bluuebunny/crossref_metadata_2025 huggingface dataset:
Calculating exit times for series Jackson networks
Authors: Massey William A.
Published: July 2016

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.
A martingale characterization of mixed Poisson processes
Authors: Pfeifer Dietmar , Heller Ursula
Published: July 2016

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.
On functionals of the adjusted range process
Authors: Hooghiemstra G.
Published: July 2016

Using a well-known transformation that preserves Wiener measure a theorem is proved that enables one to derive in a non-computational manner distributions of simple functionals of the so-called adjusted range process.
On random mappings with a single attracting centre
Authors: Mutafchiev Ljuben R.
Published: July 2016

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = P
j
, j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph G
T

with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of G
T

into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P

0
. It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.
Optimal replacement in a shock model: discrete time
Authors: Aven Terje , Gaarder Simen
Published: July 2016

A system is subject to a sequence of shocks occurring randomly at timesn= 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in timen, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.
Stochastic motions on the 3-sphere governed by wave and heat equations
Authors: Orsingher Enzo
Published: July 2016

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.
Conditions for the non-ergodicity of Markov chains with application to a communication system
Authors: Sennott Linn I.
Published: July 2016

We obtain a sufficient condition for the transience of a Markov chain, and a sufficient condition for its null recurrence. These are applied to characterize the stability of a multiple-access communication system. Performance bounds for the system are also obtained.
Smooth first-passage densities for one-dimensional diffusions
Authors: Pauwels E. J.
Published: July 2016

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density.

In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.
Joint distribution of successive zero crossing distances for stationary Gaussian processes
Authors: Rychlik Igor
Published: July 2016

As has been shown by de Maré, in a stationary Gaussian process the length of the successive zero-crossing intervals cannot be independent, except for the degenerate case of a pure cosine process. However, no closed-form expression of the distribution of these quantities is known at present. In this paper we present an accurate explicit approximative formula, derived by replacing the Slepian model process by its regression curve.
Monotone stopping games
Authors: Mamer John W.
Published: July 2016

We consider the extension of optimal stopping problems to non-zero-sum strategic settings called stopping games. By imposing a monotone structure on the pay-offs of the game we establish the existence of a Nash equilibrium in non-randomized stopping times. As a corollary, we identify a class of games for which there are Nash equilibria in myopic stopping times. These games satisfy the strategic equivalent of the classical ‘monotone case' assumptions of the optimal stopping problem.
On sojourn time in Jackson networks of queues
Authors: Lemoine Austin J.
Published: July 2016

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi
be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi
: i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti
: i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.
An urn model for cannibal behavior
Authors: Pittel B.
Published: July 2016

A sampling procedure involving an urn with red and white balls in it is studied. Initially, the urn contains n balls, r of them being white. At each step, a white ball is removed, and one more ball is selected at random, painted red (if it was white before) and put back into the urn. R. F. Green proposed this scheme in 1980 as a stochastic model of cannibalistic behavior in a biological population, with red balls interpreted as cannibals. Of primary interest is the distribution of Xnr, the terminal number of red balls. A study of R. F. Green and C. A. Robertson led them to conjecture that, for r = 1 and n →∞, Xnr
is asymptotically normal with mean ≈ n exp(–1) and variance ≈ n(3 exp(–2) –exp(–1)). In this paper we prove that the conjecture — its natural extension, in fact — is true. Namely, for r/n bounded away from 1, Xnr
is shown to be asymptotically normal with mean ≈ n exp(ρ – 1) and variance ≈ n exp[2(ρ – 1)] (ρ
2
– 3ρ + 3 – exp(l – ρ)); ρ = r/n.
Probabilities of choosing applicants of arbitrary rank in the secretary problem
Authors: Rogerson Peter A.
Published: July 2016

Gilbert and Mosteller (1966) derive the probability of obtaining the best secretary when a fixed number of potential employees of varying quality are interviewed sequentially by an employer. This paper extends these results by deriving the probability of selecting an applicant of arbitrary rank under a variety of circumstances concerning the objective of employers and the amount of information employers have about applicant quality.
Upper bounds on work in system for multichannel queues
Authors: Wolff Ronald W.
Published: July 2016

Previously derived sample path upper bounds for multi-channel work in system and work in queue are shown to be false. A new proof is given for the corresponding stochastic bounds on these quantities.
The impact of the composition of the customer base in general queueing models
Authors: Federgruen A. , Groenevelt H.
Published: July 2016

We consider general queueing models dealing with multiple classes of customers and address the question under what conditions and in what (stochastic) sense the marginal increase in various performance measures, resulting from the addition of a new class of customers to an existing system, is larger than if the same class were added to a system dealing with only a subset of its current customer base.

Our results enhance our understanding of the dependence of various performance measures with respect to the composition of the customer base. In addition they translate readily into convexity results in an (appropriately defined) arrival rate.
Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines
Authors: Shanthikumar J. George , Sumita Ushio
Published: July 2016

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.
Remarks on the maxima of a martingale sequence with application to the simple critical branching process
Authors: Pakes Anthony G.
Published: July 2016

Let where {Zn, ℱn
} is a non-negative submartingale satisfying Ei
(Zn
log Zn
) →∞. It is shown that

When {Zn
} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that

This gives an alternative proof of a result recently established by Kammerle and Schuh.
Single server queue with uniformly bounded virtual waiting time
Authors: Cohen J. W.
Published: July 2016

In a previous paper [4] the author studied the stochastic process {wn
, n = 1,2, …}, recursively defined by

with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn
, σ2 + … + σ
n
and τ1 + … + τ
n−1
has been obtained. Closely related to the process {wn
, n = 1, 2,…} is the process {un
, n = 1, 2,…} with {un
= K + [wn
+ τ
n
− K]−, n = 1,2,…; these are dual processes.

In the present paper we study the stationary distributions of the processes {wn
, n= 1,2, …} and {un
, n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn
, n = 0” and “un
= K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn
, n = 1,2, …} and {un
, n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn
, n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models.

The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.
Two queues in series with a finite, intermediate waitingroom
Authors: Neuts Marcel F.
Published: July 2016

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full.

The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.
Finite regular invariant measures for Feller processes
Authors: Beneš V. E.
Published: July 2016

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.
On the converse to the iterated logarithm law
Authors: Heyde C. C.
Published: July 2016

Let Xi, i = 1, 2, 3,… be a sequence of independent and identically distributed random variables with law ℓ(X) and write. if EX
= 0 and EX2

=
σ2
< ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that
Domains of attraction for the subcritical Galton-Watson branching process
Authors: Rubin H. , Vere-Jones D.
Published: July 2016

Let F(z) = σ fjzj
be the generating function for the offspring distribution {fj
} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function

where F

n
(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.
A stochastic model for the breaking of molecular segments
Authors: Bithell J. F.
Published: July 2016

In many chemical and biochemical situations it is of interest to know the distribution that results from the breaking up of long molecules into shorter segments under certain hypotheses. For example, Montroll and Simha (1940) assumed that polymers consist of discrete units—monomers—connected by bonds all of which have an equal chance of breaking in the depolymerization process. Charlesby (1954) used a different approach to essentially the same model and in effect obtained differential equations for the moments of the ensuing distributions. More recently, Daniels (1967) has given a more thorough mathematical exposition, especially with regard to the effect of the initial length distribution, while Blatt (1967) has considered a model in which not all links are susceptible to breakage.
On the Distribution of the Number of Zero Partial Sums
Authors: Percus Ora Engelberg , Percus Jerome K.
Published: July 2016

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.
An Age-Dependent Branching Process with Correlations Among Sister Cells
Authors: Crump Kenny S. , Mode J. Charles
Published: July 2016

In this note we consider an age-dependent branching process in which the life-spans of sister cells are correlated as well as the numbers of offspring of sister cells, but otherwise cells live and reproduce independently. One might surmise that in many populations there would be some positive correlation among siblings due to similar characteristics inherited from parents. Data of Powell (1955) seem to corroborate this conjecture. Powell's data indicate that in certain bacterial populations the life-spans of sister cells have significant positive correlations while the correlation between the life-spans of mother and daughter cells is negligible. More recently positive correlations have been observed in cell populations among the life-spans of first cousins and also among second cousins (Kubitschek (1967)). However, the model presented in this note permits correlations only among siblings.
The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology
Authors: Jagers P.
Published: July 2016

Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process.
A new family of life distributions
Authors: Birnbaum Z.W. , Saunders S.C.
Published: July 2016

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.
Note on the output of a queuing system
Authors: Boes Duane C.
Published: July 2016

Consider a queuing system that has c servers and d waiting positions. Assume that the input is Poisson with rate α and the service times are exponential with mean β
–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 time t Markov, 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.
On a quickest detection problem with costly information
Authors: Balmer D. W.
Published: July 2016

Problems of detecting the arrival of a ‘disorder' are of central importance in quality control. One such problem is formulated as an optimal stopping problem in which information about the process to be stopped may be bought at any time. A control policy will involve a determination of times at which information should be bought and of the time at which the process should be stopped. It is shown that in all circumstances there is a unique policy satisfying necessary conditions for optimality; the various circumstances and control policies are specified.
Distribution of minimum path
Authors: Tenenbein Aaron
Published: July 2016

Let Y, X
1, X
2, …, Xn
be a set of n + 1 independently and uniformly distributed random variables on the interval (0, 1). The distribution of the length of the minimum path starting at Y which covers the other n points is derived. The solution is interesting in that it involves finding the distribution of an order statistic of a function of order statistics.
The signal-noise problem — a solution for the case that signal and noise are Gaussian and independent
Authors: Driscoll Michael F.
Published: July 2016

A solution is obtained for the signal-noise problem X = M + Z in which M and Z are independent Gaussian processes. Conditions on the processes are given which insure that the best estimate under generalized square — error loss is the conditional mean of M given X = x. A sequence of approximators of the best estimate is also given.
Emptiness times of a dam with stable input and general release function
Authors: Brockwell P. J. , Chung K. L.
Published: July 2016

We investigate the nature of the set of emptiness times of a dam whose release rate depends on the content and whose cumulative input process is a pure-jump Lévy process. Detailed results are obtained for stable input processes and release functions of the form r(x) = xβ I
(o,∞)(x).
Collapsing supercritical branching processes
Authors: Daly Fergus
Published: July 2016

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.
First-passage percolation under weak moment conditions
Authors: Reh Wolfgang
Published: July 2016

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.
Matched pairs in independent Poisson processes
Authors: Oakes David
Published: July 2016

Given two independent Poisson processes 𝒜 and ℬ with rates respectivelyαandβ(α<β) we match each point of 𝒜 with the closest point of ℬ that has not already been matched. The points of 𝒜 are taken in random order. It is shown that the point process of unmatched points of ℬ is a renewal process with the same interval distribution as the busy period of anM/M/1 queue. The distribution and moments of the distance between a typical point of 𝒜 and the corresponding matched point of ℬ are obtained. Variants of the matching process in which the assigned point of ℬ must lie to the right of the point of 𝒜, and in which the matching distance must be less than a fixed tolerance are studied. The use of matched samples to control for bias in observational studies is discussed.
Stoppable families of alternative bandit processes
Authors: Glazebrook K. D.
Published: July 2016

Stoppable families of alternative bandit processes are decision processes with the property that at each decision epoch the choice is between allocating service to one of the constituent bandit processes or stopping and deciding in favour of one of them. The problem is considered of finding optimal (or good suboptimal) strategies for such processes. The theory for non-stoppable families leads us to study the performance of a simple strategy. This is shown to be optimal under certain conditions. These conditions are discussed and an example relating to research planning is given.
A note on the infinitely deep dam with a Markovian input
Authors: Phatarfod R. M.
Published: July 2016

In this paper we consider an infinitely deep dam fed by inputs which form an ergodic Markov chain and whose release M is non-unit. The extension to non-unit release follows on lines similar to the independent inputs case. We show that P(θ) – θ MΙ where P(θ) = (pijθ i
) has a maximum of N = M(M + l)/2 non-zero singularities in the unit disc, so that the general solution of the equilibrium equations has N unknown constants. We also show that these constants satisfy N linear constraints, so that the solution is fully determined.
Stability and characterisation conditions in negative programming
Authors: Whittle P.
Published: July 2016

Let F be the infinite-horizon minimal cost function, and FK
S
the minimal cost function for horizon s and terminal cost function K. In Section 2 we define the ‘stable domain' (the set of K for which ), determine some of its properties, and relate these to the questions of stability of the process (whether ) and whether a given solution of the dynamic programming equation can be identified with F These ideas are developed in Sections 3 and 5 for various strengthenings and weakenings of the hypothesis of non-negative costs. In Section 5 we derive a new sufficient condition for stability and for characterisation of F.
A diffusion process model for the optimal investment strategies of an R & D project
Authors: Zuckerman Dror
Published: July 2016

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate.

The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion.

The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.
Failure distributions of shock models
Authors: Gottlieb Gary
Published: July 2016

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.
Some analyses on the control of queues using level crossings of regenerative processes
Authors: Shanthikumar J. G.
Published: July 2016

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.
The stereology of location
Authors: Davy Pamela J.
Published: July 2016

It is shown how to estimate centroids and other measures of location of an unknown embedded subset of a bounded specimen by random sectioning.
The moments of coverage of a linear set
Authors: Greenberg Irwin
Published: July 2016

A number of points are chosen at random along a line segment and used as the left end-points of lengths of fixed size. Finite sum expressions are derived for the moments of the portion of the line segment covered by one or more of the lengths. The derivation utilizes a relationship between the coverage and the busy time in an M/D/∞ queue.
Characterizations of the exponential distribution by relevation-type equations
Authors: Grosswald E. , Kotz Samuel , Johnson N. L.
Published: July 2016

In this note characterizations of the exponential distribution are discussed, based on a generalization of the lack of memory property. The result was motivated by the notion of ‘relevation of distributions' introduced by Krakowski (1973).
On the genealogy of large populations
Authors: Kingman J. F. C.
Published: July 2016

A new Markov chain is introduced which can be used to describe the family relationships amongnindividuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.
Heritability
Authors: Wilson Susan R.
Published: July 2016

This article discusses certain aspects of determining heritability estimates from the genetic analyses of data. First, a brief review is given of the major statistical techniques for fitting models to quantitative family data. Then, by reference to a general schematic framework, the basic models that have been proposed for such data will be described. Some new environmental models are also developed. Finally, by reference to some data sets, the interactive approach to data analysis is proposed, advocating the determination of adequacy of fit of all reasonable models to available data.
Functional dependence from a descriptive viewpoint
Authors: Finch P. D.
Published: July 2016

This paper considers functional dependence from a general standpoint. Some basic ideas are explored and the extent of functional dependence studied. These are illustrated by means of simple linear regression.
Stationary point processes by Markov-dependent intervals and infinite intensity
Authors: Daley D. J.
Published: July 2016

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.
Beyond the heuristic approach to Kolmogorov-Smirnov theorems
Authors: Pollard David
Published: July 2016

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.
On the estimation of frequency in point-process data
Authors: Vere-Jones D.
Published: July 2016

Suppose that it is desired to determine the frequency, known to lie in a range 0 < σ ≦ ω ≦ Ω
T
<∞, of a periodic effect modulating the rate of occurrence of a Poisson point process. It is shown that if Ω
T
increases sufficiently slowly with T, the length of the observation period, then the frequency corresponding to the maximum of the Bartlett periodogram over this range provides a consistent estimate of the unknown frequency. The asymptotic covariance matrix is found for the maximum likelihood estimates of the parameters in a cyclic Poisson model, and the close analogue with frequency estimation for Gaussian processes is emphasized. It is pointed out, however, that in the point-process case periodogram estimates will be efficient only for very special models.
Quasi-stationary distributions in semi-Markov processes
Authors: Cheong C. K.
Published: July 2016

Our main concern in this paper is the convergence, as t → ∞, of the quantities

i, j ∈ E; where Pij
(t) is the transition probability of a semi-Markov process whose state space E is irreducible but not closed (i.e., escape from E is possible), and rj
is the probability of eventual escape from E conditional on the initial state being i. The theorems proved here generalize some results of Seneta and Vere-Jones ([8] and [11]) for Markov processes.
Composable Markov processes
Authors: Schweder Tore
Published: July 2016

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.
The multi-type Galton-Watson process with immigration
Authors: Quine M. P.
Published: July 2016

We consider the limiting behaviour of a k-type (k < ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.
The Galton-Watson process with infinite mean
Authors: Darling D. A.
Published: July 2016

Let Zn
be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z
0 = 1. If E(Z
1) < ∞ the limiting distribution of Zn
is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z
1) = ∞ (the “explosive” case) the behavior of the distribution of Zn
for large n seems not to be known. As shown by Seneta [1], there are no constants cn
such that cnZn
has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn
+ 1) has a limiting distribution.
Selective interaction between two independent stationary recurrent point processes
Authors: Srinivasan S. K. , Rajamannar G.
Published: July 2016

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).
Selective interaction of a stationary point process and a renewal process
Authors: Lawrance A. J.
Published: July 2016

The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.
On a characterization of Poisson distribution
Authors: Srivastava R. C. , Srivastava A. B. L.
Published: July 2016

Recently Rao (1963) has considered discrete models where an original observation produced by nature is subjected to a destructive process and we observe the undestroyed part of the original observation. Suppose the original observation produced by nature is distributed according to a Poisson distribution with parameter λ and the probability that the original observation n is reduced to r due to the destructive process is Now if Y denotes the resulting random variable (r.v.), then it is easily seen that

Let us call this condition the *-condition. Later Rao and Rubin ({1964),Theorem 1) proved that the *-condition is a characterizing property of the Poisson distribution.
Busy periods in vehicle queues
Authors: Shaw Leonard
Published: July 2016

Queueing models have been used to describe the slowing or stopping of a stream of vehicles due to a traffic light, a merging disturbance, or an accident. The buildup of stopped vehicles, and their eventual dispersal after the delaying factor has been removed, has been likened to the busy period of a queue, e.g., Jewell (1964). Some studies have dealt with the total delay to all drivers involved in such a queue, Daley and Jacobs (1969), Gaver (1969); or the combination of this total delay and the distance of the last vehicle from the originating incident, McNeil (1969), Shaw (1970).
The method of V. M. Popov for differential systems with random parameters
Authors: Tsokos Chris P.
Published: July 2016

The aim of this paper is to investigate the existence of a random solution and the stochastic absolute stability of the differential systems (1.0)–(1.1) and (1.2)–(1.3) with random parameters. These objectives are accomplished by reducing the differential systems into a stochastic integral equation of the convolution type of the form (1.4) and utilizing a generalized version of V. M. Popov's frequency response method.
Truncation procedures for non-negative matrices
Authors: Tweedie Richard L.
Published: July 2016

1. Let T = (tij
), i, j = 0, 1, ···, tij
≧ 0, denote an infinite non-negative matrix. We shall assume at all times that where are the matrix iterates of T; we take T
0 = I = (δij
).
Some condensation processes of McKean type
Authors: Knight Frank B.
Published: July 2016

In a short article McKean [2] inaugurated the study of a class of time-inhomogeneous Markov processes which arise from the reaction of a single particle (the “tagged” particle) with an evolving environment of other particles. Other more recent references on these processes are [1], [4] and [5], in which some existence and uniqueness theorems for the solution of the resulting non-linear equation are obtained. We confine attention here to a very particular type of these processes which arises when two interacting particles simply coalesce, and the mass is taken as the process variable.
On an inequality of Heyde
Authors: Tomkins R. J.
Published: July 2016

In [3] Heyde proposed an extended version of the well-known Hájek-Rényi inequality to include random variables without finite moments. The purpose of this note is to point out that the theorem in [3] is in error, and to prove the following theorem in its stead:

Theorem. Let X
1, X
2, ··· be any sequence of random variables, and define be independent random variables, each with mean zero and finite variance, and define Zk
= Xk
- Yk
for k ≧ 1. Let c
1, c
2, ··· be a non-increasing sequence of positive numbers.
Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test
Authors: Durbin J.
Published: July 2016

Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t
1)w(t
2)}= min (t
1, t
2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.
Customer average and time average queue lengths and waiting times
Authors: Marshall Kneale T. , Wolff Ronald W.
Published: July 2016

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.
Programming problems and changes in the stable behavior of a class of Markov chains
Authors: Evans Richard V.
Published: July 2016

This paper develops expressions for the derivatives with respect to a parameter μ of the stable probabilities of a class of Markov chains whose transition matrices are of the form Q + μW. These expressions lead to iterative schemes for calculation which in turn suggest gradient algorithms for finding locally optimal chains.
On the optimal control of stationary diffusion processes with natural boundaries and discounted cost
Authors: Morton R.
Published: July 2016

Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.
First passage to a general threshold for a process corresponding to sampling at Poisson times
Authors: Belkin Barry
Published: July 2016

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).
Moments in Markovian systems with lumped states
Authors: Taylor G. C.
Published: July 2016

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.
Goofspiel — the game of pure strategy
Authors: Ross Sheldon M.
Published: July 2016

The game of pure strategy, sometimes called Goofspiel or Gops (see [2] and [3]), is played by two players, using a normal deck of cards, as follows. The 13 clubs are first taken out of the deck and of the remaining 39 cards the 13 hearts are given to Player I, the 13 diamonds to Player II, and the 13 spades are placed face down in the center. The spades are shuffled and one is turned face up. At this point, the two players choose one of their cards and then simultaneously discard it. The one who discards the higher card (ace being low, king high) wins from the other an amount equal to the value of the upturned spade (ace = 1, king = 13). If both players discard the same card, then neither wins. The three cards are then thrown away, a new spade upturned and the game continues. After 13 plays, there are no remaining cards and the game ends.
A low traffic approximation for queues
Authors: Bloomfield P. , Cox D. R.
Published: July 2016

A general procedure is outlined for obtaining lower bounds and approximations to the amount of congestion in queues with low traffic. Some detailed formulae are given for a number of single-server systems and compared with exact solutions where available. Results are also given for a discrete time system in which departures clash with new arrivals.
Limit theorems for thinning of renewal point processes
Authors: L. Råde
Published: July 2016

Limit theorems for the thinning of renewal point processes according to two different schemes are studied. In the first scheme when a point is retained a random number of succeeding points are deleted. According to the second scheme a random number of points are deleted by an inhibitory Poisson process.
Stepping stone models for population growth
Authors: Renshaw E.
Published: July 2016

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours. An approximate solution is obtained for the probability structure of the population size, and exact results are derived for the process when immigration is introduced into a single colony from outside the system.