bluuebunny/crossref_metadata_2025 huggingface dataset:
Some counting and interval properties of the mutually-exciting processes
Authors: Adamopoulos L.
Published: April 2006

By using a cluster representation of the mutually-exciting processes their probability generating functional is derived. This is used to obtain some counting and interval properties of the process. The theoretical results are applied to a numerical example.
Emptiness times of a dam with stable input and general release function
Authors: Brockwell P. J. , Chung K. L.
Published: April 2006

We investigate the nature of the set of emptiness times of a dam whose release rate depends on the content and whose cumulative input process is a pure-jump Lévy process. Detailed results are obtained for stable input processes and release functions of the form r(x) = xβ I(o,∞)(x).
Order statistics, Poisson processes and repairable systems
Authors: Miller Douglas R.
Published: April 2006

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.
On a two-dimensional binary process
Authors: Galbraith R. F. , Walley D.
Published: April 2006

We consider a two-dimensional stochastic process of binary variablesxijdefined so that the conditional distribution ofxijgiven its predecessors depends only onxi–1jandxij–1Properties of the equilibrium distribution, when this exists, are investigated using three different representations of the process and explicit results are given in some special cases.
On joint exchangeability and conservative processes with stochastic rates
Authors: Saunders Roy
Published: April 2006

In a previous article Saunders (1975) investigated the form of transition probabilities for a generalization of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The results of that investigation are given in terms of properties of exchangeable random variables and require that the process be in a particular initial state at time zero. This article removes the restriction on the initial state by using some properties of two sequences of jointly exchangeable variables. General results analogous to those obtained previously are shown to hold for general initial states.
Mean recurrence times for k successes within m trials
Authors: Huntington Raymond J.
Published: April 2006

Given an unlimited sequence of Bernoulli trials, let E be the recurrent event that k or more successes occur within m consecutive trials in the sequence. Let μk, m denote the mean recurrence time for E. Previous work finds expressions for μk, m for limited values of k and m. The present paper derives a closed expression for μk, m for all k, m. Tables for μk, m are also presented.
On a single-server queue with group arrivals
Authors: Cohen J. W.
Published: April 2006

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.
Multivariate birth-and-death processes as approximations to epidemic processes
Authors: Griffiths D. A.
Published: April 2006

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.
Some results for dams with Markovian inputs
Authors: Phatarfod R. M. , Mardia K. V.
Published: April 2006

The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered.
Asymptotic behaviour of continuous time, continuous state-space branching processes
Authors: Grey D. R.
Published: April 2006

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.
The asymptotic theory of concomitants of order statistics
Authors: David H. A. , Galambos J.
Published: April 2006

In a random sample of n pairs (Xr, Yr), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr:n be the rth order statistic among the Xr and let Y[r:n] be the Y-variate paired with Xr:n. The Y[r:n], which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr:n. It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n–1Rt, n are determined for t such that t/n → λ with 0 < λ < 1.
Positive columns for stochastic matrices
Authors: Isaacson Dean , Madsen Richard
Published: April 2006

If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.
On two stationary distributions for the stable GI/G/1 queue
Authors: Lemoine Austin J.
Published: April 2006

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.
Dams with random outputs
Authors: Collings P. S.
Published: April 2006

The results obtained by Lehoczky on the first emptiness of a dam subject to random outputs are extended to a more general model.
On the inverses of some patterned matrices arising in the theory of stationary time series
Authors: Galbraith R. F. , Galbraith J. I.
Published: April 2006

Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ≧ p), and for the first order mixed autoregressive moving average process.
A finite dam with exponential release
Authors: Yeo G. F.
Published: April 2006

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.
A note on the first two moments of times in transient states in a semi-Markov process
Authors: Kao Edward P. C.
Published: April 2006

This paper derives results for computing the first two moments of times in transient states and times to absorption in a transient semi-Markov process. An illustrative example is presented at the end.
A multi-type critical age-dependent branching process with immigration
Authors: Weiner Howard J.
Published: April 2006

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.
First-passage time for a particular stationary periodic Gaussian process
Authors: Shepp L. A. , Slepian D.
Published: April 2006

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.
On stochastic population models in genetics
Authors: Kelly F. P.
Published: April 2006

Trajstman (1974) has shown that two different population models used to study the number of mutant forms maintained in a population have a certain marginal stationary distribution in common. In this note a general stochastic population model is proposed which subsumes these two models and shows that their transition rates are also related.
Runs of a discrete time regenerative phenomenon
Authors: Imhof J. P.
Published: April 2006

The limiting behavior of the number of runs up to time n of a regenerative phenomenon on the positive integers is related to the one of the number of regenerations. Applications are considered in fluctuation theory, and for the GI/G/1 queue.
Note on the reversible counters system of Lampard
Authors: Phatarfod R. M.
Published: April 2006

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.
Optimal and adaptive stopping based on capture times
Authors: Starr Norman
Published: April 2006

Independent exponential capture times are assumed for each of N prey. The payoff is the number of prey caught less a linear time cost. The optimal stopping time and value are obtained. When N is known an adaptive statistical procedure is proposed for which the expected payoff differs from the value by at most one-half plus a term which tends to zero (of order N–α, a < 1) as N → ∞.
The effect of intermittent exposure to risk on failure time distributions
Authors: Isham Valerie
Published: April 2006

The relation is investigated between the distributions of the total time until failure and the time of exposure to risk until failure, for individuals who are at risk only intermittently during their lifetimes. A specific example is considered in the case of computer failures.
On the transient state probabilities for a queueing model where potential customers are discouraged by queue length
Authors: Natvig Bent
Published: April 2006

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p∗i0, p∗in (n ≥ 1) being determined recursively. An exact expression for p10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.
On record values and record times
Authors: Shorrock R. W.
Published: April 2006

A correspondence between record values and independent increment point processes is established. The asymptotic behaviour of record value sequences is studied, and results on the asymptotic behaviour of record times (for continuous F) are obtained as special cases. The joint law of the kth record value and the kth record time is also derived.
The continuity of the single server queue
Authors: Kennedy Douglas P.
Published: April 2006

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.
A random walk problem with correlation
Authors: Proudfoot A. D. , Lampard D. G.
Published: April 2006

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.
Limits for the superposition ofm-dimensional point processes
Authors: Whitt Ward
Published: April 2006

To obtain a limit with independent components in the superposition ofm-dimensional point processes, a condition corresponding to asymptotic independence must be included. When this condition is relaxed, convergence to limits with dependent components is possible. In either case, convergence of finite distributions alone implies tightness and thus weak convergence in the function spaceD[0, ∞) × … ×D[0, ∞).
On Zipf's law
Authors: Woodroofe Michael , Hill Bruce
Published: April 2006

A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.
A weak convergence theorem for the empirical characteristic function
Authors: Kent John T.
Published: April 2006

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.
On the tails of waiting-time distributions
Authors: Pakes A. G.
Published: April 2006

Results are given which relate the tail behaviour of the service and limiting waiting time distributions of a GI/G/1 queue. A limit theorem for the maxima of waiting times is given.
An optimal branching migration process
Authors: Durham S. D.
Published: April 2006

We consider a population distributed over two habitats as represented by two separate one-dimensional branching processes with random environments. The presence of random fluctuation in reproduction rates in both habitats implies the possibility that neither habitat is universally superior to the other for all times and that a maximal population size is to be achieved by having population members present in both habitats. We show that optimal population growth occurs when migration between habitats occurs at a fixed rate which can be found from the environmentally determined reproduction variables of the separate habitats. The optimal processes are themselves two-type branching processes with random environments.
Inference for the diffusion branching process
Authors: Brown B. M. , Hewitt J. I.
Published: April 2006

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.
Some mean first-passage time approximations for the Ornstein-Uhlenbeck process
Authors: Thomas Marlin U.
Published: April 2006

This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons.
On the rate of convergence of normal extremes
Authors: Hall Peter
Published: April 2006

Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn)/an to the extreme value distribution exp(–e–x), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.
Level crossings and stationary distributions for general dams
Authors: Rubinovitch Michael , Cohen J. W.
Published: April 2006

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.
Transition probability matrices for correlated random walks
Authors: Nain R. B. , Sen Kanwar
Published: April 2006

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.
A continuous-time analogue of random walk in a random environment
Authors: Ritter Grant
Published: April 2006

By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.
On the Weak form of Zipf's law
Authors: Chen Wen-Chen
Published: April 2006

Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet–multinomial urn model for describing the above phenomena from a stochastic point of view.

We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.
On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1
Authors: De Meyer A. , Teugels J. L.
Published: April 2006

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.
Max-infinite divisibility
Authors: Balkema A. A. , Resnick S. I.
Published: April 2006

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if Ft is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.
On Lotka–Volterra predator prey models
Authors: Billard L.
Published: April 2006

Ever since Lotka (1925) and Volterra (1926), (1931) first considered mathematical formulations for prey-predator processes, the resultant equations have resisted attempts to solve them. However, over the intervening 50 years, standard techniques have allowed a few isolated results to be obtained for some simplified versions of the original process, but the classical equations for the stochastic model have remained unsolved. We give here solutions to the classical process for the case in which interactions occur over a sufficiently short period of time that no births occur. The technique used is one recently developed by Severo (1969a), (1969b), (1971). The approach can be easily generalised to allow solution for the case in which births do occur, as well as for the simplified versions of the original process.
A solution of the carrier-borne epidemic
Authors: Henderson W.
Published: April 2006

An alternative approach to the derivation of a general class of results in carrier-borne epidemic theory is presented. Some insight can be gained into the basic structure of carrier-borne epidemics and the method indicates which other problems of a similar type could yield to analysis.
Optimal insurance premium rates when the distribution of claims is unknown
Authors: Venezia Itzhak
Published: April 2006

We analyze here the optimal premium rates for an insurance company for the case where the claims process is a stochastic process with unknown parameters. The beliefs of the insurance company about these parameters are represented by some prior distribution function, and the premium rates are determined on the basis of this distribution. The insurer learns about the unknown parameters from the past behavior of the claims process, and revises the prior distribution whenever new information is obtained (i.e., a claims stream is observed).

The learning process introduces multi-period dynamic considerations to the problem of the rate determination, since the rates determined at any period affect the claims process, and thus the information available for future decisions. We show how the optimal rates can be obtained by using some dynamic programming techniques. We also prove that the optimal rates obtained in our case are lower than those determined by an insurance company that does not revise its distribution.
Scheduling tasks with exponential service times on parallel processors
Authors: Glazebrook K. D.
Published: April 2006

We consider the problem of how to schedulentasks on to several identical processors to meet the objective of minimising the expected flow-time. The strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.
A cyclic queueing network with dependent exponential service times
Authors: Jacobs Patricia A.
Published: April 2006

A cyclic queueing network with two servers and a finite number of customers is studied. The service times for server 1 form an earma(1,1) process (exponential mixed autoregressive moving average process both of order 1) which is a sequence of positively correlated exponential random variables; the process in general is not Markovian. The service times for the other server are independent with a common exponential distribution. Limiting results for the number of customers in queue and the virtual waiting time at server 1 are obtained. Comparisons are made with the case of independent exponential service times for server 1.
First-passage percolation under weak moment conditions
Authors: Reh Wolfgang
Published: April 2006

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.
A note on the ergodicity of Markov chains
Authors: Rosberg Zvi
Published: April 2006

For an aperiodic, irreducible Markov chain with the non-negative integers as state space, a criterion for ergodicity is given. This criterion generalizes the criteria appearing in Foster (1953), Pakes (1969) and Marlin (1973), in the sense that any test function (Liapunov function) which satisfies their conditions also satisfies ours. Applications are presented through some examples.
Extending the correlation structure of exponential autoregressive–moving-average processes
Authors: McKenzie Ed
Published: April 2006

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.
Weak convergence of the simple birth-and-death process
Authors: Ivan Constantin
Published: April 2006

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.
On snowball sampling, random mappings and related problems
Authors: Berg Sven
Published: April 2006

If a rumour is spread in a population through random contacts between its members, to how many people, on average, will the rumour be told? What can be said about the variations in the number of people who have heard the rumour? Does it matter very much whether a single person or a group of persons starts spreading the rumour? Questions such as these are discussed below, using the framework of a simple stochastic model for snowball sampling.
On stochastic scheduling with precedence relations and switching costs
Authors: Glazebrook K. D.
Published: April 2006

A collection of jobs is to be processed by a single machine. The amount of processing required by each job is a random variable with a known probability distribution. The jobs must be processed in a manner which is consistent with a precedence relation but the machine is free to switch from one job to another at any time; such switches are costly, however. This paper discusses conditions under which there is an optimal strategy for allocating the machine to the jobs which is given by a fixed permutation of the jobs indicating in which order they should be processed. When this is so, existing algorithms may be helpful in giving the best job ordering.
Numerical approximations for compound geometric distributions with applications in queueing theory
Authors: Delbrouck L. E. N.
Published: April 2006

A two-step procedure is described to approximate first come first served delay distributions in M/G/1 and GI/G/1 systems. The procedure consists in formulating a coarse initial approximation for the distribution sought, and then correcting this approximation by means of a suitable integral transformation. Examples are displayed.
On excess life in certain renewal processes
Authors: Arnold Barry C. , Groeneveld Richard A.
Published: April 2006

Excess life distributions for discrete renewal processes may be computed by using elementary discrete Markov chain concepts involving absorption probabilities. Excess life distributions in general may then be obtained by approximating the renewal process under study by a suitably chosen sequence of discrete renewal processes. The technique is illustrated in the cases of renewal processes with interarrival distributions which are a linear combination of two exponentials and uniform [0, 1]. A related algorithm is described for computer generated approximations of excess life distributions corresponding to continuous interarrival time distributions with an increasing c.d.f. Conditions for convergence of this algorithm are examined.
Some elaborations upon Gani's model for the type-token relationship
Authors: Brainerd Barron
Published: April 2006

This paper considers certain stochastic models for token and type counts in literary texts. Elaborating on some models of Gani, it is shown that reasonable fits can be obtained to some data of Yule on Macaulay's Essays.
Transient behaviour of queueing networks
Authors: Harrison P. G.
Published: April 2006

In most contemporary queueing network analysis, the assumption is made that a network is in a state of equilibrium. That is, the network's state space probabilities are assumed to be time independent. It is therefore important to be able to quantify precisely when this assumption is valid. Furthermore there are also situations in which it is desirable to model the transient behaviour of networks which occur in practice, such as computer and communication systems. For example, the immediate effects of component failure or instantaneous alteration of system status may be predicted.

In this paper an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent. From the solution, modelling of transient situations becomes possible and the time periods during which the equilibrium assumption can and should not be made may be identified; for example in terms of a time constant which is easily computed to a first-order approximation.
On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes
Authors: Heyde C. C.
Published: April 2006

This paper is concerned with a discrete-time population model in which a new individual entering the population at time t can produce offspring for the first time at time t + 2 and then subsequently at times t + 3, t + 4, ···. The numbers of offspring produced on each occasion are independent random variables each with the distribution of Z for which EZ = m <∞, and individuals have independent lines of descent. This model is contrasted with the corresponding Bienaymé-Galton-Watson one. If Xn denotes the number of individuals in the population at time n, it is shown that z–nXn almost surely converges to a random variable W, as n→∞, where Various properties of W are obtained, in particular W > 0 a.s. if and only if EZ | log Z | < ∞ Results are also given on the rate of convergence of to z when Var Z < ∞ and these display a surprising dependence on the size of z.
Poisson limits for a clustering model of strauss
Authors: Saunders Roy , Funk Gerald M.
Published: April 2006

In this article we present a limiting result for the random variable Yn(r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn(r): 0 ≦ r ≦ r∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.
On the reversibility of the input and output processes for a general birth-and-death queueing model
Authors: Natvig Bent
Published: April 2006

In this paper we consider the general birth-and-death queueing model of Natvig (1975). Define the input and output processes by the steady-state behaviour of respectively successive input and output intervals. Ignoring balking customers, two cases are considered. In the first case we treat a lost customer neither as an input nor as an output, then secondly as both. For both cases we show the input and output processes to be reverse processes. One mistake and two erroneous comments in Natvig (1975) are also corrected.
Convergence rate of perturbed empirical distribution functions
Authors: Winter B. B.
Published: April 2006

Given an i.i.d. sequenceX1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator ofFis the e.d.f.Fn;whereUois the d.f. of the unit mass at zero. Anadmissible perturbation of the e.d.f., say, is obtained ifUois replaced by a d.f., whereis a sequence of d.f.'s converging weakly toUo.Suchperturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as. It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then‘has the Chung–Smirnov property'; i.e., with probability one,1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution andHnis the d.f. of the unit mass atn–αthen the above lim sup is ≦ 1 or = ∞, depending on whetheror
Uniform bounding of probability generating functions and the evolution of reproduction rates in birds
Authors: Heyde C. C. , Schuh H.-J.
Published: April 2006

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to time t for each finite t. This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.
The autocorrelation function of the queue length process for the two-server Poisson queue
Authors: Hill James M. , Tognetti Keith P.
Published: April 2006

Using Laplace transforms an explicit analytical expression is obtained for the autocorrelation function of the number in the system for the two-server Poisson queue. The method employed may be extended to Poisson queues with more than two servers.
A limit theorem for point processes by applications
Authors: Puri Prem S.
Published: April 2006

Let 0 ≦T1≦T2≦ ·· · represent the epochs in time of occurrences of events of a point processN(t) withN(t) = sup{k:Tk≦t},t≧ 0. Besides certain mild conditions on the processN(t) (see Conditions (A1)– (A3) in the text) we assume that for everyk≧ 1, ast→∞, the vector (t – TN(t),t – TN(t)–1, · ··,t–TN(t)–k+1) converges in law to ak-dimensional distribution which coincides with that of a random vectorξk= (ξ1, · ··,ξk) necessarily satisfyingP(0 ≦ξ1≦ξ2≦ ·· ·≦ξk) = 1. LetR(t) be an arbitrary function defined fort≧ 0, satisfying 0 ≦R(t) ≦ 1, ∀0≦t<∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that

The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role ofR(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of theGI/G/∞ queue apparently not studied before.
Short distances, flat triangles and Poisson limits
Authors: Silverman Bernard , Brown Tim
Published: April 2006

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for theU-statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.
Approximations in finite-capacity multi-server queues by Poisson arrivals
Authors: Nozaki Shirley A. , Ross Sheldon M.
Published: April 2006

An approximation for the average delay in queue of an entering customer is presented for theM/G/Kqueuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.
On Bayesian models in stochastic scheduling
Authors: Gittins J. C. , Glazebrook K. D.
Published: April 2006

The D.A.I. theorem of Gittins and Jones has proved a powerful tool in solving sequential statistical problems. A generalisation of this theorem is presented. This generalisation enables us to solve certain stochastic scheduling problems where the items or jobs to be scheduled have random times to completion, the random times having distributions dependent upon parameters to which prior distributions are allocated. Such problems are of interest in many areas where scheduling is important.
A note on the inverse problem for reducible Markov chains
Authors: Pittenger A. O.
Published: April 2006

Suppose a physical process is modelled by a Markov chain with transition probability on S1 ∪ S2, S1 denoting the transient states and S2 a set of absorbing states.

If v denotes the output distribution on S2, the question arises as to what input distributions (of raw materials) on S1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.
The number of intersections of random chords to a circle — third and fourth moments
Authors: Gates John
Published: April 2006

In this paper the distributions of the number of intersections of three, four and five random chords to a circle are obtained by a reduction technique employing Ptolemy's theorem. These results are then used to obtain the skewness and kurtosis of the number of intersections of n random chords to a circle.
On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input
Authors: Vassiliou P.-C. G.
Published: April 2006

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.
The transposition replacement policy with a partial memory
Authors: Nelson Peter R.
Published: April 2006

Consider as a model for any serial list a bookshelf with books B1, ···, Bn. At each unit of time book is demanded with probability pi and is replaced one position to the left of where it was removed (or in the same position if it is already at the left-hand end). If this transposition is made only when Bi has been demanded k times in a row, we show that the average position of the next book demanded is a monotone decreasing function of k.
On the genealogy of large populations
Authors: Kingman J. F. C.
Published: April 2006

A new Markov chain is introduced which can be used to describe the family relationships amongnindividuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.
The human lifespan
Authors: Benjamin Bernard
Published: April 2006

This paper examines the distribution of deaths in life tables for the population of England and Wales as a means of exploring the way in which the length of life is steadily advancing. To assess the possibility for future extension, some extreme assumptions are made about the reduction in mortality rates for certain causes. A distinction is made between ‘anticipated' deaths and ‘senescent' deaths, the latter group falling within bounds either side of a terminal peak in the distribution of deaths by age. For males, the extreme assumptions yield a peak at age 86 with some 85 per cent of deaths within the senescent area. For females, the peak is at about age 92 with 90 per cent of deaths in the senescent area.
Learning models with continuous time parameter and multivariate point processes
Authors: Pruscha Helmut
Published: April 2006

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.
A discrete-time queueing network
Authors: Walrand Jean
Published: April 2006

A discrete-time network with multiple transitions is defined. It is characterized by the independence of the service processes at different nodes. That network is shown to have the familiar quasi-reversibility properties.
Some models for multitype spatial point processes, with remarks on analysing multitype patterns
Authors: Lotwick H. W.
Published: April 2006

Two classes of ergodic stationary multitype spatial point processes are constructed. These processes have the property that interactions between the types exist, but cannot be detected using standard second-order methods of analysis. Simulations indicate that the interactions can, however, be detected by using ‘empty space' techniques.
The dependence of sojourn times on service times in tandem queues
Authors: Cao Xi-Ren
Published: April 2006

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.
The characterization of distributions by order statistics and record values — a unified approach
Authors: Deheuvels Paul
Published: April 2006

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successivejth record valuesandassociated to an i.i.d. sequence are such thatandare independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such thatXk, nandXk+1,n–Xk, nare independent for somek≧ 2 (whereXk, nis thekth order statistic ofX1, ···,Xn) are degenerate.
The sojourn-time distribution in the M/G/1 queue by processor sharing
Authors: Ott Teunis J.
Published: April 2006

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS).

Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.
Hitting spheres by straight-line motion or Brownian motion
Authors: Enns E. G. , Smith B. R. , Ehlers P. F.
Published: April 2006

Points randomly placed inRnare independently assigned a speed and direction of motion. The time between collisions with a centraln-sphere is obtained. If instead of straight-line motion the points perform Brownian motion, the analogous inter-collision-time distribution is obtained.
Moments of the maximum in a critical branching process
Authors: Weiner Howard
Published: April 2006

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let

Martingale arguments are used to show that for some 0<a≦b<∞
Random contact processes, snowball sampling and factorial series distributions
Authors: Berg Sven
Published: April 2006

A contact process first posed as a random mapping problem is discussed together with a model for the spread of a rumour through an archaic telephone system. Snowball sampling and the well-known Reed–Frost chain binomial model are also commented on. Our main interest and emphasis is on distributional properties of discrete-time models. Simple combinatorial arguments are used along with a class of discrete distributions originally proposed for catch–recapture sampling.
A Poisson limit theorem for incomplete symmetric statistics
Authors: Berman M. , Eagleson G. K.
Published: April 2006

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.
On coupling of diffusion processes
Authors: Lindvall Torgny
Published: April 2006

The coupling method is well fitted to be used in the study of the asymptotics of one-dimensional diffusion processes. We give an elementary proof of Orey's theorem in the recurrent case, and establish rate results for tendency towards equilibrium under moment conditions on the speed measure and the initial distributions.
On the continuous-review (s, S) inventory model under compound renewal demand and random lead times
Authors: Sahin Izzet
Published: April 2006

Binomial moments of the time-dependent and limiting distributions of inventory deficit are derived for a continuous-review (s,S) inventory system under the assumption that interarrival times, demand sizes, and lead times are independent sequences of independent, identically distributed random variables. Explicit expressions for the limiting distribution are also given in some special cases of practical interest. The approach used follows that of Finch [2] who investigated the system under the additional assumption of unit demands.