bluuebunny/crossref_metadata_2025 huggingface dataset:
On Monte Carlo algebra
Authors: Gordon Richard
Published: July 2016

A Monte Carlo method is proposed and demonstrated for obtaining an approximate algebraic solution to linear equations with algebraic coefficients arising from first order master equations at steady state. Exact solutions are hypothetically obtainable from the spanning trees of an associated graph, each tree contributing an algebraic term. The number of trees can be enormous. However, because of a high degeneracy, many trees yield the same algebraic term. Thus an approximate algebraic solution may be obtained by taking a Monte Carlo sampling of the trees, which yields an estimate of the frequency of each algebraic term. The accuracy of such solutions is discussed and algorithms are given for picking spanning trees of a graph with uniform probability. The argument is developed in terms of a lattice model for membrane transport, but should be generally applicable to problems in unimolecular kinetics and network analysis. The solution of partition functions and multivariable problems by analogous methods is discussed.
Solutions for some diffusion processes with two barriers
Authors: Sweet A. L. , Hardin J. C.
Published: July 2016

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.
On the solution of differential equations arising in some attachment models of virology
Authors: Morgan Byron J. T.
Published: July 2016

(1.1) In this paper, acquaintance will be assumed with the basic mechanics of phage/bacterium, antibody/virus interactions; for background reading see, for example, Adams (1959), Bouanchaud (1970), Durham and King (1969) and Fraser (1967).
Branching processes with terminal types: application to context-free grammars
Authors: Sankoff David
Published: July 2016

In this note we consider multi-type branching processes where certain terminal types of particles, once created, are subject neither to death nor to further reproduction.
Equilibrium behavior of a stochastic system with secondary input
Authors: Şahin İzzet
Published: July 2016

Equilibrium behavior of a stochastic system with two types of input of different statistical nature and with linear continuous output is investigated. The results have applications in queueing theory, storage theory and insurance-risk theory.
On a stochastic integral equation of the Volterra type in telephone traffic theory
Authors: Padgett W. J. , Tsokos C. P.
Published: July 2016

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form

where:

(i)

ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);

(ii)

x(t; ω) is the unknown random variable for t ∊ R
+, where R
+ = [0, ∞);

(iii)

y(t; ω) is the stochastic free term or free random variable for t ∊ R
+;

(iv)

k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and

(v)

f(t, x) is a scalar function defined for t ∊ R
+ and x ∊ R, where R is the real line.
Occupation times for two state Markov chains
Authors: Pedler P. J.
Published: July 2016

Consider first a Markov chain with two ergodic states E
1 and E
2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z
0, Z
1, Z
2, ···, Zn
by

then the conditional probabilities

for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is
Weak convergence of first passage time processes
Authors: Whitt Ward
Published: July 2016

LetD = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic processXinD,let the associatedsupremum processbeS(X), wherefor anyx ∊ D. It is easy to verify thatS:D→Dis continuous in any of Skorohod's (1956) topologies extended fromD[0,1] toD[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergenceXn⇒XinDimplies weak convergenceS(Xn) ⇒S(X) inDby virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).
Stochastic inventory models with bulk demand and state-dependent leadtimes
Authors: Gross Donald , Harris Carl M. , Lechner James A.
Published: July 2016

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.
A note on linear estimation of the intensity in a doubly stochastic Poisson field
Authors: Grandell Jan
Published: July 2016

A realization of a doubly stochastic Poisson field is assumed to be observed in a Borel set S ⊂ Rk. The best linear estimate of the realization of the intensity field (at an arbitrary but fixed point x ∈ S) which generated the observation is obtained.
The transition probabilities of the extended simple stochastic epidemic model and the Haskey model
Authors: Kryscio Richard J.
Published: July 2016

We present a solution to a special system of Kolmogorov forward equations. We use this result to present a useful expression for the transition probabilities of the extended simple stochastic epidemic model and an epidemic model involving cross-infection between two otherwise isolated groups.
Some characterizations based on the Bhattacharya matrix
Authors: Shanbhag D. N.
Published: July 2016

Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.
Chebyshev-like inequalities for dam models
Authors: Turnbull Bruce W.
Published: July 2016

A dam model is proposed for which it is assumed that the conditional distributions of the inputs, given the past, are known only to lie in some class
M

. For selected
M
, bounds are derived on various quantities of interest such as the mean time to first emptiness. The case of normal inputs is treated in greater detail and a release rule is discussed. The techniques used are similar to those used in the theory of gambling as developed by Dubins and Savage (1965).
On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times
Authors: Loris-Teghem Jacqueline
Published: July 2016

A generalized queueing system with (N+ 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.

The standardKl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.
A comparison of linear versus non-linear prediction for polynomial functions of the Ornstein-Uhlenbeck process
Authors: III John Donelson , Maltz Fred
Published: July 2016

The general minimum mean square error prediction problem is studied for the class of processes defined by polynomial functions of the Ornstein-Uhlenbeck process. In particular, the relative error difference for non-linear prediction compared with optimal linear prediction is calculated. Limiting values are determined, including an upper bound for the maximal relative error difference. These results show no difference over the entire class for both long and short prediction lead times. For a single Hermite polynomial, the relative error difference is zero. An analytical solution for the maximal difference for lead times below a given threshold is derived. This latter result has been verified numerically on a digital computer for polynomials up to 17th degree.
A note on boundary - crossing probabilities for the Brownian motion
Authors: Smith C. S.
Published: July 2016

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob.
8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.
Solving a singular diffusion equation occurring in population genetics
Authors: Jensen Louis
Published: July 2016

A technique for solving a partial differential equation is presented. The technique is based upon the known solution of a similar equation. The method is used to attempt to solve the equation describing the change in the frequency of an allele in the presence of selection and random drift in a finite population. Two cases can be solved within a reasonable degree of approximation: (a) the viabilities are additive and (b) heterozygotes are symmetrically overdominant to the homozygotes. The solutions in both cases are compared with the exact discrete solutions found by powering the re evant transition matrix.
Un théorème d'unicité pour les hyperplans poissoniens
Authors: Matheron G.
Published: July 2016

A stationary Poisson process of hyperplanes in Rn
is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.
A two-dimensional random walk in the presence of a partially reflecting barrier
Authors: Cressie Noel
Published: July 2016

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.
A limit theorem with applications in order statistics
Authors: Galambos János
Published: July 2016

Let A
1, A
2, ···, An
be events on a given probability space and let Br, n
be the event that exactly r of the A's occur. Let further Sk
(n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n
), as n→ + ∞, in terms of limits of the Sk
(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X
1, X
2, ···, Xn
by choosing Aj
= {Xj
≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.
Asymptotic growth of a class of size-and-age-dependent birth processes
Authors: Waugh W. A. O'N.
Published: July 2016

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.
Random collision processes and their limiting distribution using the discrimination information
Authors: Shōichi Nishimura
Published: July 2016

By analogy with statistical mechanics random collision processes are considered. In two cases we derive their limiting distributions using discrimination information. In one case the limiting distribution is unique and in the other one the family of limiting distributions is an exponential family.
Limiting diffusions for the conditioned M/G/1 queue
Authors: Kennedy Douglas P.
Published: July 2016

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.
Some inequalities on variances
Authors: Goldstein Matthew
Published: July 2016

Sharp upper and lower bounds for the variance of a non-negative function of a non-negative random variable are obtained under rather weak hypotheses regarding the function. Comparisons between bounds are made and some specific examples are considered.
Un modele probabiliste utile en biologie moleculaire
Authors: Joffe Anatole
Published: July 2016

Let X
1 ··· Xn
be i.i.d.r.v.'s uniformly distributed on [0, l]. Let X
(1) ··· X
(n) be the ordered r.v.'s 0≦ X
(1) ≦ X
(2)···≦ X
(n) ≦ l. We obtain the n + 1 intervals (X
(i), X
(i+1))
n

i=0, with X
(0) = 0, X
(n+1)= l of length Li
= X
(i+1) – X
(i). Let N(c) be the number of intervals of length L
i < c. We have where

The study of Nc
is of some interest in molecular biology and was studied in [3] where the author found a complicated expression for the probability for a randomly chosen segment to have a length less than or equal to c; he then obtains an approximation. From the experimental point of view, it seems more reasonable to work with the mean since one can obtain an unbiased estimator that way.
Multivariate monotone likelihood ratio and uniform conditional stochastic order
Authors: Whitt Ward
Published: July 2016

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn
These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn
. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.
On the evaluation of suboptimal strategies for families of alternative bandit processes
Authors: Glazebrook K. D.
Published: July 2016

Families of alternative bandit processes have been used as models for problems in a variety of areas. Optimal strategies for these decision processes are determined by dynamic allocation indices. These indices are here shown to play an important role in the evaluation of suboptimal strategies.
Non-negative matrices, dynamic programming and a harvesting problem
Authors: Grey D. R.
Published: July 2016

Existing results on the asymptotic behaviour of solutions to two dynamic programming equations involving non-negative matrices are reviewed and strengthened in certain directions. The results are then applied to strategies for harvesting of a small population so as to optimise its survival potential in a limited environment.
Rarefactions of compound point processes
Authors: Serfozo Richard F.
Published: July 2016

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.
Limit laws for kth order statistics from strong-mixing processes
Authors: Dziubdziela W.
Published: July 2016

We present necessary and sufficient conditions for the weak convergence of the distributions of the kth order statistics from a strictly stationary strong-mixing sequence of random variables to limit laws which are represented in terms of a compound Poisson distribution. The obtained limit laws form a class larger than that occurring in the independent case.
Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space
Authors: Hall Peter
Published: July 2016

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an
→ 0. The formulae separate naturally into three cases, corresponding to nan
→ 0, nan → a (0 < a <∞) and nan
→∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.
Continuum structures I
Authors: Baxter Laurence A.
Published: July 2016

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.
The non-ergodic Jackson network
Authors: Goodman Jonathan B. , Massey William A.
Published: July 2016

We generalize Jackson's theorem to the non-ergodic case. Here, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does. We do so by formulating and algorithmically solving a new non-linear throughput equation. These results, together with the ergodic results and the ones for closed networks, completely characterize the large-time behavior of any Jackson network.
A note on extinction criteria for bisexual Galton-Watson processes
Authors: Bruss F. Thomas
Published: July 2016

This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).
On non-singular Markov renewal processes with an application to a growth–catastrophe model
Authors: Niemi Seppo
Published: July 2016

The paper is concerned with Markov renewal processes satisfying a certain non-singularity condition. The relation of this condition to irreducibility, Harris recurrence and regularity of the associated forward Markov process is studied. This enables one to prove limit theorems of a total variation type for Markov renewal processes and semi-regenerative processes by applying Orey's theorem to the forward process. The results are applied to a GI/G/1 queue and a growth-catastrophe population model.
A note on first-passage time and some related problems
Authors: Nobile A. G. , Ricciardi L. M. , Sacerdote L.
Published: July 2016

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.
A rearrangement inequality for the longest run, with an application to network reliability
Authors: Tong Y. L.
Published: July 2016

Let X
l, · ··, Χ n
be independent binary variables with parameters θ
l, · ··, θ n
respectively, and let R denote the length of the longest run of 1's. This note concerns a new expression for and a rearrangement inequality. The inequality is applied to solve an optimal permutation problem for consecutive-k-out-of-n: F networks, and its implications on a recent conjecture of Derman et al. (1982) are discussed.
A stress–strength system
Authors: Ebrahimi Nader
Published: July 2016

Suppose Y is the strength of a system subjected to a sequence of stresses X
1, X
2, X
3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by

where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.
On reinforcement-depletion compartmental urn models
Authors: Donnelly Peter , Whitt Ward
Published: July 2016

We verify and extend a conjecture of Purdue (1981) concerning the stochastic monotonicity of absorption times in a class of compartmental urn models. We also describe the effect of increased variability in the reinforcement sizes. Finally, we investigate variability in the content process for large populations. In many applications, compartmental models substantially under-represent the variability observed in the data, so that there has been considerable interest in modifying the model to increase the variability. We show that the squared coefficient of variation of the content is not asymptotically negligible when both the size and the variability of the reinforcements are of the same order as the initial population.
Optimal inspection policy for a multi-unit machine
Authors: Zuckerman Dror
Published: July 2016

We consider a machine which generates income at rate I during its operating time. The machine is composed of N independent stochastically failing units. A failure of each one of the units causes a breakdown of the machine.

The machine's status (good or failed) is observed continuously, by a controller, at zero cost. In the event of a breakdown, exactly one unit is failed, and a series of inspections is performed, in order to identify the failed unit and the reasons for its failure. At any time along the inspection process only one unit can be tested.

An inspection of a given unit is characterized by its cost rate and inspection time. At the end of the inspection process, the failed unit is repaired at a known cost, and a new operating cycle is started.

Our objective is to formulate an optimal inspection strategy under two optimality criteria: long-run average net income, and total expected discounted net income.
Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications
Authors: Shanthikumar J. George , Sumita Ushio
Published: July 2016

We consider a discrete-time stochastic process {Wn
, n≧0} governed by i.i.d random variables {ξ n
} whose distribution has support on (–∞,∞) and replacement random variables {Rn
} whose distributions have support on [0,∞). Given Wn, Wn

+ 1 takes the value Wn
+ ζ n

+ 1 if it is non-negative. Otherwise Wn

+ 1 takes the value Rn +

1 where the distribution of Rn

+ 1 depends only on the value of Wn + ζn +

1
. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn
= 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.
Stationary and stability of fork-join networks
Authors: Konstantopoulos Panagiotis , Walrand Jean
Published: July 2016

We consider a fork-join network with arrival and service times forming a stationary and ergodic process. The usual stability condition, namely that the input rate is strictly less than all the service rates, is proved to be valid in this general case. Finally we extend the result to the case where there is random routing.
Quasi-ergodicity for non-homogeneous continuous-time Markov chains
Authors: Zeifman A. I.
Published: July 2016

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l
1
. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.
Stationary increments in the accumulated work process in processor-sharing queues
Authors: Foley R. D. , Klutke Georgia-Ann
Published: July 2016

We present a new approach to the processor-sharing queue that allows us to study the accumulated work process of a job that requires an amount of processing time x. Our approach simplifies the proofs of some earlier results on expected conditional response times and extends them to the M/G/φ (·) class. The approach illuminates some of the paradoxical features of these systems.
Spatial patterns arising from plant dispersal as modelled by a correlated random walk
Authors: Routledge R. D.
Published: July 2016

A correlated random walk in the plane is studied for which the direction of a step depends on the direction of the previous step. Both step direction and step length are continuous random variables. Such a random walk has been used to model the vegetative dispersal of certain plant populations. The analysis provides general conclusions about dependencies on parameters, an efficient scheme for generating numerical results, and testable predictions for plant populations.
Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains
Authors: Bucklew James A , Ney Peter , Sadowsky John S.
Published: July 2016

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.
Stationary distributions for piecewise-deterministic Markov processes
Authors: Costa O. L. V.
Published: July 2016

In this paper we show that the problem of existence and uniqueness of stationary distributions for piecewise-deterministic Markov processes (PDPs) is equivalent to the same problem for the associated Markov chain, so long as some mild conditions on the parameters of the PDP are satisfied. Our main result is the construction of an invertible mapping from the set of stationary distributions for the PDP to the set of stationary distributions for the Markov chain. Some sufficient conditions for existence are presented and an application to capacity expansion is given.
First-order autoregressive models for gamma and exponential processes
Authors: Sim C. H.
Published: July 2016

In this paper we propose an autoregressive representation for a particular type of stationary Gamma(θ
–1, v) process whose n-dimensional joint distributions have Laplace transform |In
+ θSnVn
|
–v
, where Sn = diag(s
1, · ··, sn
), Vn
is an n × n positive definite matrix with elements υ ij = p|i–j|i

2, i, j = 1, ···, n and p is the lag-1 autocorrelation of the gamma process. We also generalize the two-parameter NEAR(1) model of Lawrance and Lewis (1981) to an exponential first-order autoregressive model with three parameters. The correlation structure and higher-order properties of the two proposed models are also given.
Optimal R&D programs in a random environment
Authors: Posner M. J. M. , Zuckerman D.
Published: July 2016

Our study examines a stochastic R&D model with flexible termination time and without rivalry. Specifically, we assume a stochastic relationship between expenditures rate and the project's status. Furthermore, the termination time of the project is incorporated into the R&D model as a decision variable by allowing the controller to ‘sell' the obtained technology from the project at any point of time. The proposed framework extends the classical approach in the R&D literature.

The main purpose of our study is to determine the optimal stopping time of the project and to characterize qualitatively the firm's expenditure strategy. We show that under certain realistic conditions, the optimal stopping strategy is a control limit policy. Furthermore, the research effort increases monotonically over the development time of the project.
On information-based minimal repair and the reduction in remaining system lifetime due to the failure of a specific module
Authors: Natvig Bent
Published: July 2016

The first part of this paper is inspired by a somewhat surprising result in Arjas and Norros (1989). Here we give some results comparing remaining system lifetime just after a ‘black box' minimal repair of a system and after a natural minimal repair based on information on the component level. In the second part we consider the reduction in remaining system lifetime due to the failure of a specific module and explore the relation to the reduction in remaining system lifetime due to the failure of a component inside the module. This former reduction also equals the increase in remaining system lifetime due to a minimal repair of the module at its time of failure. The expected value of this reduction/increase is proportional to the so-called Natvig measure of the importance of the module.
Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy
Authors: Rhee Hahn-Kyou , Sivazlian B. D.
Published: July 2016

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.
A note on sojourn times in queuing networks with multiserver nodes
Authors: Hemker Josef
Published: July 2016

Time-reversal arguments can be used to re-derive (and slightly generalize) previous results for sojourn-time distributions in product-form queuing networks.
Interparticle dependence in a linear death process subjected to a random environment
Authors: Lefèvre Claude , Michaletzky György
Published: July 2016

Recently, Ball and Donnelly (1987) investigated the nature of the interparticle dependence in a death process with non-linear rates. In this note, after some remarks on their result, a similar problem is examined for a linear death process where the death rate per particle is a monotone function of the current state of a random environment. It is proved that if the exterior process involved is a homogeneous birth-and-death process valued in ℕ, then the survival times of any subset of particles are positively upper orthant dependent. A simple example shows that this property is not valid for general exterior processes.
On the maximum and its uniqueness for geometric random samples
Authors: Bruss F. Thomas , O'cinneide Colm Art
Published: July 2016

Given n independent, identically distributed random variables, let ρ n
denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n
's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.
A unification of some approaches to Poisson approximation
Authors: Witte Hans-Jürgen
Published: July 2016

Let Sn
be a sum of independent random variables. For the approximation of Sn
by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn
and Y. These results are applied to some special Sn
counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.
Periodic strong ergodicity in non-homogeneous Markov systems
Authors: Gerontidis Ioannis I.
Published: July 2016

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
Strong stochastic convexity: closure properties and applications
Authors: Shanthikumar J. George , Yao David D.
Published: July 2016

A family of random variables {X(θ)} parameterized by the parameter θ satisfies stochastic convexity (SCX) if and only if for any increasing and convex function f(x), Ef[X(θ)] is convex in θ
. This definition, however, has a major drawback for the lack of certain important closure properties. In this paper we establish the notion of strong stochastic convexity (SSCX), which implies SCX. We demonstrate that SSCX is a property enjoyed by a wide range of random variables. We also show that SSCX is preserved under random mixture, random summation, and any increasing and convex operations that are applied to a set of independent random variables. These closure properties greatly facilitate the study of parametric convexity of many stochastic systems. Applications to GI/G/1 queues, tandem and cyclic queues, and tree-like networks are discussed. We also demonstrate the application of SSCX in bounding the performance of certain systems.
Rate conservation for stationary processes
Authors: Ferrandiz Josep M. , Lazar Aurel A.
Published: July 2016

We derive a rate conservation law for distribution densities which extends a result of Brill and Posner. Based on this conservation law, we obtain a generalized Takács equation for the G/G/m/B queueing system that only requires the existence of a stochastic intensity for the arrival process and the residual service time distribution density for the G/GI/1/B queue. Finally, we solve Takács' equation for the N/GI/1/∞ queueing system.
The equality of the workload and total attained waiting time in average
Authors: Yamazaki Genji , Miyazawa Masakiyo
Published: July 2016

It has recently been shown that, for the FCFS G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. We show that, for a general queueing system setting, the workload and total attained waiting time of customers in service are identical in average but the equality of the distributions is not true in general except for the FCFS G/G/1 queue.
Some estimates of the rate of convergence for birth and death processes
Authors: Zeifman A. I.
Published: July 2016

The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations.
A multivariate reward process defined on a semi-Markov process and its first-passage-time distributions
Authors: Masuda Yasushi , Sumita Ushio
Published: July 2016

A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.
Availability of an R-out-of-N system with spares and repairs
Authors: Fawzi B. B. , Hawkes Alan G.
Published: July 2016

We obtain the equilibrium solutions of an R-out-of-N system subject to random breakdown. There are M spares and a single repairman, who instals good spares into the system when breakdowns occur and also repairs the failed items. Installation has pre-emptive repeat priority over repairs. Arbitrary distributions of installation and repair times are allowed. Equilibrium availability and downtime distributions are obtained from the equilibrium state distribution.
Convex ordering of the attained waiting times in single-server queues and related problems
Authors: Miyazawa Masakiyo , Yamazaki Genji
Published: July 2016

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.
On the virtual waiting time in an M/G/1 retrial queue
Authors: Falin G. , Fricker C.
Published: July 2016

This paper deals with the stationary distribution of the virtual waiting time, i.e. the time between the arrival and the beginning of service of a customer in a single-server queue that operates as follows. If the server is busy at an arrival time, the customer is rejected. This customer attempts service again after some random delay and continues to do so until the first time at which the server is idle. At this time, the customer is served and leaves the system after service completion. Interarrival times and delays are assumed to be two independent sequences of i.i.d. exponentially distributed random variables. Service times are also i.i.d., generally distributed, and independent of the previous sequences.
Adventures in applied probability
Authors: Gani J.
Published: July 2016

This article attempts to provide a historical perspective of the author's involvement in applied probability. Several research problems in this area are described, and three organizational experiences of importance to the author outlined. After a brief account of the development of the Applied Probability journals, the article concludes with some philosophical remarks.
Some thoughts on stationary processes and linear time series analysis
Authors: Heyde C. C.
Published: July 2016

In this paper we trace the development of the asymptotic analysis of autocorrelations for stationary purely non-deterministic time series. We emphasize the interplay between mathematical requirements and modelling philosophy. We then proceed to extend the theory to the case where only a certain weak form of asymptotic independence of the linear prediction errors is needed rather than the earlier martingale difference or independence requirements.
Mesoadditive processes and the specific conductivity of lattices
Authors: Hammersley J. M.
Published: July 2016

The hypercubic lattice with bonds of random electrical resistance affords a model for the specific conductivity of microscopically irregular material. The resulting stochastic process lies between a subadditive and a superadditive process: mesoadditive processes of this type provide several unsolved problems for pure mathematicians.
Coin tossing, revisited
Authors: Percus O. E. , Percus J. K.
Published: July 2016

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P
1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P
2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P
1 = 0. In the region p′ < p, P
1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.
On passage and conditional passage times for Markov chains in continuous time
Authors: Kijima Masaaki
Published: July 2016

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let
kTij
(
jTik
) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain
HX(t) constructed by making the set of states absorbing. It will be shown that the densities of
kTij
and
jTik
for any birth-death process are unimodal and the modes
kmij
(
jmik
) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of
kTij
and
jTik
for a time-reversible Markov chain are presented. Symmetry among
kTij, jTik
, and is also discussed, where , and are conditional passage times of the reversed process of X(t).
Autoregressive processes in optimization
Authors: Cipra Tomáš
Published: July 2016

Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.
The distributional structure of finite moving-average processes
Authors: McKenzie ED
Published: July 2016

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.
Optimal stopping in the parking problem with U-turn
Authors: Tamaki Mitsushi
Published: July 2016

A motorist drives his car toward his destination along a street and looks for a motor pool. Motor pools are assumed to occur independently, with probabilityp.Observing whether there exists a motor pool or not, the driver decides either to stop (i.e., return to the latest motor pool observed so far and park there) or continue driving. Once the driver stops, he walks the remaining distance to his destination. Letr, 0 <r< 1, be the relative speed of driving a car compared with that on foot. Then the time duration required to reach the destination is measured byr· (distance driven) + (distance on foot) and the objective of the driver is to find a parking policy which minimizes the expected time duration. It is shown that, under an optimal policy, a U-turn never occurs before the destination, but may occur beyond the destination. Moreover, the expected time is computed and some comparisons are made between our problem and the classical parking problem.
Regeneration in tandem queues with multiserver stations
Authors: Sigman Karl
Published: July 2016

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.
An improved stochastic model allowing interaction among individuals
Authors: Cerrito P. B.
Published: July 2016

The purpose of this paper is to explore the properties of a new model which can describe the interaction among individuals. Various other models have been proposed, but all assume that each individual acts independently of all others. Our model permits the assumption that individuals can be influenced and dominated by other individuals. We also demonstrate how this model can describe the interaction of various species in an ecosystem. This model will lead naturally to the definition of a population index of diversity. A means of estimating the values of the model will also be given.