bluuebunny/crossref_metadata_2025 huggingface dataset:
Exponential ergodicity in derived Markov chains
Authors: Teugels Jozef L.
Published: April 2006

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.
On a property of the variance of the waiting time of a queue
Authors: Tambouratzis D. G.
Published: April 2006

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.
On random sequential packing in the plane and a conjecture of palasti
Authors: Blaisdell B. Edwin , Solomon Herbert
Published: April 2006

The random packing of geometric objects in one-, two- or three-dimensions may afford useful insights into the structure of crystals, liquids, absorbates on crystals, and in higher dimensions, into problems of pattern recognition. Random packing has accordingly received increasing attention in recent years. Two principal packing procedures have been formulated and each gives rise to different packing ratios. In one case, all possible configurations of a sphere-packed volume are assumed to be equally likely. In the other and most widely reported case, there is random sequential addition of spheres to the volume until it is packed. This is the situation we study in this paper. Most of the work to date has been limited to the theoretical study of the one-dimensional lattice or to continuous cases particularly in the limit for long lines. The higher dimensional cases have resisted theoretical attack but have been studied by computer simulation by Palasti [12] and Solomon [14] and by physical simulation by Bernal and Scott (see [14]).
Finite dams with inputs forming a Markov chain
Authors: Ali Khan M.S.
Published: April 2006

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.
Work-conserving priorities
Authors: Wolff Ronald W.
Published: April 2006

In many situations, it is reasonable to assume that a priority rule does not affect the total time spent in service of any job. Rules with this property are said to bework-conserving.This concept unifies and simplifies the analysis of a variety of priority queues. Some results are obtained for rules applied to theGI/G/1 queue. Some special properties of Poisson arrivals are discussed, and a new proof of the equivalence of averaging over all time with averaging over arrival epochs is presented. In this case, explicit results for particular rules are obtained in examples. In another example, the optimal rule (from a very restrictive class) is determined without specializing the arrival stream.
A note on the queueing systems GI/D/1 and D/G/1
Authors: Pakes A. G.
Published: April 2006

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has

For the queueing systems GI/D/1 and D/G/1 we shall show that it is possible to make use of the special form of the service time and inter-arrival time distributions, respectively, to evaluate the right hand side of (1). A similar evaluation applies to the limiting distribution

when it exists. The results obtained could also be obtained from those of Finch (1969) and Henderson and Finch (1970) by using suitable limiting arguments.
A characterization of stable processes
Authors: Lukacs Eugene
Published: April 2006

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t1, t2 ɛ T, t1 〈 t2; the random variable X(t2) – X(t1) is called the increment of the process X(t) over the interval [t1, t2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].
Recent results in information theory
Authors: Kotz Samuel
Published: April 2006

Information theory, in the strict sense, is a rapidly developing branch of probability theory originating from a paper by Claude E. Shannon in theBell System Technical Journalin 1948,in which anew mathematical model ofcommunications systems was proposed and investigated.

One of the central innovations of this model was in regarding the prime components of a communications system (the source of messages and the communication channel) as probabilistic entities. Shannon also proposed a quantitative measure of the amount of information based on his notion of entropy and proved the basic theorem of this theory concerning the possi bility of reliable transmission of information over a particular class of noisy channels.
Applications of the age distribution in age dependent branching processes
Authors: Weiner Howard J.
Published: April 2006

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.
The single server queue with Poisson input and semi-Markov service times
Authors: Neuts Marcel F.
Published: April 2006

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.

The results for theM/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.
Equilibrium probability distributions for low density highway traffic
Authors: Newell G. F.
Published: April 2006

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.
Genetic correlation in the stepping stone model with non-symmetrical migration rates
Authors: Maruyama Takeo
Published: April 2006

Kimura (1953) proposed a “stepping stone” model for the study of the genetic structure of a subdivided population. In this model, it was assumed that a population consists of infinitely many colonies located at grid points of ann-dimensional lattice and that each colony exchanges individuals with neighboring colonies in each generation, and also receives immigrants as random samples from the whole population. The former type of migration has been called “short range migration” and the latter type “long range migration”. Later, Kimura and Weiss (1964), and Weiss and Kimura (1965) have obtained formulas for the genetic correlation and variance between colonies for general cases of the model, assuming that the short range migration is symmetrical in each fixed direction. The purpose of the present report is to extend the results of Kimura and Weiss to cover situations where the restriction of symmetry of short range migration is removed, so that migration rates in opposite directions need not be equal. I believe that in nature there are cases which require the model presented in this report. For example, consider a plant population distributed along a river, or on a plain where the wind at the time when the seeds are scattered is stronger in one direction than in others. For animals and plants it is often true that the centre of habitat is more densely populated than the marginal regions where the environment is less suitable for the species. In such a case the migration rate toward the outside from the centre is larger than that in the opposite direction. As in other theories of population genetics, we will assume that the size of each colony is determined by the carrying capacity of the environment and is not affected by the migration rate. Thus we assume that in each generation each colony produces many more gametes than those which contribute to the next generation, and, among those many gametes, a certain number, say 2N,are chosen from various colonies to form the individuals of a particular colony.
The departure process from the GI/G/1 Queue
Authors: Vlach Thomas L. , Disney Ralph L.
Published: April 2006

The departure process from the GI/G/1 queue is shown to be a semi-Markov process imbedded at departure points with a two-dimensional state space. Transition probabilities for this process are defined and derived from the distributions of the arrival and service processes. The one step transition probabilities and a stationary distribution are obtained for the imbedded two-dimensional Markov chain.
A dialysis system with one absorbing and one semi-reflecting state
Authors: Kastenbaum Marvin A.
Published: April 2006

The random walk between an absorbing and a reflecting or semi-reflecting barrier has been discussed in the literature. Indeed, a biological example of such a random walk, arising from a system of countercurrent dialysis, was described by this author. In this earlier paper a technique was suggested for obtaining the probability of being in a specific state after a given number of transitions, if the probability is unity of being in the reflecting state at time zero. The present paper extends this technique to yield the full matrix of higher transition probabilities.
On age dependent branching processes
Authors: Weiner Howard
Published: April 2006

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].
Stationary joint distributions arising in the analysis of theM/G/1 queue by the method of the imbedded markov chain
Authors: Jenkins J. H.
Published: April 2006

Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.
Maximal branching processes and ‘long-range percolation’
Authors: Lamperti John
Published: April 2006

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form

where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.
Two results in the theory of queues
Authors: Ali H.
Published: April 2006

In this paper it is shown that the distribution of the instant of service of a customer is symmetric as between the distributions of service and interarrival time. AlsoU(t),the expected number of departures in (0,t), is a delayed renewal function for theGI/M/1 queue.
Diffusion out of a triangle
Authors: Smith W. , Watson G. S.
Published: April 2006

A problem in molecular genetics called for the following result. If a particle is dropped at random on a right-angled isosceles triangle, and thus allowed to execute symmetrical Brownian motion, the chance that it will first leave the triangle via the hypotenuse is 0.41062. An inequality and two methods of deriving this probability are illustrated. The most useful source of solutions to diffusion problems of this kind is the torsion problem of elasticity.
On a three-state sojourn time problem
Authors: Gibson A. E. , Conolly B. W.
Published: April 2006

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define

which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t0 at which the visit to state T begins.
The construction of limiting distributions of response probabilities
Authors: Marliss G. S. , McGregor J. R.
Published: April 2006

As stated elsewhere [9], the two experimenter-controlled events learning model of Bush and Mosteller [1] may be described by a Markov process X0,X1, ···satisfying the following conditions.

(i)

Xo has an arbitrary distribution on (0, 1).

(ii)

If Xn is given, then with probability π1 and with probability π2

(ii)
On the busy periods for the M/G/1 queue with finite and with infinite waiting room
Authors: Cohen J. W.
Published: April 2006

The Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.
Some asymptotic properties of a two-dimensional periodogram
Authors: Pagano M.
Published: April 2006

The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory forp-dimensional periodograms (p> 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for thep-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.
Passage-time generating functions for continuous-time finite Markov chains
Authors: Darroch J. N. , Morris K. W.
Published: April 2006

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).
Markov population processes
Authors: Kingman J.F.C.
Published: April 2006

The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is represented by the vectorn= (n1,n2, …,nk),where ntis the number of individuals in theith colony, and the random evolution ofnis supposed to be that of a continuous-time Markov chain. The jumps of the chain may be of three types, corresponding to the arrival of a new individual, the departure of an existing one, or the transfer of an individual from one colony to another.
Highway Delays Resulting From Flow-Stopping Incidents
Authors: Donald P. Gayer
Published: April 2006

Modern highways, particularly the freeways of large cities, carry a considerable volume of traffic during certain times of day. Thus if any interruption or retardation of flow occurs, a large reaction in the shape of a monumental and time-consuming traffic jam soon appears. For example, when an accident or mechanical breakdown gives rise to a severe flow restriction or stoppage, many other vehicles may be quickly halted, and remain stopped until the impediment is cleared away. In addition, the flow of traffic may be slowed considerably even after the original stoppage is removed owing to the existence of a queue. Consequently, vehicles that arrive long after the original restriction is removed experience prolonged, and sometimes seemingly inexplicable, delay. Our purpose is to develop a probability model for the situation described, with the aim of estimating the consequence of a temporary flow restriction. Various measures of (in)effectiveness are worth considering. We consider primarily the total vehicle-hours waited while the jam dissipates; the latter may roughly measure total social cost. The total number of vehicles involved in the jam is also of interest, as are other figures of merit.
Divergent population processes and mammal outbreaks
Authors: Holgate P.
Published: April 2006

Birth and death processes, i.e. Markov chains {X(t)}, 0 ≦t≦ ∞, taking values in the setIof non-negative integers, for which the infinitesimal transition probabilities arehave found wide application in mathematical ecology . In order that the infinite system of differential Equations (1) should have solutionssuch thatPij(t) ≦ 0, alltandit is necessary and sufficient that. The conditions ensure that with probability one, the sample paths ofX(t) have no discontinuities worse than jumps, and in this case the solution is unique. If (3) does not hold then for somet1at least, there is a set of sample paths of positive probability, each of which hast1as a limit point of jumps. For pure birth processes, (in which μj= 0) this clearly impliesX(t)→ ∞ ast→t1.
A limit theorem for random walks with drift
Authors: Heyde C. C.
Published: April 2006

LetXi,i= 1, 2, 3, … be a sequence of independent and identically distributed random variables. Writeand forx≧ 0 defineM(x) + 1 is then the first passage time out of the interval (– ∞,x] for the random walk processSn.
On the work load process in a general preemptive resume priority queue
Authors: Schassberger R.
Published: April 2006

Consider the following queuing system: A sequence of customers arrive at a service unit in a recurrent stream. A customer is of priority k with probability πk, k = 1, …, n. A class i customer preempts service of class k, k > i. Interrupted service is resumed without loss or gain in service time. Service is FIFO within classes. Service times for class k are drawn from a general distribution function Bk(t).

Using the method of phases and a resolution technique from the theory of Markov processes we obtain Laplace transforms of various distributions.
On the numerical solution of Brownian motion processes
Authors: Anderssen R. S. , Hoog F. R. De , Weiss R.
Published: April 2006

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.
Maximum likelihood estimation for stochastic rth-order reactions. II
Authors: Mullooly John P.
Published: April 2006

In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.
The divided pooled output of several strictly-periodic point processes
Authors: Proudfoot A. D. , Lampard D. G.
Published: April 2006

The coefficients of variation and serial correlation are given for the inter-event intervals of the process obtained by selecting everykth event of the pooled output of several periodic point processes. Relevance to physiological research is explained.
On characterization of a generalized inaccuracy measure in information theory
Authors: Sharma Bhu Dev , Autar Ram
Published: April 2006

Sharma and Autar [6] have formed a functional equation in two variables containing two parameters which, as a particular case, includes Daróczy's [1] generalized functional equation used in information theory. The solutions of our functional equation, under suitable boundary conditions, lead to an inaccuracy function of a new type. A characterization of this inaccuracy function and the convexity of Daróczy's entropy of type-ßhave been given in this paper.
Multi-channel queues in heavy traffic
Authors: Loulou Richard
Published: April 2006

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.
A central limit theorem for exchangeable variates with geometric applications
Authors: Moran P. A. P.
Published: April 2006

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.
On quasi-stationary distributions for multi-type Galton-Watson processes
Authors: Buiculescu Mioara
Published: April 2006

In this paper we consider the long term behaviour till absorption of multitype Galton-Watson processes using the theory of quasi-stationary distributions.
On the distribution of the inter-record times in an increasing population
Authors: Yang Mark C. K.
Published: April 2006

It is shown in this note that if the population increases geometrically, then the asymptotic distribution for the inter-record times is also geometric. The records in Olympic games are used as an example. Also, it is noted that the rapid breaking of Olympic records is not due mainly to the increase in population.
Properties of the bivariate delayed Poisson process
Authors: Lawrance A. J. , Lewis P. A. W.
Published: April 2006

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called thebivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.
Limits of compound and thinned point processes
Authors: Kallenberg Olav
Published: April 2006

Letη=Σjδτjbe a point process on some spaceSand letβ,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent ofη.We can then form the compound point processξ= Σjβjδτjwhich is a random measure onS.The purpose of this paper is to study the limiting behaviour ofξas. In the particular case whenβtakes the values 1 and 0 with probabilitiespand 1 –prespectively,ξbecomes ap-thinning ofηand our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.
Autoregression for discrete processes mod 2
Authors: Kanter Marek
Published: April 2006

Given a ‘black box’ of logical elements, a sequence of independent random pulses is fed in and the output from the black box is observed. A procedure is given for estimating the parameters of the black box, in the case when the black box is constructed in a manner analogous to the classical autoregressive scheme.
An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes
Authors: Heyde C. C.
Published: April 2006

The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autoregression of small order, detailed rate results for each autocorrelation and for the estimators of all parameters can be obtained. A rate result for the convergence of the sample mean is given in the general case.
A note on random walks. II
Authors: Speed T. P.
Published: April 2006

The various duality results existing for two-barrier random walks are unified. Particular cases in the literature involving reflecting and reversing are deduced as corollaries.
A heavy traffic result for the finite dam
Authors: Blomqvist Nils
Published: April 2006

The steady state contentZof a finite dam in discrete time is investigated for small absolute values of the expected net input and correspondingly large values of the dam capacity. It is shown that under general conditionsZhas, asymptotically, a truncated exponential distribution, a result that supplements previous knowledge in queueing theory.
Projected thick sections through multi-dimensional particle aggregates
Authors: Davy Pamela
Published: April 2006

Formulae relating properties of a two-phase structure to properties of a section through that structure are well known in stereology. An exact description of the error introduced into these formulae when the projection of a thick slice is observed rather than a true section (i.e. a slice of zero thickness) is presented for a model of opaque particles distributed according to a homogeneous Poisson point process in En.
On a result of Rubin and Vere-Jones concerning subcritical branching processes
Authors: Hoppe Fred M.
Published: April 2006

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.
Computing optimal quality control policies — two actions
Authors: Wang Robert C.
Published: April 2006

We study a Markov quality control model in which we may either reset the machine or keep the machine producing items. Both discounted and average costs are considered. We shall show that a monotone policy is optimal and show how to compute the optimal critical point. Two special models are studied along with some numerical results.
Mean-square and almost-sure convergence of supercritical age-dependent branching processes
Authors: Ganuza Edgar Z. , Durham S. D.
Published: April 2006

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.
On supercritical Galton-Watson processes allowing immigration
Authors: Pakes A. G.
Published: April 2006

We establish the absolute continuity of the limit random variables of two supercritical Galton-Watson branching processes, one allowing unrestricted immigration and the other having a state dependent immigration component.
Queues in transportation systems, II: An independently-dispatched system
Authors: Crane Michael A.
Published: April 2006

We consider a transportation system consisting of a linear network ofN+ 1 terminals served bySvehicles of fixed capacity. Customers arrive stochastically at terminali, 1 ≦i≦N, seeking transportation to some terminalj, 0 ≦j≦i− 1, and are served as empty units of vehicle capacity become available ati. The vehicle fleet is partitioned intoNservice groups, with vehicles in theith group stopping at terminalsi,i− 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.
The quasi-stationary distributions of queues in heavy traffic
Authors: Kyprianou E. K.
Published: April 2006

This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn} of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.
The waiting time until first duplication
Authors: Arnold Barry C.
Published: April 2006

Balls are drawn with replacement from an urn containing m distinguishable balls until a match is noted. The distribution of the number of drawings required is considered in the case where only the last k balls drawn are remembered. The asymptotic behavior of this distribution, as m and k become large, is investigated. Two further variants of the problem are suggested.
Pólya sequences, binomial convolution and the union of random sets
Authors: Walkup David W.
Published: April 2006

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.
Some properties of line segment processes
Authors: Parker Philip , Cowan Richard
Published: April 2006

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for BorelA⊂R2, for the first moments ofM(A), the number of segment mid-points inA;N(A), the number of segments which intersect with convexA;S(A), the total length withinAof segments crossingA; andC(A) the number of segment-segment crossings withinA. In the case of Poisson mid-points, the distribution of therth nearest line-segment to a given point is found.
Failure rate — a unified approach
Authors: Lee Larry , Thompson W. A.
Published: April 2006

Aging, of biological and mechanical systems, is well described as ‘deterioration of the power to withstand destruction.’ The failure-rate concept is the mathematical way of describing aging. Failure rates have been defined for discrete and continuous time and used extensively, particularly in actuarial science and reliability. This paper assumes an arbitrary scale on which an object encounters stress, and develops a theory of failure rate with respect to ‘stress-time’ in analogy with the discrete- and continuous-time cases.
Two extreme value processes arising in hydrology
Authors: Karr Alan F.
Published: April 2006

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.
On the distribution of inanimate marks over a linear birth-and-death process
Authors: Morgan Byron J. T.
Published: April 2006

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived.
A population birth-and-mutation process, I: explicit distributions for the number of mutants in an old culture of bacteria
Authors: Mandelbrot Benoit
Published: April 2006

Luria and Delbrück (1943) have observed that, in old cultures of bacteria that have mutated at random, the distribution of the number of mutants is extremely long-tailed. In this note, this distribution will be derived (for the first time) exactly and explicitly. The rates of mutation will be allowed to be either positive or infinitesimal, and the rate of growth for mutants will be allowed to be either equal, greater or smaller than for non-mutants. Under the realistic limit condition of a very low mutation rate, the number of mutants is shown to be a stable-Lévy (sometimes called “Pareto Lévy”) random variable, of maximum skewnessß, whose exponentαis essentially the ratio of the growth rates of non-mutants and of mutants. Thus, the probability of the number of mutants exceeding the very large valuemis proportional tom–α–1(a behavior sometimes referred to as “asymptotically Paretian” or “hyperbolic”). The unequal growth rate casesα≠ 1 are solved for the first time. In theα= 1 case, a result of Lea and Coulson is rederived, interpreted, and generalized. Various paradoxes involving divergent moments that were encountered in earlier approaches are either absent or fully explainable.

The mathematical techniques used being standard, they will not be described in detail, so this note will be primarily a collection of results. However, the justification for deriving them lies in their use in biology, and the mathematically unexperienced biologists may be unfamiliar with the tools used. They may wish for more details of calculations, more explanations and Figures. To satisfy their needs, a report available from the author upon request has been prepared. It will be referred to as Part II.
Convergence of general branching processes and functionals thereof
Authors: Jagers Peter
Published: April 2006

With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation.
A cluster process representation of a self-exciting process
Authors: Hawkes Alan G. , Oakes David
Published: April 2006

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.
Analysing binary lattice data with the nearest-neighbor property
Authors: Strauss David J.
Published: April 2006

The probability density of binary variables on a lattice with the nearest-neighbor condition is given by the Gibbs Random Field formula. This paper examines some consequences of the result. Approximate formulae for the maximum likelihood estimator of the persistence parameter are derived and discussed, with an example. A comparison is made between two test statistics for independence.
Ergodicity and identifiability for random translations of stationary point processes
Authors: Mori Toshio
Published: April 2006

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.
TheG/M/mqueue with finite waiting room
Authors: Hokstad Per
Published: April 2006

TheG/M/mqueue with onlyswaiting places is studied. We start by studying the joint distribution of the number of customers present at timetand the time elapsing until the next arrival aftert.This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For theG/M/1 queue also the transient solution is given. Finally the cases= ∞ is considered.
Inaccuracy and a coding theorem
Authors: Autar Ram , Soni Raminder Singh
Published: April 2006

Kerridge introduced a measure known as inaccuracy for complete probability distributions which is the generalisation of Shannon's entropy. In this paper we study a grouping property of the inaccuracy. Also we have established a coding theorem for personal codes by considering inaccuracy of orderaand generalised mean length of ordertunder the condition.
Strong mixing properties of linear stochastic processes
Authors: Chanda K. C.
Published: April 2006

Let {Zt; t = 0, ± 1, ···} be a pure white noise process with γ = E{|Z1|δ}< ∞ for some δ > 0. Assume that the characteristic function (ch.f.) ϕ0 of Z1 is Lebesgue-integrable over (—∞, ∞). Let {gv;v = 0, 1, 2, ···, g0 = 1} be a sequence of real numbers such that where λ = δ(1 + δ)−1. Define , where the identity is to be understood in the sense of convergence in distribution. Then {Xt; t = 0, ± 1, ···} is a strongly mixing stationary process in the sense that if is the σ-fìeld generated by the random variables (r.v.) Xa, ···, Xb then for any where M is a finite positive constant which depends only on ϕ0 and
A model for queues in series
Authors: Sharma O. P.
Published: April 2006

This paper studies the stationary behaviour of a finite space queueing model consisting of r queues in series with multi-server service facilities at each queue. Poisson input and exponential service times have been assumed. The model is suitable for phase-type service as well as service with waiting allowed before the different phases. In the case of single-server queues explicit expressions for certain probability distributions, parameters and a steady-state solution for infinite queueing space have been obtained.
The total number of heterozygotes before fixation
Authors: Holgate P.
Published: April 2006

This paper is about the total number of individuals who are heterozygotic for a specified allele, before it is either lost or fixed. The exact distribution is found for small populations, and two limiting processes are investigated.
On the parity of individuals in a branching process
Authors: Gani J. , Saunders I. W.
Published: April 2006

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.
The second-order analysis of stationary point processes
Authors: Ripley B. D.
Published: April 2006

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.
Some exact results for dams with Markovian inputs
Authors: Tin Pyke , Phatarfod R. M.
Published: April 2006

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.
Recurrent emptiness in a finite queue with increasing arrival rate
Authors: Rosenlund Stig I.
Published: April 2006

For a single-server queue with one waiting place and increasing arrival rate some necessary and sufficient conditions for infinitely many returns to emptiness with probability one are given.
The transition probabilities of the general stochastic epidemic model
Authors: Kryscio Richard J.
Published: April 2006

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.
Networks of queues with customers of different types
Authors: Kelly F. P.
Published: April 2006

The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution.

Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.
A note on the optimal control of carrier-borne epidemics
Authors: Wickwire K.
Published: April 2006

A model is proposed for the control of a deterministic carrier-borne epidemic by immunising susceptibles. An optimal control is of bang-bang type and it is proved that it can have at most one switch.
Inequalities for queues with dependent interarrival and service times
Authors: O'Brien G. L.
Published: April 2006

Consider two queues with k servers. Inequalities between their sample paths and thence between the distribution functions of their waiting times and queue sizes are deduced from inequalities between the conditional distribution functions, given previous interarrival and service times, for the two queues.