bluuebunny/crossref_metadata_2025 huggingface dataset:
A theoretical analysis of congestion in a two-lane freeway
Authors: Mcneil Donald R.
Published: April 2006

To current workers in the mathematical theory of road traffic, one of the most challenging and important problems is to describe and analyze the congestion suffered by motorists on freeways. While there is, as yet, no satisfactory unified theory, several different models have been suggested, each of which is reasonable for a limited class of situations. These approaches fall largely into two categories. The first may be described as looking at traffic “in the large” and its principal exponents were Lighthill and Whitham (1955) and Prigogine (1961). The theory of Lighthill and Whitham involves treating the system of vehicles as a fluid subject to a rule of conservation of mass (that is, vehicles) and an assumed relation between the flow and concentration at each point. This theory is completely deterministic. Prigogine's approach is to regard the system of vehicles as behaving like the particles in a gas and subject to the laws of statistical mechanics.
A theorem on discrete time branching processes allowing immigration
Authors: Reynolds John F.
Published: April 2006

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn(x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn–1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn the population size in the nth generation. We write
The minimum of a stationary Markov process superimposed on a U-shaped trend
Authors: Daniels H.E.
Published: April 2006

1.This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locallyU-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)
Note on the output of a queuing system
Authors: Boes Duane C.
Published: April 2006

Consider a queuing system that hascservers anddwaiting positions. Assume that the input is Poisson with rate α and the service times are exponential with meanβ–1. Further assume the following: (i) a customer arriving when all servers are busy and all waiting positions are occupied is “cleared” from the system; (ii) a customer arriving when all servers are busy and not all waiting positions are occupied waits with probability 1 – ζ and “balks” or “clears” with probability ζ; (iii) a customer arriving when not all servers are busy commences service immediately (never balks); and, (iv) a customer who is waiting for service may “defect”, the distribution of time until a waiting customer defects being given by an exponential distribution with mean γ–1. Also, the usual independence assumptions, which make the process that is described by the number in the system at timetMarkov, are assumed. An “output” of this queuing system is defined to occur whenever a service completion occurs, or whenever an arrival “clears” or “balks”, or whenever a waiting customer “defects”. Thus the output is a pooling of service completion epochs, the epochs when arrivals are cleared, the epochs when arrivals balk, and the defection epochs.
Prediction of a noise-distorted, multivariate, non-stationary signal
Authors: Sobel Eugene
Published: April 2006

This paper deals with the problem of predicting or extracting a certain type of non-stationary sequence or signal from another sequence composed of the signal and a stationary noise. The noise is assumed to be uncorrelated with the signal. Although all sequences in the body of this paper are multivariate, for purposes of discussion, this introduction will deal with only the univariate case.
A mathematical study of the founder principle of evolutionary genetics
Authors: Holgate P.
Published: April 2006

Some comparisons are made between various characteristics of the genetic structures of populations of the same size and age, which have (i) evolved from a small founder population, and (ii) evolved from a population which has been of constant size throughout the period considered.
On the offspring number distribution in a genetic population
Authors: Feldman M. W.
Published: April 2006

We consider a dioecious diploid population ofNindividuals,N1males andN2=N–N1females. The alleles will be represented byaandA, and the population reproduces according to the Wright scheme, that is, by random mating with non-overlapping generations.
On the survival of a gene in a subdivided population
Authors: Pollak Edward
Published: April 2006

A classical type of problem in population genetics is that of calculating the probability that a line descended from a particular gene will become extinct. In one problem of this sort, dealt with by Fisher (1922) and Haldane (1927), it is assumed that the population being studied is very large and that initially the number of genes of a particular type, say typeA, is small. These authors obtained the solution by the use of the theory of branching processes.
Random mating in a population of cyclic size
Authors: Chia A. B.
Published: April 2006

The well-known Wright's genetic model is extended to a population whose size varies cyclically. Explicit expressions are obtained for the absorption probability and the rate of approach to homozygosity in the case where there is no selection. When selection operates, diffusion approximations are obtained for the various quantities of interest.
Infinite dams with inputs forming a Markov chain
Authors: Khan M. S. Ali , Gani J.
Published: April 2006

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].
Gap-acceptance in road traffic
Authors: Hawkes A. G.
Published: April 2006

We find the distribution of delay to minor road vehicles waiting to merge or cross a single stream of major road traffic. The decision to cross is taken on the basis of a gap-acceptance function. The model turns out to be a simple queueing problem in which a customer finding an empty queue has a different service time distribution from queueing customers. The key to this representation is given in Section 3. Some numerical results in Section 6 indicate that in most circumstances a simple model will give adequate results.
On the superposition ofm-dimensional point processes
Authors: Çinlar Erhan
Published: April 2006

Considernindependent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions asn →∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.
Testing for uniformity on a compact homogeneous space
Authors: Beran R. J.
Published: April 2006

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.
Integral representations of transition probabilities and serial covariances of certain markov chains
Authors: Daley D. J.
Published: April 2006

This note is concerned with integral representations of some transition probabilities of countable state space Markov chains embedded in birth and death processes, of the serial covariances of functions defined on such chains when stationary, and finally the properties of the spectral measure of stationary processes with monotonically decreasing or completely monotonic serial covariances.
A bound for the maximum of a number of random variables
Authors: Gallot S.
Published: April 2006

In this note, we refine an upper bound, established by P. Whittle, for the probability that the maximum of a number of random variables x1, …, xn is less than a given value C when

and

are given probabilities in terms of which the bound will be expressed.
The expected number and angle of intersections between random curves in a plane
Authors: Morton R.R.A.
Published: April 2006

Consider a plane containing a set of curves. Suppose a separate set of curves is constrained to lie at random on a particular region of this plane. In this note an expression is obtained for the expected number of intersections between the two sets of curves, and it is shown that the angles between the two sets at their points of intersection are distributed as ½ sin θ.
Extreme value theory for a class of discrete distributions with applications to some stochastic processes
Authors: Anderson C. W.
Published: April 2006

Letξnbe the maximum of a set ofnindependent random variables with common distribution functionFwhose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply toξnfor suchFand this paper attempts to find weaker ones which give some description of the behaviour ofξnasn→ ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.
A stochastic model for the formation of a planetary system
Authors: Galley S. J.
Published: April 2006

The stochastic model consists of a two-dimensional random walk with unit steps, each step being a Bernoulli trial. Two sets of possible conditions arising from astronomical theory give rise to two different barriers in the plane of the walk which cause a return to the origin. Another barrier indicates the formation of a planet and also causes a return to the origin. Consideration of the composition of the planets formed at this barrier allows their orbital distances and direction of rotation to be calculated. The predictions of one of the two models were found to be in close agreement with the observed planetary distances and masses in the solar system.
Traffic light queues as a generalization to queueing theory
Authors: Mine Hisashi , Ohno Katsuhisa
Published: April 2006

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.
Moments in Markovian systems with lumped states
Authors: Taylor G. C.
Published: April 2006

In dealing with Markov chains, we are sometimes forced by circumstances to make observations on lumped states. We may wish, however, to deal with a random variable which is a function of the numbers in the various (unlumped) states at various times and whose moments, therefore, will involve the probabilities of transition between the unlumped states.

This paper demonstrates that, although these probabilities are immeasurable directly, we can, provided that we restrict our attention to a certain well-defined class of functions as our random variables, reduce all moments and product-moments of such random variables to functions of lumped state probabilities.
Stochastic approach to chemical kinetics
Authors: McQuarrie Donald A.
Published: April 2006

In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction isHe foundempiricallythat the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., ifS(t) is the concentration of sucrose, thenThe constant of proportionality is called therate constantof the reaction. IfSois the initial concentration of sucrose, thenSince then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called themechanismof the reaction.
Queues with moving average service times
Authors: Pearce C.
Published: April 2006

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) resultP(a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.
A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows
Authors: Lloyd Emlyn H.
Published: April 2006

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).
Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses
Authors: Lawrance A. J.
Published: April 2006

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.
The two-dimensional Poisson process and extremal processes
Authors: Pickands James
Published: April 2006

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.
Point processes arising in vehicular traffic flow
Authors: Brill Edward A.
Published: April 2006

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of aGI/G/squeue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the casesM/M/s(s= 1, 2, ···, ∞) andM/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for theM/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady stateM/G/∞queue are independent at any fixed time; this is a result I have been unable to find in the literature.
On the single server queue with preemptive service interruptions
Authors: Şahin İzzet
Published: April 2006

In [4], the limiting behaviour of a stochastic system with two types of input was investigated by reducing the problem to the solution of an integral equation. In this note we use the same approach to study the equilibrium waiting time problem for the general single server queue with preemptive service interruptions. (For a comprehensive account of the existing literature on queues with service interruptions we refer to [2] and [3].)
Some new results in storage theory
Authors: Prabhu N. U.
Published: April 2006

In the storage model proposed by Kendall [6], the input is assumed to be a stochastic process with stationary independent increments, and the release is continuous and is at a unit rate. The storage process arising from this model is known to reach a steady state if the expected net input per unit time is negative. In this paper we consider the situation where this expected net input is nonnegative and derive limiting distributions for the wet period, total dry period during a given time interval and the dam content.
Some Duality Results in the Theory of Queues
Authors: Greenberg Irwin
Published: April 2006

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to theM/G/1 –G/M/l queues.
A Correlated Queue
Authors: Conolly B.W. , Hadidi N.
Published: April 2006

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.
Measurement of a wandering signal amid noise
Authors: Hannan E. J.
Published: April 2006

A formula is given for the response function of a filter which extracts a signal generated by a non-stationary process from amid noise. The non-stationarity is due to the presence of zeros on the unit circle of the function characterising the difference equation generating the signal. The signal is broken into a trend component and a stochastic integral in which t occurs only through the factor exp it λ and it is shown that the filter which optimally extracts the latter perfectly represents the former. The considerations cover the case of a vector series. Applications to problems in seasonal variation measurement are indicated.
Waitingtimes between record observations
Authors: Neuts Marcel F.
Published: April 2006

If Δrdenotes the waitingtime between the (r− 1)st and therth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δrsatisfies the weak law of large numbers and a central limit theorem.

This theorem supplements those of Foster and Stuart and Rényi, who investigated the indexVrof therth upper record.

Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVris very inadequate for approximation purposes—Barton and Mallows.
Hierarchical population models with poisson recruitment
Authors: Pollard J. H.
Published: April 2006

The analysis of an organization having various grades of employee has been undertaken in many different contexts. In 1961, for example, Young and Almond considered an institution (of undisclosed type) having six grades of staff. Gani, in 1963, considered ‘Australian Universities’ as an organization and students were graded according to their stage of study. The author used techniques similar to those of this paper in 1964 to analyze deterministically the age structure of the Australian Academy of Science. Indeed, the analyses have usually been deterministic.
Limiting behaviour of the distributions of the maxima of partial sums of certain random walks
Authors: Emery D. J.
Published: April 2006

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.
Correlation functions in queueing theory
Authors: Srinivasan S. K. , Subramanian R. , Vasudevan R.
Published: April 2006

The object of this paper is to study the actual waiting time of a customer in aGI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer thanwin a given time interval (0,T) are calculated.
Embedded renewal processes in theGI/G/squeue
Authors: Whitt Ward
Published: April 2006

The stableGI/G/squeue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for theGI/G/1 queue, but a simpleGI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in theGI/G/1 andGI/G/squeues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class ofGI/G/squeues in light traffic are implied.
Factorial moments and probabilities for the general stochastic epidemic
Authors: Billard L.
Published: April 2006

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.
Diffusion limits for congestion models
Authors: Mcneil Donald R.
Published: April 2006

Heuristic methods yield simple limiting results for the behaviour of input-output systems under conditions of heavy traffic. It is assumed that the input is time-dependent, while the output rate depends entirely on the content of the system. It is shown that the content, when suitably normalized and scaled, behaves like a (non-stationary) Ornstein-Uhlenbeck process.
Observations on a result in sequential analysis by P. Whittle
Authors: Gittins J. C.
Published: April 2006

In a series of papers Whittle has proposed locating the optimal boundary between the continuation region and the stopping region, in the no-overshoot approximation, for a class of sequential analysis problems by using the criterion that the gradient of the objective function must be continuous at the boundary. It is shown here that the result originally proposed by Whittle is not true in the generality which he supposed, and an indication is given of some of the circumstances in which it does hold.
Some comments on spectral representations of non-stationary stochastic processes
Authors: Tong H.
Published: April 2006

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.
Limiting second moments for transient states of Markov chains
Authors: Wynn H. P.
Published: April 2006

The set of transient states of a Markov chain is considered as a system. If numbers of arrivals to the system at discrete time points have constant mean and covariance matrix then there is a limiting distribution of numbers in the states. Necessary and sufficient conditions are given for this distribution to yield zero correlations between states.
Asymptotic distributions for the Ornstein-Uhlenbeck process
Authors: Beekman John A.
Published: April 2006

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.
Multivariate exponential distributions based on hierarchical successive damage
Authors: Arnold Barry C.
Published: April 2006

A class of multivariate exponential distributions is suggested which includes those presented by Marshall and Olkin, Downton, and Hawkes. It is based on a concept of hierarchical successive damage. Recursive expressions for the Laplace transforms and for first and second moments are derived in the bivariate case. Related multivariate geometric distributions are described.
A supplement to the strong law of large numbers
Authors: Heyde C. C.
Published: April 2006

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n−1 Σi=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.
Further results on Kendall's autoregressive series
Authors: Hardin Jay C. , Brown Thomas J.
Published: April 2006

Theoretical expressions for the cross correlation and cross spectra of the input and output variables in the difference equation Xt=aXt−1 + bXt−2+ Yt are derived. These expressions are compared with estimates of these expectations obtained by employing a Fast Fourier Transform technique on digitally generated series.
Generalized state-dependent Erlangian queues: Speculations about calculating measures of effectiveness
Authors: Conolly B. W.
Published: April 2006

The provision of easily calculable measures of effectiveness for generalised Erlang queues (state-dependent parameters λn and μn of arrival and of service) motivates speculation about the applicability of renewal theory. The application envisaged is justified by known results for certain models and its extension to an operationally more promising system is proposed. Use of the formula ‘L = λW' with ‘effective’ λ calculated by foregoing methods is likewise shown to be justified by known results for certain models and hence its wider applicability is conjectured. Mechanisms are discussed which may provide improved models, and investigation is made of choices of λn and μn which may lead to time dependent solutions having a prescribed form. The example of panic buying is considered as an example.
Limit theorems for a branching process with disasters
Authors: Athreya K. B. , Kaplan N.
Published: April 2006

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.
Asymptotic probabilities in a critical age-dependent branching process
Authors: Weiner Howard J.
Published: April 2006

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.

The method is also applied to the total progeny in the critical process.
Asymptotic inference for an Ising lattice
Authors: Pickard D. K.
Published: April 2006

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.
Limit theorems for processes of randomly displaced regular events
Authors: Collings P. S.
Published: April 2006

Two types of limit theorems are proved for processes of randomly displaced regular events. Firstly, as the displacements tend to infinity, the counting process is shown to converge weakly to a Poisson process and secondly, as the interval between events tends to zero, convergence of the finite-dimensional distributions of the associated storage process to a diffusion is proved.
Erlang's formula and some results on the departure process for a loss system
Authors: Shanbhag D. N. , Tambouratzis D. G.
Published: April 2006

The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean λ–1. It is also seen that in the limit interdeparture times are mutually independent.
The stationary distribution of the infinitely-many neutral alleles diffusion model
Authors: Watterson G. A.
Published: April 2006

An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct.
A spatial stochastic model of inter-plant competition
Authors: Diggle Peter J.
Published: April 2006

A stochastic model for inter-plant competition in a single-species population is presented, in which the probability that a given plant will be eliminated is defined in terms of the sizes and relative spatial locations of rival plants. The properties of the model are then investigated and illustrations of particular cases given, with particular emphasis on the distribution of the sizes of surviving plants. Empirical comparisons are made between the properties of the model and some experimental results due to Ford (1975b). Some possible extensions to the basic model are suggested.
Motion of a particle in a velocity-dependent random force
Authors: Karmeshu
Published: April 2006

The motion of a particle is investigated in the presence of a velocity-dependent random force assumed to be proportional to velocity. Two different possibilities are considered, namely, the presence and absence of random driving force. In the absence of random driving force, the velocity and displacement auto-correlation function are calculated. The probability distribution in velocity space is also evaluated. It is found that in the absence of intrinsic damping, the energy of the particle increases without limit. The condition for the energetic stability of the particle in the presence of random driving force is obtained. The Fokker-Planck equation for the probability distribution in velocity space is derived from the stochastic Liouville equation for delta-correlated velocity-dependent random force.
Genetic drift with polygamy and arbitrary offspring distribution
Authors: Cannings C.
Published: April 2006

The rate of genetic drift at an autosomal locus for a bisexual, diploid population of fixed size is studied. The generations are non-overlapping. The model encompasses a variety of mating systems, including random monogamy, random polygamy in one sex and random mating. The rate of drift is shown for several models to depend on the expected number of parents that two randomly selected individuals have in common. The male and female offspring are assigned to families in a fairly general way, which permits the study of a model in which each family has offspring of one sex only. The equation arising in this last case is identical to one of Jacquard for a system in which sib-mating is excluded.
On the subcritical Bellman-Harris process with immigration
Authors: Pakes A. G. , Kaplan Norman
Published: April 2006

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrivalGI/G/∞queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.
Some extensions of Takács's limit theorems
Authors: Shanbhag D. N.
Published: April 2006

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing systemGI/G/swith a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider aGI/G/ssystem with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.
Busy period of queues with state dependent arrival and service rates
Authors: Hadidi Nasser
Published: April 2006

A technique is derived providing for the analysis of busy period processes in the queueing models M/Mn/1 and Mn/M/1; queueing models with state dependent service and arrival rates respectively. A single treatment covers both models and no separate analysis is required.
Optimal control of finite dams: discrete (2-stage) output procedure
Authors: Faddy M. J.
Published: April 2006

The problem discussed is that of determining optimal release rates for finite dams when the release rate may take one of two values and specific costs are involved in increasing and decreasing this release rate. From another point of view, this is a problem of controlling the output simply by switching on and off. A diffusion model is suggested and by considering a family of plausible output policies an optima procedure, which minimises the long-term average cost of operating the dam, is established. Various properties of this output policy are given.
Galton-Watson processes in varying environments
Authors: Jagers Peter
Published: April 2006

Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.
More bounds on the expectation of a convex function of a random variable
Authors: A. Ben-Tal , Hochman E.
Published: April 2006

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).

The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.
On a characterization of Poisson distributions
Authors: Aczél J.
Published: April 2006

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.
On branching processes allowing immigration
Authors: Yang Y. S.
Published: April 2006

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.
The rate of convergence in limit theorems for service systems with finite queue capacity
Authors: Tomko Joseph
Published: April 2006

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First anM/M/msystem is considered and the rate of approximation is given. Then the case of theM/G/1 system is studied for traffic intensityρ> 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums.
Exact fluctuation results for Markov-dependent coin-tossing
Authors: Darroch J. N. , Whitford Heather J.
Published: April 2006

Feller (1968) showed that the probability functions of each of three random variables associated with coin-tossing (independent, simple, symmetric random walk) are (a) the same (except at the end points), (b) approximately arc-sine. In this paper these two properties are shown to hold for Markov-dependent coin-tossing (Markov-dependent, simple, symmetric random walk).
A modified erlang approach to approximating GI/G/1 queues
Authors: Marchal William G. , Harris Carl M.
Published: April 2006

This paper presents a modification to the traditional technique of approximating general queueing problems by the Erlang distributions. It presents both theory and the results of some numerical testing.
Heavy traffic theory for queues with several servers. I
Authors: Köllerström Julian
Published: April 2006

Queues with several servers are examined here, in which arrivals are assumed to form a renewal sequence and successive service times to be mutually independent and independent of the arrival times. The first-come-first-served queue discipline only is considered. An asymptotic formula for the equilibrium waiting time distribution is obtained under conditions of heavy traffic.
On infinite dams with inputs forming a stationary process
Authors: Tin Pyke , Phatarfod R. M.
Published: April 2006

This paper considers a dam of infinite capacity with a discrete-valued stationary input process and a unit release whenever possible. It is shown how, by suitable manipulations of the equation governing the dam content process, the stationary distribution of the dam being empty can be obtained, as also can (with a few additional assumptions) the expected value of the dam content in the stationary case. Results obtained are applied to particular cases of input — independent and identical, Markov, Bivariate Markov and moving-average.
A bivariate distribution in regeneration
Authors: Chung Kai Lai
Published: April 2006

The joint distribution of the time since last exit, and the time until next entrance, into a unique boundary point is given in Formula (1) below. The boundary point may be replaced by a regenerative phenomenon.
A diffusion process model for the optimal operation of a reservoir system
Authors: Pliska Stanley R.
Published: April 2006

The water level in a reservoir is modelled as a controlled diffusion process on a compact interval of the real line. The problem is to control the water discharge rate so as to minimise the expected costs, which depend upon the histories of the water levels and release rates. The form of the optimal control is studied for two general classes of reservoir control problems.
Modes of growth of counting processes with increasing arrival rates
Authors: Waugh W. A. O'N.
Published: April 2006

Jumping processes which grow by unit jumps, with decreasing sojourn times between successive jumps, are studied. Markovian birth processes, non-Markovian branching processes, and some generalizations of these are special cases. Three classes are described, in one of which growth is explosive, in the second asymptotically continuous, and in the third oscillatory. A theorem is proved which gives an explicit functional expression in the asymptotically continuous case, and borderline cases between the classes are investigated.
Bounds on the delay distribution inGI/G/1 queues
Authors: Ross Sheldon M.
Published: April 2006

Bounds are obtained for the limiting distribution of the delay in queue for aGI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the service distribution is either IFR, DFR, NBU or NWU.
Queues in transportation systems, I: a Markovian system
Authors: Crane Michael A.
Published: April 2006

We study a transportation system consisting ofSvehicles of unit capacity andNpassenger terminals. Customers arrive stochastically at terminali, 1 ≦i≦N, seeking transportation to a terminalj, 1 ≦j≦N, with probabilityPij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a relatedGI/G/Squeue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.
The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue,M/M/1
Authors: Jenkins John H.
Published: April 2006

A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.
On dams with additive inputs and a general release rule
Authors: Çinlar E. , Pinsky M.
Published: April 2006

An infinite capacity dam subject to an additive input process and a general release rule is considered. The input process can have infinitely many jumps in any finite interval, and the rate of release is r(x) when the dam content is x. The content is constructed as the increasing limit of a sequence of Markov processes and the convergence is shown to be almost surely uniform over finite intervals. The limit process is shown to be a standard Markov process and its characteristic equation is computed.