bluuebunny/crossref_metadata_2025 huggingface dataset:
Heavy traffic approximations for the Galton-Watson process
Authors: Fahady K. S. , Quine M. P. , Vere-Jones D.
Published: April 2006

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).
Why the Shannon and Hartley entropies are ‘natural’
Authors: Aczél J. , Forte B. , Ng C. T.
Published: April 2006

The following properties of entropies, as measures of expected information, seem natural. The amount of information expected from an experiment does not change if we add outcomes of zero probability (expansibility). The expected information is symmetric in the (probabilities of the) outcomes. The information expected from a combination of two experiments is less than or equal to the sum of the informations expected from the single experiments (subadditivity); equality holds here if the two experiments are independent (additivity).

In this paper it is shown that linear combinations of the Shannon and Hartley entropies and only these have the above properties. The Shannon and the Hartley entropies are also individually characterized.
Optimal resource allocation in chemical research
Authors: Gittins J. C.
Published: April 2006

The problem of allocating scientific effort between a number of competing projects is considered. A stochastic mathematical model is set up, and for the case of just two competing projects a number of results are proved which enable the best policy to be determined. An indication is given of the ways in which these results might be extended so as to deal with more general situations.
Optimal immunisation policies for epidemics
Authors: Abakuks Andris
Published: April 2006

Policies of immunising the susceptibles in the general stochastic epidemic are considered. With costs assigned to the infection and immunisation of susceptibles, an optimal policy is found which at any stage minimises the expected future cost. Optimal policies are also found for the general deterministic epidemic and for an epidemic with carriers model. The three policies are compared and numerical examples are provided. An optimal policy is described in the situation when both isolation of infectives and immunisation of susceptibles are allowed.
Stochastic difference equations with non-integral differences
Authors: Robinson P. M.
Published: April 2006

As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squares applied to an approximate Fourier-transformed model yields estimators of the coefficients that are consistent with respect to thetruemodel under some conditions. The conditions are weak when the model contains predetermined variables that obey an “aliasing condition”; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic difference-differential equations are also available.
Bounds on the extinction time distribution of a branching process
Authors: Agresti Alan
Published: April 2006

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.
Quadratic transformations: a model for population growth. II
Authors: Kesten Harry
Published: April 2006

In this last part theFn(i) andMn(i) are considered as random variables whose distributions are described by the model and various mating rules of Section 2. Several convergence results will be proved for those specific mating rules, but we begin with the more general convergence theorem 6.1. The proof of this theorem brings out the basic idea of this section, namely that whenFnandMnare large,Fn + 1(i) andMn + 1(i) will, with high probability, be close to a certain function ofFn(·) andMn(·) (roughly the conditional expectation ofFn+1(i) andMn + 1(i) givenFn(·) andMn(·)). As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equalityfor some transformationTof the form (1.4), (1.5). More generally for fixedkexcept on a set whose probability is small whenFnandMnare large. If the theorems of Section 3 or 4 apply, Tk(fn(·),mn(·)) will be close to a fixed vector ζ whenkis large and thus there is hope thatfn(·) andmn(·) will converge, onceFnandMnbecome large. We therefore have to put on some conditions which will makeFnandMngrow. This is the role of (6.34) and, to some extent, also of (6.17). The main difficulty is that the expected size of the (n + 1)th generation, given the nth generation, depends on the frequencies of the different types present in the nth generation. Even if (6.34) holds, the conditional expected size of the (n + 1)th generation, given thenth generation, may actually be smaller than the size of thenth generation for certain directionsfn(·),m(·).
Multiple channel queues in heavy traffic. III: random server selection
Authors: Whitt Ward
Published: April 2006

As in [4] and [5], we study service facilities with r arrival channels and s service channels. However, here we assume that customers, immediately upon arrival, randomly select one of the s service channels. Successive customers make this choice independently, choosing server i with probability pi, p1 + · · · + ps = 1. Customers are then served by the servers they select in order of their arrival without defections. The average processing rates as well as the server selection probabilities may vary from server to server, but again we assume the r arrival channels are independent and independent of the service channels. The service channels are not independent, however, because of the random server selection. For simplicity, we only consider a single queueing system; the extension to sequences follows immediately using the argument of [5].
Inheritance
Authors: Stirzaker David
Published: April 2006

We consider a population of reproducing individuals who inherit, earn, consume, and bequeath wealth. A model is constructed to describe the wealth of an individual selected from the nth generation by following a random line of descent from the initial individual. It is shown that bequests are commonly a convex function of wealth. Considering a linear approximation to the bequest function enables us to obtain estimates of the limiting distribution of wealth as the number of generations increases, when earnings of parent and offspring are independent. More generally when earnings of parent and offspring are not independent we obtain upper bounds for the tail of the wealth distribution using a martingale inequality.
The recurrence and transience of two-dimensional linear birth and death processes
Authors: Hutton J.
Published: April 2006

A two-dimensional linear birth and death process is a continuous-time Markov chain Y(·) with state space (Z+)2 which can jump from the point (n, m) to one of its four neighbors, with rates that are linear functions of n and m. Criteria are extended for determining whether such a process has a positive probability or zero probability of escaping to infinity. In the transient case considered, the projections of the imbedded Markov chain {Xn} of the successive states visited by Y(·) on a suitable pair of orthonormal vectors v and w are shown to be regularly varying sequences with index 1. Specifically, (Xn, v)∽δn and (Xn, w)∽ kn/log n for positive constants δ and k.
Renewal processes decomposable into i.i.d. components
Authors: Ito Yoshifusa
Published: April 2006

Let N be a stationary renewal process with a probability density function f(t). Suppose that N can be expressed as the superposition of a finite number of i.i.d. stationary components N(1), …, N(p) (p≧2). Then, under a supplementary condition on f(t), N and N(1), …, N(p) are all Poisson. This is proved by using recurrence relations given in Ito (1978) for the probability distribution of i.i.d. components of a superposition process.
Characterization of stable processes by identically distributed stochastic integrals
Authors: Riedel M.
Published: April 2006

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.
Filtering of Markov renewal queues, II: Birth-death queues
Authors: Hunter Jeffrey J.
Published: April 2006

In this part we extend and particularise results developed by the author in Part I (pp. 349–375) for a class of queueing systems which can be formulated as Markov renewal processes. We examine those models where the basic transition consists of only two types: ‘arrivals' and ‘departures'. The ‘arrival lobby' and ‘departure lobby' queue-length processes are shown, using the results of Part I to be Markov renewal. Whereas the initial study focused attention on the behaviour of the embedded discrete-time Markov chains, in this paper we examine, in detail, the embedded continuous-time semi-Markov processes. The limiting distributions of the queue-length processes in both continuous and discrete time are derived and interrelationships between them are examined in the case of continuous-time birth–death queues including the M/M/1/M and M/M/1 variants. Results for discrete-time birth–death queues are also derived.
The queue GI/M/s with customers of different types or the queue GI/Hm/s
Authors: De Smit Jos H. A.
Published: April 2006

We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability pi and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter bi. This model is equivalent to the queue GI/Hm/s, where Hm denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.
The depletion time forM/G/1 systems and a related limit theorem
Authors: Keilson Julian , Sumita Ushio
Published: April 2006

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.
The exact order of normal approximation in bivariate renewal theory
Authors: Ahmad Ibrahim A.
Published: April 2006

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.
Parametric estimators for stationary time series with missing observations
Authors: Dunsmuir W. , Robinson P. M.
Published: April 2006

Three related estimators are considered for the parametrized spectral density of a discrete-time processX(n),n= 1, 2, · · ·, when observations are not available for all the valuesn= 1(1)N. Each of the estimators is obtained by maximizing a frequency domain approximation to a Gaussian likelihood, although they do not appear to be the most efficient estimators available because they do not fully utilize the information in the processa(n) which determines whetherX(n) is observed or missed. One estimator, called M3, assumes that the second-order properties ofa(n) are known; another, M2, lets these be known only up to an unknown parameter vector; the third, M1, requires no model fora(n). Under representative sets of conditions, which allow for both deterministic and stochastica(n), the strong consistency and asymptotic normality of M1, M2, and M3 are established. The conditions needed for consistency whenX(n) is an autoregressive moving-average process are discussed in more detail. It is also shown that in general M1 and M3 are equally efficient asymptotically and M2 is never more efficient, and may be less efficient, than M1 and M3.
Inbreeding with multiple loci
Authors: Fielden Christa
Published: April 2006

The paper develops a formula for inbreeding probabilities in the multi-locus case, this formula being analogous to that given by Malecot for single-locus chromosomes. A detailed example provides a comparison between the present approach and that by Malecot and it is found that the two methods produce slightly different results. This slight difference is then shown to be due to certain paths in pedigrees which seem to have been overlooked by Malecot. An indication of calculations necessary for higher-locus cases concludes the paper.
Further investigations of incompatibility systems in angiosperm plants
Authors: Heuch Ivar
Published: April 2006

In the incompatibility system operating in Lythrum salicaria, the three incompatibility types Long, Mid and Short will be equally frequent in a population of infinite size. But in small populations the complete system cannot be maintained indefinitely because of random loss of incompatibility types. A simulation model for such populations shows that this may be an important factor for populations consisting of less than 20 plants. When loss actually occurs, it will in most cases involve the Short type and in some cases the Mid type, but only rarely the Long type. This result may explain why the Long type appears to be the most common one in nature and Short the most infrequent one.
Diffusion approximations of Markov chains with two time scales and applications to population genetics
Authors: Ethier S. N. , Nagylaki Thomas
Published: April 2006

ForN= 1, 2, ···, let {(XN(k),YN(k)),k= 0, 1, ···} be a homogeneous Markov chain in ℝmx ℝn. Suppose that, asymptotically asN→ ∞, the ‘infinitesimal’ covariances and means ofXN([·/∊N]) areaij(x, y) andbi(x, y), and those ofYN([·/δN]) are 0 andcl(x, y). Assume limN→∞δN= limN→∞∊N/δN= 0 and the zero solution ofý=c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i)XN([·/∊N]) converges weakly to a diffusion process with coefficientsaij(x, 0) andbi(x, 0), and (ii)YN([t/∊N]) → 0 in probability for everyt> 0. (The case limN→∞δN= δ∞> 0 = limN→∞∊Nis also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.

The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.
Polymerisation processes with intrapolymer bonding. II. Stratified processes
Authors: Whittle P.
Published: April 2006

We consider a polymerisation process stratified in that space is divided into regions, between which migration occurs, but with bonding occurring only within a region. In the case of a process whose specification is spatially homogeneous, criticality (gelation) is then easily detectable as the point at which statistical equidistribution over regions becomes unstable. Stratification does import a new feature, however, in that the equipartition solution can become metastable below criticality; local fluctuations of density can induce ‘gelational collapse’ at a density below the critical value. We derive also detailed results for the inhomogeneous case, both below and above criticality. Statistics of polymer size and structure are also easily determined in the stratified case, although one can locate the critical point without recourse to these. Finally, one can to a large extent treat the case in which inter- and intrapolymer reaction rates differ, and show that such difference affects the onset of metastability rather than of instability.
Mutant substitutions at linked nucleotide sites
Authors: Watterson G. A.
Published: April 2006

Approximate results are obtained for the probability distribution of the number of mutant substitutions over a given number of generations. The distribution is a compound Poisson distribution having a variance which is greater than the mean. The substitutions occur in clusters. The most extreme case of clustering is studied in this paper, which assumes that there is no recombination between mutant nucleotides.
The number of alleles and segregating sites in a sample from the infinite-alleles model
Authors: Griffiths R. C.
Published: April 2006

A study is made of the joint distribution of the number of alleles and segregating sites in a random sample of genes from the infinite-alleles model of population genetics. Tables of the joint distribution are given for representative values of the mutation parameter. A conclusion is that each segregating site has a high probability of delineating an allele type.
The solution of certain two-dimensional Markov models
Authors: Fayolle G. , King P. J. B. , Mitrani I.
Published: April 2006

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.
The penultimate form of approximation to normal extremes
Authors: Cohen Jonathan P.
Published: April 2006

Let Yn denote the largest of n independent N(0,1) random variables. It is shown that the error in approximating the distribution of Yn by the type III extreme value distribution exp {– (–Ax + B)k}, k > 0, is uniformly of order (log n)–2 if and only if the constants A, B and k satisfy certain conditions. In particular, this holds for the penultimate form of Fisher and Tippett (1928). Furthermore, two sufficient conditions are given so that these results can be extended to a stationary Gaussian sequence.
Full-information best-choice problems with recall of observations and uncertainty of selection depending on the observation
Authors: Petruccelli Joseph D.
Published: April 2006

n i.i.d. random variables with known continuous distribution function F are observed sequentially with the object of choosing the largest. After any observation, say the kth, the observer may solicit any of the first k observations. If the (k – t)th is solicited, the probability of a successful solicitation may depend on t, the number of observations since the (k – t)th, and on the quantile of the (k – t)th observation. General properties of optimal selection procedures are obtained and the optimal procedures and their probabilities of success are derived in some special cases.
Operator-geometric stationary distributions for markov chains, with application to queueing models
Authors: Tweedie R. L.
Published: April 2006

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] who took E to be finite; they also include chains studied in queueing theory, such as (Nn, Sn) where Nn is the number of customers in a GI/G/1 queue immediately before, and Sn the remaining service time immediately after, the nth arrival.

We show that the stationary distribution Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions.

G1/G/1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS
Deterministic limit of the stochastic model of chemical reactions with diffusion
Authors: Arnold L. , Theodosopulu M.
Published: April 2006

Conditions are given for which the Markov jump process describing the stochastic model of chemical reactions with diffusion converges to the solution of the corresponding deterministic reaction–diffusion equation.
The random tangential projection of a surface
Authors: Miles R. E.
Published: April 2006

Suppose a smooth surfaceSis exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency withS, there is a corresponding point registration in an image planeHplaced perpendicular to the beam. The registering rays form a cylindrical surface which registers inHas a smooth curveC(except possibly for a number of cusps). Properties ofCwhen the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in whichScontains curved edges and in which any boundary curves ofSalso register. Finally, various related types of random tangent to a surface are considered.
A limit theorem for statistics of spatial data
Authors: Baddeley Adrian
Published: April 2006

A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate ‘ergodic’ distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the ‘true’ one. Examples in mathematical morphology are given.
The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam
Authors: Le Minh Do
Published: April 2006

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.
Correlation functions in the G/M/1 system
Authors: Srinivasan S. K. , Pillai C. Chudalaimuthu
Published: April 2006

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.
New better than used processes
Authors: Marshall Albert W. , Shared Moshe
Published: April 2006

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) > x} satisfies P{TX > s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.
Quasi-stationary distributions of birth-and-death processes
Authors: Cavender James A.
Published: April 2006

Letqn(t) be the conditioned probability of finding a birth-and-death process in statenat timet,given that absorption into state 0 has not occurred by then. A family {q1(t),q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.
Locating bright spots in a point process
Authors: Saunders I. W.
Published: April 2006

As an approach to modelling the ‘matching’ of optical receptors in animals to the objects they are designed to see, we study the problem of locating regions of high intensity in a point process on the real line, using the counts of points in a movable interval of fixed length. We define performance measures analogous to statistical size and power for this procedure and, for points forming a renewal process, give conditions on the quantiles of the convolutions of the interpoint distribution which ensure that the optimal length for the ‘detector’ is close to that of the ‘object’ to be detected. We show that these conditions are satisfied for a Poisson process. Similar conditions ensure that the optimal length is close to zero, and we give a class of distributions satisfying these conditions. Finally we show that the results can be extended to simple two-dimensional models.
On sets of countable non-negative matrices and Markov decision processes
Authors: Kennedy Douglas P.
Published: April 2006

Consider a set S of countable non-negative matrices satisfying the property that for any two indices i, j, for some n ≧ 1 there are matrices M1, M2, · · ·, Mn in S with (M1M2 · · · Mn)ij >0. For non-negative vectors x set Tx = supM∈SMx, where the supremum is taken separately in each coordinate. Assume that for each x with Tx finite in each coordinate there is a matrix in S which achieves the supremum simultaneously for all coordinates. With these two assumptions on S, the R-theory for a countable irreducible matrix is extended to the operator T. The results are used to consider the existence of stationary optimal policies for Markov decision processes with multiplicative rewards.
Adaptation of the service capacity in a queueing system which is subjected to a change in the arrival rate at unknown epoch
Authors: Yadin M. , Zacks S.
Published: April 2006

The paper studies the problem of optimal adaptation of an M/M/1 queueing station, when the arrival rate λ0 of customers shifts at unknown epoch, τ, to a known value, λ1. The service intensity of the system starts at μ0 and can be increased at most N times to μ1 < μ2 < · · · < μN. The cost structure consists of the cost changing μi to μj (i + 1 ≦ j ≦ N); of maintaining service at rate μ (per unit of time) and of holding customers at the station (per unit of time). Adaptation policies are constrained by the fact that μ can be only increased. A Bayes solution is derived, under the prior assumption that τ has an exponential distribution. This solution minimizes the total expected discounted cost for the entire future.
Fluctuating interest rates
Authors: Pollard J. H.
Published: April 2006

The annual premium for a life assurance contract is obtained by equating the expected discounted value of the sum payable on death to the expected discounted value of the series of premiums, and loading the resultant net premium for expenses and contingencies. The approach is essentially deterministic. An adequate stochastic model of mortality is readily available, but it is of limited practical value. A life office writes a large number of policies on independent lives and, apart from the effects of a few very large policies, the overall mortality behaviour of its portfolio is effectively deterministic.
A relaxation view of a genetic problem
Authors: Seneta E.
Published: April 2006

Suppose where S is a compact set (of say). Let Φ (mapping S into S) and Ψ be continuous on S and such that Ψ(xk) is monotone (non-decreasing, say) as k increases, and xk+1 = Φ(xk). Put Ψ∗ = lim (k → ∞)Ψ(xk); if {xk(i)}∞i=0 is a convergent subsequence of {xk}, with a limit-point α, then Ψ∗ = Ψ(α), and

and
On parametric density estimators
Authors: Heathcote C. R.
Published: April 2006

There is an extensive literature on estimating a probability density (or some other appropriate curve) f using statistics of the form

Here X1, X2, · · ·, Xn is a sample from the population, the weight function w is constrained by suitable regularity conditions, and the sequence {bn} of band-widths satisfies bn → 0, nbn → ∞ as n → ∞. Rosenblatt (1971) presents details of this method of estimation and provides an extensive list of references.
Insensitivity of steady-state distributions of generalized semi-Markov processes by speeds
Authors: Schassberger R.
Published: April 2006

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.
The structure of Gaussian fields near a level crossing
Authors: Wilson Richard J. , Adler Robert J.
Published: April 2006

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.
Second-order corrections for Brownian motion approximations to first-passage probabilities
Authors: Yuh Yih-Shyh
Published: April 2006

Correction terms are obtained for the Brownian motion approximation to one- and two-barrier first-passage probabilities. These approximations are given in terms of their Laplace transforms, which are formally (and non-rigorously) inverted. Applications to the one-sample Kolmogorov-Smirnov statistic are discussed.
The throughput of a series of buffers
Authors: Kelly F. P.
Published: April 2006

Messages are to be transmitted through a series ofnnodes linked by communication channels. The lengths of successive messages are independent identically distributed random variables, and the time taken to transmit a message through a channel is determined by its length. Each node has a finite buffer, and when the number of messages at a node reaches the buffer size transmission from the preceding node is interrupted. This paper is concerned with the maximum rate at which messages can pass through the system, called the throughput. We investigate the asymptotic behaviour of throughput as the series length increases, and determine the rate at which buffer sizes should grow to ensure that throughput does not decline to 0.
Some distributional approximations in Markovian queueing networks
Authors: Brown T. C. , Pollett P. K.
Published: April 2006

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.
Bond percolation on honeycomb and triangular lattices
Authors: Wierman John C.
Published: April 2006

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.
Optimal strategies for repeated games
Authors: Finkelstein Mark , Whitley Robert
Published: April 2006

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let Fn∗ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that Fn/Fn∗ is a supermartingale with E(Fn/Fn∗) ≤ 1 and, consequently, E(lim Fn/Fn∗) ≤ 1. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that E(Fn/Fn∗) = 1 for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that Fn∗/Fn is a submartingale with E(Fn∗/Fn) ≤ 1 and E(lim Fn∗/Fn) ≤ 1; and E(Fn∗/Fn) = 1 if and only if at each time period j, 1 ≤ j ≤ n, the strategies leading to Fn and Fn∗ are ‘the same’.
Stochastic geometry from the standpoint of integral geometry
Authors: Ambartzumian R. V.
Published: April 2006

This two-part paper surveys some recent developments in integral and stochastic geometry. Part I surveys applications of integral geometry to the theory of euclidean motion-invariant random fibrefields (a fibrefield is a collection of smooth arcs on the plane), involving marked point processes, Palm distribution theory and vertex pattern analysis. Part II develops the more sophisticated theory of Buffon sets in stochastic geometry and the characterisation of measures of lines, giving applications to problems concerning random triangles and colourings, line processes and fixed convex sets.
A simple model with applications in structural reliability, extinction of species, inventory depletion and urn sampling
Authors: El-Neweihi E. , Proschan F. , Sethuraman J.
Published: April 2006

This paper is devoted to a model which has applications in the study of reliability, extinction of species, inventory depletion, and urn sampling, among others. A series–parallel system consists of (k + 1) subsystems C0, C1, · · ·, Ck, also called cut sets. Cut set Ci contains ni components arranged in parallel, i = 0, 1, · · ·, k. No two cut sets have a component in common. It is assumed that after t components have failed, each of the remaining components is equally likely to fail, t = 0, 1, · · ·. It is also assumed that the components fail one at a time. In Part 1 of this paper we study the probability that the system fails because a specified cut set C0, say, fails first. We obtain several alternative expressions and recurrence relations for this probability. Some of these formulae are useful for computation, while others permit us to derive qualitative features such as monotonicity, Schur-concavity, asymptotic limits, etc. These results are extended to cover the case in which some cut set first drops to level a, where a is a specified positive integer. In Part 2, we compute the probability distribution, frequency function, and failure rate of the lifelengths of series–parallel systems. We also obtain corresponding recurrence relations and finite and asymptotic properties.
The time constant of first-passage percolation on the square lattice
Authors: Cox J. Theodore
Published: April 2006

Letμ(F) be the time constant of first-passage percolation on the square lattice with underlying distribution functionF. Two theorems are presented which show, under some restrictions, thatμvaries continuously inFwith respect to weak convergence. These results are improvements of existing continuity theorems.
Nested renewal processes
Authors: Ansell J. , Bendell A. , Humble S.
Published: April 2006

A class of stochastic processes useful in the investigation of the deterioration and replacement of equipment, as well as in the study of group arrival/batch service queues, storage systems, epidemics and computer software, is introduced. These so-called nested renewal processes consist of ordinary and cumulative renewal processes sequentially nested within a hierarchy. The main properties of these processes are discussed with emphasis on the asymptotic results. An example of their application to tyre wear is considered.
Fluctuation identities for lévy processes and splitting at the maximum
Authors: Greenwood Priscilla , Pitman Jim
Published: April 2006

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.
On Poisson traffic processes in discrete-state Markovian systems by applications to queueing theory
Authors: Melamed Benjamin
Published: April 2006

This paper considers a regular Markov process with continuous parameter, countable state space, and stationary transition probabilities, and defines a class of traffic processes over it. The possibility that multiple traffic processes constitute mutually independent Poisson processes is investigated.

A variety of independence conditions on a traffic process and the underlying Markov process are shown to lead to Poisson-related properties; these conditions include weak pointwise independence, and pointwise independence. Some examples of queueing-theoretic applications are given.

For the class of traffic processes considered here in a queueing-theoretic context, Muntz'sMMproperty, Gelenbe and Muntz's notion of completeness, and Kelly's notion of quasi-reversibility are shown to be essentially equivalent to pointwise independence of traffic and state. The relevance of the theory to queueing network decomposition is pointed out.
An infinite particle system with additive interactions
Authors: Durrett Richard
Published: April 2006

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set as t → ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.
Randomization of intensities in a Markov chain
Authors: Yadin M. , Syski R.
Published: April 2006

The matrix of intensities of a Markov process with discrete state space and continuous time parameter undergoes random changes in time in such a way that it stays constant between random instants. The resulting non-Markovian process is analyzed with the help of supplementary process defined in terms of variations of the intensity matrix. Several examples are presented.
Comparing multi-server queues with finite waiting rooms, I: Same number of servers
Authors: Sonderman David
Published: April 2006

We compare two queueing systems with the same number of servers that differ by having stochastically ordered service times and/or interarrival times as well as different waiting room capacities. We establish comparisons for the sequences of actual-arrival and departure epochs, and demonstrate by counterexample that many stochastic comparisons possible with infinite waiting rooms no longer hold with finite waiting rooms.
Corrected diffusion approximations in certain random walk problems
Authors: Siegmund D.
Published: April 2006

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.
Optimal selection based on relative ranks with a random number of individuals
Authors: Gianini-Pettitt Jacqueline
Published: April 2006

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.
On queues in discrete regenerative environments, with application to the second of two queues in series
Authors: Balagopal K.
Published: April 2006

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.

In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.
Vector linear time series models: corrections and extensions
Authors: Deistler M. , Dunsmuir W. , Hannan E. J.
Published: April 2006

In this paper some theorems of Dunsmuir and Hannan [2] are corrected. The topological structure of spaces of (likelihood) equivalence classes of arma structures is more closely investigated and its relation to the statistical problem is elucidated. The formulation is also made more general.
Some properties of the crossings process generated by a stationary χ2 process
Authors: Sharpe Ken
Published: April 2006

The process generated by the crossings of a fixed level, u, by the process Pn(t) is considered, where

and the Xi(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of Pn(t) above u tends to a Rayleigh distribution as u→ ∞.
Regenerative multivariate point processes
Authors: Berman Mark
Published: April 2006

A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.
Coefficients of ergodicity: structure and applications
Authors: Seneta E.
Published: April 2006

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.
On a tandem queueing model with identical service times at both counters, I
Authors: Boxma O. J.
Published: April 2006

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.

Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.

In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.
On analytical methods for incomplete Markov random fields
Authors: Ravaska Timo
Published: April 2006

It is shown that many properties of Markov processes can be extended to the case where the time parameter is multidimensional (a partially ordered set). This includes Kolmogorov's backward and forward equations, and semi-group treatment of homogeneous processes.
Two types of mutually annihilating particles
Authors: Holley Richard
Published: April 2006

We consider two types of particles performing simple random walks on the integers. When two particles of different types collide they are immediately annihilated. All particles of one type are started off to the left of the origin, and all particles of the other type are started off to the right of the origin. We determine the asymptotic rate of annihilations and the fluctuations, around this rate, of the numbers annihilated at large times.