bluuebunny/crossref_metadata_2025 huggingface dataset:
An optimal hunting policy for a stochastic logistic model
Authors: Abakuks Andris
Published: July 2016

A stochastic version of the logistic model for population growth is considered, and the general form of an optimal policy is found for hunting the population so as to maximise the long-term average number of captures per unit time. This optimal policy is described by a critical population size x∗such that it is optimal to hunt if and only if the population size is greater than or equal to x∗. Methods of determining x∗for given parameter values are provided, and some properties of the optimal policy as the population size tends to infinity are proved.
A formal approach to queueing processes in the steady state and their applications
Authors: Miyazawa Masakiyo
Published: July 2016

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in G/G/c queues. As applications, the finiteness of their moments is discussed for G/G′/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)).
Semi-Markovian manpower models in continuous time
Authors: Mehlmann Alexander
Published: July 2016

This paper presents the general properties of the semi-Markovian manpower model in continuous time. The asymptotic relation for the population numbers in various grades is based on the forces of transition zij
(u) from state i to state j at duration u.
The discrete asymptotic behaviour of a simple batch epidemic process
Authors: Billard L. , Lacayo H. , Langberg N. A.
Published: July 2016

A simple epidemic process in which the number of individuals who can become infected at any point in time is itself a random variable is described. The discrete asymptotic behaviour of such a process is discussed. In particular, the associated marginal distribution of the limiting process is considered.
A strong variational principle for continuous spin systems
Authors: Pirlot Marc
Published: July 2016

We prove in the setting of the system of unbounded spins developed by Lebowitz and Presutti, a strong variational principle analogous to Lanford and Ruelle's for discrete spins. We establish this result for finite-range regular and superstable potentials satisfying a condition which enables us to define the thermodynamic energy of the system.
A representation for discrete distributions by equiprobable mixtures
Authors: Peterson Arthur V. , Kronmal Richard A.
Published: July 2016

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.
Ergodic properties of a two-dimensional binary process
Authors: Galbraith R. F. , Walley D.
Published: July 2016

Sufficient conditions are derived for ergodicity of the two-dimensional process of binary variables discussed by Galbraith and Walley (1976).
On the minimum of gaps generated by one-dimensional random packing
Authors: Itoh Yoshiaki
Published: July 2016

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, t]. We have

with

Using this equation the asymptotic behaviour of P(L(x)≧h) is discussed.
On the transient waiting times for a GI/M/1 priority queue
Authors: Dalen G. , Natvig B.
Published: July 2016

In this paper we consider the GI/M/1 queueing model with infinite waiting-room capacity. The customer arriving at t = 0 will find k — 1 customers waiting. The latter customers belong to a second priority class, whereas the ones arriving in [0,∞) belong to a first priority class and have the higher priority. Within each class we have a first-in-first-out queueing discipline. A customer, once at the service-point, remains there until his service is completed. Then the next customer for service is the one of highest priority among those queueing.

For this model we derive the transient waiting times for customers belonging to both priority classes. The results are of special interest in appointment systems where customers may not turn up.
On a stopping time of Starr and its use in estimating the number of transmission sources
Authors: Vardi Yehuda
Published: July 2016

Consider N sources, each of which transmits signals according to a renewal process with a common, known, underlying distribution. Sequential point estimators for N, which are asymptotically efficient in comparison to the best fixed stopping strategy, are derived. Confidence statements for N, and applications to the problem of estimating finite populations are discussed.

The exact limiting distribution, and moments, of the stopping time we use, which was suggested by Starr, are derived.
A continuous-time analogue of random walk in a random environment
Authors: Ritter Grant
Published: July 2016

By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.
On functional equations and measures of information. II
Authors: Kannappan PL. , Ng C. T.
Published: July 2016

We derive the representation of additive measures of information with the sum property under a regular generating function. This family of measures includes the Shanon entropy, directed/symmetric divergence and inaccuracy.
The probabilistic significance of the rate matrix in matrix-geometric invariant vectors
Authors: Neuts Marcel F.
Published: July 2016

Many queueing models have embedded Markov chains, whose invariant probability vector is of a (modified) matrix-geometric form. The rate matrix R is shown to have the following probabilistic interpretation:

The element Rvj
is the expected number of visits to the state (i + 1, j), before the first return to the set i = {(i, 1), …,(i, m)}, in a chain starting in the state (i, v).
The convergence of a branching Brownian motion used as a model describing the spread of an epidemic
Authors: Wang Frank J. S.
Published: July 2016

A spatial epidemic process where the individuals are located at positions in the Euclidean space R
2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R
2 according to a Brownian motion with a diffusion coefficient σ
2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.
On a simple formula for random mappings and its applications
Authors: Burtin Y. D.
Published: July 2016

A random mapping of a countable set N into itself is studied. An explicit formula is found for the probability that a subset of N will not contain cycles of the mapping. This formula is applied then for a detailed asymptotic analysis of two special mappings.
High-Level exceedances of regenerative and semi-stationary processes
Authors: Serfozo Richard
Published: July 2016

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.
On the variance of the maximum of partial sums of n-exchangeable random variables with applications
Authors: Anis A. A. , Gharib M.
Published: July 2016

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.
Representation of the characteristic function of a stochastic integral
Authors: Riedel M.
Published: July 2016

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form

(where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.
Remarks on the relation between fractional moments and fractional derivatives of characteristic functions
Authors: Laue G.
Published: July 2016

We consider fractional derivatives of characteristic functions. We use these fractional derivatives for the formulation of new conditions for the existence of non-integer moments. We also compare the known conditions for the existence of moments of arbitrary random variables with our new conditions. As a consequence many conditions can be written in a unified terminology.
The lifetime of a standby system of units having the Birnbaum and Saunders distribution
Authors: Raaijmakers F. J. M.
Published: July 2016

In this paper a formula is derived for the probability density function of the lifetime for a system of identical units. The lifetime of the system is the sum of the lifetimes of the independent units when the units have lifetimes distributed independently by the law proposed by Birnbaum and Saunders (1969).
Conditions for evolutionarily stable strategies
Authors: Abakuks Andris
Published: July 2016

It is pointed out that the conditions given by Haigh (1975) for finding evolutionarily stable strategies corresponding to a given matrix are sufficient, but not always necessary. An example is given, and an amended version of the necessary and sufficient conditions is stated.
On a characterization of gamma distributions via exponential mixtures
Authors: Bhattacharjee Manish C.
Published: July 2016

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.
An inequality for a metric in a random collision process
Authors: Nishimura Shōichi
Published: July 2016

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distributionG(x) [9]. We define a metrice1[Fn;G], which is analogous to the functionale[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove thate1[Fn;G] is monotone non-increasing asn→ ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for somea >1 the initial distribution has anath moment.
The exponential rate of convergence of the distribution of the maximum of a random walk
Authors: Veraverbeke N. , Teugels J. L.
Published: July 2016

Let Gn
(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn
(x) — G(x) is asymptotically equal to c.H(x)n
−3/2
γn
as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.
Extremal processes and random measures
Authors: Shorrock R. W.
Published: July 2016

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.
Decidability of α (pk) > 0 for some k
Authors: Madsen Richard W.
Published: July 2016

Paz (1963), (1971) has shown that for an n × n stochastic matrix P the property that some power of P has a positive ergodic coefficient is decidable. In particular he shows that α (Pk) > 0 for some k, only if it is positive for k = ½n(n – 1). However he states that it is not known whether this bound is sharp. In this paper a sharp bound is given, namely k =½(n – 1)2 + ½ or ½(n − 1)2 + 1 depending on whether n is even or odd. The proof of this is based on some results from number theory.
A note on the supercritical generalized age-dependent branching process
Authors: Kaplan Norman
Published: July 2016

A sufficient condition is given for a supercritical generalized age-dependent branching process, properly normalized, to converge to zero. This condition is the strongest one possible, for if it does not hold, then it is known that the normalized process does indeed converge to a legitimate random variable.
Order statistics of partial sums of mutually independent random variables
Authors: Pollaczek Felix
Published: July 2016

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.
Characterization theorems using conditional expectations
Authors: Kotlarski Ignacy I. , Hinds William E.
Published: July 2016

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn
with n ≧ 1.
A note on the Lévy distance
Authors: Thompson J. W.
Published: July 2016

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.
On a point process with independent locations
Authors: Isham Valerie
Published: July 2016

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.
A simple random walk on parallel axes moving at different rates
Authors: Barnett V.
Published: July 2016

Prompted by a rivulet model for the flow of liquid through packed columns we consider a simple random walk on parallel axes moving at different rates. A particle may make one of three transitions at each time instant: to the right or to the left on the axis it was on at the previous time instant, or across to the other axis. Results are obtained for the unrestricted walk, and for the walk with absorbing, or reflecting, end-points.
A multinomial model for transition probability matrices
Authors: Pegram G. G. S.
Published: July 2016

A model for the transition probability matrices (t.p.m.'s) of finite discrete Markov chains is suggested which may help those who wish to use a larger number of states than would seem reasonable with the data available in the current estimation situation. The model is especially useful in that a finite t.p.m. of arbitrary size can be specified by as few as two parameters. An example of the model's estimation and use is presented, showing it in a fair light in comparison with the conventional method of t.p.m. specification.
The reverse Galton-Watson process
Authors: Esty Warren W.
Published: July 2016

Consider the following path, Zn
(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN
= i converge, as N → ∞ to a probability function of a Markov process, Xn
, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN
= 0 but also ZN

–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ
2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.
An absorption probability for the Ornstein-Uhlenbeck process
Authors: Dirkse John P.
Published: July 2016

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.
Most stringent bounds on the probability of the union and intersection of m events for systems partially specified by s
1, s
2, …sk
, 2 ≦ k < m
Authors: Kwerel Seymour M.
Published: July 2016

For dependent probability systems of m events partially specified by the quantities Sv, the sum of the probabilities of each of the combinations of v events, v = 1, 2, …, 2 ≦ k < m; this paper develops the most stringent upper and lower bounds on P
1, the probability of the union of the m events; and on P
[m], the probability of the simultaneous occurrence of the m events.
A further note on duality of random walks
Authors: Barnett N. S.
Published: July 2016

A multi-dimensional generalisation of a well-known duality relation in random walk theory is given.
Inequalities for queues with dependent interarrival and service times
Authors: O'Brien G. L.
Published: July 2016

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.
Analysing binary lattice data with the nearest-neighbor property
Authors: Strauss David J.
Published: July 2016

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.
Uniform coupling of non-homogeneous Markov chains
Authors: Griffeath David
Published: July 2016

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.
Evacuation of a Yule process with immigration
Authors: Brown Mark , Ross Sheldon , Shorrock Richard
Published: July 2016

Individuals arrive at a geographical area that is initially empty, in accordance with a pure birth process with birth parameters λj
= jλ + θ, j ≧ 0. Due to contamination, this geographical area is unsafe for its population and at some fixed time T in the future everyone in the area will be evacuated and no further immigration will be allowed.

Suppose now than an intermediate evacuation time τ, 0 ≦ τ ≦ T, at which time everyone present in the area would be evacuated, is to be chosen. The area would then again fill up with individuals between times τ and T, and, at T, the final evacuation would be made and the area would be permanently sealed off. The problem is to choose τ so as to minimise the total expected cost incurred by time T, where a cost g(x) is incurred for each individual that spends a time x in the area.
Inaccuracy and a coding theorem
Authors: Autar Ram , Soni Raminder Singh
Published: July 2016

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 order a and generalised mean length of order t under the condition .
Inequalities with application in retail inventory analysis
Authors: Agnew Robert A.
Published: July 2016

Simple bounds on service level and turnover velocity are obtained for a periodic-review inventory system with a stationary order-up-to-level stocking policy and no backordering. Exact computational formulas are given for Poisson demand. An illustrative numerical example is presented, and the application of these bounds to retail inventory analysis is discussed.
A diffusion process model for the optimal operation of a reservoir system
Authors: Pliska Stanley R.
Published: July 2016

The water level in a reservoir is modelled as a controlled diffusion process on a compact interval of the real line. The problem is to control the water discharge rate so as to minimise the expected costs, which depend upon the histories of the water levels and release rates. The form of the optimal control is studied for two general classes of reservoir control problems.
Some limit theorems for a supercritical branching process allowing immigration
Authors: Pakes A. G.
Published: July 2016

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.
The inverse problem in reducible Markov chains
Authors: Ray W. D. , Margo F.
Published: July 2016

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.
The joint probability density function of the occupation time of a three-state problem
Authors: Hsia Wei Shen
Published: July 2016

The joint probability density function of the occupation time of a three-state stochastic process with constant transition matrix: a
ij
≠ 0, i ≠ j, i, j = 1, 2, 3 but αii
= 0, i = 1, 2, 3 is studied.
Some properties of line segment processes
Authors: Parker Philip , Cowan Richard
Published: July 2016

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R
2, for the first moments of M(A), the number of segment mid-points in A; N(A), the number of segments which intersect with convex A; S(A), the total length within A of segments crossing A; and C(A) the number of segment-segment crossings within A. In the case of Poisson mid-points, the distribution of the rth nearest line-segment to a given point is found.
The rates of growth of the Galton-Watson process in varying environment
Authors: Foster James H. , Goettge Robert T.
Published: July 2016

It is shown that a Galton-Watson process in varying environment, unlike a classical Galton-Watson process, can have an infinite (though at most countable) number of distinct rates of growth.
Functional and random central limit theorems for the Robbins-Munro process
Authors: McLeish D. L.
Published: July 2016

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.
Partial attraction of maxima
Authors: Green Richard F.
Published: July 2016

There exist three classes of probability laws that are stable for maxima. A number of well-known distributions lie in the domains of attraction of these laws. This fact is sometimes exploited by fitting the distribution of maxima with one of the stable laws. Such a procedure may well be misguided, however, since distributions exist which produce maxima having any desired distribution and not just a stable type. In this paper partial attraction of maxima is defined and it is shown that all distributions have a non-empty domain of partial attraction of maxima. In fact, there exists a distribution that lies simultaneously in the domain of partial attraction of maxima of all distributions.
Failure rate — a unified approach
Authors: Lee Larry , Thompson W. A.
Published: July 2016

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.
Optimal issuing policies for non-identical units with stochastic lifetimes
Authors: Enns E. G. , Ehlers P. F.
Published: July 2016

Two non-identical units are to be issued sequentially. The optimal policy determines the issuing sequence which maximizes the expected lifetime of the pair of units. The units each have non-decreasing hazard rates which depend on whether the unit is in service or in storage. Sufficient conditions are obtained for optimal issuing policies in a variety of examples.
The total number of heterozygotes before fixation
Authors: Holgate P.
Published: July 2016

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.
The dispersion of a neutral allele considered as a branching process
Authors: Crump Kenny S. , Gillespie John H.
Published: July 2016

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data fromDrosophilapopulations are used to show the improbability of the same neutral allele occurring at widely separated localities.
Evaluations of barrier-crossing probabilities of Wiener paths
Authors: Park C. , Schuurmann F. J.
Published: July 2016

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.
The distribution of the maximum Brownian excursion
Authors: Kennedy Douglas P.
Published: July 2016

The distribution of the maximum of the unsigned scaled Brownian excursion process and of a modification of that process are derived. These distributions are related to the one-dimensional Brownian bridge.
A bound for bivariate probability of large deviations
Authors: Goldstein Matthew
Published: July 2016

Suppose (X
1, Y
1), (X
2, Y
2), …, (Xn
, Yn
) are independent random vectors such that a ≦ Xi
≦ b and a ≦ Yi
≦ b, i = 1, 2, …, n. An upper bound which exponentially converges to zero is derived for the probability Pr{Sx
– nμ
x ≧ nt
1;SY – nμY
≧ nt
2} where Sx
= Σ Xi
, SY = Σ Yi
,EYi = μY, EXi = μx
and t
1 > 0, t2 > 0. The bound is a function of the difference b — a, the correlation between Xi
and Yi, μx
and μY
and t
1 and t
2
.
On a multivariate generalized occupancy model
Authors: Johnson N. L. , Kotz Samuel
Published: July 2016

A generalized occupancy distribution obtained by Uppuluri and Carpenter (1971) is derived by elementary methods, and extended to a multivariate situation.
On exponential bounds for the waiting-time distribution function in GI/G/1
Authors: Bergmann R. , Stoyan D.
Published: July 2016

Exponential bounds for the stationary waiting-time distribution of the type ae–θt
are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.
Recurrent emptiness in a finite queue with increasing arrival rate
Authors: Rosenlund Stig I.
Published: July 2016

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.
Exact fluctuation results for Markov-dependent coin-tossing
Authors: Darroch J. N. , Whitford Heather J.
Published: July 2016

Feller (1968) showed that the probability functions of each of three random variables associated with coin-tossing (independent, simple, symmetric random walk) are (a) the same (except at the end points), (b) approximately arc-sine. In this paper these two properties are shown to hold for Markov-dependent coin-tossing (Markov-dependent, simple, symmetric random walk).
Complements to heavy traffic limit theorems for the GI/G/1 queue
Authors: Whitt Ward
Published: July 2016

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.
On the limiting behaviour of a basic stochastic process
Authors: Şahin İzzet , Achou Oussama
Published: July 2016

Determination of the limiting distributions for a class of mixed-type stochastic processes with state-dependent rates of decline is reduced to the solution of a class of integral equations. For the case where the rate of decline is proportional to the state, some results are obtained by solving the integral equation of the process through Fuchs' method.
On linearly regressive processes
Authors: Reynolds John F.
Published: July 2016

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.
Processes with conditional stationary independent increments
Authors: Serfozo Richard F.
Published: July 2016

We study a class of processes which are essentially processes with stationary independent increments whose basic parameters are allowed to vary randomly over time. These processes are equivalent to random time transformations of processes with stationary independent increments where the time process is independent of the original process. Several limiting theorems are presented including weak and strong laws of large numbers and a functional central limit theorem.
The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue, M/M/1
Authors: Jenkins John H.
Published: July 2016

A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.
Nearest-neighbour systems: a lemma with application to Bartlett's global solutions
Authors: Besag J. E.
Published: July 2016

Bartlett has proposed two alternative forms of spatial-temporal Markov model, each leading to the standard spatial equilibrium distribution for nearest-neighbour systems on a multi-dimensional lattice. The validity of one of these forms is discussed with the aid of a lemma.
Inequalities with application in economic risk analysis
Authors: Agnew Robert A.
Published: July 2016

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.
Factorial moments and probabilities for the general stochastic epidemic
Authors: Billard L.
Published: July 2016

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.
Some results for general cell-size-dependent branching processes
Authors: Sudbury Aidan , Clifford Peter
Published: July 2016

A general model for the growth and division of cells in which the growth rate and division probability at any instant depend only on their size at that time is introduced. Conditions under which (a) the distribution of cell-size at division converges ergodically, (b) the sizes tend to 0 or ∞, are exhibited, and bounds to the correlation between the sizes at division of sister cells are given in a wide class of cases.
Some limit theorems for Markov branching processes
Authors: Leslie J. R.
Published: July 2016

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.
First-emptiness problems in applied probability under first-order dependent input and general output conditions
Authors: Lehoczky J. P.
Published: July 2016

Results for the first-emptiness time of a semi-infinite reservoir and the integral functional of the process up to first-emptiness time are derived under Markov chain input conditions and general output conditions. The results are further extended to allow an input process which is the sum of k consecutive elements of the Markov chain, k ≧ 1.
Some results on regular variation for distributions in queueing and fluctuation theory
Authors: Cohen J. W.
Published: July 2016

For the distribution functions of the stationary actual waiting time and of the stationary virtual waiting time of the GI/G/l queueing system it is shown that the tails vary regularly at infinity if and only if the tail of the service time distribution varies regularly at infinity.

For sn
the sum of n i.i.d. variables xi, i = 1, …, n it is shown that if E {x
1} < 0 then the distribution of sup, s
1
s
2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x
1 varies regularly at infinity and conversely, moreover varies regularly at + ∞.

In the appendix a lemma and its proof are given providing necessary and sufficient conditions for regular variation of the tail of a compound Poisson distribution.
An iterated logarithm law for maxima of nonstationary Gaussian processes
Authors: Deo Chandrakant M.
Published: July 2016

An iterated logarithm type theorem is obtained for maxima of nonstationary discrete-parameter Gaussian processes.
On characterization of a generalized inaccuracy measure in information theory
Authors: Sharma Bhu Dev , Autar Ram
Published: July 2016

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-ß have been given in this paper.
An application of the minimum discrimination information estimate to compute log-likelihood ratios
Authors: Melnick E. L. , Kullback S.
Published: July 2016

In this paper the minimum discrimination information estimate is used to compute the log-likelihood ratio or logarithm of the Radon-Nikodym derivative In (dP
1/dP
2) when the stochastic process {x(t), t∈T) has either the probability measure P
1 or P
2. One example tests the mean value function of Gaussian processes. The other tests the mean value function of a continuous time Poisson process.
A generalised incomplete beta function and its application to multi-line stock control
Authors: Gittins J. C. , Maher M. J.
Published: July 2016

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.
Multi-channel queues in heavy traffic
Authors: Loulou Richard
Published: July 2016

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 note on the MX/GY/1, K bulk queueing system
Authors: Bagchi Tapan P. , Templeton J. G. C.
Published: July 2016

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.
A population birth-and-mutation process, I: explicit distributions for the number of mutants in an old culture of bacteria
Authors: Mandelbrot Benoit
Published: July 2016

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.
Numerical solution of Fisher's equation
Authors: Gazdag Jenö , Canosa José
Published: July 2016

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.
Some asymptotic results on multiple matching
Authors: Hafner R. , Sendler W.
Published: July 2016

We considers≦nrandomly chosen permutations of the numbers 1, 2, …,n, and write them under each other, thus forming ans×nmatrix, called “random-batch”. A rule, prescribing how many elements of a column may occur exactly one time, how many may occur exactly two times, etc., is called a fixpoint-structure. Assuming each possible permutation to be chosen with equal probability, the number of columns having a certain fixpoint-structureFis a random variableX(F). The limiting distribution ofX(F) forn→ ∞is considered for different cases ofs=s(n). The main result (Theorem 4) says, that the common distribution of a finite numbertof given fixpoint-structures tends to the product oftPoisson-laws,n→ ∞.
A limit theorem for semi-Markov processes
Authors: Szász Domokos
Published: July 2016

A limit theorem is proved for semi-Markov processes, which depend on a small parameter, tending to 0, in the case when the processes have an absorbing state and some asymptotically non-essential states and one asymptotically essential state. The application of the theorem is illustrated by an example from reliability theory.
The periodicity of endemic malaria
Authors: Radcliffe J.
Published: July 2016

In many endemic areas the level of infection of malaria exhibits a regular cycle. A stochastic model is formulated to describe endemic malaria and is used to justify a set of deterministic equations. The model incorporates the properties peculiar to malaria, i.e., in endemic areas malaria is not usually fatal and an individual can be and usually is reinfected. The theory of small oscillations is applied to the deterministic equations, and under the assumption of stable populations of hosts and vectors an expression is obtained for the period of oscillation of the infection.
The recurrent epidemic
Authors: Billard L.
Published: July 2016

The recurrent epidemic model has hitherto proved mathematically intractable. It is shown here that by imposing the mild condition that the source of new susceptibles is initially of a constant size M for M sufficiently large that there is zero probability of more than M new susceptibles being added to the population in the course of the epidemic, then state probabilities for the stochastic recurrent epidemic model can be found.
Information flow in some classes of Markov systems
Authors: Dawson D. A.
Published: July 2016

The information flow in discrete Markov systems provides a method for determining that such a system is ergodic. Estimates are obtained for the information flow in some classes of Markov systems and using these estimates criteria for the ergodicity of the systems are established.
Characterizations of the exponential distribution by order statistics
Authors: Huang J. S.
Published: July 2016

Let X
1,n
≦ … ≦ Xn, n
be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX
1,n
is identically distributed as X
1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.
A single-server queue with limited virtual waiting time
Authors: Takács Lajos
Published: July 2016

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.