bluuebunny/crossref_metadata_2025 huggingface dataset:
Stability of finite queue, tandem server systems
Authors: Hildebrand David K.
Published: July 2016

One of the troublesome problems in the extension of queueing theory to more complex systems of servers is the development of models suitable for the analysis of assembly lines and similar structures. Results in special cases are given in [2], [5], and [6]. A reasonable model would assume:
Conditional limit theorems for general branching processes
Authors: Green P. J.
Published: July 2016

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.
A two-dimensional ‘immigration–branching' model with application to earthquake occurrence times and energies
Authors: Lai C. D.
Published: July 2016

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.
A class of non-identifiable stochastic models
Authors: Cane Violet R.
Published: July 2016

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.
Almost sure asymptotic likelihood theory for diffusion processes
Authors: Lee T. S. , Kozin F.
Published: July 2016

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.
Stochastic analysis of the raw silk reeling process
Authors: Miura M.
Published: July 2016

We consider the variation in size of raw silk in the fixed size raw silk reeling process, and the relations between occurrences of feeding ends and some properties of cocoons from the stochastic viewpoint. The results obtained by stochastic analysis appear reasonable for the reeling experiment using three groups of cocoons.
Limit theorems for sums of a sequence of random variables defined on a Markov chain
Authors: Wolfson David B.
Published: July 2016

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.
A note on the inverse problem for reducible Markov chains
Authors: Pittenger A. O.
Published: July 2016

Suppose a physical process is modelled by a Markov chain with transition probability on S
1 ∪ S
2, S
1 denoting the transient states and S
2 a set of absorbing states.

If v denotes the output distribution on S
2, the question arises as to what input distributions (of raw materials) on S
1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.
The volume of a random simplex in an n-ball is asymptotically normal
Authors: Ruben Harold
Published: July 2016

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.
On the length distribution of stochastic subarcs of a rectifiable arc
Authors: Mann J. E.
Published: July 2016

Suppose that A
1, A
2, · ··, An
are disjoint subarcs of a rectifiable arc A ⊆ Rk. The location and length of each subarc are assumed to be continuous random variables. An expression for the expected fraction of A covered by subarcs each of length greater than τ ≧ 0 is obtained and applied to two conceptually distinct problems associated with light penetration of an absorbing medium.
Some aspects of the Smoluchowski process
Authors: McDunnough Philip
Published: July 2016

This article deals with the generalized Smoluchowski process, {
n
(t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process {
n
(t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.
Strong ergodicity for continuous-time Markov chains
Authors: Isaacson Dean , Arnold Barry
Published: July 2016

The concept of strong ergodicity for discrete-time homogeneous Markov chains has been characterized in several ways (Dobrushin (1956), Lin (1975), Isaacson and Tweedie (1978)). In this paper the characterization using mean visit times (Huang and Isaacson (1977)) is extended to continuous-time Markov chains. From this it follows that for a certain subclass of continuous-time Markov chains, X(t), is strongly ergodic if and only if the associated embedded chain is Cesaro strongly ergodic.
Random arcs on the circle
Authors: Siegel Andrew F.
Published: July 2016

Place n arcs of equal lengths randomly on the circumference of a circle, and let C denote the proportion covered. The moments of C (moments of coverage) are found by solving a recursive integral equation, and a formula is derived for the cumulative distribution function. The asymptotic distribution of C for large n is explored, and is shown to be related to the exponential distribution.
A note on simultaneous recurrence conditions on a set of denumerable stochastic matrices
Authors: Federgruen A. , Hordijk A. , Tijms H. C.
Published: July 2016

In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems.
Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities
Authors: Mertens Jean-François , Samuel-Cahn Ester , Zamir Shmuel
Published: July 2016

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi
→ ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk
< y
o, · · ·, yN

–1 for some k ≧ N, is necessary and sufficient for transience.
A characterization of the uniform distribution on the circle in the analysis of directional data
Authors: Bingham M. S.
Published: July 2016

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.
Some new results in the mathematical theory of phage-reproduction
Authors: Puri Prem S.
Published: July 2016

In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT
and ZT
, the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT
is then fitted to some observed data of Delbrück [2].
The total progeny in a branching process and a related random walk
Authors: Dwass Meyer
Published: July 2016

This paper is a continuation of [1]. The techniques of [1] are used to get specific information about the distribution of the total progeny in a branching process. This distribution is also related to one which arises in a random walk problem.
Solutions of ordinary differential equations as limits of pure jump markov processes
Authors: Kurtz Thomas G.
Published: July 2016

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation

used to describe a number of processes including radioactive decay and population growth.
On some problems in operations research
Authors: Samuel Ester
Published: July 2016

Consider an airplane which is at a distance t ≧ 0 from its destination (the distance being measured in time-units) and has n anti-aircraft missiles. The enemy sends its aircraft according to a Poisson process with known parameter, which by redefining the time-scale may, without loss of generality, be taken to be one. When meeting an enemy aircraft, the plane can use, simultaneously, some of its missiles, and it is assumed that the hits are independent random events with known probability 1 > 1 – q > 0. If no missile hits the enemy aircraft, the airplane is shot down with known probability 1 ≧ 1 – u > 0. The case u= 0 is of particular interest. (We have excluded the values q = 0, q = 1 and u= 1 since these yield trivial solutions.)
Bunching in a semi-Markov process
Authors: Hawkes A. G.
Published: July 2016

In the type II counter with constant deadtime, particles which arrive within some constant time τ following another particle are unrecorded. We can think of this process as an alternating sequence of gaps and bunches of events. Gaps have duration > τ, while the intervals between any successive pair of events within a bunch are all ≦ τ. Counter theory is usually concerned with the distribution of intervals between recorded events (i.e., the first event of each bunch) and the distribution of the number of recorded events in a given time interval. In the case where the events form a renewal process this has been studied intensively by Pyke [2], Smith [5] and Takács [6].
Queueing with alternating priorities, treated as random walk on the lattice in the plane
Authors: Yadin Micha
Published: July 2016

The queueing system with alternating priorities has been discussed recently in numerous articles (Maxwell [1], Avi Itzhak, Maxwell, Miller [2], Neuts, Yadin [3], and others). Consider a system which consists of several queueing units. Each unit is independent in the sense that it has its own waiting line which is generated by an independent Poisson stream of customers. However, all of the units are served by a single server who allocates his time between the units.
Stochastic models as functionals: some remarks on the renewal case
Authors: Grübel Rudolf
Published: July 2016

We regard the standard stochastic model of renewal theory as a functional which associates with a lifetime distribution μ the corresponding renewal measure v = Σ0
∞
μ ∗
n
. The behaviour of this functional near exponential distributions is investigated. The literature mainly deals with the asymptotic behaviour of v([0, t]) and v[t, t + 1]) as t → ∞— this new approach leads to expansions which are valid uniformly in t ∈ R.
Random circles in the d-dimensional unit ball
Authors: Affentranger Fernando
Published: July 2016

This note gives the solution of the following problem concerning geometric probabilities. What is the probabilityp(Bd; 2) that the circumference determined by three pointsP, P1andP2chosen independently and uniformly at random in the interior of ad-dimensional unit ballBdin Euclidean spaceEd(d≧ 2) is entirely contained inBd? From our result we conclude thatp(Bd; 2) →π /(3√3) asd→∞.
Optimal strategies for discovering new species in continuous time
Authors: Alsmeyer Gerold , Irle Albrecht
Published: July 2016

Consider a population of distinct species Sj
, j∈J, members of which are selected at different time points T
1
, T
2,· ··, one at each time. Assume linear costs per unit of time and that a reward is earned at each discovery epoch of a new species. We treat the problem of finding a selection rule which maximizes the expected payoff. As the times between successive selections are supposed to be continuous random variables, we are dealing with a continuous-time optimal stopping problem which is the natural generalization of the one Rasmussen and Starr (1979) have investigated; namely, the corresponding problem with fixed times between successive selections. However, in contrast to their discrete-time setting the derivation of an optimal strategy appears to be much harder in our model as generally we are no longer in the monotone case.

This note gives a general point process formulation for this problem, leading in particular to an equivalent stopping problem via stochastic intensities which is easier to handle. Then we present a formal derivation of the optimal stopping time under the stronger assumption of i.i.d. (X
1
, A
1) (X2, A2
), · ·· where Xn
gives the label (j for Sj
) of the species selected at Tn
and An
denotes the time between the nth and (n – 1)th selection, i.e. An
= Tn – Tn–

1. In the case where even Xn
and An
are independent and An
has an IFR (increasing failure rate) distribution, an explicit solution for the optimal strategy is derived as a simple consequence.
A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes
Authors: Giorno V. , Nobile A. G. , Ricciardi L. M.
Published: July 2016

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.
A spatial Markov property for nearest-neighbour Markov point processes
Authors: Kendall W. S.
Published: July 2016

Nearest-neighbour Markov point processes were introduced by Baddeley and Møller (1989) as generalizations of the Markov point processes of Ripley and Kelly. This note formulates and discusses a spatial Markov property for these point processes.
The job search problem as an employer–candidate game
Authors: Mcnamara J. M. , Collins E. J.
Published: July 2016

In the standard job search problem a single decision-maker (say an employer) has to choose from a sequence of candidates of varying fitness. We extend this formulation to allow both employers and candidates to make choices. We consider an infinite population of employers and an infinite population of candidates. Each employer interviews a (possibly infinite) sequence of candidates for a post and has the choice of whether or not to offer a candidate the post. Each candidate is interviewed by a (possibly infinite) sequence of employers and can accept or reject each offer. Each employer seeks to maximise the fitness of the candidate appointed and each candidate seeks to maximise the fitness of their eventual employer. We allow both discounting and/or a cost per interview. We find that there is a unique pair of policies (for employers and candidates respectively) which is in Nash equilibrium. Under these policies each population is partitioned into a finite or countable sequence of subpopulations, such that an employer (candidate) in a given subpopulation ends up matched with the first candidate (employer) encountered from the corresponding subpopulation. In some cases the number of non-empty subpopulations in the two populations will differ and some members of one population will never be matched.
The exact and asymptotic formulas for the state probabilities in simple epidemics with m kinds of susceptibles
Authors: Lacayo H. , Langberg Naftali A.
Published: July 2016

A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.
The saddlepoint approximation for a general birth process
Authors: Daniels H. E.
Published: July 2016

Saddlepoint approximations to the probabilities for a general time-homogeneous birth process are derived from the known Laplace transform with respect to time. Numerical computations for the simple (logistic) epidemic show the approximation to have a remarkably low relative error over the useful range of times for populations as small as 5. The merit or otherwise of renormalising the approximate probabilities is discussed. To study the accuracy for moderately large population sizes, a special form of birth rate is used, for which the exact probabilities can be computed directly.
On Coupling of continuous-time renewal processes
Authors: Lindvall Torgny
Published: July 2016

This paper shows how the coupling method applies to renewal theory in continuous time. Old and new rate results for tendency towards equilibrium and forgetfulness of initial delay are obtained. The coupling approach also provides a new way of analysing the asymptotics of functions of renewal processes which avoids the route via a renewal equation.
Shock models leading to increasing failure rate and decreasing mean residual life survival
Authors: Ghosh Malay , Ebrahimi Nader
Published: July 2016

Shock models leading to various univariate and bivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) distributions are discussed. For proving the IFR properties, shocks are not necessarily assumed to be governed by a Poisson process.
On ross's conjectures about queues with non-stationary poisson arrivals
Authors: Heyman D. P.
Published: July 2016

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.
Estimation for diffusion processes under misspecified models
Authors: McKeague Ian W.
Published: July 2016

The asymptotic behavior of the maximum likelihood estimator of a parameter in the drift term of a stationary ergodic diffusion process is studied under conditions in which the true drift function and true noise function do not coincide with those specified by the parametric model.
Lumpability for non-irreducible finite markov chains
Authors: Abdel-Moneim Atef M. , Leysieffer Frederick W.
Published: July 2016

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.
On the waiting time of a two-stage queueing system with blocking
Authors: Langaris Christos , Conolly Brian
Published: July 2016

An analysis is given of the first-come-first-served waiting-time process in stages 1 and 2 of a two-stage service system with k and n parallel service channels in the first and second stages respectively, and m intermediate waiting places (k, n ≧ 1, m ≧ 0).

Although the method of analysis is straightforward the details are intricate and require careful study of the location of the zeros of a high-degree polynomial.

The analysis paves the way for an extensive study of the numerical effect on waiting time of blocking in commonly encountered systems of this nature. ‘Effective service time' in stage 1, defined so as to include blocked time, is considered separately.
On the asymptotic distribution of multivariate renewal processes
Authors: Niculescu Ştefan P.
Published: July 2016

Using a result of Bikjalis (1971) concerning the rate of convergence in the multidimensional central limit theorem we obtain informations about some limit distributions in multivariate renewal theory.
On some classes of population-size-dependent Galton–Watson processes
Authors: Höpfner Reinhard
Published: July 2016

Some classes of population-size-dependent Galton–Watson processes {Z(t)}
t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.
The renewal-process stationary-excess operator
Authors: Whitt Ward
Published: July 2016

This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).
Partial balance and insensitivity
Authors: Whittle P.
Published: July 2016

A direct and general proof is given of the equivalence of partial balance and insensitivity.
A multivariate IFR class
Authors: Savits Thomas H.
Published: July 2016

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.
A note on strong unimodality and dispersivity
Authors: Droste W. , Wefelmeyer W.
Published: July 2016

A distribution R is ‘more dispersed' than a distribution P if any two quantiles of R are more widely separated than the corresponding quantiles of P. Lewis and Thompson (1981) prove that a non-degenerate distribution P is strongly unimodal if and only if it is ‘dispersive', i.e. if P∗ Q
1 is more dispersed than P∗ Q
2 whenever Q1 is more dispersed than Q
2
. For P admitting a positive Lebesgue density, we prove that P is strongly unimodal if and only if P ∗ Q is more dispersed than P for every distribution Q. Hence the latter property also characterizes the dispersivity of P.
Stochastic flowshop no-wait scheduling
Authors: Frostig E. , Adiri I.
Published: July 2016

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji
on machine Mj
is an independent random variable Ti
. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti
, (
i
are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.
Records from improving populations
Authors: Ballerini Rocco , Resnick Sidney
Published: July 2016

We consider records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied. If {Xj
} is stationary the record rate is asymptotically constant and if {Xj
} is i.i.d. the record rate is, in addition, asymptotically normal. These results are illustrated by analysis of the times in the mile run.
Parametric inference in Markov branching processes with time-dependent random immigration rate
Authors: Pruscha Helmut
Published: July 2016

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.
A limit theorem which clarifies the ‘Petersburg Paradox'
Authors: Martin-Löf Anders
Published: July 2016

The total gain, SN
, in N successive Petersburg games is considered, and a limit theorem for SN/N – n when N = 2
n
and n → ∞is proved. The limit distribution can be determined numerically with good accuracy, and this fact provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.
An order statistic characterization of the poisson renewal process
Authors: Liberman Uri
Published: July 2016

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.
Optimal design of a storage system with constant or linear release rate
Authors: De Morais Paulo Renato
Published: July 2016

This paper is concerned with the determination of the optimal release rate for a continuous storage system. The input to the store is assumed to be a compound Poisson process (with non-negative jumps), and its content is released either at a constant rate or at a linear rate. Rewards are collected at an output-dependent rate and are continuously discounted at a constant rate. The problem consists in finding the optimal release rate which maximizes the infinite-horizon expected discounted return. This problem is solved by deriving an explicit solution for the expected discounted return, from which the optimal release rate can be found.
Approximations for the GI/G/c queue
Authors: Van Hoorn M. H. , Seelen L. P.
Published: July 2016

In this paper, we give good-quality approximations for multiserver queues with general service-time distributions. For the interarrival-time distribution we consider hyperexponential distributions and mixtures of Erlang distributions with the same scale parameters. We give an algorithm to compute various operating characteristics such as the delay probability and the mean queue length.

The approximations are obtained by making assumptions regarding the residual service times of services in progress at service completion epochs, whereas the arrival process is modelled exactly. As the operating characteristics are far more sensitive to changes in the parameters of the arrival process, the latter is very important.

The quality of the approximations is extensively tested by using the exact results of more than 2000 cases. In heavy traffic the average relative error is up to 1% and in moderate traffic about 4%.
Periodic and sequential preventive maintenance policies
Authors: Nakagawa Toshio
Published: July 2016

This paper considers periodic and sequential preventive maintenance (PM) policies for the system with minimal repair at failure: the PM is done (i) at periodic timeskxand (ii) at constant intervalsxk(k= 1, 2, ···, N). The system has a different failure distribution between PM'S and is replaced at the Nth PM. The optimal policies which minimize the expected cost rates are discussed. The optimalxand N of periodic PM and {xk} of sequential PM are easily computed in a Weibull distribution case.
A stochastic population projection system based on general age-dependent branching processes
Authors: Mode Charles J. , Jacobson Marc E. , Pickens Gary T.
Published: July 2016

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.
Moving-average models with bivariate exponential and geometric distributions
Authors: Langberg Naftali A. , Stoffer David S.
Published: July 2016

Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.
Uniformly best invariant stopping rules
Authors: Engen Steinar , Seim Eva
Published: July 2016

The class of stopping rules for a sequence of i.i.d. random variables with partially known distribution is restricted by requiring invariance with respect to certain transformations. Invariant stopping rules have an intuitive appeal when the optimal stopping problem is invariant with respect to the actual gain function. Uniformly best invariant stopping rules are derived for the gamma distribution with known shape parameter and unknown scale parameter, for the uniform distribution with both endpoints unknown, and for the normal distribution with unknown mean and variance. Some comparisons with previously published results are made.
Multivariate Lomax distribution: properties and usefulness in reliability theory
Authors: Nayak Tapan Kumar
Published: July 2016

A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.
The range of the mean-value quantities of planar tessellations
Authors: Kendall Wilfrid S. , Mecke Joseph
Published: July 2016

Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.
On the probability of covering the circle by random arcs
Authors: Huffer F. W. , Shepp L. A.
Published: July 2016

Arcs of length lk, 0 < lk
< 1, k = 1, 2, ···, n, are thrown independently and uniformly on a circumference having unit length. Let P(l
1
, l
2, · ··, ln
) be the probability that is completely covered by the n random arcs. We show that P(l
1
, l
2
,· ··, ln
) is a Schur-convex function and that it is convex in each argument when the others are held fixed.
Uniformization for semi-Markov decision processes under stationary policies
Authors: Beutler Frederick J. , Ross Keith W.
Published: July 2016

Uniformization permits the replacement of a semi-Markov decision process (SMDP) by a Markov chain exhibiting the same average rewards for simple (non-randomized) policies. It is shown that various anomalies may occur, especially for stationary (randomized) policies; uniformization introduces virtual jumps with concomitant action changes not present in the original process. Since these lead to discrepancies in the average rewards for stationary processes, uniformization can be accepted as valid only for simple policies.

We generalize uniformization to yield consistent results for stationary policies also. These results are applied to constrained optimization of SMDP, in which stationary (randomized) policies appear naturally. The structure of optimal constrained SMDP policies can then be elucidated by studying the corresponding controlled Markov chains. Moreover, constrained SMDP optimal policy computations can be more easily implemented in discrete time, the generalized uniformization being employed to relate discrete- and continuous-time optimal constrained policies.
Some remarks on regression with autoregressive errors and their residual processes
Authors: Kulperger R. J.
Published: July 2016

We consider some linear regression models Y = Σα lfl
(z) + X, where X is an autoregressive (AR) process. The residuals estimate the i.i.d. innovations sequence which drives the AR process. We then consider the partial sum process of the residuals and show they converge to Brownian bridges in certain cases. Some remarks are also made on similar processes when differencing is first applied to remove trends. When an AR process is differenced the residual partial sum can be asymptotically a random polynomial.
On two integral equations of queueing theory
Authors: Cohen J. W.
Published: July 2016

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {
v

n
, n = 1,2, …}, recursively defined by

K
is a positive constant, τ
1, τ
2, …; Σ
1, Σ
2, …; are independent, non-negative variables, with τ
1, τ
2,…, identically distributed, similarly, the variables Σ
1, Σ
2, …, are identically distributed.
Dams in series with continuous release
Authors: Moran P.A.P.
Published: July 2016

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.
A note on passage times and infinitely divisible distributions
Authors: Miller H. D.
Published: July 2016

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T
1, T
1 + T
2, … it undergoes jumps ξ
1, ξ
2, …, where the time intervals T
1, T
2, … between successive jumps are mutually independent random variables each following the exponential density e–t
while the jumps, which are independent of the τi
, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.
Carrier-borne epidemics with immigration. I — Immigration of both susceptibles and carriers
Authors: Dietz K. , Downton F.
Published: July 2016

Much of the theory of epidemics (see Bailey, 1957) has been concerned with models for their behaviour in closed populations. In such models the epidemic ultimately dies out, and interest has been concentrated on, for example, the ultimate size of the epidemic and its duration in time. In practice a population is rarely completely closed, and for many diseases an endemic model rather than an epidemic model is appropriate. To create models for endemic diseases it is necessary to introduce both new persons susceptible to the disease into the population and new sources of infection. For the so-called general stochastic epidemic, Ridler-Rowe (1967) has obtained certain limiting properties of the population where immigration of both susceptibles and infectives into the population takes place, but much work remains to be done to obtain, for example, the general equilibrium behaviour of this model.
Gap-acceptance in road traffic
Authors: Hawkes A. G.
Published: July 2016

We find the distribution of delay to minor road vehicles waiting to merge or cross a single stream of major road traffic. The decision to cross is taken on the basis of a gap-acceptance function. The model turns out to be a simple queueing problem in which a customer finding an empty queue has a different service time distribution from queueing customers. The key to this representation is given in Section 3. Some numerical results in Section 6 indicate that in most circumstances a simple model will give adequate results.
Highway Delays Resulting From Flow-Stopping Incidents
Authors: Donald P. Gayer
Published: July 2016

Modern highways, particularly the freeways of large cities, carry a considerable volume of traffic during certain times of day. Thus if any interruption or retardation of flow occurs, a large reaction in the shape of a monumental and time-consuming traffic jam soon appears. For example, when an accident or mechanical breakdown gives rise to a severe flow restriction or stoppage, many other vehicles may be quickly halted, and remain stopped until the impediment is cleared away. In addition, the flow of traffic may be slowed considerably even after the original stoppage is removed owing to the existence of a queue. Consequently, vehicles that arrive long after the original restriction is removed experience prolonged, and sometimes seemingly inexplicable, delay. Our purpose is to develop a probability model for the situation described, with the aim of estimating the consequence of a temporary flow restriction. Various measures of (in)effectiveness are worth considering. We consider primarily the total vehicle-hours waited while the jam dissipates; the latter may roughly measure total social cost. The total number of vehicles involved in the jam is also of interest, as are other figures of merit.
The distribution of radioactive recovery in randomly cut and sedimented DNA
Authors: Litwin Samuel
Published: July 2016

Viral DNA is studied by the technique of radioactive labelling and subsequent ultra centrifugation which separates it from reference DNA according to length or density. One hypothesis that frequently is of service is that the strands of DNA being studied are randomly cut along their lengths. Often the average number of cuts, λ, per unit length is either known by the nature of the experiment or is desired as a statistic. If radioactive phosphorous, P32, is used to label the DNA, e.g., 6 P32 atoms per unit length of 200,000 nucleotides (each of which is connected to its neighbor by two phosphorous-containing links) then within two weeks about 3 P32 atoms will have decayed and the DNA obtained, once separated into single stranded form (denatured), will form a predictable distribution of sizes. Of course 3 decays may be obtained more quickly by starting with more than 6 P32 atoms per unit length.
An identity for Markovian replacement processes
Authors: Mandl Petr
Published: July 2016

A Markovian process with rewards is considered. The interference into its development consists in instantaneous changes of state called replacements. It is shown that, for any replacement policy not depending on the future, the expected reward is a linear function of the expected replacement numbers. The application of this result is illustrated by examples.
On the maximum of sums of random variables and the supremum functional for stable processes
Authors: Heyde C.C.
Published: July 2016

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S
0= 0, S

n
= Σ
i=1

n

Xi
, n ≧ 1, and Mn
= max0 ≦ k ≦ n

Sk
. In the case where the Xi
are such that Σ1
∞
n
−1Pr(Sn
> 0) < ∞, we have lim
n→∞M
n
= M which is finite with probability one, while in the case where Σ1
∞
n
−1Pr(Sn
< 0) < ∞, a limit theorem for Mn
has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1
∞
n
−1Pr(Sn
< 0) < ∞, Σ1
∞
n
−1Pr(Sn
> 0) < ∞ (the case of oscillation of the random walk generated by the Sn
) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi
themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn
in the case of oscillation.
The joint asymptotic distribution of the k-smallest sample spacings
Authors: Weiss Lionel
Published: July 2016

Suppose Q
1
⋆, … Q
n
⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).
A note on stochastic processes with independent increments taking values in an abelian group
Authors: Bingham M.S.
Published: July 2016

It is well known that any stochastically continuous real valued stochastic process with independent increments defined on a compact time interval can be decomposed into a sum of independent processes, one of which is Gaussian with continuous sample paths, and the remainder of which have sample paths which are continuous except at a finite number of points with the discontinuities occurring at Poisson time points. The purpose of this note is to announce a proof of the above theorem in the case where the process takes values in an abelian group G. The detailed proof will appear elsewhere. The basic ideas of the proof in the case when G is finite dimensional Euclidean space are contained in Chapter VI of Gikhman and Skorohod (1965).
An invariance property of Poisson processes
Authors: Brown Mark
Published: July 2016

In this paper we shall investigate point processes generated by random variables of the form 〈gi
(Ti
]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti
〉.
A linear birth and death process under the influence of another process
Authors: Puri Prem S.
Published: July 2016

Let {X
1 (t), X
2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X
2(t), t ≧ 0} may influence the growth of the process {X
1(t), t ≧ 0}, while the process X
2 (·) itself grows without any influence whatsoever of the first process. The process X
2 (·) is taken to be a simple linear birth and death process with λ
2 and µ
2 as its birth and death rates respectively. The process X
1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X
2 (·) in the following manner: λ (t) = λ
1 (θ + X
2 (t)); µ(t) = µ
1 (θ + X
2 (t)). Here λ i, µi
and θ are all non-negative constants. By studying the process X
1 (·), first conditionally given a realization of the process {X
2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X
2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X
1 (t) always exists as t→∞, except only when both λ
1 > µ
1 and λ
2 > µ
2. Also, certain problems concerning moments of the process, regression of X
1 (t) on X
2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.
General stochastic epidemic with recovery
Authors: Billard L.
Published: July 2016

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.
On the extinction times of varying and random environment branching processes
Authors: Agresti Alan
Published: July 2016

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.
Some counting and interval properties of the mutually-exciting processes
Authors: Adamopoulos L.
Published: July 2016

By using a cluster representation of the mutually-exciting processes their probability generating functional is derived. This is used to obtain some counting and interval properties of the process. The theoretical results are applied to a numerical example.