bluuebunny/crossref_metadata_2025 huggingface dataset:
Series expansions for the properties of a birth process of controlled variability
Authors: Cox D. R. , Isham V.
Published: April 2006

A birth process is studied in which the birth rate at any time is a function of the difference between the current population size and a target corresponding to unit growth rate. If this controlling function is a decreasing function of its argument a stabilizing effect is to be expected. By supposing that the controlling function varies very slowly, series expansions for the properties of the process are obtained, the leading term corresponding to a diffusion approximation. The sequence of births considered as a point process of controlled variability is examined and approximations to the interval distribution and covariance density obtained.
A continuous review (s, S) inventory system in a random environment
Authors: Feldman Richard M.
Published: April 2006

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.
Continuity of limit random variables in the branching random walk
Authors: Biggins J. D. , Grey D. R.
Published: April 2006

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.
A Poisson limit theorem for weakly exchangeable events
Authors: Eagleson G. K.
Published: April 2006

LetY1,Y2, · ·· be a sequence of independent, identically distributed random variables,gsome symmetric 0–1 function ofmvariables and setSilverman and Brown (1978) have shown that under certain conditions the statisticis asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of theXil…im.
Recognizing both the maximum and the second maximum of a sequence
Authors: Tamaki M.
Published: April 2006

We consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants. The optimal stopping rule and the maximum probability of choosing both of them are derived.
Dispersive distributions, and the connection between dispersivity and strong unimodality
Authors: Lewis Toby , Thompson J. W.
Published: April 2006

Two continuous distributions, G, H so related that any two quantiles of H are more widely separated than the corresponding quantiles of G may be said to be ‘ordered in dispersion'; Saunders and Moran have given examples. It is shown here that distributions F (called ‘dispersive' distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, F ∗H. The class of dispersive distributions is determined, and shown to coincide with the class of strongly unimodal distributions.
The supremum distribution of another Gaussian process
Authors: Cressie Noel , Davis Robert W.
Published: April 2006

A formula is derived for the supremum of a stationary Gaussian process which has a correlation function that is tent-like in shape, until it flattens out at a constant negative value. Examples and graphs are presented in the last section.
Clustering of events in a stochastic process
Authors: Glaz Joseph
Published: April 2006

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.
A random graph
Authors: Ross Sheldon M.
Published: April 2006

We consider an unordered graph where there is one arc emanating from each node. We suppose that the arc that emanates from i will go to j with probability Pj. The probability that the resultant graph is connected and a recursive formula for the distribution of the number of components it possesses are derived.
A simultaneous characterization of the Poisson and bernoulli distributions
Authors: Kimeldorf George , Plachky Detlef , Sampson Allan R.
Published: April 2006

LetN, X1,X2, · ·· be non-constant independent random variables withX1,X2, · ·· being identically distributed andNbeing non-negative and integer-valued. It is shown that the independence ofandimplies that theXi's have a Bernoulli distribution andNhas a Poisson distribution. Other related characterization results are considered.
Success probabilities for second guessers
Authors: Pittenger A. O.
Published: April 2006

Two people independently and with the same distribution guess the location of an unseen object in n-dimensional space, and the one whose guess is closer to the unseen object is declared the winner. The first person announces his guess, but the second modifies his unspoken idea by moving his guess in the direction of the first guess and as close to it as possible. It is shown that if the distribution of guesses is rotationally symmetric about the true location of the unseen object, ¾ is the sharp lower bound for the success probability of the second guesser. If the distribution is fixed and the dimension increases, then for a certain class of distributions, the success probability approaches 1.
A characterization of the gamma distribution by the negative binomial distribution
Authors: Engel Jan , Zijlstra Mynt
Published: April 2006

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.
An exponential moving-average sequence and point process (EMA1)
Authors: Lawrance A. J. , Lewis P. A. W.
Published: April 2006

A construction is given for a stationary sequence of random variables {Xi} which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {εi}. The joint and trivariate exponential distributions ofXi−1,XiandXi+ 1are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi} sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.
Superposition of distinguishable point processes
Authors: Ito Yoshifusa
Published: April 2006

The properties of frequency distributions of inter-event intervals in the pooled output of several point processes where the points can be distinguished are investigated. In the case where the component processes are mutually independent, the p.d.f.'s of inter-event intervals in the pooled output are explicitly expressed, and the reciprocal relations on the p.d.f.'s and on the numbers of inter-event intervals are revealed.
An example of phase transition in countable one-dimensional Markov random fields
Authors: Cox Ted
Published: April 2006

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).
Predicting the benefit of screening for disease
Authors: Dubin Neil
Published: April 2006

To evaluate the benefits and risks associated with screening for disease, a model is developed to characterize the changes in incidence and survival distributions effected by a screening program. Screening is presumed to increase survival by resulting in diagnosis of disease at earlier stages. All disease states in the model are observable, thus facilitating application to empirical data. An example of such an application using data from a breast cancer detection project is given for the case of one screening for two-stage disease having Weibull-distributed diagnosis times.
Non-linear threshold autoregressive models for non-linear random vibrations
Authors: Ozaki Tohru
Published: April 2006

Time series models for non-linear random vibrations are discussed from the viewpoint of the specification of the dynamics of the damping and restoring force of vibrations, and a non-linear threshold autoregressive model is introduced. Typical non-linear phenomena of vibrations are demonstrated using the models. Stationarity conditions and some structural aspects of the model are briefly discussed. Applications of the model in the statistical analysis of real data are also shown with numerical results.
On a generalized storage model with moment assumptions
Authors: Puri Prem S. , Woolford Samuel W.
Published: April 2006

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn),n =0, 1, ·· ·}, with 0 =T0<T1< · ··, almost surely, whereXntakes values in the set {1, 2, ·· ·}. If atTn, Xn=j, then there is a random ‘input'Vn(j) (a negative input implying a demand) of ‘type'j, having distribution functionFj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn(k)}, fork≠j, and thatVn(j) has first and second moments. Here the random variablesVn(j) represent instantaneous ‘inputs' (a negative value implying a demand) of typejfor our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t),L(t)), whereZ(t) (defined in (2)) is the level of storage at timetandL(t) (defined in (3)) is the demand lost due to shortage of supply during [0,t]. Different limit distributions are obtained for the cases when the ‘average stationary input'ρ, as defined in (5), is positive, zero or negative.
The algebraic equivalence of two measures of genetic distance
Authors: Dawson Jeremy E.
Published: April 2006

A formal proof is given of the equivalence, conjectured by Rostron, of two measures of the genetic difference between two populations.
The average noise from a Poisson stream of vehicles
Authors: Andersen Per K. , Andersen Søren , Lauritzen Steffen L.
Published: April 2006

The distribution of the average noise power from a Poisson stream of vehicles is, properly normalised, shown to converge to a normal distribution although the corresponding stationary process is deterministic. The speed of convergence is estimated. Finally the asymptotic efficiency of a sampling procedure is discussed.
Estimation for multitype branching processes
Authors: Quine M. P. , Durham P.
Published: April 2006

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.
A note on simple branching processes with infinite mean
Authors: Hudson Irene L. , Seneta E.
Published: April 2006

We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn} say, conditions are given ensuring that there exists a sequence of positive constants, {ρn}, such that {ρnU(Zn + 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.
A note on a condition satisfied by certain random walks
Authors: Doney R. A.
Published: April 2006

The problem considered is to elucidate under what circumstances the condition

holds, where and Xi are independent and have common distribution function F. The main result is that if F has zero mean, and (*) holds with F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases are also treated. (The case cannot arise in these circumstances.) A partial result is also given for the case when and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn, the number of positive terms in (S1, S2, …, Sn). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.
An approximation to a two-dimensional binary process
Authors: Welberry T. R. , Miller G. H.
Published: April 2006

We further consider a two-dimensional stochastic process of binary variables previously discussed by Galbraith and Walley (1976). A method of estimating properties of the equilibrium distribution is described which considerably extends the region of probability space that can satisfactorily be explored.
On the relation between offered, critical and accepted gap distributions
Authors: Thedeen Torbjörn
Published: April 2006

We consider the situation when vehicles in a minor road try to cross a major road. The relation between the distributions of the critical, offered and accepted gaps is studied. Estimates of the critical gap distribution from observations of offered and accepted gaps are given and also studied in some simulation examples.
A new look at the simple epidemic process
Authors: Billard L. , Lacayo H. , Langberg N. A.
Published: April 2006

Classical epidemic models have invariably proved to be mathematically intractable. By considering the distribution of the number of infectives in a simple epidemic process as a convolution of exponential waiting times, the solution to the classical model is obtained easily giving more insight into the underlying structure. The idea can be extended to other simple epidemic models.
Invariant measures of time-reversible Markov chains
Authors: Suomela P.
Published: April 2006

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.
An example of Wold's point processes by Markov-dependent intervals
Authors: Lai C. D.
Published: April 2006

Wold's point process is briefly introduced and its forward equation is derived in terms of an integro-differential equation which is used to obtain the ‘renewal' function. An example of this family of processes is given in which each interval is exponentially distributed. The theory of diagonal expansion for a bivariate distribution is used to obtain the numerical estimate of the spectrum of counts.
Sequential random packing in the plane
Authors: Weiner Howard J.
Published: April 2006

The Palasti conjecture on the asymptotic mean proportion of coverage is verified for the sequential random packing of rectangular cars with sides parallel to rectangular boundaries in the models of Rényi and Solomon. The extension to n dimensions is given. An extension to a random car size model is indicated.
A class of non-identifiable stochastic models
Authors: Cane Violet R.
Published: April 2006

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.
A note on the branching random walk
Authors: Kaplan Norman
Published: April 2006

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj <∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.
Control of grade sizes in a partially stochastic Markov manpower model
Authors: Davies G. S.
Published: April 2006

A manpower model is examined in which there are fixed promotion rates, no demotions and stochastic wastage. We discuss the probability of attainability of structures and structural paths using recruitment control.
Birth-and-death processes on the integers with phases and general boundaries
Authors: Hajek Bruce
Published: April 2006

The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.
Strategy stability in complex randomly mating diploid populations
Authors: Hines W. G. S.
Published: April 2006

A class of Lyapunov functions is used to demonstrate that strategy stability occurs in complex randomly mating diploid populations. Strategies close to the evolutionarily stable strategy tend to fare better than more remote strategies. If convergence in mean strategy to an evolutionarily stable strategy is not possible, evolution will continue until all strategies in use lie on a unique face of the convex hull of available strategies.

The results obtained are also relevant to the haploid parthenogenetic case.
Heritability
Authors: Wilson Susan R.
Published: April 2006

This article discusses certain aspects of determining heritability estimates from the genetic analyses of data. First, a brief review is given of the major statistical techniques for fitting models to quantitative family data. Then, by reference to a general schematic framework, the basic models that have been proposed for such data will be described. Some new environmental models are also developed. Finally, by reference to some data sets, the interactive approach to data analysis is proposed, advocating the determination of adequacy of fit of all reasonable models to available data.
Functional dependence from a descriptive viewpoint
Authors: Finch P. D.
Published: April 2006

This paper considers functional dependence from a general standpoint. Some basic ideas are explored and the extent of functional dependence studied. These are illustrated by means of simple linear regression.
A finite algorithm for the rank regression problem
Authors: Osborne M. R.
Published: April 2006

A new approach to the minimization of polyhedral convex functions is applied to give a finite algorithm for the rank regression problem. Numerical results for the Daniel and Wood example are presented.
On the probability generating function of the sum of Markov Bernoulli random variables
Authors: Gani J.
Published: April 2006

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.
On the interpretation of random fluctuations in competing chemical systems
Authors: Hall Peter
Published: April 2006

Several stochastic models have been proposed to describe the kinetic theory of reversible chemical reactions. However, in macroscopic systems the effects of stochastic variability are often outweighed by mean effects. In the present paper we show that some observed phenomena can be explained quite adequately by a stochastic model in which the stochastic variability is not negligible in comparison with mean effects. Our argument involves approximations to a stochastic model for competing chemical reactions.
Relationships between order statistics and record values and some characterization results
Authors: Gupta Ramesh C.
Published: April 2006

The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially.
On conditional passage time structure of birth-death processes
Authors: Sumita Ushio
Published: April 2006

LetN(t) be a birth-death process onN= {0,1,2,· ··} governed by the transition rates λn> 0 (n≧ 0) andμη> 0 (n≧ 1). LetmTmbe the conditional first-passage time from r to n, given no visit tomwherem <r < n.The downward conditional first-passage timenTmis defined similarly. It will be shown that, foranyλn> 0 andμη> 0. The limiting behavior ofis considerably different from that of the ordinary first-passage timewhere, under certain conditions, exponentiality sets in asn→∞. We will prove that, when λn→ λ > 0 andμη→μ> 0 asn→ ∞withρ= λ/μ <1, one hasasr→ ∞whereTBP(λ,μ)is the server busy period of anM/M/1 queueing system with arrival rate λand service rateμ.
On the expected width function for topologically random channel networks
Authors: Troutman Brent M. , Karlinger Michael R.
Published: April 2006

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.
On the exact order of normal approximation in multivariate renewal theory
Authors: Niculescu Ştefan P. , Omey Edward
Published: April 2006

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).
The intensity conservation law for queues with randomly changed service rate
Authors: Miyazawa Masakiyo
Published: April 2006

König et al. (1978) have derived the so-called intensity conservation law in a stationary process connected with a marked point process (PMP). That law has been shown to be useful in obtaining invariance relations in queues (cf. Franken et al. (1981)). In this paper, somewhat different versions of the intensity conservation laws are derived for a stationary process with jump points. These laws are applied to queues with randomly changed service rate. As special cases, most of equations obtained by König et al.'s law can be derived from this law. Also, we derive some inequalities between characteristic quantities in a queue with a simple type of randomly changed service rate.
A note on conditioned random walk
Authors: Doney R. A.
Published: April 2006

This note is concerned with N, the time at which a random walk first exits from [0,∞), and M, the maximum of the random walk up to time N. In the case that the random walk has zero mean and finite variance, simple proofs are given of asymptotic estimates for P{M > x}, P{N ≦ ux2|M > x} and P{M ≦ v √n|N > n}.
The control of a finite dam
Authors: Attia F. A. , Brockwell P. J.
Published: April 2006

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).
A necessary condition for extinction in those bisexual Galton-Watson branching processes governed by superadditive mating functions
Authors: Hull David M.
Published: April 2006

Mating functions considered by Asmussen (1980) and Daley (1968) for the class of bisexual Galton-Watson branching processes (GWBP) are shown to be superadditive. Consideration of a process that allows only sibling mating leads to a necessary condition for almost sure extinction in bisexual GWBP governed by superadditive mating functions. A simple example shows that the condition is not sufficient.
A note on extended block replacement policy with used items
Authors: Murthy D. N. P. , Nguyen D. G.
Published: April 2006

The extended block replacement policy with used items as advocated by Tango (1978) uses used items of age T only although the policy creates used items of lower age. In this note we modify Tango's policy so as to make use of all used items created by the policy.
Weak and strong law results for a function of the spacings
Authors: McCormick William P.
Published: April 2006

Letbe i.i.d. uniform on (0,1) random variables and defineSi,n=Ui,n–1Ui–1,n–1,i= 1, · ··,nwhere theUi–n–1are the order statistics from a sample of sizen –1 andU0,n–1=0 andUn,n–1= 1. TheSi,nare called the spacings divided byU1,· ··,Un–1. For a fixed integerl, set. Exact and weak limit results are obtained for theMl,n.Further we show that with probability 1This extends results of Cheng.
A limit theorem which clarifies the ‘Petersburg Paradox'
Authors: Martin-Löf Anders
Published: April 2006

The total gain, SN, in N successive Petersburg games is considered, and a limit theorem for SN/N – n when N = 2n and n → ∞is proved. The limit distribution can be determined numerically with good accuracy, and this fact provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.
On the roles of the Bessel and Poisson distributions in chemical kinetics
Authors: Hall Peter
Published: April 2006

We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.
Bounded Recurrence times and lead time in the design of a repetitive screening program
Authors: Prorok Philip C.
Published: April 2006

An unsolved problem in the design of a repetitive screening trial is the determination of a procedure which yields the optimal number of screens for efficient evaluation. An approach to this problem described in this paper is to use lead time properties of cases detected at screening to define a stopping rule. Previous results are generalized by allowing the intervals between screens to be arbitrary, and the probability of detecting preclinical disease to vary among screens.
On Coupling of continuous-time renewal processes
Authors: Lindvall Torgny
Published: April 2006

This paper shows how the coupling method applies to renewal theory in continuous time. Old and new rate results for tendency towards equilibrium and forgetfulness of initial delay are obtained. The coupling approach also provides a new way of analysing the asymptotics of functions of renewal processes which avoids the route via a renewal equation.
Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes
Authors: Chandramohan J. , Liang Lung-Kuang
Published: April 2006

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.
A simple construction of an upper bound for the mean of the maximum of n identically distributed random variables
Authors: Gravey A.
Published: April 2006

This note gives a method of finding an upper bound for the mean of the maximum of n identically distributed non-negative random variables. The bound is explicitly given and numerically compared with the exact value of the mean of the maximum for some classical distributions (geometric, Poisson, Erlang, hyperexponential).
Boundaries with negative jumps for the Brownian motion
Authors: Park C.
Published: April 2006

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.
A sharp upper probability bound for the occurrence of at least m out of n events
Authors: Platz Ole
Published: April 2006

Consider a fixed set of n events. Let S1 denote the sum of the probabilities of occurrence of each event and S2 the sum of the probabilities of occurrence of each of the pairs of events. Using a dual linear programming method a sharp upper bound is derived for the probability of occurrence of at least m out of the n events when S1 and S2 are given. The bound improves one of the inequalities obtained by Sathe et al. (1980).
Honest bernoulli excursions
Authors: Smith Laurel , Diaconis Persi
Published: April 2006

For simple random walk on the integers, consider the chance that the walk has traveled distance k from its start given that its first return is at time 2n. We derive a limiting approximation accurate to order 1/n. We give a combinatorial explanation for a functional equation satisfied by the limit and show this yields the functional equation of Riemann's zeta function.
Generalised mixtures of exponential distributions
Authors: Roy Dilip , Mukherjee S. P.
Published: April 2006

An attempt has been made to generalise the concept of exponential mixture to the multivariate case through the random environment model. Some important properties of this class of mixture distributions have been studied, along with characterising properties. A further generalisation for a type of dependent exponential distribution has also been made.
Oil exploration: sequential decisions in the face of uncertainty
Authors: Benkherouf L. , Bather J. A.
Published: April 2006

A simple Bayesian model for oil exploration is suggested to investigate strategies for drilling. A condition on the way successes and failures affect the prior distribution implies a certain form of the detection mechanism. It is shown that the problem of finding strategies for drilling reduces to an optimal stopping problem. Two new families of distributions are obtained with generating functions related to classical work on partitions of integers. By using such distributions and simple mixtures of them as priors, the stopping problem can be solved explicitly. This leads to the construction of simple strategies and their effectiveness is demonstrated by evaluating suitable operating characteristics.
Asymptotic final size distribution of the multitype Reed–Frost process
Authors: Scalia-Tomba Gianpaolo
Published: April 2006

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.
Waiting time and workload in queues with periodic Poisson input
Authors: Lemoine Austin J.
Published: April 2006

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].
Stochastic monotonicity in general queueing networks
Authors: Shanthikumar J. George , Yao David D.
Published: April 2006

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).
Birth and death processes in random environments with feedback
Authors: Cornez Richard
Published: April 2006

A generalization of a birth and death chain in a random environment (Yn, Zn) is developed allowing for feedback to the environmental process (Yn). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Zn) process goes extinct almost surely or has strictly positive probability of non-extinction.
Moving-average models with bivariate exponential and geometric distributions
Authors: Langberg Naftali A. , Stoffer David S.
Published: April 2006

Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.
On stochastic comparison of random vectors
Authors: Shanthikumar J. George
Published: April 2006

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.
Optimal replacement in a shock model: discrete time
Authors: Aven Terje , Gaarder Simen
Published: April 2006

A system is subject to a sequence of shocks occurring randomly at times n = 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in time n, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.
Stochastic inequalities for an overflow model
Authors: Hordijk Arie , Ridder Ad
Published: April 2006

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.
The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline
Authors: Fakinos D.
Published: April 2006

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.
On the best order of observation in optimal stopping problems
Authors: Gilat David
Published: April 2006

For optimal stopping problems in which the player is allowed to choose the order of the random variables as well as the stopping rule, a notion of order equivalence is introduced. It is shown that different (non-degenerate) distributions cannot be order-equivalent.

This result unifies and generalizes two theorems of a similar nature recently obtained by Hill and Hordijk (1985).
On transient regenerative processes
Authors: Murphree Emily , Smith Walter L.
Published: April 2006

A transient cumulative process based upon a sequence of possibly infinite lifetimes is defined, and examples of such a process are described. Given a mild condition on the improper lifetime distribution and given that all lifetimes observed by timetare finite, the expected value of this transient process attis related to the expected value of a cumulative process based upon proper lifetimes. This relationship is exploited to show that the conditional behavior of the transient process is analogous to that of a proper process and, in particular, the transient process is asymptotically normal.
Return probabilities for correlated random walks
Authors: Iossif Gillian
Published: April 2006

A correlated random walk on ad-dimensional integer lattice is studied in which, at any stage, the probabilities of the next step being in the various possible directions depend on the direction of the previous step. Using a renewal argument, asymptotic estimates are obtained for the probability of return to the origin afternsteps.
On the single-server queue with the preemptive-resume last-come–first-served queue discipline
Authors: Fakinos D.
Published: April 2006

The paper considers theGI/G/1 queueing system under the assumption of a last-come–first-served queue discipline, where each customer begins service immediately upon his arrival. At the next arrival, the previous service is interrupted but no loss of service is involved. It has been shown that when the system is considered exclusively at arrival epochs or exclusively at departure epochs, then the equilibrium distribution of the queue-size is geometric, while the remaining durations of the corresponding services are independent random variables each one distributed as the idle period in the dual (inverse) queue. In this paper alternative simpler proofs of the above results are given.
Finiteness of moments of partial busy periods for M/G/C queues
Authors: Ghahramani Saeed
Published: April 2006

Conditions for finiteness of moments of the following quantities have been found: the duration of a busy period of an Μ /G/∞ system; the duration of a partial busy period of an M/G/C loss system, and the duration of a partial busy period of an M/G/C queue.
Estimation of inbreeding from population samples
Authors: Smith Cedric A. B. , Thomson Robert
Published: April 2006

Some simple methods of estimating inbreeding from tests on a random sample of individuals are discussed. These are mostly iterative procedures, converging to the maximum likelihood estimate, with algorithms for finding standard errors. But there are problems of interpretation, both from the statistical and biological points of view.
On the limit of a supercritical branching process
Authors: Bingham N. H.
Published: April 2006

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.
Invariant measures for Markov chains with no irreducibility assumptions
Authors: Tweedie R. L.
Published: April 2006

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such asϕ-irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measureπfor general space chains, and forπto have a finitef-moment ∫π (dy)f(y), wherefis a general function. In the casef≡ 1 these cover the question of finiteness ofπitself.

In this paper we show that the same conditions imply the same conclusionswithout any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.

The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.
On entrance times of a homogeneous N-dimensional random walk: an identity
Authors: Cohen J. W.
Published: April 2006

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.
Rates of convergence to the stationary distribution for k-dimensional diffusion processes
Authors: Davies P. L.
Published: April 2006

Using a coupling technique, rates of convergence in total variation of certain k -dimensional diffusion processes to the stationary distribution are obtained. The results place assumptions only on the coefficients of the elliptic differential operator governing the diffusion.
Multivariate distributions for the life lengths of components of a system sharing a common environment
Authors: Lindley Dennis V. , Singpurwalla Nozer D.
Published: April 2006

In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.
An adaptive multistage queueing system
Authors: Viscolani Bruno
Published: April 2006

The system provides each customer with a service made up of a random number of stages in sequence. Arrivals are Poisson, and the stage times are independent exponential random variables. The number of stages of a particular service depends on the customer's random demand and on the arrival process, in a way which is aimed at preventing the queue from growing fast while matching the customer's demand as well as possible.
Modeling the effect of erosion on crop production
Authors: Todorovic P. , Gani J.
Published: April 2006

This paper is concerned with a model for the effect of erosion on crop production. Crop yield in the yearnis given byX(n) =YnLn,whereis a sequence of strictly positive i.i.d. random variables such thatE{Y1} <∞, andis a Markov chain with stationary transition probabilities, independent of. When suitably normalized,leads to a martingale which converges to 0 almost everywhere (a.e.) asn →∞. In addition, for largen,the distribution ofLnis approximately lognormal. The conditional expectations and probabilities of, given the past history of the process, are determined. Finally, the asymptotic behaviour of the total crop yieldis discussed. It is established that under certain regularity conditionsSnconverges a.e. to a finite-valued random variableSwhose Laplace transform can be obtained as the solution of a Volterra-type linear integral equation.
On the critical infection rate of the one-dimensional basic contact process: numerical results
Authors: Ziezold Herbert , Grillenberger Christian
Published: April 2006

Instead of the basic contact process on with infection rate λ we consider for m ≧ 0 the Markov process starting with ξ0(k) = 1 for k ≧ 0 and ξ0(k)= 0 for k < 0 and with changing only those k which are at most m places to the right of the left-most infected cell. For m = 0, 1,· ··, 14 direct computations give critical values which are lower bounds for the critical value of the original basic contact process.