bluuebunny/crossref_metadata_2025 huggingface dataset:
Stochastic control of geometric processes
Authors: Aase Knut K.
Published: July 2016

Stochastic optimization of semimartingales which permit a dynamic description, like a stochastic differential equation, leads normally to dynamic programming procedures. The resulting Bellman equation is often of a very genera! nature, and analytically hard to solve. The models in the present paper are formulated in terms of the relative change, and the optimality criterion is to maximize the expected rate of growth. We show how this can be done in a simple way, where we avoid using the Bellman equation. An application is indicated.
On estimating the diffusion coefficient
Authors: Dohnal Gejza
Published: July 2016

Random processes of the diffusion type have the property that microscopic fluctuations of the trajectory make possible the identification of certain statistical parameters from one continuous observation. The paper deals with the construction of parameter estimates when observations are made at discrete but very dense time points.
Bivariate processes with positive or negative dependent structures
Authors: Ebrahimi Nader
Published: July 2016

In this paper various notions of positive and negative dependence for bivariate stochastic processes are introduced and their interrelationship is studied. Examples are given to illustrate these concepts.
Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol
Authors: Fayolle Guy , Iasnogorodski Rudolph
Published: July 2016

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn
), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn
) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.
Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts
Authors: Hanschke Thomas
Published: July 2016

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.
Sojourn times in queuing networks with multiserver modes
Authors: Schassberger R. , Daduna H.
Published: July 2016

This paper generalizes previous results for sojourn-time distributions along so-called overtake-free routes in product-form networks of queues.
Characterizing the geometric distribution using expectations of order statistics
Authors: Dallas A. C.
Published: July 2016

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.
Bounding the stochastic performance of continuum structure functions. II
Authors: Baxter Laurence A. , Kim Chul
Published: July 2016

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.
The convexity of a general performance measure for multiserver queues
Authors: Harel Arie , Zipkin Paul
Published: July 2016

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications.

We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.
Bounds on the expected waiting time in a GI/G/1 queue: upgrading for low traffic intensity
Authors: Rosberg Zvi
Published: July 2016

For a GI/G/1 queue we derive new lower and uper bounds on the expected stationary waiting time which are a function of more than the first two moments of the inter-arrival and service times. When the probability for a delay is low, the bounds tend to be better than other bounds. This is illustrated in an example of a Uniform/Uniform/1 queue.
Epidemics with carriers: A note on a paper of Dietz
Authors: Downton F.
Published: July 2016

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.
A generalized bivariate exponential distribution
Authors: Marshall Albert W. , Olkin Ingram
Published: July 2016

In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age.

The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.
Linear processes are nearly Gaussian
Authors: Mallows C. L.
Published: July 2016

LetUdenote the set of all integers, and suppose thatY= {Yu;u∈U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.)G(·). Leta= {au;u∈U} be a sequence of real numbers with Σuau2= 1. ThenXu= ΣwawYu–wdefines a stationary linear processX= {Xu; u ɛ U} withE(Xu) = 0,E(Xu2) = 1 foru∊U. LetF(·) be the d.f. ofX0.We prove that if maxu|au| is small, then (i) for eachw, Xwis close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy≦gmaxu|au| where Φ(·) is the standard Gaussian d.f., andgdepends only onG(·); (ii) for each finite set (w1, …wn), (Xw1, …Xwn) is close to Gaussian in a similar sense; (iii) theprocess Xis close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filterais studied.
Statistical relativistic particle
Authors: Gilson J. G.
Published: July 2016

This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.
A diffusion model for the control of a dam
Authors: Bather J. A.
Published: July 2016

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.
A Note on the First Emptiness Problem of a Finite Dam with Poisson Type Inputs
Authors: Phatarfod R. M.
Published: July 2016

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation

where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.
A branching process with disasters
Authors: Kaplan Norman , Sudbury Aidan , Nilsen Trygve S.
Published: July 2016

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.
A note on the age-dependent branching process with immigration
Authors: Kaplan Norman
Published: July 2016

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} <∞. This result is a direct generalization of a theorem proven for the branching process without immigration.
The lengths of fragments covering a fixed marker in randomly cut linear or circular DNA
Authors: Litwin Samuel
Published: July 2016

A linear DNA molecule may be labelled at a fixed locus by a minute complementary radioactive molecule. A collection of identical molecules is to be so labelled and each one independently cut at a random number, N, of random places, X (N Poisson, X uniform). Fragments containing the label are to be collected and assayed by length. It is shown that the recovery pattern (fragment length distribution) contains a jump discontinuity at the fixed locus and may be used to determine the distance between the attachment site and the nearest end of the molecule.

The recovery pattern under the hypothesis that the collection of molecules are circularly permuted, i.e. the labelled locus is uniformly distributed over the length of the molecule, does not contain such discontinuities. The case where the labelling molecule has a non-negligible extent is also treated.
On the Weak form of Zipf's law
Authors: Chen Wen-Chen
Published: July 2016

Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet–multinomial urn model for describing the above phenomena from a stochastic point of view.

We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.
On the lengths of the pieces of a stick broken at random
Authors: Holst Lars
Published: July 2016

Consider the pieces of a randomly broken stick. How long is the j th longest piece? How many breaks are necessary for getting all pieces less than a given length? These and related questions are studied in particular when the number of pieces is large. Using simple properties of the exponential distribution new proofs are given of old results and new results are obtained.
On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean
Authors: Cohn H. , Schuh H.-J.
Published: July 2016

It is shown that the limiting random variable W(si
) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si
) < ∞}. This implies that for all branching processes (Zn
) with infinite mean there exists a function U such that the distribution of V = lim
n

U(Zn
)e–n
a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si
) is derived.
On the Kesten–Spitzer algorithm for minimal hitting time in controlled Markov chains
Authors: Bhattacharjee Manish C. , Basu Sujit K.
Published: July 2016

For a Markov chain with optional transitions, except for those to an arbitrary fixed state accessible from all others, Kesten and Spitzer proved the existence of a control policy which minimized the expected time to reach the fixed state and for constructing an optimal policy, proposed an algorithm which works in certain cases. For the algorithm to work they gave a sufficient condition which breaks down if there are countably many states and the minimal hitting time is bounded. We propose a modified algorithm which is shown to work under a weaker sufficient condition. In the bounded case with countably many states, the proposed sufficient condition is not necessary but a similar condition is. In the unbounded case as well as when the state space is finite, the proposed condition is shown to be both necessary and sufficient for the original Kesten–Spitzer algorithm to work. A new iterative algorithm which can be used in all cases is given.
On the continuity of the time constant of first-passage percolation
Authors: Cox J. Theodore , Kesten Harry
Published: July 2016

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n
be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk
)→ μ(U) whenever Uk
converges weakly to U.
On the approximation of convex, closed plane curves by multifocal ellipses
Authors: Vincze P. Erdös I.
Published: July 2016

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.
On the distribution of Hotelling's one-sample T
2 under moderate non-normality
Authors: Davis A. W.
Published: July 2016

The asymptotic distribution of Hotelling's one-sample T
2 in multivariate Edgeworth populations is expanded to terms of the first order. Comparison with published simulation results indicates that the result is quite useful, even in cases where the underlying population is not well represented by an Edgeworth expansion.
Finite dams with inputs forming a Markov chain
Authors: Ali Khan M.S.
Published: July 2016

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.
The finite dam
Authors: Roes P. B. M.
Published: July 2016

Denote by X(t) the total quantity of water flowing into a weir of capacity K in the interval (0, t). Unless the weir is empty, there is a continuous release of water at unit rate; if it becomes full, the excess water overflows instantaneously.
A theorem on discrete time branching processes allowing immigration
Authors: Reynolds John F.
Published: July 2016

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn
(x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn–

1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn
the population size in the nth generation. We write
Two classification theorems of states of Markov chains
Authors: Kim C. W.
Published: July 2016

Two theorems on Markov chains, both of which already appear in the literature: the classification of the states into the set of all non-recurrent (transient) states and recurrent classes, and the corresponding classification for idempotent Markov chains due to Doob, are proved from the viewpoint of ergodic theory.
Physical nearest-neighbour models and non-linear time-series
Authors: Bartlett M. S.
Published: July 2016

A general class of spatial-temporal Markov processes is defined leading to the standard spatial equilibrium distribution for nearest-neighbour models on a multi-dimensional lattice. Physical properties are obtainable from the marginal spatial spectral function. However, only the simplest one-dimensional case corresponds to a linear model with a readily derived spectrum. Non-linear models corresponding to two- and three-dimensional lattices are presented in their simplest terms, and a preliminary discussion of approximate solutions is included.
On the comparison of a theoretical and an empirical distribution function
Authors: Takács Lajos
Published: July 2016

Let ξ
1
, ξ2, ···, ξm
be mutually independent random variables having a common distribution function
P
{ξ
r
≦x} = F(x)(r = 1, 2, ···, m). Let Fm
(x) be the empirical distribution function of the sample (ξ
1, ξ
2
, ···, ξm), that is, Fm
(x) is defined as the number of variables ≦x divided by m.
A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case
Authors: Puri Prem S.
Published: July 2016

The subject of this paper is the study of the distribution of integrals of the type

where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15]), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3], Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19], it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y(t) with f(X(t), t) = bX(t), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).
On generating a random sequence
Authors: Hwang F. K. , Lin S.
Published: July 2016

Suppose there are k kinds of balls identical except for their colours. The number of balls of the ith colour is ri
(i = 1, 2, ···, k) and Put down the R balls in a sequence (a line), then an acceptable sequence is one which does not have two adjacent balls of the same colour. The problem is to generate a random sample of the set of acceptable sequences S(r
1, r
2, ···, rk
).
On certain unrestricted, linear, unit step, continuous time random walks
Authors: Gibson A. E. , Conolly B. W.
Published: July 2016

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt
and B(t) = 1 – e–μt
, we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt
and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt
. M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.
Denumerable Markov processes with bounded generators: a routine for calculating pij
(∞)
Authors: Jensen Arne , Kendall David
Published: July 2016

1. Let the (honest) Markov process with transition functions (pij
(0)) have transition rates (qij
) and suppose that, for some M,

so that the matrix Q = (qij
) determines a bounded operator on the Banach space l
1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and Ω
F
will be a closed restriction of Q with dense domain, so that Ω
F
= Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt
= (pij
(t)) is given by

where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P
∞ = (lim
t→∞
pij
(t)).
Study of a special M/M/1 queue
Authors: Tambouratzis D. G.
Published: July 2016

To optimize the performance of an M/M/s queue, Moder and Phillips [1] have studied the problem in which there are always n
1 open counters ready to offer service. The actual number of open counters increases by one, up to a maximum n
2, every time the number of waiting customers exceeds a certain critical number; it decreases by one every time the queue length becomes smaller than another critical number.
Optimal stopping of constrained Brownian motion
Authors: Föllmer Hans
Published: July 2016

We discuss the potential theory of optimal stopping for a standard process and an unbounded reward function. This is applied to Brownian motion constrained to aN(m,σ2) distribution at time 1. Boyce [2] has discovered, via computer, various interesting features of this example. We provide direct proofs for some of them, in particular for the qualitative jump of the optimal strategy as the varianceσ2passes the critical value 1.
On the work load process in a general preemptive resume priority queue
Authors: Schassberger R.
Published: July 2016

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

Using the method of phases and a resolution technique from the theory of Markov processes we obtain Laplace transforms of various distributions.
A theorem on Markov branching processes
Authors: Reynolds John F.
Published: July 2016

This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value.
Additional quasi-stationary distributions for semi-Markov processes
Authors: Flaspohler David C. , Holmes Paul T.
Published: July 2016

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits:

where Xj
(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.
The linear cell-size-dependent branching process
Authors: Clifford Peter , Sudbury Aidan
Published: July 2016

In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.
General solidarity theorems for semi-Markov processes
Authors: Cheong Choong K. , Teugels Jozef L.
Published: July 2016

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij
(t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij
(t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij
(t). Among others it is shown that f(t) can be taken regularly varying.
The quasi-stationary distributions of queues in heavy traffic
Authors: Kyprianou E. K.
Published: July 2016

This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn
} of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.
A note on the number of busy servers in a GI/G/s queue in light traffic
Authors: Kennedy Douglas P.
Published: July 2016

It is proved that in a GI/G/s queue with the traffic intensity ρ < (S – m + l)/S, m ≧ 1, at least m servers are simultaneously idle infinitely often with probability one.
Analytic solution of a finite dam governed by a general input
Authors: Srinivasan S. K.
Published: July 2016

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.
An approximation for the busy period of the M/G/1 queue using a diffusion model
Authors: Heyman D. P.
Published: July 2016

A diffusion model for the M/G/1 queue due to D. P. Gaver is used to obtain an approximation for the density function of the busy period. The approximation has the same mean and variance as the exact density function, and can be given explicitly when the service time is constant, or has a negative exponential or gamma distribution, or is a mixture of these types.
Anisotropically random mosaics
Authors: Moore Marc
Published: July 2016

The concept of isotropically random mosaic introduced by Pielou, is recalled. The notion of anisotropically random mosaic with respect to a point is defined. An example is given.
Geometric and exponential decay in derived Markov chains
Authors: Callaert Herman
Published: July 2016

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.
On a constrained optimal stopping problem
Authors: Kennedy D. P.
Published: July 2016

For a sequence of random variables {Xn
, n ≧ 0}, optimal stopping is considered over stopping times T constrained so that ET ≦ α, for some fixed α > 0. It is shown that under certain circumstances a Lagrangian approach may be used to reduce the problem to an unconstrained optimal stopping problem of a conventional type. The optimal value of the natural dual problem is shown to be equal to the optimal value of the original (primal) problem when certain randomised stopping times are permitted. Two examples are considered in detail.
On the stochastic ordering of waiting times for patterns in sequences of random digits
Authors: Blom Gunnar
Published: July 2016

Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game.
Comparison of self-organizing linear search rules
Authors: Lam K.
Published: July 2016

Consider a file with n records denoted by R
1, R
2
, …, Rn
. At each access of a record, the file has to be searched sequentially and it is assumed that the search cost is proportional to the number of probes needed to retrieve the records. The access probabilities p
1, p
2, …, pn
are assumed to be unknown constants and accesses are assumed to be independent. A move-forward self-organizing rule moves a record accessed in the ith position to the li
th position without changing the relative ordering of other records where li
, = 1, li
, < i for i = 2, …, n and l

i+1≧li
. A move-forward rule R is said to be ≦ another move-forward rule R′ if l′
i
≦li
for all i. It is shown that when p
2 = p
3 = ··· = p

n
, R ≦ R
′, implies cost R ≦ cost R′
. This is a generalization of some known results. A new consequence is that the move-up-k + 1 rule is more costly than the move-up-k rule and the move-to-front rule is the most costly of all move-forward rules.
The steady-state behaviour of multistate monotone systems
Authors: Natvig Bent , Streller Arnfried
Published: July 2016

In this paper the steady-state behaviour of multistate monotone systems of multistate components is considered by applying the theory for stationary and synchronous processes with an embedded point process. After reviewing some general results on stationary availability, stationary interval availability and stationary mean interval performance probabilities, we concentrate on systems with independently working and separately maintained components. For this case an explicit formula is given for the mean time which the system in steady state sojourns in states not below a fixed critical level.
On the expected width function for topologically random channel networks
Authors: Troutman Brent M. , Karlinger Michael R.
Published: July 2016

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.
Continuous majorisation and randomness
Authors: Hickey Raymond J.
Published: July 2016

Majorisation is used to compare continuous distributions in terms of randomness. General results on randomness in the continuous case are given and these are used to investigate the connection between randomness and parameter values in some well-known families of distributions including the normal and gamma.
Non-strong mixing autoregressive processes
Authors: Andrews Donald W. K.
Published: July 2016

Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.
An optimal sequential policy for controlling a Markov renewal process
Authors: McNamara J. M.
Published: July 2016

This paper discusses a renewal process whose time development between renewals is described by a Markov process. The process may be controlled by choosing the times at which renewal occurs, the objective of the control being to maximise the long-term average rate of reward. Let γ ∗ denote the maximum achievable rate. We consider a specific policy in which a sequence of estimates of γ ∗ is made. This sequence is defined inductively as follows. Initially an (a priori)estimate γo is chosen. On making the nth renewal one estimates γ ∗ in terms of γ
o, the total rewards obtained in the first n renewal cycles and the total length of these cycles. γ n
then determines the length of the (n + 1)th cycle. It is shown that γ n
tends to γ ∗ as n tends to∞, and that this policy is optimal.

The time at which the (n + 1)th renewal is made is determined by solving a stopping problem for the Markov process with continuation cost γ n
per unit time and stopping reward equal to the renewal reward. Thus, in general, implementation of this policy requires a knowledge of the transition probabilities of the Markov process. An example is presented in which one needs to know essentially nothing about the details of this process or the fine details of the reward structure in order to implement the policy. The example is based on a problem in biology.
Next-fit bin packing with random piece sizes
Authors: Halfin Shlomo
Published: July 2016

Investigations into time-slotted communication channels for transmission of data packets led us to analyze the stochastic behavior of the next-fit bin packing algorithm. In this paper we obtain results for general piece-size distributions and truncated distributions, we calculate explicit solutions for the case of the truncated exponential, and we apply the results to calculate the maximal throughput of the time-slotted communication channel.
Random sequential packing simulations in three dimensions for aligned cubes
Authors: Cooper Douglas W.
Published: July 2016

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L
3/NS
3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L
3/NS)⅓
. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.
A linear random growth model
Authors: Quine M. P. , Robinson J.
Published: July 2016

Points start to form on an ‘uncovered' unit interval according to a Poisson process with parameter λ. From newly formed points a covering region grows in both directions at velocity v, while new points continue to form on uncovered parts of the interval. Eventually the whole interval will be covered. Let N ≧ 1 denote the total number of points formed. We derive integral expressions for E(N) and Var(N) and give precise asymptotic expressions for these moments as ρ = λ/v →∞. Asymptotic normality of N is also established.
The Mardia–Dryden shape distribution for triangles: a stochastic calculus approach
Authors: Kendall David G.
Published: July 2016

A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.
Computer experimentation in applied probability
Authors: Neuts Marcel F.
Published: July 2016

This is a discussion of the place of empirical methods in the study of probability models. It is argued that, for models of great complexity, careful experimentation may be a legitimate source of factual information on the behavior of such models. The need for rigorous standards of empirical methodology and of appropriate reporting is stressed. A simple, but interesting question on the distribution of the maximal eigenvalue of positive matrices with random elements is used as an example of an experimental study. In the course of the discussion of that example, several operational rules of sound computer experimentation are stated.
A simple proof of the multivariate random time change theorem for point processes
Authors: Brown Timothy C. , Nair M. Gopalan
Published: July 2016

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.
On the range of a constrained random walk
Authors: Den Hollander W. Th. F. , Weiss G. H.
Published: July 2016

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.
The optimal value of markov stopping problems with one-step look ahead policy
Authors: Yasuda Masami
Published: July 2016

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.
Some results for repairable systems with general repair
Authors: Kijima Masaaki
Published: July 2016

In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age Vn –

1 = y immediately after the (n – l)th repair, the nth failure-time Xn
is assumed to have the survival function where is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An
taking a value between 0 and 1, where An
denotes the degree of the nth repair. For the extremal values 0 and 1, An
= 1 means a minimal repair and An= 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn = Vn

– 1
+ AnXn
as Model 1 and Vn = An
(Vn

– 1 + Xn
) as Model II. Various monotonicity properties of the process with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn
] when a general repair is used is derived.
Stochastic formation of hierarchies
Authors: Helms L. L.
Published: July 2016

In this paper, N particles el
, · ··, eN
occupying positions in finite state spaces S
1, · ··, SN
, respectively, are considered along with an element of a finite space S N+

1 of hierarchies which describe the organization of the particles into pairs, pairs of pairs, etc. A stochastic process {X
t: t
0} on a probability measure space (Ω, , P) is a hierarchic process if it is a right-continuous Markov jump process taking on values in the state space
Sj
and having the property that the (N + 1)th component of Xt
can jump from a hierarchy to a successor or antecedent of that hierarchy. Asymptotic distributions of perturbed hierarchic processes, bilateral processes, and unilateral processes are determined in terms of an interaction function and the asymptotic distributions of the particles in the absence of any interaction.
Characterizations of the normal distribution by suitable transformations
Authors: Beer S. , Lukacs E.
Published: July 2016

Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero.

In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.
The distribution of the maximum of a random number of random variables with applications
Authors: Galambos Janos
Published: July 2016

The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group.
On the busy period of the modified GI/G/1 queue
Authors: Pakes A. G.
Published: July 2016

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of aGI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/M/1 queue.
Some remarks on a result of Blomqvist
Authors: Speed T. P.
Published: July 2016

An alternative proof of a result of Blomqvist on the “heavy-traffic” problem for the two-barrier random walks using weak convergence methods is indicated.
A boundary-crossing result for Brownian motion
Authors: Scheike Thomas H.
Published: July 2016

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.