bluuebunny/crossref_metadata_2025 huggingface dataset:
Multivariate exponential distributions based on hierarchical successive damage
Authors: Arnold Barry C.
Published: July 2016

A class of multivariate exponential distributions is suggested which includes those presented by Marshall and Olkin, Downton, and Hawkes. It is based on a concept of hierarchical successive damage. Recursive expressions for the Laplace transforms and for first and second moments are derived in the bivariate case. Related multivariate geometric distributions are described.
Some remarks on probability inequalities for sums of bounded convex random variables
Authors: Goldstein M.
Published: July 2016

Let X
1, X
2, · ··, Xn
be independent random variables such that ai
≦ Xi
≦ bi
, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi
), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.
Truncation approximation of the limit probabilities for denumerable semi-Markov processes
Authors: Tweedie Richard L.
Published: July 2016

It is shown that methods used by the author to approximate limit probabilities for Markov processes from their Q-matrices extend to semi-Markov processes. The limit probabilities for semi-Markov processes can be approximated using only truncations of the embedded Markov chain transition matrix and the vector of mean holding times.
Two dependent Poisson processes whose sum is still a Poisson process
Authors: Jacod Jean
Published: July 2016

We show that there exists a pair of Poisson processes which are not stochastically independent, and whose sum is still a Poisson process.
On Gordin's central limit theorem for stationary processes
Authors: Eagleson G. K.
Published: July 2016

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.
Further results on Kendall's autoregressive series
Authors: Hardin Jay C. , Brown Thomas J.
Published: July 2016

Theoretical expressions for the cross correlation and cross spectra of the input and output variables in the difference equation Xt
=aXt

−1 + bXt−

2
+ Yt
are derived. These expressions are compared with estimates of these expectations obtained by employing a Fast Fourier Transform technique on digitally generated series.
On a certain type of network of queues
Authors: Madan K. C.
Published: July 2016

The paper studies a network of queues in which units arrive singly in a Poisson stream at a service channel S from which they branch out into k parallel channels S
1
, S
2, …, Sk. After having been serviced at S
1, S
2, … Sk
, the units converge again into a single channel S′. The service times of units at each of the channels are assumed to be exponential. Units finally serviced at S’ may leave the system or may again join S. This has been considered by taking two models denoted as Model A and Model B. Steady-state probabilities giving the number of units present in the system have been obtained explicitly for both the models. The expressions for mean queue lengths have also been arrived at.
A finite dam with variable release rate
Authors: Yeo G. F.
Published: July 2016

This note considers a finite dam fed by independently and identically distributed (i.i.d.) inputs, being either (i) of at least size β (> 0) or (ii) negative exponentially distributed, occurring in a Poisson process. The instantaneous release rate may be a function r(·) of the content; additional and numerical results are given for the special case where r(x) = µxα
(0 ≦ α<∞, 0 < µ <∞) is proportional to the αth power of the content. The basic method used in [7] for the special case r(x) = µx for obtaining the distribution of the number of steps and of the time to first overflowing is shown to carry over almost completely in case (i), but only partially so in case (ii).
A versatile Markovian point process
Authors: Neuts Marcel F.
Published: July 2016

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.

Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.
A Poisson limit theorem for weakly exchangeable events
Authors: Eagleson G. K.
Published: July 2016

Let Y
1, Y2
, · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set

Silverman and Brown (1978) have shown that under certain conditions the statistic

is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the X
il…i
m

.
Repeated matrix games
Authors: Kamerud Dana B.
Published: July 2016

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn
) of such rewards, to which we assign the payoff lim supn (R
1 + · ·· + Rn
). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).
Strategy stability in complex populations
Authors: Hines W. G. S.
Published: July 2016

In biological populations, several strategies for engaging in intraspecific contests may be present. Investigations of stability show that the mean strategy in use in the population can prove stable but that neutrally stable and gradually changing polymorphisms in strategy are to be expected when the contest affects reproductive success.
The fair charge on a car ferry
Authors: Grey D. R.
Published: July 2016

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.
Strong consistency of a modified maximum likelihood estimator for controlled Markov chains
Authors: Doshi Bharat , Shreve Steven E.
Published: July 2016

A controlled Markov chain with finite state space has transition probabilities which depend on an unknown parameter α lying in a known finite set A. For each α, a stationary control law ϕ α
is given. This paper develops a control scheme whereby at each stage t a parameter α t
is chosen at random from among those parameters which nearly maximize the log likelihood function, and the control ut
is chosen according to the control law ϕ αt. It is proved that this algorithm leads to identification of the true α under conditions weaker than any previously considered.
Stochastic inequalities between customer-stationary and time-stationary characteristics of queueing systems with point processes
Authors: König D. , Schmidt V.
Published: July 2016

By means of a general intensity conservation principle for stationary processes with imbedded marked point processes (PMP) stochastic inequalities are proved between customer-stationary and time-stationary characteristics of queueing systemsG/G/s/r.
Linked loci in finite populations
Authors: Holgate P.
Published: July 2016

A finite population of gametes is studied, classified according to the alleles present atklinked loci. A canonical method of following the joint probability distribution of the gametic types from generation to generation is developed. It is shown how the investigation of the rate of first fixation can be systematised. Explicit results are given fork= 2, and, although not so complete, fork= 3.
Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine
Authors: Meilijson Isaac
Published: July 2016

Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovič iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.
θ-stationary point processes and their second-order analysis
Authors: Byth Karen
Published: July 2016

The concept of θ-stationarity for a simple second-order point process in R2
is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.
On queues with periodic Poisson input
Authors: Lemoine Austin J.
Published: July 2016

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.
Stationary waiting-time distributions in the GI/PH/1 queue
Authors: Neuts Marcel F.
Published: July 2016

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.
Characterization of the optimal class of output policies in a control model of a finite dam
Authors: Zuckerman Dror
Published: July 2016

In this paper we characterize the optimal class of output policies in a control model of a dam having a finite capacity. The input of water into the dam is determined by a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with a cost of K, (K ≧ 0) or decreased from M to 0 with zero cost, any such changes taking effect instantaneously. There is a reward of A monetary units for each unit of output, (A ≧ 0). The problem is to formulate an optimal output policy which maximizes the long-run average net reward per unit time.
Aspects of population genetics theory
Authors: Ewens W. J.
Published: July 2016

Population genetics and evolution have long supplied a stimulus to mathematical and statistical research. In this note a brief survey of current theory is given, with special reference to the work of P. A. P. Moran. Finally, an analysis of various biometrical questions is presented in the spirit of Moran's contributions to mathematical genetics.
The basic structures of Voronoi and generalized Voronoi polygons
Authors: Miles R. E. , Maillardet R. J.
Published: July 2016

For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified.

Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱1, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱
n
, showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱
n
relative to a homogeneous Poisson process.

Unlike 𝒱 no 𝒱
n
contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱
n
tend to circularity, so that the sums of their opposite interior angles tend to π.
Plane polygons revisited
Authors: Neumann B. H.
Published: July 2016

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.
The scale of perfection
Authors: Kendall David G.
Published: July 2016

A survey is given of the attitude to and use of the concepts of deficient, perfect, and abundant number from the time of Nicomachus of Gerasa to that of David Anakht and Alcuin of York. Alcuin's ‘Letter to Dafnin', the source of an anecdote frequently mentioned in mathematical texts, is included as an appendix.

The statistics of deficient, perfect, and abundant numbers over the range 1–50000 are studied and presented graphically in several novel ways and compared with the work of Davenport and others on the distribution-law for the values of z = σ (n)/(2n).

Some queries are raised concerning observations which ancient writers might have been expected to make; for example, did they notice that about one half of all the even numbers are abundant?
Linear regression by functional least squares
Authors: Heathcote C. R.
Published: July 2016

The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.
Distributions on the circle and sphere
Authors: Watson Geoffrey S.
Published: July 2016

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.
Another look at the finite mean supercritical Bienaymé–Galton–Watson process
Authors: Cohn Harry
Published: July 2016

Let {Z
n} be a finite mean supercritical Bienaymé– Galton–Watson process. It is known that there exist norming constants {C
n} such that {Z
n
/C
n} converges almost surely to a limit W. Also there is a whole literature concerning properties of {C
n} and W. We attempt a new approach to the limit theory of {Z
n} by relating it to the theory of sums of independent and identically distributed random variables.
Inference for general Ising models
Authors: Pickard David K.
Published: July 2016

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models.

In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.
Age dependent stochastic models for phage reproduction
Authors: Srinivasan S. K. , Rangan A.
Published: July 2016

The problem of age specific bacteriophage growth is dealt with, using the product density approach. Expressions for the product densities, taking into account the intermediate stages of transformations from the vegetative phage to mature phage, are also given.
Stochastic population processes in the theory of radiative transfer
Authors: Brockwell P. J.
Published: July 2016

Starting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t
= 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω
1, ω
2, …, ω

n
of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.
A theoretical analysis of congestion in a two-lane freeway
Authors: Mcneil Donald R.
Published: July 2016

To current workers in the mathematical theory of road traffic, one of the most challenging and important problems is to describe and analyze the congestion suffered by motorists on freeways. While there is, as yet, no satisfactory unified theory, several different models have been suggested, each of which is reasonable for a limited class of situations. These approaches fall largely into two categories. The first may be described as looking at traffic “in the large” and its principal exponents were Lighthill and Whitham (1955) and Prigogine (1961). The theory of Lighthill and Whitham involves treating the system of vehicles as a fluid subject to a rule of conservation of mass (that is, vehicles) and an assumed relation between the flow and concentration at each point. This theory is completely deterministic. Prigogine's approach is to regard the system of vehicles as behaving like the particles in a gas and subject to the laws of statistical mechanics.
Work-conserving priorities
Authors: Wolff Ronald W.
Published: July 2016

In many situations, it is reasonable to assume that a priority rule does not affect the total time spent in service of any job. Rules with this property are said to be work-conserving. This concept unifies and simplifies the analysis of a variety of priority queues. Some results are obtained for rules applied to the GI/G/1 queue. Some special properties of Poisson arrivals are discussed, and a new proof of the equivalence of averaging over all time with averaging over arrival epochs is presented. In this case, explicit results for particular rules are obtained in examples. In another example, the optimal rule (from a very restrictive class) is determined without specializing the arrival stream.
A rate of convergence result for the super-critical Galton-Watson process
Authors: Heyde C. C.
Published: July 2016

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 < m = EZ
1 < ∞. The Galton-Watson process evolves in such a way that the generating function Fn
(s) of Zn
is the nth functional iterate of F(s). The convergence problem for Z
n, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn
= m –nZn
converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn
to W.
A note on Belyaev's limiting distribution of the intervals between losses in an n-server system
Authors: Tambouratzis D. G.
Published: July 2016

The aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.
Further existence theorems for the optimal inventory equation
Authors: Boylan Edward S.
Published: July 2016

For given functions, g(x), h(x), F(x) and a constant, a, f(x) is said to be a solution of the optimal inventory equation if

where f(y - z) = f(0) when z > y. (The same convention holds in similar equations below.) A method of deriving a solution to (1) is to define inductively a sequence of approximating functions, fn(x), by:
On the waiting time distribution in a generalized GI/G/1 queueing system
Authors: Loris-Teghem Jacqueline
Published: July 2016

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N
1 phases of random duration and that a customer 𝒞
n
who arrives while the station is being closed down must wait a random time
idn
(i = 1, ···, N
1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables
idn
are identically distributed.) A customer 𝒞
n
arriving when the closing-down of the station is already accomplished has to wait a random time (N
1 + 1)
dn
corresponding to the set up time of the station. Besides, a customer 𝒞
n
who arrives when the station is busy has to wait an additional random time 0
dn. We thus have (N
1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N
2 + 2) types of service time and (N
3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution.

The process {wn
}, where wn
denotes the waiting time of customer 𝒞
n
if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn
} related to {wn
}, and then derive the distribution of the variables wn.
A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain
Authors: Lehoczky J. P.
Published: July 2016

The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized.

The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.
Sequential decisions in the control of a space-ship (terminal cost proportional to magnitude of miss distance)
Authors: Feder Paul I. , Stroud Thomas
Published: July 2016

This is an extension of the paper [1] by Bather and Chernoff, where approximations are obtained to the solution of a sequential control problem involving the decision of when to use fuel to alter course during a space-ship's approach to a target. It is desired to come as close as possible to the target. If fuel is used too early, it may be used incorrectly due to imprecise information about the location of the target; however if the fuel is used too late it may be ineffective in reducing the miss distance.
Some results on Markovian replacement processes
Authors: Mandl Petr
Published: July 2016

A replacement of type (jk) in a Markov process with rewards means an instantaneous shift of the trajectory from state j into state k. A formula is obtained for the characteristic function of the total reward from the process under an arbitrary non-anticipative replacement policy. From the formula results are deduced concerning Markov policies, optimal policies and policies minimizing the variance of the reward.
A stochastic model for the formation of a planetary system
Authors: Galley S. J.
Published: July 2016

The stochastic model consists of a two-dimensional random walk with unit steps, each step being a Bernoulli trial. Two sets of possible conditions arising from astronomical theory give rise to two different barriers in the plane of the walk which cause a return to the origin. Another barrier indicates the formation of a planet and also causes a return to the origin. Consideration of the composition of the planets formed at this barrier allows their orbital distances and direction of rotation to be calculated. The predictions of one of the two models were found to be in close agreement with the observed planetary distances and masses in the solar system.
On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals
Authors: Kyprianou E. K.
Published: July 2016

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable

denote the first busy period initiated by a waiting time u at time t = 0.
On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting
Authors: Morton R.
Published: July 2016

Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.
A note on a functional equation arising in Galton-Watson branching processes
Authors: Athreya Krishna B.
Published: July 2016

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.
A relation between stationary queue and waiting time distributions
Authors: Haji Rasoul , Newell Gordon F.
Published: July 2016

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.
A note on the estimation of the mean of a homogeneous random field
Authors: Skalsky Michael
Published: July 2016

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x
1, ···, xn
) in Rn
by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.
On the estimation of intensities in a stochastic process generated by a stochastic intensity sequence
Authors: Grandell Jan
Published: July 2016

A stationary doubly stochastic Poisson sequence is considered. Asymptotic properties of the best linear estimates of the intensities are investigated.
Correlation functions in queueing theory
Authors: Srinivasan S. K. , Subramanian R. , Vasudevan R.
Published: July 2016

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.
Embedded renewal processes in the GI/G/s queue
Authors: Whitt Ward
Published: July 2016

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.
On an epidemic in a stratified population
Authors: Watson R. K.
Published: July 2016

Most epidemic models previously studied have assumed a homogeneously mixing population. Instead of making this assumption, a population divided into classes is considered; and it is assumed that the degree of mixing between classes is less than that within classes. The stochastic model in this form is intractable and approximations are made, yielding results in reasonable agreement with simulation trials.
On the maximum of a stationary independent increment process
Authors: Ross Sheldon M.
Published: July 2016

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.
On the distribution of the number of cyclically occurring random events
Authors: Podkaminer Leon
Published: July 2016

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.
Non-linear regression for multiple time-series
Authors: Robinson P. M.
Published: July 2016

A general multivariate non-linear regression model is considered, including as special cases linear regression when the regression matrix is of less than full rank, simultaneous equations systems and regression on an unobservable predetermined variable. Given a time-series of observations at unit intervals we consider the estimation of the parameters, subject to non-linear constraints, by minimizing a criterion based on the Fourier-transformed model. We allow the residuals to be generated by a stationary, linear, process and establish asymptotic properties of our estimates.
Crossing problems for non-constant thresholds and certain non-Markov processes
Authors: Belkin Barry , Stone Lawrence D. , Snyder Martin Avery
Published: July 2016

A result in Stone, Belkin, and Snyder ((1970) J. Math. Anal. Appl.
30, 448–470) gave a method for finding the Laplace-Stieltjes transform of the distribution of certain non-negative, homogeneous, additive functionals of a Markov process with stationary transition measure. By considering certain two dimensional Markov processes and applying this result, a method is obtained for finding time above a threshold and first passage distributions for a one dimensional process either when (1) the process is Markovian and the threshold is possibly non-constant, or (2) the threshold is constant and the process is the indefinite integral of a Markov process. Specific process-threshold combinations are considered in several examples including the case of a linear threshold for a Wiener process and a for compound Poisson process with exponential (either one-sided or bilateral) after-jump distribution. In addition, the first passage distribution to a constant threshold is computed for an integrated Poisson sampling process.
Recursive matrix games
Authors: Orkin Michael
Published: July 2016

“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.
The waiting time until first duplication
Authors: Arnold Barry C.
Published: July 2016

Balls are drawn with replacement from an urn containing m distinguishable balls until a match is noted. The distribution of the number of drawings required is considered in the case where only the last k balls drawn are remembered. The asymptotic behavior of this distribution, as m and k become large, is investigated. Two further variants of the problem are suggested.
A single server queue in discrete time with customers served in random order
Authors: Balmer D. W.
Published: July 2016

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).
A correction to “The area under the infectives trajectory of the general stochastic epidemic”
Authors: Downton F.
Published: July 2016

In a recent paper (Downton (1972)) an attempt was made to use the combinatorial approach of Downton (1967) to obtain an expression for the Laplace transform (and hence the moments) of the area under the infectives trajectory of the general stochastic epidemic. The basic property of that area, which makes this possible, is that it is composed of a random number of rectangles, each of whose area has a distribution depending on the infection rate and the number of susceptibles present at the appropriate stage in the development of the epidemic, but which is independent of the number of infectives. However, Mr A. Abakuks has pointed out that in one respect the argument used was faulty. It was argued that the contribution to the area under the infectives curve of an interval ending in a reduction in the number of infectives from i to i – 1 had an exponential distribution with parameter p (the infection rate); and if the interval ended in a reduction in the number of susceptibles from s to s – 1 the distribution was exponential with parameter s. While this is true unconditionally, the combinatorial situation described was essentially a conditional one, in which each of these contributions to the total area had an exponential distribution, independent of i, but with parameter (ρ + s) regardless of whether the interval ended in a reduction in the number of infectives or of susceptibles.
Unrooted trees for numerical taxonomy
Authors: Dobson Annette J.
Published: July 2016

It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree.

This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.
Asymptotic properties of the number of replications of a paired comparison
Authors: Uppuluri V. R. R. , Blot W. J.
Published: July 2016

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.
On the inverses of some patterned matrices arising in the theory of stationary time series
Authors: Galbraith R. F. , Galbraith J. I.
Published: July 2016

Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ≧ p), and for the first order mixed autoregressive moving average process.
Searching for a one-dimensional random walker
Authors: McCabe Bernard J.
Published: July 2016

Let {xk
}
k ≧ − r
be a simple Bernoulli random walk with x
–r
= 0. An integer valued threshold ϕ = {ϕ
k
}
k≧1 is called a search plan if |ϕ
k+1−ϕ
k
|≦1 for all k ≧ 1. If ϕ is a search plan let τϕ
be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ
1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ
has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.
Optimal control of finite dams: discrete (2-stage) output procedure
Authors: Faddy M. J.
Published: July 2016

The problem discussed is that of determining optimal release rates for finite dams when the release rate may take one of two values and specific costs are involved in increasing and decreasing this release rate. From another point of view, this is a problem of controlling the output simply by switching on and off. A diffusion model is suggested and by considering a family of plausible output policies an optima procedure, which minimises the long-term average cost of operating the dam, is established. Various properties of this output policy are given.
Success runs in a two-state Markov chain
Authors: Rajarshi M. B.
Published: July 2016

Success runs of a fixed length in a two-state Markov chain are discussed. The results are analogous to those in the case of independent Bernoulli trials. The generating function of the waiting time for the first occurrence of a success run is obtained from the theory of recurrent events. Under certain conditions, the distribution of the number of long runs in a large number of trials is shown to be Poisson.
A note on the first two moments of times in transient states in a semi-Markov process
Authors: Kao Edward P. C.
Published: July 2016

This paper derives results for computing the first two moments of times in transient states and times to absorption in a transient semi-Markov process. An illustrative example is presented at the end.
An elementary proof for the Rao-Rubin characterization of the Poisson distribution
Authors: Shanbhag D. N.
Published: July 2016

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970).
Resource allocation in speculative chemical research
Authors: Gittins J. C.
Published: July 2016

A stochastic theory is given for speculative industrial chemical research. It is shown how the theory may be used to suggest a suitable balance between long-term and short-term projects.
Optimal feedback control of stochastic McShane differential systems
Authors: Ahmed N. U. , Teo K. L.
Published: July 2016

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.
The number of meetings in free Poisson traffic
Authors: Unkelbach Hans D. , Wegmann Helmut
Published: July 2016

Using Rényi's model of free Poisson traffic the distribution of the number of meetings of vehicles on a highway section during a given time interval is investigated. An integro-differential equation for the generating function of that variable is deduced and the first moments are calculated. The generating function is given explicitly in simple cases and approximately in cases of practical interest.
Some approximate results for storage systems with continuous inputs
Authors: Herbert H. G.
Published: July 2016

Some approximate results for infinite capacity storage systems in which the input rate follows a birth and death process are given in this paper. In particular, an immigration-emigration process and an Ehrenfest process are considered. The almost-null-recurrent content distribution is derived when the demand rate is constant. Further, an approximation for the content distribution during a period of excessive over-supply is obtained for the cases of constant demand and demand proportional to the content.
Strong mixing properties of linear stochastic processes
Authors: Chanda K. C.
Published: July 2016

Let {Zt
; t = 0, ± 1, ···} be a pure white noise process with γ = E{|Z
1
|
δ}< ∞ for some δ > 0. Assume that the characteristic function (ch.f.) ϕ
0 of Z
1 is Lebesgue-integrable over (—∞, ∞). Let {gv
;v = 0, 1, 2, ···, g
0 = 1} be a sequence of real numbers such that where λ = δ(1 + δ)−1. Define , where the identity is to be understood in the sense of convergence in distribution. Then {Xt
; t = 0, ± 1, ···} is a strongly mixing stationary process in the sense that if is the σ-fìeld generated by the random variables (r.v.) Xa
, ···, Xb
then for any where M is a finite positive constant which depends only on ϕ
0 and
How many random digits are required until given sequences are obtained?
Authors: Blom Gunnar , Thorburn Daniel
Published: July 2016

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.
Random coverage of the circle and asymptotic distributions
Authors: Hüsler J.
Published: July 2016

Place n arcs of equal length an
uniformly at random on the circumference of a circle. We discuss the limiting distributions of the number of gaps, the length of the maximum gap and the uncovered proportion of the circle, depending on the asymptotic behaviour of an
→ 0.
Approximate variances associated with random configurations of hard spheres
Authors: Girling A. J.
Published: July 2016

The Percus–Yevick approximation of classical liquid theory is employed to obtain variances associated with random configurations of equal spheres in ℝ3. The idea is illustrated by considering the number of spheres, and also the amount of hard material, contained within a fixed cylinder — an application of some practical value.