: April 2006

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.
Conservative processes with stochastic rates
Authors: Saunders Roy
Published: April 2006

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.
On platoon formation on two-lane roads
Authors: Barzily Zeev , Rubinovitch Michael
Published: April 2006

An approximate model for the study of platoon formation on two-lane highways is discussed in detail. The model assumes that the two-lane highway is divided in each traffic direction into alternating road sections of fixed lengths. The passing in one type of section is unrestricted and the passing in the other one is prohibited. It is assumed that there are slow and fast vehicles on the highway and that inputs follow independent Poisson processes. The results include the distribution of the number of vehicles in a platoon and the average speed of a typical fast vehicle.
The front velocity of the simple epidemic
Authors: Wierman John C.
Published: April 2006

The propagation rate of the one-dimensional stochastic simple epidemic converges almost surely to a front velocity for the epidemic. Percolation methods are used in the proof.
Time reversal and age distributions, I. Discrete-time Markov chains
Authors: Tavaré S.
Published: April 2006

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.
Non-linear time series models for non-linear random vibrations
Authors: Ozaki T.
Published: April 2006

Non-linear time series models for non-linear vibrations are presented. Some typical behaviour of non-linear vibrations generated from Duffing's equation or van der Pol's equation are explained through the models.
A representation for discrete distributions by equiprobable mixtures
Authors: Peterson Arthur V. , Kronmal Richard A.
Published: April 2006

We obtain a representation of an arbitrary discrete distribution withnmass points by an equiprobable mixture of r distributions, each of which has no more thana(≧2) mass points, where r is the smallest integer greater than or equal to (n –1)/(a –1).An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.
On the minimum of gaps generated by one-dimensional random packing
Authors: Itoh Yoshiaki
Published: April 2006

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, t]. We have

with

Using this equation the asymptotic behaviour of P(L(x)≧h) is discussed.
Correlation structure of teletraffic measurements from randomly-scanned calls arrival data
Authors: Lee P. A.
Published: April 2006

In teletraffic measurements, a call arrival process is commonly studied using a method with time-uniform or periodic scanning. The information recorded is the number of calls arrived between the scannings, from which data the number of scans between two successive calls is obtained. These later numbers are used as a measure of the interarrival times.

For an exponential call arrival process, except in the case of Poisson scanning, all other scanning schemes yield the number of scans which are not independent in any two interarrival intervals. By treating the problem as an interaction of two stationary stochastic point processes, we determine the exact joint probability distribution of the number of scans in two adjacent and non-adjacent interarrival intervals. An explicit expression for the correlation coefficient is also obtained.
Moments of ladder heights in random walks
Authors: Doney R. A.
Published: April 2006

A well-known result in the theory of random walks states that E{X2} is finite if and only if E{Z+} and E{Z_} are both finite (Z+ and Z_ being the ladder heights and X a typical step-length) in which case E{X2} = 2E{Z+}E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z+ and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.
The probabilistic significance of the rate matrix in matrix-geometric invariant vectors
Authors: Neuts Marcel F.
Published: April 2006

Many queueing models have embedded Markov chains, whose invariant probability vector is of a (modified) matrix-geometric form. The rate matrixRis shown to have the following probabilistic interpretation:

The elementRvjis the expected number of visits to the state (i+ 1,j), before the first return to the seti= {(i, 1), …,(i, m)}, in a chain starting in the state (i,v).
On the lengths of the pieces of a stick broken at random
Authors: Holst Lars
Published: April 2006

Consider the pieces of a randomly broken stick. How long is thejth longest piece? How many breaks are necessary for getting all pieces less than a given length? These and related questions are studied in particular when the number of pieces is large. Using simple properties of the exponential distribution new proofs are given of old results and new results are obtained.
Limit theorems for Markov processes via a variant of the Trotter-Kato theorem
Authors: Rosenkrantz Walter A. , Dorea C. C. Y.
Published: April 2006

A variant of the Trotter–Kato theorem due to Kurtz (1969) is used to give new and simpler proofs of functional central limit theorems for Markov processes. Applications include theorems of Bellman and Harris (1951), Stone (1961), Karlin and McGregor (1965), Gihman and Skorokhod (1972) and Rosenkrantz (1975). In addition our methods yield a novel counterexample to the so-called ‘diffusion approximation'.
Bounds for the expected delays in some tandem queues
Authors: Niu Shun-Chen
Published: April 2006

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case ofnqueues in tandem where all the servers except the last one have constant service times.
Bounds on the velocity of spread of infection for a spatially connected epidemic process
Authors: Faddy M. J. , Slorach I. H.
Published: April 2006

The simple (non-spatial) stochastic epidemic is generalised to allow infected individuals to move forward through a system of spatially connected coloniesC1,C2,C3, ·· ·each containing susceptible individuals. Upper and lower bounding processes are considered, to establish bounds on the asymptotic velocity of forward spread of the infection through these spatially connected colonies. These bounds are shown to be asymptotically equivalent under certain conditions, and some simulations reveal other features of the process.
Characterizations of the exponential distribution by relevation-type equations
Authors: Grosswald E. , Kotz Samuel , Johnson N. L.
Published: April 2006

In this note characterizations of the exponential distribution are discussed, based on a generalization of the lack of memory property. The result was motivated by the notion of ‘relevation of distributions' introduced by Krakowski (1973).
On a generalization of gamma processes
Authors: Çinlar Erhan
Published: April 2006

We introduce a class of increasing processes with independent increments that reduce to gamma processes under the further condition of stationarity. Each such process can be reduced to a simple gamma process by a stochastic integral transformation. Applications to deformation laws of materials such as concrete are mentioned.
Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals
Authors: Ramaswami V. , Neuts Marcel F.
Published: April 2006

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.
Linked loci in finite populations
Authors: Holgate P.
Published: April 2006

A finite population of gametes is studied, classified according to the alleles present atklinked loci. A canonical method of following the joint probability distribution of the gametic types from generation to generation is developed. It is shown how the investigation of the rate of first fixation can be systematised. Explicit results are given fork= 2, and, although not so complete, fork= 3.
On the continuity of the time constant of first-passage percolation
Authors: Cox J. Theodore , Kesten Harry
Published: April 2006

LetUbe the distribution function of the non-negative passage time of an individual edge of the square lattice, and leta0nbe the minimal passage time from (0, 0) to (n, 0). The processa0n/nconverges in probability asn→ ∞to a finite constantμ(U) called the time constant. It is proven thatμ(Uk)→ μ(U) wheneverUkconverges weakly toU.
θ-stationary point processes and their second-order analysis
Authors: Byth Karen
Published: April 2006

The concept of θ-stationarity for a simple second-order point process inR2is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.
Likelihood and nearest-neighbor distance properties of multidimensional Poisson cluster processes
Authors: Baudin Michel
Published: April 2006

The probability generating functional representation of a multidimensional Poisson cluster process is utilized to derive a formula for its likelihood function, but the prohibitive complexity of this formula precludes its practical application to statistical inference. In the case of isotropic processes, it is however feasible to compute functions such as the probabilityQ(r) of finding no point in a disc of radius r and the probabilityQ(r| 0) of nearest-neighbor distances greater thanr, as well as the expected numberC(r| 0) of points at a distance less than r from a given point. Explicit formulas and asymptotic developments are derived for these functions in then-dimensional case. These can effectively be used as tools for statistical analysis.
On queues with periodic Poisson input
Authors: Lemoine Austin J.
Published: April 2006

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.
On estimation of parameters of Gaussian stationary processes
Authors: Taniguchi Masanobu
Published: April 2006

In fitting a certain parametric family of spectral densitiesfθ(x) to a Gaussian stationary process with the true spectral density g (x), we propose two estimators ofθ, sayby minimizing two criteriaD1(·),D2(·) respectively, which measure the nearness offθ(x) to g (x). Then we investigate some asymptotic behavior ofwith respect to efficiency and robustness.
A note on the expected number of survivors in supercritical carrier-borne epidemics
Authors: Saunders Roy , Lefèvre Claude , Kryscio Richard J.
Published: April 2006

We provide a formal proof of a conclusion due to Abakuks (1974) which states that the expected number of survivors in Downton's carrier-borne epidemic model approaches the limit (ρ /π)δ as the initial number of susceptibles tends to infinity. Here ρ denotes the relative removal rate for carriers, π denotes the conditional probability that an infected susceptible will become a carrier, δ denotes the Kronecker delta function and denotes the initial number of carriers.
Exponential growth of a branching process usually implies stable age distribution
Authors: Berndtsson Bo , Jagers Peter
Published: April 2006

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.
The simple branching process: a note on convergence when the mean is infinite
Authors: Davies P. L.
Published: April 2006

Letdenote the simple branching process withZ0= 1 and letGdenote the distribution function ofZ1.SupposeGsatisfiesx−α−γ(x)≦1 −G(x) ≦x−α+γ(x)for largex, where (i) 0 < α < 1, (ii)γ(x) is non-negative and non-increasing, (iii)xγ(x)is non-decreasing and (iv)Then limn→∞αnlog (Zn+ 1) converges almost surely to a non-degenerate finite random variableWsatisfyingP(W= 0) =q= probability of extinction of theprocess.
Filtered renewal processes with a two-sided impact function
Authors: Andersen Per Kragh
Published: April 2006

An expression for the characteristic function of a stationary filtered renewal process with a two-sided impact function is derived, and formulae for the moments are obtained.
Convergence to equilibrium in a traffic model with restricted passing
Authors: Unkelbach Hans Dieter
Published: April 2006

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.

The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.
Cluster shock models
Authors: Thall Peter F.
Published: April 2006

The survival distribution of a device subject to a sequence of shocks occurring randomly over time is studied by Esary, Marshall and Proschan (1973) and by A-Hameed and Proschan (1973), (1975). The present note treats the case in which shocks occur according to a homogeneous Poisson cluster process. It is shown that if[the device surviveskshocks] =zk, 0 <z< 1, then the device exhibits a decreasing failure rate. A DFR preservation theorem is proved for completely monotonic. A counterexample to the IFR preservation theorem is given in whichis strictly IFR while the failure rate is initially decreasing and then increasing.
An upper bound for the velocity of first-passage percolation
Authors: Janson Svante
Published: April 2006

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.
Normal approximations for distributions arising in the stochastic approach to chemical reaction kinetics
Authors: Dunstan F. D. J. , Reynolds J. F.
Published: April 2006

Earlier stochastic analyses of chemical reactions have provided formal solutions which are unsuitable for most purposes in that they are expressed in terms of complex algebraic functions. Normal approximations are derived here for solutions to a variety of reactions. Using these, it is possible to investigate the level at which the classical deterministic solutions become inadequate. This is important in fields such as radioimmunoassay.
Lower bounds for the critical probability in percolation models with oriented bonds
Authors: Gray Lawrence , Wierman John C. , Smythe R. T.
Published: April 2006

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.
On the time to first overflow in dams with inputs forming a Markov chain
Authors: Kennedy Douglas P.
Published: April 2006

A finite dam is studied in which the net input in each period, i.e. the excess of input over demand, is a function of a Markov chain. By using a martingale, the joint distribution of the time to first overflow and the cumulative unsatisfied demand until overflow is investigated. The probability of overflow before the unsatisfied demand exceeds a fixed level is also considered.
Towards consensus: some convergence theorems on repeated averaging
Authors: Chatterjee S. , Seneta E.
Published: April 2006

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.
On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process
Authors: Liebetrau A. M.
Published: April 2006

The second-moment structure of an estimatorV*(t) of the variance-time curveV(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the formTβ(V*(tTα)– V(tTα)),0 <α< 1, to a non-stationary Gaussian process. Similar results are shown to hold whenα= 0 and in the case whereV(tTα) is replaced by a suitable estimator.
Clustering on coloured lattices
Authors: Strauss David J.
Published: April 2006

This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example.
Infinite dams with discrete additive inputs
Authors: Ali Khan M. S.
Published: April 2006

This paper considers an infinite dam fed by a discrete inputXtduring the time interval [t,t+ 1),t= 0, 1, 2, ···. At timet –0 there is an outputYt= min(Zt–1, +Xt–1,r) from the dam leaving behind the amountZt= max(0,Zt–1, +Xt–1,r). The probability Pr(Zt=i),i= 0, 1, 2, ··· is discussed under the strict assumption thatr> 1 and the given initial condition thatZ0=u, u= 1, 2, ···. The generating function technique has been used throughout the paper.
Monte Carlo simulation of the renewal function
Authors: Brown Mark , Solomon Herbert , Stephens Michael A.
Published: April 2006

The problem of Monte Carlo estimation of M(t) = EN(t), the expected number of renewals in [0, t] for a renewal process with known interarrival time distribution F, is considered. Several unbiased estimators which compete favorably with the naive estimator, N(t), are presented and studied. An approach to reduce the variance of the Monte Carlo estimator is developed and illustrated.
A fixed-point property for Galton–Watson processes with generation dependence
Authors: Fearn Dean H.
Published: April 2006

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.
Survival functions induced by stochastic covariate processes
Authors: Myers Lawrence E.
Published: April 2006

A vectorXof patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients. The instantaneous force of mortality (hazard function) operating on the patient is assumed to be a time-dependent, continuous quadratic functional of the prognostic vector. Conditional on initial dataX0, the probability of survivingTunits of time is expressed in terms of the solution of a Riccati equation, which can be evaluated in closed form if the coefficients of the process and the hazard are constant. This conditional expectation does not preserve proportional hazards.
A note on the normalised moments of distributions with non-monotonic hazard rate
Authors: MacGillivray H. L.
Published: April 2006

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.
Lognormal genesis
Authors: Brown Gavin , Sanders J. W.
Published: April 2006

Models have been proposed in many diverse areas to generate a lognormal distribution and the underlying idea has always been some form of the law of proportionate effect. In a sense any model must resemble this recipe: take logs and apply the central limit theorem. Our model is no exception. However our formulation is designed to encompass the previous models and demonstrate that the fundamental concept is one of classification. This supersedes a multitude of models which incorporate a mechanism peculiar to a specific application in order to use the law of proportionate effect; we illustrate with applications of the model to ecology, econometrics and geostatistics.
A note on filtered Markov renewal processes
Authors: Andersen Per Kragh
Published: April 2006

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.
The advancing wave in a spatial birth process
Authors: Daniels H. E.
Published: April 2006

It is known that, in the deterministic model of a one-dimensional spatial linear birth process, a finite initial distribution of individuals develops into an advancing wave with the minimum velocity possible for such waves. In this paper, the spatial covariance density of the process is studied in the vicinity of a point moving outwards with a given velocity. Its behaviour casts doubt on the applicability of the deterministic result to individual realisations of the process.
Almost sure convergence in Markov branching processes with infinite mean
Authors: Grey D. R.
Published: April 2006

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.
Algebraic characterization of infinite Markov chains where movement to the right is limited to one step
Authors: Samuel-Cahn Ester , Zamir Shmuel
Published: April 2006

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.
Characterizations via conditional distributions
Authors: Berk Robert H.
Published: April 2006

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.
Least absolute deviation estimates in autoregression with infinite variance
Authors: Gross S. , Steiger W. L.
Published: April 2006

We consider anL1analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α>1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.
Poisson mixtures and quasi-infinite divisibility of distributions
Authors: Puri Prem S. , Goldie Charles M.
Published: April 2006

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.
An improved model for signalised intersections with vehicle-actuated control
Authors: Cowan Richard
Published: April 2006

Explicit results are found for average delays and average lengths of red and green phases for a realistic continuous-time model of a two-street intersection operating with vehicle-actuated controls. The assumed arrival processes satisfactorily model both the random ‘bunch-gap' structure of traffic and the fact that vehicles have non-zero length.
Multistate coherent systems
Authors: El-Neweihi E. , Proschan F. , Sethuraman J.
Published: April 2006

The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.
A Markov chain model for the kinetics of reactant isolation
Authors: Olson William H.
Published: April 2006

In the chemistry of some chain molecules, adjacent monomeric side groups can form dimers. Here a Markov chain model is developed for irreversible formation of dimers. Closed form expressions for the finite time expected numbers of sequences of unreacted monomers of various lengths (number of monomers) are given. The relation between this model and the parking problem is discussed.
An extension of the Rao-Rubin characterization of the Poisson distribution
Authors: Shanbhag D. N.
Published: April 2006

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.
Local limit theorems for non-critical Galton–Watson processes by or without immigration
Authors: Höpfner R.
Published: April 2006

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.
Extreme values, range and weak convergence of integrals of Markov chains
Authors: Brockwell P. J. , Resnick S. I. , Pacheco-Santiago N.
Published: April 2006

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.
Further properties for unilateral binary processes
Authors: Galbraith R. F. , Walley D.
Published: April 2006

Results and methods discussed in two previous papers are extended to other cases. Comparison is made with recent work by Pickard (1980) and an earlier conjecture is disproved.
Weak convergence of an autoregressive process used in modeling population growth
Authors: Cumberland W. G. , Sykes Z. M.
Published: April 2006

Under simple limiting conditions a first-order autoregressive process is shown to converge weakly to an Ornstein-Uhlenbeck process. The result is discussed in the context of modeling vital rates for biological populations in randomenvironments.
Product autoregression: a time-series characterization of the gamma distribution
Authors: Mckenzie Ed
Published: April 2006

A non-linear stationary stochastic process {Xt} is derived and shown to have the property that both the processes {Xt} and {logXt} have the same correlation structure, viz. the Markov or first-order autoregressive correlation structure. The generation of such processes is discussed briefly and a characterization of the gamma distribution is obtained.
Generalized semi-Markov schemes and open queueing networks
Authors: Barbour A. D.
Published: April 2006

Generalized semi-Markov schemes were devised to give a versatile general model embracing queueing networks and similar systems of practical importance, and they have proved particularly successful in uniting many disparate results on insensitivity. However, it turns out that, although closed queueing networks are expressible as GSMS, open networks are not, and that the insensitivity results for such networks are not therefore strictly within their scope. In this paper, it is shown that, as one might hope, open networks can be realized as limits of a suitable sequence of closed networks in such a way that the insensitivity properties of the GSMS are transferred to the open network in the limit, and thus that open networks too can, in a sense, be considered to be GSMS. However, it appears from the technical nature of the arguments involved that, despite this close relationship between GSMS and open networks, it may nonetheless be simpler to treat them separately when constructing the proofs of theorems.
Almost sure limit results for the supercritical Bellman-Harris process
Authors: Kuczek Thomas
Published: April 2006

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.
Multivariate monotone likelihood ratio and uniform conditional stochastic order
Authors: Whitt Ward
Published: April 2006

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.
On the evaluation of suboptimal strategies for families of alternative bandit processes
Authors: Glazebrook K. D.
Published: April 2006

Families of alternative bandit processes have been used as models for problems in a variety of areas. Optimal strategies for these decision processes are determined by dynamic allocation indices. These indices are here shown to play an important role in the evaluation of suboptimal strategies.
Plane polygons revisited
Authors: Neumann B. H.
Published: April 2006

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.
The ‘ideal’ estimation equation
Authors: Bartlett M. S.
Published: April 2006

The Cramér–Rao inequality for the variance of an unbiassed estimator is first recalled, and the value of the ‘ideal' equation of estimation existing if the minimum bound is reached illustrated by examples. When this estimation equation is not available, the more general inequality due to Kiefer is more relevant. The analogous estimation equation corresponding to the attaining of the more general minimum bound is again illustrated by examples; and the general theory, which is also available for more than one parameter, is indicated for any location and/or scaling problem.
Linear regression by functional least squares
Authors: Heathcote C. R.
Published: April 2006

The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.
Approximations to hard-core models and their application to statistical analysis
Authors: Westcott Mark
Published: April 2006

This paper derives upper and lower bounds to the distribution functions of nearest-neighbour and minimum nearest-neighbour distances between N points generated by a hard-core model on the surface of a sphere. The use of these bounds in statistical inference is discussed.
Inference for general Ising models
Authors: Pickard David K.
Published: April 2006

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models.

In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.
The outcome of a general spatial epidemic on the line
Authors: Faddy M. J.
Published: April 2006

The general (non-spatial) stochastic epidemic is extended to allow infective individuals to move forward through a system of spatially connected locations · ··,L1,L2, · ·· (on the line) each containing susceptible individuals and the outcome of the epidemic in each of these locations is then considered. In the deterministic case, a (spatial) equilibrium solution and threshold behaviour are discussed. In the stochastic case, a (spatial) quasi-equilibrium behaviour (conditional on sufficient numbers of infectives present) is discussed; numerical results suggest some correspondence between this stochastic quasi-equilibrium and the deterministic equilibrium.
On restricted and pseudo-contagious occupancy distributions
Authors: Charalambides Ch. A.
Published: April 2006

Considermdistinguishable urns withkdistinguishable cells each and suppose thatnindistinguishable balls are randomly allocated into these urns. When the capacity of each cell is limited to one ball (restricted occupancy) or unlimited (pseudo-contagious occupancy) and empty urns are not permitted, the probability function and factorial moments of the numberMt=Mt(n, m, k) of urns containing exactlytballs are expressed in terms of numbers related to the Stirling numbers. The limiting distributions ofMt= Mt(n, m, k) whenk→ 0 ork →∞ are derived; these distributions are expressed in terms of the Stirling numbers of the first and second kind respectively. Moreover a modification of the occupancy models of Barton and David is shown to lead to the preceding distributions.
Radial generation of n-dimensional poisson processes
Authors: Quine M. P. , Watson D. F.
Published: April 2006

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.
A note on the volume of small random simplexes
Authors: Gates John
Published: April 2006

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.
Random motion on binary trees
Authors: Frank Douglas H. , Durham Stephen
Published: April 2006

This paper presents a model for random motion on binary trees. The symmetric model is defined first, and then a model with drift is given. Formulas for absorption probabilities and expected absorption times are presented in both cases.
On the determination of a distribution by its median residual life function: a functional equation
Authors: Gupta Ramesh C. , Langford Eric S.
Published: April 2006

In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f0(t)k(log f0 (t)), where f0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.
A service model in which the server is required to search for customers
Authors: Neuts Marcel F. , Ramalhoto M. F.
Published: April 2006

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.

Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.
Random polytopes in a ball
Authors: Buchta C. , Müller J.
Published: April 2006

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.
Life distribution properties of devices subject to a pure jump damage process
Authors: Abdel-Hameed Mohamed
Published: April 2006

A device is subject to damage. The damage occurs randomly in time according to a pure jump process. The device has a threshold and it fails once the damage exceeds the threshold. We show that life distribution properties of the threshold right tail probability are inherited as corresponding properties of the survival probability, under suitable conditions on the parameters of the damage process. Moreover we discuss an optimal replacement problem for such devices.
On spàtial general epidemics and bond percolation processes
Authors: Kuulasmaa Kari , Zachary Stan
Published: April 2006

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.