bluuebunny/crossref_metadata_2025 huggingface dataset:
An uncertainty function arising in sequential search theory
Authors: Posner Edward C. , Rumsey Howard
Published: April 2006

This note presents a new “uncertainty function” of finite probability distributions that arises in sequential search theory. The function arises as the finite term in an asymptotic expression for an expected search time as the allowable “error probability” approaches zero. This function is not a function of entropy; it may be of more relevance than entropy to workers in certain fields for describing the complexity of a finite distribution.
The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1
Authors: Cohen J. W.
Published: April 2006

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.
An application of the generating function to the solution of a genetic problem
Authors: Diamantis Basil
Published: April 2006

An application of the generating function to genetics has provided the probability distribution of homozygotes obtained by self-fertilization, in the case of two linked genetic loci. The individual probabilities are functions of the frequency of genetic recombination between the two loci.
Simpler conditions for Wald equations
Authors: Brown B. M. , Eagleson G. K.
Published: April 2006

If T is a stopping time on a martingale {St}, the mth order moments of T and ST (for integers m ≧ 2) are related by an identity known as a Wald equation of order m. Wald equations hold under certain conditions, which are simplified in this paper by showing that among the set of previously known conditions, some are implied by the others.
A limit theorem for priority queues in heavy traffic
Authors: Harrison J. Michael
Published: April 2006

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.
Distribution of minimum path
Authors: Tenenbein Aaron
Published: April 2006

Let Y, X1, X2, …, Xn be a set of n + 1 independently and uniformly distributed random variables on the interval (0, 1). The distribution of the length of the minimum path starting at Y which covers the other n points is derived. The solution is interesting in that it involves finding the distribution of an order statistic of a function of order statistics.
Fractional integrals of stationary processes and the central limit theorem
Authors: Rosenblatt M.
Published: April 2006

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.
Queues with random service output: the case of Poisson arrivals
Authors: Grinstein Jacob , Rubinovitch Michael
Published: April 2006

A general class of single server queueing models is formulated. They distinguish between two factors that may influence the duration of service times: variability in the service requirements of customers, and variability (over time) in the service output of the server. Accordingly, we assume that the demands for service of successive customers form a sequence of independent, identically distributed random variables and that the amount of service produced by a busy server in a time interval is determined by the increment of a process with stationary independent increments over that interval. The results include the distribution of the busy period and the limiting distribution of the queue length. We also investigate the potential waiting process which is an extension of virtual waiting time process in existing queueing models.
Limiting distributions for the general branching process with immigration
Authors: Durham S. D.
Published: April 2006

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.
The number of departures from a semi-Markov queue
Authors: McNickle D. C.
Published: April 2006

In this note an identity due to Arjas (1972) is used to express the distribution of the number of departures from a single server queue in which both service and interarrival times may depend on customer type in terms of the busy period and busy cycle processes.
On linearly regressive processes
Authors: Reynolds John F.
Published: April 2006

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.
Some limit theorems for a supercritical branching process allowing immigration
Authors: Pakes A. G.
Published: April 2006

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.
Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region
Authors: Chan Arthur H. C.
Published: April 2006

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.
A stochastic model for time lag in reporting of claims
Authors: Karlsson Jan-Erik
Published: April 2006

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics.
On a fixed point theorem and its stochastic equivalent
Authors: Yahav Joseph A.
Published: April 2006

A discrete-time Markov process on the interval [0, 1] is considered. Sufficient conditions for the existence of a unique stationary limiting distribution are given.
A note on Harris's condition for recurrence in random walk processes
Authors: Thomson P. J. , Wilkinson K. M.
Published: April 2006

This note corrects a short and transparent proof of Harris's condition for recurrence in random walk processes.
An optimal hunting policy for a stochastic logistic model
Authors: Abakuks Andris
Published: April 2006

A stochastic version of the logistic model for population growth is considered, and the general form of an optimal policy is found for hunting the population so as to maximise the long-term average number of captures per unit time. This optimal policy is described by a critical population size x∗such that it is optimal to hunt if and only if the population size is greater than or equal to x∗. Methods of determining x∗for given parameter values are provided, and some properties of the optimal policy as the population size tends to infinity are proved.
On functional equations and measures of information. II
Authors: Kannappan PL. , Ng C. T.
Published: April 2006

We derive the representation of additive measures of information with the sum property under a regular generating function. This family of measures includes the Shanon entropy, directed/symmetric divergence and inaccuracy.
A note on the moments of the final size of the general epidemic model
Authors: Dunstan Ross
Published: April 2006

In the general epidemic model we study the first two moments of the final size. Beginning with the backwards equation, algebraic methods are used to find their asymptotic series expansions as the population size increases.
The random walk associated by the game of roulette
Authors: Hill James M. , Gulati Chandra M.
Published: April 2006

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.
On the limit of the Markov binomial distribution
Authors: Wang Y. H.
Published: April 2006

Let X1X2, · ·· be a Markov Bernoulli sequence with initial probabilities p of success and q = 1 – p of failure, and probabilities 1 – (1 – π) p, (1 – π) p in the first row and (1 – π) (1 – p), (1 – π) p + πin the second row of the transition matrix. If we define Sn = Σi=1nXi, then the limit distribution P{Sn = k} is obtained when n →∞, np →λ.
An invariance principle in k-dimensional extended renewal theory
Authors: Csenki Attila
Published: April 2006

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.
A characterization of the exponential distribution by spacings
Authors: Ahsanullah M.
Published: April 2006

Let X1, X2, ···, Xn be a random sample of size n from a population with probability density function f(x), x > 0, and let X1, n < X2, n < ··· < Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − Xi−1, n) and (n − i)(Xi+1, n − Xi, n) for one i and one n with 2 ≦ i ˂ n.
Calculating extinction probabilities for the birth and death chain in a random environment
Authors: Torrez William C.
Published: April 2006

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.
Estimation of the criticality parameter of a superitical branching process with random environments
Authors: Nanthi K.
Published: April 2006

For a supercritical branching process x = {xn; n ≧ 0, x0 = 1} with random environments, define when xn > 0; and = 1 when xn = 0. When x is assumed to satisfy the standard regularity assumptions, under the non-extinction hypothesis, is a strongly consistent and asymptotically unbiased estimator for the criticality parameter π and is asymptotically normal. A strongly consistent estimator,

is also proposed for the associated variance, σ2.
A note on the infinitely deep dam with a Markovian input
Authors: Phatarfod R. M.
Published: April 2006

In this paper we consider an infinitely deep dam fed by inputs which form an ergodic Markov chain and whose release M is non-unit. The extension to non-unit release follows on lines similar to the independent inputs case. We show that P(θ) – θ MΙ where P(θ) = (pijθ i) has a maximum of N = M(M + l)/2 non-zero singularities in the unit disc, so that the general solution of the equilibrium equations has N unknown constants. We also show that these constants satisfy N linear constraints, so that the solution is fully determined.
The stochastic geyser problem for first-passage times
Authors: Steinebach Josef
Published: April 2006

Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).
Time-dependent solution of the logistic model for population growth in random environment
Authors: Prajneshu
Published: April 2006

The exact time-dependent solution as well as the stationary solution of the logistic model for population growth with varying carrying capacity is worked out in both the Stratonovich and Ito calculi by solving the forward Kolmogorov equation.
A stochastic model of lexical change
Authors: Dobson Annette J.
Published: April 2006

A model is given to describe the concurrent evolution of vocabularies of different languages belonging to the same family. Lexical change involves the replacement of old words by new words acquired by inheritance, borrowing or some other form of innovation. This process is described by a set of simultaneous differential equations involving the probabilities that, at any time, languages share similar words for any particular meaning. Various special cases of the model are compared using data for some Indo-European languages.
Reconstruction of a stationary Gaussian process from its sign-changes
Authors: de Maré Jacques
Published: April 2006

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.
A characterization of the negative exponential distribution with application to reliability theory
Authors: Phillips M. J.
Published: April 2006

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.
The covariance of two type II counters
Authors: Berman Mark
Published: April 2006

Consider three independent Poisson processes of point events of rates λ1, λ 2 and λ12. Suppose there are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters are subject to dead times of Type II, i.e. where all events, whether recorded or not, cause dead times. The covariance of the numbers of events recorded by the two counters in a suitable time period is derived. This extends earlier work of Cox and Isham (1977) and Kingman (1977) who considered the same problem for Type I counters, i.e. where only recorded events cause dead times.
A model for blue-green algae and gorillas
Authors: Morgan Byron J. T. , Leventhal B.
Published: April 2006

The linear birth-and-death process is elaborated to allow the elements of the process to live as members of linear clusters which have the possibility of breaking up. For the supercritical case, expressions, based on an approximation, are derived for the mean numbers of clusters of the various sizes as time → ∞. These expressions compare very well with exact solutions obtained by the method of Runge-Kutta. Exact solutions for the mean values for all time are given for when the death rate is zero.
Optimal strategies in measurable learning systems on metric spaces
Authors: Doberkat Ernst–Erich
Published: April 2006

A dynamic programming approach for the investigation of learning systems is taken. Making use of one-stage decision models and dynamic programs, respectively, two learning models are formulated and the existence of optimal strategies for learning in the respective models is proven.
The volume of an isotropic random parallelotope
Authors: Ruben Harold
Published: April 2006

The p-content of the p-parallelotope ∇p, n determined by p independent isotropic random points z1, …, zp in ℝn (1 < p ≦ n) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇p, n. This is done explicitly in several cases. The results also have interest in multivariate statistics.
Some asymptotic results for the induced selection differential
Authors: Nagaraja H. N.
Published: April 2006

We define induced selection differential and discuss asymptotic distribution theory for this quantity. We also obtain the asymptotic joint distribution of the selection differential and the induced selection differential. These are used as measures of improvement in genetic selection programs. We consider the linear regression model set up in detail to obtain various possible limit laws for the induced selection differential.
Linear birth and death processes with killing
Authors: Karlin Samuel , Tavaré Simon
Published: April 2006

We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form λ i = bi + θ, µi = ai, γ i = ci, where γ i is the killing rate. Then P{killed in (t, t + h) | X(t) = i} = γ ih + o(h), h ↓ 0. A variety of explicit results are found, and an example from population genetics is given.
On the reduction in remaining system lifetime due to the failure of a specific component
Authors: Natvig Bent
Published: April 2006

In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.
On the distribution of Hotelling's one-sample T2 under moderate non-normality
Authors: Davis A. W.
Published: April 2006

The asymptotic distribution of Hotelling's one-sample T2 in multivariate Edgeworth populations is expanded to terms of the first order. Comparison with published simulation results indicates that the result is quite useful, even in cases where the underlying population is not well represented by an Edgeworth expansion.
Comparison of two treatments when there may be an initial effect
Authors: Scott Elizabeth L.
Published: April 2006

Consider situations where the treatment may cause an initial effect and may also cause a long-range effect. We want to evaluate the treatment, or to compare two treatments, when the effect of treatment may result from the two distinct mechanisms, M1 and M2. We may wish to evaluate M1 and M2 separately, but we may also want to evaluate their combined effect M3. Examples are given and the general results are applied to the special case arising in weather modification studies and elsewhere: the possible effects are multiplicative and the distribution of non-zero variables is gamma with at most the scale parameter affected by treatment. An example demonstrates that the two components may be too weak to be judged significant while their sum is large and significant. The locally optimum C(α) test is used.

There is a brief discussion of the power function of the tests. The asymptotic power agrees well, in general, by the results of the Monte Carlo simulation for the test Z3 of the combined effect. If the zero values are discarded and then Z2 employed, there is large bias in the power. The bias is more pronounced if the Wilcoxon, Mann–Whitney test is employed. Notice that the two effects under study may be acting in the same direction or they may be in opposition.
A secretary problem with backward solicitation and uncertain employment
Authors: Choe K. I. , Bai D. S.
Published: April 2006

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.
The autostrada queueing problem
Authors: Conolly B. W.
Published: April 2006

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.
On the M/G/1 queue by additional inputs
Authors: Ott Teunis J.
Published: April 2006

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system.

The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.
On the number of clumps resulting from the overlap of randomly placed figures in a plane
Authors: Kellerer A. M.
Published: April 2006

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.
A joint characterization of the multinomial distribution and the Poisson process
Authors: Kimeldorf George , Thall Peter F.
Published: April 2006

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.
Reversibility of Markov chains with applications to storage models
Authors: Ōsawa Hideo
Published: April 2006

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.
First-passage times with PFr densities
Authors: Assaf David , Shared Moshe , shanthikumar J. George
Published: April 2006

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.
Stochastic flowshop no-wait scheduling
Authors: Frostig E. , Adiri I.
Published: April 2006

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti, (i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.
A note on the geometric/probabilistic relationship in a Markov manpower model
Authors: Davies G. S.
Published: April 2006

The probability of attaining a goal structure in a partially stochastic manpower model is discussed in relation to the position of the structure described in Euclidean space. It is found that, within the regions of structures attainable with non-zero probability, the probabilities decrease on lines emanating from the structure attainable with probability 1.
Parametric inference in Markov branching processes with time-dependent random immigration rate
Authors: Pruscha Helmut
Published: April 2006

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.
Limit diffusions of some stepping-stone models
Authors: Sato Ken-Iti
Published: April 2006

A Markov chain model of a population consisting of a finite or countably infinite number of colonies withNparticles at each colony is considered. There aredtypes of particle and transition from thenth generation to the (n+ 1)th is made up of three stages: reproduction, migration, and sampling. Natural selection works in the reproduction stage. The limiting diffusion operator (asN→∞) for the proportion of types at colonies is found. Convergence to the diffusion is proved under certain conditions.
The Palm-type grain size distribution in stationary grain models
Authors: König Dieter , Schmidt Volker
Published: April 2006

In this paper a point-process approach is given for determining the Palm-type (number-weighted) distribution of the size factor of the grains of a stationary grain model in the plane with non-overlapping, identically shaped and identically orientated convex grains starting from a suitably chosen characteristic of the grain model observed in fixed points of the plane.
Maintainability of manpower structures — counterexamples, results and conjectures
Authors: Haigh John
Published: April 2006

The question as to what grade structures are maintainable in Markov manpower models, with or without the constraint that the total size be kept constant, is examined. Known results are collected together, examples and further results are given, and an old conjecture is generalised.
An improved stochastic model allowing interaction among individuals
Authors: Cerrito P. B.
Published: April 2006

The purpose of this paper is to explore the properties of a new model which can describe the interaction among individuals. Various other models have been proposed, but all assume that each individual acts independently of all others. Our model permits the assumption that individuals can be influenced and dominated by other individuals. We also demonstrate how this model can describe the interaction of various species in an ecosystem. This model will lead naturally to the definition of a population index of diversity. A means of estimating the values of the model will also be given.
Multitype finite mean supercritical age-dependent branching processes
Authors: Cohn Harry
Published: April 2006

Let {Zφ (t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c–1(t)Zφ (t)} converges almost surely to a non-degenerate limit Wφ as t →∞. Characterizations of {c(t)} and Wφ are also given.
On extreme-value theory in the presence of a trend
Authors: De Haan L. , Verkade E.
Published: April 2006

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.
On characterizations of the geometric distribution by independence of functions of order statistics
Authors: Srivastava R. C.
Published: April 2006

Let X1, · ··, Xn, n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y1 ≦ Y2 ≦ · ·· ≦ Yn denote the corresponding order statistics. Define Rn = Yn – Y1 and Zn = Σj=2n (Υj – Y1). Then it is well known that (i) Y, and Rn are independent and (ii) Y1 and Zn are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.
The variance of a Poisson process of domains
Authors: Kellerer A. M.
Published: April 2006

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair-distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region.
Characterizations of inverse problems in queueing and their stability
Authors: Kalashnikov V. V. , Rachev S. T.
Published: April 2006

Two inverse problems in queueing are studied, namely: (i) a restoration of system characteristics, and (ii) a restoration of model classes. These problems are discussed in the framework of a general concept of characterization problems given by Kalashnikov and Rachev (1985).
A family of bounds for the transient behavior of a Jackson network
Authors: Massey William A.
Published: April 2006

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.
Expected population size in the generation-dependent branching process
Authors: Biggins J. D. , Götz Thomas
Published: April 2006

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.
Mixed Cox processes, with an application to accident statistics
Authors: Bartlett M. S.
Published: April 2006

A doubly stochastic switching model previously used by the author to check against doubly stochastic ‘white-noise' models is generalized in the case of Poisson processes to any distribution of fluctuating intensity rates. The resulting Cox process is used in place of a simple Poisson process as the basis of a mixed model, which is then fitted to some data on automobile accidents previously analysed by Seal (1980) in terms of a mixed Poisson process. It is shown that there is just significant evidence in favour of some additional heterogeneity ‘within' each year for individual drivers; the nature of the data analysed (yearly time intervals) does not, however, allow discrimination between heterogeneity which is definitely attributable to a common fluctuating environment, and that which is in effect independent for each driver.
Optimal strategies for discovering new species in continuous time
Authors: Alsmeyer Gerold , Irle Albrecht
Published: April 2006

Consider a population of distinct species Sj, j∈J, members of which are selected at different time points T1, T2,· ··, one at each time. Assume linear costs per unit of time and that a reward is earned at each discovery epoch of a new species. We treat the problem of finding a selection rule which maximizes the expected payoff. As the times between successive selections are supposed to be continuous random variables, we are dealing with a continuous-time optimal stopping problem which is the natural generalization of the one Rasmussen and Starr (1979) have investigated; namely, the corresponding problem with fixed times between successive selections. However, in contrast to their discrete-time setting the derivation of an optimal strategy appears to be much harder in our model as generally we are no longer in the monotone case.

This note gives a general point process formulation for this problem, leading in particular to an equivalent stopping problem via stochastic intensities which is easier to handle. Then we present a formal derivation of the optimal stopping time under the stronger assumption of i.i.d. (X1, A1) (X2, A2), · ·· where Xn gives the label (j for Sj) of the species selected at Tn and An denotes the time between the nth and (n – 1)th selection, i.e. An = Tn – Tn–1. In the case where even Xn and An are independent and An has an IFR (increasing failure rate) distribution, an explicit solution for the optimal strategy is derived as a simple consequence.
Optimal pest control through catastrophes
Authors: Kyriakidis E.G. , Abakuks Andris
Published: April 2006

This paper is concerned with the problem of controlling a simple immigration–birth process, which represents a pest population, by the introduction of catastrophes which, when they occur, reduce the population size to zero. The optimality criterion is that of minimising the long-term average cost per unit time of the process. Firstly, an optimal policy is found within a restricted class of stationary policies, which introduce catastrophes if and only if the population size is greater than or equal to some critical value x. The optimality of this policy within the wider class of all stationary policies is then verified by applying the general results of Bather (1976).
On the compound Poisson limit for high level exceedances
Authors: Cohen Jonathan
Published: April 2006

Let ∊1∊2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un(τ)) of [3] for every τ > 0, where nP(∊1> un(τ))→ τ. Assume only that P (there are j exceedances of un(τ) by ∊1, ∊2, · ··, ∊n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.
Some further results on random splitting of an interval
Authors: Lloyd Christopher J.
Published: April 2006

The random splitting model of Lloyd and Williams (1988) is generalised, to allow first beta splitting distributions and secondly, discrete splitting distributions. Analogues of some earlier results are developed. In particular, the equality of the average value of the largest split and the probability that this is achieved at the first split continues to hold.
On reinforcement-depletion compartmental urn models
Authors: Donnelly Peter , Whitt Ward
Published: April 2006

We verify and extend a conjecture of Purdue (1981) concerning the stochastic monotonicity of absorption times in a class of compartmental urn models. We also describe the effect of increased variability in the reinforcement sizes. Finally, we investigate variability in the content process for large populations. In many applications, compartmental models substantially under-represent the variability observed in the data, so that there has been considerable interest in modifying the model to increase the variability. We show that the squared coefficient of variation of the content is not asymptotically negligible when both the size and the variability of the reinforcements are of the same order as the initial population.
Transient solution to the many-server Poisson queue: a simple approach
Authors: Parthasarathy P. R. , Sharafali M.
Published: April 2006

An elegant time-dependent solution for the number in the M/M/c queueing system is derived in a direct way.
The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline
Authors: Keilson Julian , Ramaswamy Ravi
Published: April 2006

The vacation model studied is an M/G/1 queueing system in which the server attends iteratively to ‘secondary' or ‘vacation' tasks at ‘primary' service completion epochs, when the primary queue is exhausted. The time-dependent and steady-state distributions of the backlog process [6] are obtained via their Laplace transforms. With this as a stepping stone, the ergodic distribution of the depletion time [5] is obtained. Two decomposition theorems that are somewhat different in character from those available in the literature [2] are demonstrated. State space methods and simple renewal-theoretic tools are employed.
Queueing networks by negative customers and negative queue lengths
Authors: Henderson W.
Published: April 2006

A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.
Time-dependent process of M/G/1 vacation models with exhaustive service
Authors: Takagi Hideaki
Published: April 2006

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.
Stochastically maximizing the number of successes in a sequential assignment problem
Authors: Righter Rhonda
Published: April 2006

In the classical sequential assignment problem as introduced by Derman et al. (1972) there are n workers who are to be assigned a finite number of sequentially arriving jobs. If a worker of value p is assigned a job of value x the return is px, where we interpret the return as the probability that the given worker correctly completes the given job. The job value is a random value that is observed upon arrival, and jobs must be assigned or rejected when they arrive. Each worker can only do one job. Derman et al. showed that when the objective is to maximize the expected return, i.e., the expected number of correctly completed jobs, the optimal policy is a simple threshold policy, which does not depend on the worker values. Their result was extended by Albright (1974) to allow job arrivals according to a Poisson process and a single random deadline for job completion (which is equivalent to discounting). Righter (1987) further extended the result to permit workers to have independent random deadlines for job completions. Here we show that when there are independent deadlines a simple threshold policy that is independent of the worker values stochastically maximizes the number of correctly completed jobs, and therefore maximizes the expected number of correctly completed jobs. We also show that there is no policy that stochastically maximizes the number of correctly completed jobs when there is a single deadline. However, when there is single deadline and the objective is to maximize the probability that n jobs are done correctly by n workers, then the optimal policy is determined by a single threshold that is independent of n and of the worker values.
On information-based minimal repair and the reduction in remaining system lifetime due to the failure of a specific module
Authors: Natvig Bent
Published: April 2006

The first part of this paper is inspired by a somewhat surprising result in Arjas and Norros (1989). Here we give some results comparing remaining system lifetime just after a ‘black box' minimal repair of a system and after a natural minimal repair based on information on the component level. In the second part we consider the reduction in remaining system lifetime due to the failure of a specific module and explore the relation to the reduction in remaining system lifetime due to the failure of a component inside the module. This former reduction also equals the increase in remaining system lifetime due to a minimal repair of the module at its time of failure. The expected value of this reduction/increase is proportional to the so-called Natvig measure of the importance of the module.
Invariant stochastic properties of a class of directed circuits
Authors: Kalpazidou S.
Published: April 2006

Invariant stochastic properties are studied for circuit processes, i.e. processes whose sample paths can be represented by a denumerable set of overlapping directed circuits c in a denumerable set S and a set of positive weights , when the family of the weights varies.
Change of measure up to a random time: theory
Authors: Mortimer T. M. , Williams David
Published: April 2006

Change of measure up to fixed times or stopping times is the theme of the famous Cameron–Martin–Girsanov formula. The paper studies change of measure up to random times which are not stopping times of the natural filtration. The ultimate aim is to build up a family of interesting models for physics and chemistry.