bluuebunny/crossref_metadata_2025 huggingface dataset:
On identifiability in the autopsy model of reliability theory
Authors: Antoine Robin , Doss Hani , Hollander Myles
Published: July 2016

A coherent system is observed until it fails. At the instant of system failure, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. We consider whether the component life distributions can be determined from the distributions of the observed data.

Meilijson (1981) gave a condition on the structure of the system that was sufficient for the identifiability of the component distributions, under the assumption that the component life distributions are continuous and have common essential extrema. Nowik (1990) gave necessary and sufficient conditions for identifiability under the more restrictive condition that the component distributions have atoms at their common essential infimum and are mutually absolutely continuous. We give a necessary condition for identifiability, which we show to be equivalent to Nowik's condition, under the assumption that the distributions are continuous and strictly increasing. We derive a sufficient condition for identifiability, more general than Meilijson's, for the case in which the component distributions are assumed to be analytic. We also show that our necessary condition for identifiability is both necessary and sufficient when the component life distributions are assumed to belong to certain parametric families.
Closure of the NBUC class under formation of parallel systems
Authors: Hendi M. I. , Mashhour A. F. , Montasser M. A.
Published: July 2016

The class of new better than used in convex ordering (NBUC) is shown to be closed under formation of parallel systems with independent and identically distributed components.
A model for a system subject to random shocks
Authors: Lee Eui Yong , Lee Jiyeon
Published: July 2016

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.
The Beňes equations for the distribution of excursions for general storage processes
Authors: Guillemin Fabrice , Mazumdar Ravi
Published: July 2016

In this paper we obtain the Beňes equation for the evolution of the probability distribution of the excursion process associated with the level crossings of a general storage process. We then show that under stationarity and ergodicity assumptions on the process we can recover the well-known rate conservation law (RCL). Using the stationary solution we then show that the existence of an invariant solution can be studied in terms of an operator equation and we show how this characterization leads to a very simple explicit computation of the stationary distribution.
Preservation of certain classes of life distributions under Poisson shock models
Authors: Fagiuoli Enrico , Pellerey Franco
Published: July 2016

Recently defined classes of life distributions are considered, and some relationships among them are proposed. The life distribution H of a device subject to shocks occurring randomly according to a Poisson process is also considered, and sufficient conditions for H to belong to these classes are discussed.
Likelihood ratios for the infinite alleles model
Authors: Joyce Paul
Published: July 2016

The stationary distribution for the population frequencies under an infinite alleles model is described as a random sequence (x1,x2, · ··) such that Σxi= 1. Likelihood ratio theory is developed for random samples drawn from such populations. As a result of the theory, it is shown that any parameter distinguishing an infinite alleles model with selection from the neutral infinite alleles model cannot be consistently estimated based on gene frequencies at a single locus. Furthermore, the likelihood ratio (neutral versus selection) converges to a non-trivial random variable under both hypotheses. This shows that if one wishes to test a completely specified infinite alleles model with selection against neutrality, the test will not obtain power 1 in the limit.
Ergodicity of a Jackson network by batch arrivals
Authors: Borovkov A. A. , Schassberger R.
Published: July 2016

The Jackson network under study receives batch arrivals at i.i.d. intervals and features Markovian routing and exponentially distributed service times. The system is shown to be stable, in the sense of not being overloaded, if and only if, for each node, the total arrival rate of external and internal customers is less than the service rate. The method of proof is of more general interest.
Distortion of certain Voronoi tesselations when one particle moves
Authors: Orive Luis-M. Cruz
Published: July 2016

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.
A note on seasonal Markov chains with gamma or gamma-like distributions
Authors: Lloyd E. H. , Saleem S. D.
Published: July 2016

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt
} has a Laplace transform (LT) L(Xt

+1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t
for this class, and for a seasonal generalization of it.

A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968).

A seasonal version of this process is developed, valid for any number of seasons.

Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.
Poisson mixtures and quasi-infinite divisibility of distributions
Authors: Puri Prem S. , Goldie Charles M.
Published: July 2016

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.
Convergence rate of perturbed empirical distribution functions
Authors: Winter B. B.
Published: July 2016

Given an i.i.d. sequenceX1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator ofFis the e.d.f.Fn;whereUois the d.f. of the unit mass at zero. Anadmissible perturbation of the e.d.f., say, is obtained ifUois replaced by a d.f., whereis a sequence of d.f.'s converging weakly toUo.Suchperturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as. It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then‘has the Chung–Smirnov property'; i.e., with probability one,1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution andHnis the d.f. of the unit mass atn–αthen the above lim sup is ≦ 1 or = ∞, depending on whetheror
An embedded level crossing technique for dams and queues
Authors: Brill P. H.
Published: July 2016

The new concept of embedded level crossings is combined with the old principle of stationary set balance to produce an alternative approach for obtaining the steady-state distribution of the level in a dam with general release rule. The method yields the steady state distribution of the customer waiting time in the GI/G/1 queue as a special case. Results for a dam in which the instantaneous release rate is proportional to the level, and for the M/G/1, GI/M/1, Ek/M/1 and D/M/1 queues are derived using the new technique.
Invariant measures of time-reversible Markov chains
Authors: Suomela P.
Published: July 2016

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.
A solution to the generalized birthday problem with application to allozyme screening for cell culture contamination
Authors: Gail M. H. , Weiss G. H. , Mantel N. , O'Brien S. J.
Published: July 2016

This paper gives the probability that no common allozyme signature is found among n randomly selected cell culture lines when the s mutually exclusive and exhaustive signatures have arbitrary known probabilities. This result is useful in detecting cell culture contamination by a single cell line. This problem is a generalization of the classic problem of finding the probability of no common birthday among n individuals when it is assumed that each individual has a chance 1/s = 1/365 of a birthday on a given day. Both an exact recursive solution and an approximate solution are given for the general problem.
Count distributions, orderliness and invariance of Poisson cluster processes
Authors: Ammann Larry P. , Thall Peter F.
Published: July 2016

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.
The expected first-passage time for a sufficient statistic arising in the Poisson disorder problem
Authors: Wickwire K.
Published: July 2016

In the Poisson disorder problem the probability that the parameter of a Poisson process y, has increased by a constant amount, given observations of ys, s ≦ t, is a Markov process of mixed type with jumps of variable (state-dependent) magnitude superimposed upon a drift which satisfies an ordinary differential equation. Using a likelihood-ratio transformation, one can reduce the backward equation satisfied by the expected first-passage time to a constant level for the mixed process to a differential-difference equation with a constant retardation. We discuss a method for solving this equation and present some numerical results on its solution. The accuracy of some approximations which are easier to calculate is investigated.
A singular property of multiserver systems with constant service time
Authors: Halfin S.
Published: July 2016

The behavior of queuing systems with two servers and one waiting room is investigated. It is shown that if the service time is constant, then the difference between the times to service completion of the two servers (phase difference) tends to a constant, for increasing input intensities. This phenomenon holds for a wide class of arrival processes, but not when the service time has even a small variability. These results imply that the delay is not stochastically monotone in the input intensity. In general, we conjecture that the behavior of the phase differences, and delays, depends on whether the size of the waiting room is even or odd.
The uncontrolled traffic merge
Authors: Cowan Richard
Published: July 2016

The merging of n independent traffic streams is studied. The arrival processes comprise random sized bunches of cars, the bunches being separated in time by random sized gaps. Within each bunch, cars are spaced at one time unit apart. Under a distributional assumption on the inter-bunch gaps exact formulae are found for (a) the mean delay to cars, (b) the first two moments of the post-merge bunch size, and (c) the distribution of the inter-bunch gaps in the post-merge process. It is shown that the distributional assumption is realistic and moreover that the distribution of (c) is of the same form. The arrival bunches have general size distribution with finite mean and variance.
Heavy traffic theory for queues with several servers. II
Authors: Köllerström Julian
Published: July 2016

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.
Another martingale bound on the waiting-time distribution in GI/G/1 queues
Authors: Tan Harry H.
Published: July 2016

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.
Time reversal and age distributions, I. Discrete-time Markov chains
Authors: Tavaré S.
Published: July 2016

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.
Maximum likelihood estimates of incorrect Markov models for time series and the derivation of AIC
Authors: Ogata Yosihiko
Published: July 2016

The asymptotic behavior of the maximum likelihood estimators of Markov models or autoregressive models are given when the true distribution is not a member of the assumed parametric family. The derivation of Akaike's Information Criterion is reviewed for this case.
Reliability analysis of complex repairable systems by means of marked point processes
Authors: Franken Peter , Streller Arnfried
Published: July 2016

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.
Inspection and replacement policies
Authors: Zuckerman Dror
Published: July 2016

In this article we examine a breakdown model in which the system's status can be determined only by a test. Upon detection of failure the system must be replaced by a new identical one. The costs incurred include cost of inspection, operating costs, failure cost and a cost associated with planned replacements. Throughout the paper we restrict attention to replacement rules in which the time interval between two successive inspections is regarded as a fixed quantity. The decision variables include the inspection interval and the scheduling for preventive (planned) replacements. The problem is to specify a replacement rule which minimizes the long-run average cost per unit time. We show that under certain monotone conditions there is a natural candidate for an optimal replacement rule.
On the optimality of stationary replacement strategies
Authors: Bergman Bo
Published: July 2016

In this paper it is shown that for a large class of replacement problems the class of stationary replacement strategies is complete, i.e. in order to minimize the average long run cost per unit time it suffices to consider replacement rules which are equal for each new unit irrespectively of what has been observed from earlier units. The main result is based on a version of the law of large numbers for martingale differences proved in the appendix.
Moments of ladder heights in random walks
Authors: Doney R. A.
Published: July 2016

A well-known result in the theory of random walks states that E{X
2} is finite if and only if E{Z+
} and E{Z_} are both finite (Z
+ and Z_ being the ladder heights and X a typical step-length) in which case E{X
2} = 2E{Z+
}E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z
+ and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.
Linear processes and bispectra
Authors: Rosenblatt M.
Published: July 2016

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.
Percolation on subsets of the square lattice
Authors: McDiarmid Colin
Published: July 2016

We adapt arguments from a paper of Seymour and Welsh concerning percolation probabilities on the infinite square lattice L to show that for certain regions R in L, if there is a positive probability of having an infinite open path in L starting at the origin then there is also a positive probability of having such a path within R.
A useful random time-scale transformation for the standard epidemic model
Authors: Watson Ray
Published: July 2016

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.
On the distribution of time to first system failure
Authors: Chiang Dalen T. , Niu Shun-Chen
Published: July 2016

For a maintained reliability system, i.e., components may be repaired upon failures, we prove that the distribution of time to first system failure starting with all components new is NBU under the following conditions: (a) non-repairable components have NBU uptime distributions, (b) repairable components have exponential uptime and DFR downtime distributions. This is a slight generalization of a theorem of Barlow and Proschan (1976). However, our method of proof is different from theirs. We also show that the initial up-period of the system is stochastically larger than any other system up-period. Finally, a method of finding a lower bound for the expected time until first system failure is outlined using our main result.
Two characterizations of the geometric distribution
Authors: Arnold Barry C.
Published: July 2016

Let X
1, X
2, …, Xn
be independent identically distributed positive integer-valued random variables with order statistics X
1:n
, X
2:n
, …, Xn
:n. If the Xi
's have a geometric distribution then the conditional distribution of Xk

+1:n
– Xk

:n
given Xk+

1:n
– Xk

:n
> 0 is the same as the distribution of X
1:n–k
. Also the random variable X
2:n
– X
1:n
is independent of the event [X
1:n
= 1]. Under mild conditions each of these two properties characterizes the geometric distribution.
On the optimal assignment of servers and a repairman
Authors: Derman C. , Lieberman G. J. , Ross S. M.
Published: July 2016

We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.
Stationary measures for multitype branching processes
Authors: Hoppe Fred
Published: July 2016

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.
Stochastic pursuit-evasion games with information lag. I. Perfect observation
Authors: Schreuer Mynda
Published: July 2016

A discrete time, scalar, pursuit-evasion game is presented in which an evader, moving according to a stationary stochastic process, is continually being followed by a pursuer. Both players have perfect observations of the evader's positions, but the observations of the pursuer are subject to a time lag. It is assumed that the strategy of the evader can be represented as an infinite moving average, and that he is restricted by a velocity constraint. The pursuer is limited to strategies linear in his information, and the payoff is taken to be the mean-square distance between pursuer and evader. Under these conditions it is shown that the game does not have a value, and subsequently the lower and upper values and corresponding strategies are found.
Limit theorems for processes such as the Markov branching process
Authors: Barbour Andrew D.
Published: July 2016

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.
The asymptotic behaviour of the minimal total expected cost for the denumerable state Markov decision model
Authors: Hordijk Arie , Schweitzer Paul J. , Tijms Henk
Published: July 2016

This paper considers the discrete time Markov decision model with a denumerable state space and finite action space. Under certain conditions it is proved that the minimal total expected cost for a planning horizon of n epochs minus n times the minimal long-run average expected cost per unit time has a finite limit as n → ∞ for each initial state.
On a characterization of information improvement
Authors: Autar Ram
Published: July 2016

The object of this paper is to characterize an information theoretic measure associated with three probability distributions known as ‘Information Improvement’ through a functional equation which arises by considering the additive property of the measure. This measure has been extensively used in economic analysis.
An optimal branching migration process
Authors: Durham S. D.
Published: July 2016

We consider a population distributed over two habitats as represented by two separate one-dimensional branching processes with random environments. The presence of random fluctuation in reproduction rates in both habitats implies the possibility that neither habitat is universally superior to the other for all times and that a maximal population size is to be achieved by having population members present in both habitats. We show that optimal population growth occurs when migration between habitats occurs at a fixed rate which can be found from the environmentally determined reproduction variables of the separate habitats. The optimal processes are themselves two-type branching processes with random environments.
Conditional moments in a critical age-dependent branching process
Authors: Weiner Howard J.
Published: July 2016

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.

The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn
(αt) | X(t) > 0), E(Nn
(t) |X(αt) > 0), is obtained.
On a fixed point theorem and its stochastic equivalent
Authors: Yahav Joseph A.
Published: July 2016

A discrete-time Markov process on the interval [0, 1] is considered. Sufficient conditions for the existence of a unique stationary limiting distribution are given.
A note on Harris's condition for recurrence in random walk processes
Authors: Thomson P. J. , Wilkinson K. M.
Published: July 2016

This note corrects a short and transparent proof of Harris's condition for recurrence in random walk processes.
Time-reversibility of linear stochastic processes
Authors: Weiss Gideon
Published: July 2016

Time-reversibility is defined for a process X(t) as the property that {X(t
1), …, X(tn
)} and {X(– t
1), …, X(– tn
)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes.
Bounds for the extinction probability of a simple branching process
Authors: Quine M. P.
Published: July 2016

The extinction probability q of a supercritical simple branching process is well known to be less than unity. Intuitively, it is apparent that when the offspring mean is close to one, so, usually, will q be. This notion is made rigorous, and simple bounds are given for q in terms of the second and third factorial moments, which are applicable when the offspring mean is close to unity. A comparison is made of various upper bounds for q. The note contains some numerical examples.
First-passage time for a particular stationary periodic Gaussian process
Authors: Shepp L. A. , Slepian D.
Published: July 2016

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x
0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.
Waiting times without memory
Authors: Durrett Richard T. , Ghurye S. G.
Published: July 2016

A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y > 0, given that T >x, the conditional probability of T >x +y is the same as the unconditional probability of T >y; in other words, the physical process operates as if it has no memory concerning the successive occurrences of a certain event. The paper investigates the consequences of defining the property of lack of memory on more general time-domains than the positive reals. As a side issue, there is discussion of a stochastic variation of Cauchy's functional equation.
Geometrical measures of the smoothness of random functions
Authors: Switzer Paul
Published: July 2016

For stationary isotropic random functions on a Euclidean space, we characterize and compare the mean values of certain geometric measures of the smoothness of realizations. In particular we examine mean properties of the contours and gradients of the random function, and the effect of local averaging on smoothness in special cases.
On the distribution of hits
Authors: Rudolfer S. M.
Published: July 2016

A marksman fires at a target which consists of s zones. At each stage, he shoots independently of his previous efforts and has a probability p
0 of missing the target which is a function of the probabilities p
1
…, ps
of hitting zones 1, …, s, respectively. The distribution of the number of hits Xn
on the target after n shots is obtained and used to study the marksman's optimal strategy for the case and α
0 ≦ 1.
Diffusion limits of critical branching processes conditioned on extinction in the near future
Authors: Esty Warren W.
Published: July 2016

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.
The second-order analysis of stationary point processes
Authors: Ripley B. D.
Published: July 2016

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.
Some exact results for dams with Markovian inputs
Authors: Tin Pyke , Phatarfod R. M.
Published: July 2016

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj
) where p
ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.
A continuous-time population model with poisson recruitment
Authors: McClean Sally I.
Published: July 2016

A continuous-time model of a multigrade system is developed, which includes Poisson arrivals, interaction between grades and a leaving process. It therefore constitutes a continuous-time analogue of Pollard's hierarchical population model with Poisson recruitment. An expression is found for the first and second moments of grade size at any time. A general formulation of the joint probability generating function of the numbers in each grade is given, and the limiting distribution of grade size is shown to be Poisson.
Reliability systems in random environment
Authors: Råde L.
Published: July 2016

A parallel reliability system is subject to shocks which are generated by a renewal point process. The shocks cause the components to fail with certain probabilities. First we study the case when the components fail independently of each other with the same probability. For this case simple closed expressions for the Laplace-Stieltjes transform and expectation of time to system failure are derived. Then a case of dependent components is studied: the probability of failure of a component depends on the number of functioning components. Some extensions are mentioned.
A theorem on compositions of random probability generating functions and applications to branching processes with random environments
Authors: Kaplan Norman
Published: July 2016

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject.
The optimal admission policy to a multiserver queue with finite horizon
Authors: Emmons H.
Published: July 2016

An M/M/s queueing system with a simple cost structure is considered, under the assumption that the system will close in a finite time after which any remaining customers will require extra overtime service costs. We determine the optimal policy for admitting customers to the queue, as a function of the time, t, to closing and the current queue length, i. It is shown to have the form: admit if and only if f
1(t) ≦ i ≦ f
2(t). The bounds f
1(t) and f
2(t) are specified, and it is shown under what conditions f
1(t) = 0 (a control limit rule) or f
2(t) = ∞ (an inverse control limit rule).
The stationary distribution of an interesting Markov chain
Authors: Hendricks W. J.
Published: July 2016

In a single-shelf library of N books we suppose that books are returned by placing them at one end of the shelf. The probability of selecting each book is assumed to be known, and only one book is removed at a time. The N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.
On the inverse of the first passage time probability problem
Authors: Capocelli R. M. , Ricciardi L. M.
Published: July 2016

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.
Some problems of stochastic resource allocation
Authors: Gittins J. C.
Published: July 2016

Suppose that a number of jobs are to be carried out by a single processing unit orserver. The server can process any number of jobs at the same time but the time taken to finish all the jobs is the same no matter what scheduling policy is used, provided the server is never left idle. It is, however, possible to minimise the total delays to individual jobs by adopting a suitable policy. Optimal policies are derived here for the situation when the processing times are random variables. If the completion rates (analogous to hazard rates) for every job are increasing it is optimal to process the jobs in sequence. If they are decreasing the optimal policy involves simultaneous processing. These results are applied to a more general resource allocation problem arising in industrial chemical research.
An infinite discrete dam with dependent inputs
Authors: Herbert H. G.
Published: July 2016

This paper is concerned with an infinite discrete dam fed by inputs which form a moving average sequence. Generating functions are derived for the joint time dependent distribution of the content, accumulated input and total dry time, the distribution of first emptiness, and the stationary content distribution. Also investigated is the problem of first emptiness before overflow for the finite dam.
A random walk problem with correlation
Authors: Proudfoot A. D. , Lampard D. G.
Published: July 2016

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.
A characterization of the exponential distribution
Authors: Ahsanullah M. , Rahman M.
Published: July 2016

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.
The random volume of interpenetrating spheres in space
Authors: Moran P. A. P.
Published: July 2016

The distribution of the volume occupied by random spheres in a cube is considered, both when the number of spheres is fixed and when their centres form a Poisson field. The mean and variance are obtained and in the latter case the distribution is proved to converge to normality. The probability of complete coverage is also obtained heuristically.
Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary
Authors: Paranjape S. R. , Park C.
Published: July 2016

Let D = [0, S] × [0, T] be a rectangle in E2
and X(s, t), (s, t)∈D, be a two parameter Yeh-Wiener process. This paper finds the probability distribution of the supremum of X(s, t) on the boundary of D by taking the limit of the probability distribution of the supremum of X(s, t) along certain paths as these paths approach the boundary of D. The probability distribution of the supremum of X(s, t) on the boundary of D gives a nice lower bound for the probability distribution of the supremum of X(s, t) on D, which is unknown.
A cluster process representation of a self-exciting process
Authors: Hawkes Alan G. , Oakes David
Published: July 2016

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.
An example of a perturbation-type extremum controller
Authors: Titterington D. M.
Published: July 2016

Controls of the perturbation type are evolved for the purpose of monitoring the random movement of a response curve using noise-corrupted observations of the response. Dynamic programming methods are used to obtain approximations to the optimum control system when the response is quadratic. Further results are briefly described.
Notes on “Estimation theory for growth and immigration rates in a multiplicative process”
Authors: Heyde C. C. , Seneta E.
Published: July 2016

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.
A characterization of the geometric distribution
Authors: Dallas A. C.
Published: July 2016

The geometric distribution is characterized. The lack of memory property is replaced by the constancy of the conditional variance. Then the characterization is obtained.
On coupling of diffusion processes
Authors: Lindvall Torgny
Published: July 2016

The coupling method is well fitted to be used in the study of the asymptotics of one-dimensional diffusion processes. We give an elementary proof of Orey's theorem in the recurrent case, and establish rate results for tendency towards equilibrium under moment conditions on the speed measure and the initial distributions.
Moments of particle size distributions under sequential breakage with applications to species abundance
Authors: Siegel Andrew F. , Sugihara George
Published: July 2016

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.
A joint characterization of the multinomial distribution and the Poisson process
Authors: Kimeldorf George , Thall Peter F.
Published: July 2016

It has been recently proved that if N, X
1, X
2, … are non-constant mutually independent random variables with X
1,X
2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X
1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.
A characterization of the geometric distribution
Authors: Sreehari M.
Published: July 2016

Let X
1, X
2, …, Xn
be independent identically distributed positive integer-valued random variables with order statistics X
1:n
, X
2:n
, …, X
n:n
. We prove that if the random variable X2:n
– X
1:n
is independent of the events [X1:n = m] and [X1:n
= k], for fixed k > m > 1, then the Xi
's are geometric. This is related to a characterization problem raised by Arnold (1980).
A state-dependent lifetime process of individuals subject to external perturbations
Authors: Mergenthaler Wolfgang
Published: July 2016

We consider an individual which ultimately dies or divides, and whose state is subject to drift and jumps caused by external perturbations. The mortality and division rates being state-dependent, the present paper deals with the time-dependent distribution of the individual's position in the state-space and with its lifetime distribution. The results are applied to a model of a biological cell which is exposed to ionizing radiation. Under certain conditions on the parameters of the type of perturbation one can show that the division probability decreases and the mean regeneration time increases with increasing frequency and ‘effect' of the perturbations.
Scheduling spares with exponential lifetimes in a two-component parallel system
Authors: Pinedo Michael
Published: July 2016

Consider a two-component parallel system. There is a stockpile of n spares, spare i having lifetime Xi
, exponentially distributed with rate µi
. The system breaks down when neither component functions. Every time a spare expires it has to be replaced, to avoid system breakdown. We investigate the policies which maximize the expected lifetime of the system using these n spares.
A diffusion approximation for correlation in queues
Authors: Woodside C. M. , Pagurek B. , Newell G. F.
Published: July 2016

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in anyGI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.

Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system inM/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.
Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers
Authors: Lavenberg S. S. , Reiser M.
Published: July 2016

We consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form. A customer can change its class but not its type as it traverses the network. We show that the stationary state probabilities at instants at which customers of a particular type arrive at a particular service center and enter a particular class are equal to the stationary state probabilities at arbitrary times for the network with one less customer of that type. Applications of this result are given.
A comparison criterion for explosions in point processes
Authors: Haezendonck J. , de Vylder F.
Published: July 2016

It is proved that under certain conditions the probability of explosion of a first point process is not larger than the probability of explosion of a second point process if the intensities of both processes fulfil the same inequality.
A characterization of the gamma distribution by the negative binomial distribution
Authors: Engel Jan , Zijlstra Mynt
Published: July 2016

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.
The expected population size in a cell-size-dependent branching process
Authors: Sudbury Aidan
Published: July 2016

In cell-size-dependent growth the probabilistic rate of division of a cell into daughter-cells and the rate of increase of its size depend on its size. In this paper the expected number of cells in the population at time t is calculated for a variety of models, and it is shown that population growths slower and faster than exponential are both possible. When the cell sizes are bounded conditions are given for exponential growth.
Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes
Authors: Tweedie R. L.
Published: July 2016

For regular Markov processes on a countable space, we provide criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process. An application to birth-death processes is given.
Random secants of a convex body generated by surface randomness
Authors: Ehlers P. Frank , Enns Ernest G.
Published: July 2016

Length distributions for random secants through a convex region K are derived for three types of randomness. The results are formulated in terms of geometric properties of K, e.g. the overlap surface content of K with its translated self. The distribution of distance between two random points in K, expressed in terms of the overlap volume, is shown to extend to non-convex (including disjoint) regions.
Search for a moving target under minimax estimation
Authors: Corwin Thomas L.
Published: July 2016

A target is assumed to choose its starting position in a search at an unknown position in a finite search space. No prior probability distribution for the target's initial location is assumed. During the search the target is assumed to move from position to position in the search space according to a Markov process. A search is defined to be the observation of a sequence of random variables. Representations for the minimax estimator for target location at any stage of the search, the least favorable prior distribution for the target, and the value of the estimation game are presented. An example is computed in which Bayes estimators are compared with minimax estimators for target location.