bluuebunny/crossref_metadata_2025 huggingface dataset:
Rates of Poisson convergence for some coverage and urn problems using coupling
Authors: Holst L. , Kennedy J. E. , Quine M. P.
Published: July 2016

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.
A note on a randomized occupancy problem
Authors: Swensen Anders Rygh
Published: July 2016

Consider N urns into which n balls are dropped independently with equal probability of hitting each urn and constant probability p of staying in the urn. We find the characteristic function of the joint distribution of KN
and LN
where KN
is the number of urns that have not been hit, and LN
is the number of urns where all balls have fallen through. Furthermore, we study the asymptotic distribution of (KN, LN
) as n, N → ∞ at various rates.
The high-order autocovariance structure of the telegraph wave
Authors: Renshaw Eric
Published: July 2016

Exact expressions are developed for the nth order autocovariance structure of the telegraph wave, the integral of which defines a biased correlated random walk.
Scheduling jobs with exponential processing times on parallel machines
Authors: Lehtonen Tapani
Published: July 2016

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.
A cost relationship between age and block replacement policies
Authors: Savits Thomas H.
Published: July 2016

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c
1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.
A method for approximating the probability functions of a Markov chain
Authors: Crank Keith N.
Published: July 2016

This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.
A stochastic explosive reaction system with sampling
Authors: Dawson Donald A. , Fleischmann Klaus
Published: July 2016

Large stochastic systems of marked particles are considered. These ‘populations' grow according to pairwise mutually catalytic reactions and in addition particles may exchange their type (mark) by a sampling procedure. We are interested in the explosive behavior of the model and its high density properties (law of large numbers).
Commande optimale du processus de wiener
Authors: Lefebvre Mario
Published: July 2016

The problem of the optimal control of the Wiener process in Rn
is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.
M/G/1 queueing systems with returning customers
Authors: Greenberg Betsy S.
Published: July 2016

Single-channel queues with Poisson arrivals, general service distributions, and no queue capacity are studied. A customer who finds the server busy either leaves the system for ever or may return to try again after an exponentially distributed time. Steady-state probabilities are approximated and bounded in two different ways. We characterize the service distribution by its Laplace transform, and use this characterization to determine the better method of approximation.
Structure functions with finite minimal vector sets
Authors: Baxter Laurence A. , Lee Seung Min
Published: July 2016

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα
is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.
On a system of components with joint lifetimes distributed as a mixture of independent exponential laws
Authors: Lefèvre Claude , Malice Marie-Pierre
Published: July 2016

A system of n non-renewable components sharing a common environment is considered. The joint lifetimes of the components are taken distributed as a mixture of n independent exponential laws. It is shown how the mixing distribution can affect the number of components functioning and the reliability of a k-out-of-n system. The analysis is carried out by using different partial orderings for distribution functions.
Obituary: some memories of Pat Moran
Authors: Hannan E. J.
Published: July 2016

In May 1953, I was sent for the remainder of that year to the Australian National University (ANU) in Canberra by Australia's central bank, where I was then a junior research officer. Feeling rather lost and lonely, I was reading in the library when Pat Moran looked over my shoulder and, noticing what I was reading, asked me to come and see him. This event altered my life, for I then came into close contact with a man of firstclass mind and of generous scientific spirit.
Multivariate birth-and-death processes as approximations to epidemic processes
Authors: Griffiths D. A.
Published: July 2016

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.
The asymptotic theory of linear time-series models
Authors: Hannan E. J.
Published: July 2016

A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.
On a functional equation for general branching processes
Authors: Doney R. A.
Published: July 2016

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW
) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W|
y
) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0
β < 1, where L is one of a class of functions of slow variation.
Some results for quasi-stationary distributions of birth-death processes
Authors: Kijima Masaaki , Seneta E.
Published: July 2016

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).
Stochastic ordering of random processes with an imbedded point process
Authors: Baccelli François
Published: July 2016

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.
On the first zero of an empirical characteristic function
Authors: Bräker H. U. , Hüsler J.
Published: July 2016

We deal with the distribution of the first zero Rn
of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn
in this case, which clarifies some recent results on Rn
in relation to the behaviour of the characteristic function.
On approximately optimal index strategies for generalised arm problems
Authors: Fay N. A. , Walrand J. C.
Published: July 2016

Nash has extended Gittins' work to describe optimal strategies for a class of generalised bandit problems. Here we use a forwards induction argument to analyse ε -optimal strategies for generalised bandit problems. An evaluation procedure for such problems is described; this may be used to analyse models in research planning and stochastic scheduling.
On a buffer problem for packetized voice with an N-periodic strongly interchangeable input process
Authors: Ott Teunis J. , Shanthikumar J. George
Published: July 2016

Consider a slotted communication channel which carries packetized voice and which can transmit exactly one packet every timeslot. Assume that every conversation routed over this channel generates exactly one packet every N timeslots. We study, for the case of an infinite buffer and an N-periodic strongly interchangeable input process, buffer behavior and packet delays as functions of the number of calls routed over the channel, as long as that number is less than or equal to N. Among our results are a simple algorithm which computes the marginal distribution of the buffer content and an algorithm with complexity of order N
4 which computes the distribution of the maximal buffer content (over a period of N timeslots).
Two sufficient properties for the insensitivity of a class of queueing models
Authors: Haviv Moshe
Published: July 2016

For indivisible strong work-conserving queueing models with a Poisson arrival process, each of the following two properties is a sufficient condition for insensitivity. (1) The completed workload of a job receiving service is independent of the number of jobs in the system. (2) Independently of the completed workloads of the jobs in the system, they all are equally likely to be in service. For models which additionally belong to the class described by two families of parameters each of these properties is also necessary for insensitivity.
On the cut-off phenomenon in some queueing systems
Authors: Konstantopoulos P. , Baccelli F.
Published: July 2016

A number of stochastic queueing systems exhibit an interesting phenomenon known as the cut-off phenomenon. A properly scaled version of the distance between the transient process and the stationary one converges to a step function as the initial load converges to infinity. The purpose of this paper is to promote the idea that this phenomenon is a direct consequence of the coupling between the two processes, being thus generalizable to systems lacking any kind of Markovian structure.
Modelling heterogeneity in survival data
Authors: Hougaard Philip
Published: July 2016

Ordinary survival models implicitly assume that all individuals in a group have the same risk of death. It may, however, be relevant to consider the group as heterogeneous, i.e. a mixture of individuals with different risks. For example, after an operation each individual may have constant hazard of death. If risk factors are not included, the group shows decreasing hazard. This offers two fundamentally different interpretations of the same data. For instance, Weibull distributions with shape parameter less than 1 can be generated as mixtures of constant individual hazards. In a proportional hazards model, neglect of a subset of the important covariates leads to biased estimates of the other regression coefficients. Different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences. For instance, the heterogeneity distribution can be either identifiable or unidentifiable. Both mathematical and interpretational consequences of the choice of distribution are considered. Heterogeneity can be evaluated by the variance of the logarithm of the mixture distribution. Examples include occupational mortality, myocardial infarction and diabetes.
Some bounds on reliability of coherent systems of IFRA components
Authors: Chaudhuri Gopal , Deshpande Jayant V. , Dharmadhikari Avinash D.
Published: July 2016

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.
Winding angle and maximum winding angle of the two-dimensional random walk
Authors: Bélisle Claude , Faraway Julian
Published: July 2016

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10msteps, form= 2, 3, 4, 5, 6, 7.
Compensator conditions for stochastic ordering of point processes
Authors: Kwieciński A. , Szekli R.
Published: July 2016

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞
+ Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.
On rate conservation for non-stationary processes
Authors: Mazumdar Ravi , Kannurpatti Raghavan , Rosenberg Catherine
Published: July 2016

This paper extends the rate conservation principle to cadlag processes whose jumps form a non-stationary point process with a time-dependent intensity. It is shown that this is a direct consequence of path integration and the strong law of large numbers for local martingales. When specialized to mean rates a non-stationary version of Miyazawa's result is obtained which is recovered in the stationary case. Some applications of the result are also given.
A new continuous-time search model
Authors: Stadje Wolfgang
Published: July 2016

We study a search model in which offers of random size are received randomly over time. The arrival times form a point process of a certain type, and the offer size distribution may depend on the corresponding arrival time. The search costs have a time-dependent cost rate. The objective is to stop the search process such that the expected discounted net reward (associated with the maximal offer received so far) is maximized. A stopping time σ is suggested, and conditions are specified under which σ turns out to be optimal.
Bounds for discounted stochastic scheduling problems
Authors: Glazebrook K. D.
Published: July 2016

Suppose that π is a policy for resource allocation in a stochastic environment and π ∗ is an optimal policy. Two existing procedures for policy evaluation are described and compared. Both of these evaluate π by means of upper bounds on R(π ∗) – R(π), the total reward lost when making resource allocations according to π rather than π∗. The bounds developed by these two methods are called Type 1 and Type 2. We demonstrate by example that neither of these procedures dominates the other in the sense of always yielding tighter bounds. A modification to Type 2 bounds is proposed resulting in an improved procedure which always dominates the Type 1 approach.
A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems
Authors: Zheng Yu-Sheng
Published: July 2016

The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.
Some monotonicity properties of the delayed renewal function
Authors: Hansen B. G. , Frenk J. B. G.
Published: July 2016

Analogues of some theorems of Brown (1980) concerning renewal measures with DFR or IMRL underlying distributions are proved for delayed renewal measures. Some related results, such as partial converses of the main theorems, are also presented.
Reliability estimation of a complex renewable system with an unbounded number of repair units
Authors: Solovyev A. D. , Konstant D. G.
Published: July 2016

In this study an asymptotical analysis of the reliability of a complex renewable system with an unbounded number of repair units is provided. The system state is given by a binary vector e(t) = [e
1(t), · ··, en
(t)], ei
(t) = 0(1), if at moment t the ith element is failure-free (failed). We assume that at the state e the ith element has failure intensity λ
i
(e). At the instant of failure of every element the renewal work begins and the renewal time has distribution function Gi
(t). Let E_ be the set of failed system states. The goal of this study is the asymptotic estimation of the distribution of the time until the first system failure, .
Transient and busy period analysis of the GI G/1 Queue as a Hilbert factorization problem
Authors: Bertsimas Dimitris J. , Keilson Julian , Nakazato Daisuke , Zhang Hongtao
Published: July 2016

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of theGI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for theGI/R/1,R/G/1 queues, whereRis the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.
A maximum sequence in a critical multitype branching process
Authors: Spåtaru Aurel
Published: July 2016

Let (Zn
) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn
) consider the right eigenvector u = (u
1, · ··, up
) > 0, and set . It is shown that lim inf, lim sup whenever Z
0 = i, where .
Change of measure up to a random time: theory
Authors: Mortimer T. M. , Williams David
Published: July 2016

Change of measure up to fixed times or stopping times is the theme of the famous Cameron–Martin–Girsanov formula. The paper studies change of measure up to random times which are not stopping times of the natural filtration. The ultimate aim is to build up a family of interesting models for physics and chemistry.
How to compare two systems
Authors: Ebrahimi Nader
Published: July 2016

Consider two systems with identical components but different structure functions. In this paper we propose a method to compare these two systems when the marginal behaviors of components are specified but when the complete joint behavior of components is unspecified. We also give two useful theorems for applying our method. We apply our techniques to two examples.
A note on extended uniformization for non-exponential stochastic networks
Authors: Van Dijk Nico M.
Published: July 2016

The standard uniformization technique for continuous-time Markov chains is generalized to non-exponential stochastic networks.
Biometric and chromosome algebras
Authors: Holgate Philip
Published: July 2016

This note continues the development of the infinite-dimensional genetic algebra approach to problems of population genetics. Two algebras are studied. One describes the familiar problem of a quantitative characteristic, and the other provides a way of treating the whole chromosome as an entity.
A convergence theorem for symmetric functionals of random partitions
Authors: Joyce Paul , Tavaré Simon
Published: July 2016

This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.
The first-passage density of the Brownian motion process to a curved boundary
Authors: Durbin J. , Williams D.
Published: July 2016

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.
On the number of times where a simple random walk reaches its maximum
Authors: Katzenbeisser W. , Panny W.
Published: July 2016

Let Qn
denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn
, are considered and their asymptotic behavior is studied.
Processes with associated increments
Authors: Glasserman Paul
Published: July 2016

We derive conditions under which the increments of a vector process are associated — i.e. under which all pairs of increasing functions of the increments are positively correlated. The process itself is associated if it is generated by a family of associated and monotone kernels. We show that the increments are associated if the kernels are associated and, in a suitable sense, convex. In the Markov case, we note a connection between associated increments and temporal stochastic convexity.

Our analysis is motivated by a question in variance reduction: assuming that a normalized process and its normalized compensator converge to the same value, which is the better estimator of that limit? Under some additional hypotheses we show that, for processes with conditionally associated increments, the compensator has smaller asymptotic variance.
Transition probability density of a certain diffusion process concentrated on a finite spatial interval
Authors: Milian A.
Published: July 2016

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.
Estimation of the derivative of a stationary measure with respect to a control parameter
Authors: Vázquez-Abad Felisa J. , Kushner Harold J.
Published: July 2016

The paper deals with a problem which arises in the Monte Carlo optimization of steady state or ergodic systems which can be modelled by Markov chains. The transition probability depends on a parameter, and one wishes to find the parameter value at which some performance function is minimum. The only available data are obtained from either simulation or actual operating information. For such a problem one needs good statistical estimates of the derivatives. Conditions are given for the existence of the derivative of the stationary measure with respect to the parameter, in the sense that the derivative is a signed measure, and is the limit of the natural approximating sequence. Some properties and a useful characterization of the derivative are obtained. It is also shown that, under appropriate conditions, the derivative of the n-step transition function converges to the derivative of the stationary measure as n tends to ∞. This latter result is of particular importance whether one is simply estimating or is actually optimizing via some sequential Monte Carlo procedure, since the basic observations are always taken over a finite time interval.
On strict stationarity and ergodicity of a non-linear ARMA model
Authors: Liu Jian , Susko Ed
Published: July 2016

Two recent papers by Petruccelli and Woolford (1984) and Chan et al. (1985) showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie (1974), (1975) in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie's results. In this paper, instead of pursuing the ideas in Tweedie's results, we shall develop a criterion similar in spirit to the technique used by Beneš (1967) in the context of continuous-time Markov chains. Consequently, we derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1, q) model to yield a sufficient condition for strict stationarity which is identical to the condition given by Petruccelli and Woolford (1984) for the threshold AR(1). Hence, the conjecture that the moving average component does not affect stationarity is partially verified. Furthermore, under an additional irreducibility assumption, ergodicity of a non-linear ARMA model is established. The paper then concludes with a necessary condition for stationarity of the threshold ARMA(1, q) model.
Time-dependent process of M/G/1 vacation models with exhaustive service
Authors: Takagi Hideaki
Published: July 2016

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.
Eigenstructure of the infinitely-many-neutral-alleles diffusion model
Authors: Ethier S. N.
Published: July 2016

The complete set of eigenvalues is found for the (unlabeled) infinitely-many-neutral-alleles diffusion model. The transition density for the process, originally derived by Griffiths, is rederived as an eigenfunction expansion.
On the limiting distribution of a supercritical branching process in a random environment
Authors: Hambly Ben
Published: July 2016

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.
Non-homogeneous semi-Markov systems and maintainability of the state sizes
Authors: Vassiliou P.-C. G. , Papadopoulou A. A.
Published: July 2016

In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structure in the state is studied. It is also proved that all maintainable expected duration structures by recruitment control belong to a convex set the vertices of which are specified. Finally an illustration is provided of the present results in a manpower system.
On the total heights of random rooted trees
Authors: Takacs Lajos
Published: July 2016

Denote by Sn
the set of all distinct rooted trees with n labeled vertices. Define τn
as the total height of a tree chosen at random in the set Sn
, assuming that all the possible nn

–1 choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that is, the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn
and their asymptotic behavior as n → ∞.
On record values and the exponent of a distribution with regularly varying upper tail
Authors: Berred Mohamed
Published: July 2016

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.
The Wiener process between a reflecting and an absorbing barrier
Authors: Schwarz Wolfgang
Published: July 2016

We consider a Wiener process between a reflecting and an absorbing barrier and derive a series solution for the transition density of the process and for the density of the time to absorption. If the drift is towards the reflecting barrier, the variance is not too large, and the distance of the barriers is not too small, the leading term of the series derives from imaginary solutions of the basic eigenvalue equation of this problem. It is shown that these leading terms often make the dominant contribution to the complete series. Finally, we consider previous attempts by Fürth [3], Cox and Miller [1], and by Goel and Richter-Dyn [5] to solve the stated problem and point out, in some detail, why their solutions are wrong.
Cost rate heuristics for semi-Markov decision processes
Authors: Glazebrook K. D. , Bailey Michael P. , Whitaker Lyn R.
Published: July 2016

In response to the computational complexity of the dynamic programming/backwards induction approach to the development of optimal policies for semi-Markov decision processes, we propose a class of heuristics resulting from an inductive process which proceeds forwards in time. These heuristics always choose actions in such a way as to minimize some measure of the current cost rate. We describe a procedure for calculating such cost rate heuristics. The quality of the performance of such policies is related to the speed of evolution (in a cost sense) of the process. A simple model of preventive maintenance is described in detail. Cost rate heuristics for this problem are calculated and assessed computationally.
On distributions having the almost-lack-of-memory property
Authors: Chukova S. , Dimitrov B.
Published: July 2016

It is shown that random variables X exist, not exponentially or geometrically distributed, such that

P{X – b ≧ x | X ≧ b} = P{X ≧ x}

for all x > 0 and infinitely many different values of b. A class of distributions having the given property is exhibited. We call them ALM distributions, since they almost have the lack-of-memory property. For a given subclass of these distributions some phenomena relating to service by an unreliable server are discussed.
Distribution of the number of alleles in multigene families
Authors: Griffiths R. C.
Published: July 2016

The distribution of the number of alleles in samples from r chromosomes is studied. The stochastic model used includes gene conversion within chromosomes and mutation at loci on the chromosomes. A method is described for simulating the distribution of alleles and an algorithm given for computing lower bounds for the mean number of alleles.

A formula is derived for the expected number of samples from r chromosomes which contain the allele type of a locus chosen at random.
The Gillis–Domb–Fisher correlated random walk
Authors: Chen Anyue , Renshaw Eric
Published: July 2016

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of then-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive then-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.
Approximating kth-order two-state Markov chains
Authors: Wang Y. H.
Published: July 2016

In this paper, we consider kth-order two-state Markov chains {Xi
} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σ
x
|𝐏(Sn
= x) − 𝐏(Y = x)|, where S
n = X
1 + · ··+ Xn
and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.
Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators
Authors: Lawrance A. J.
Published: July 2016

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk
-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.
On closed ring queueing networks
Authors: Bambos Nicholas
Published: July 2016

In this paper we first study ring structured closed queueing networks with distinguishable jobs. Under assumptions of periodicity and ergodicity of the service times, essentially the most general, it is shown that the limits defining the average flows of the jobs exist almost surely, and methods for their estimation by simulation are given. However, it turns out that the values of the flows depend on the initial positions of the jobs, due to the emergence of distinct persistent blocking modes. The effect of these modes on the behavior of general networks with queueing loops is examined.

For independent and identically distributed service times, conditions are specified for the network to asymptotically approach a steady state at large times.

Finally, we study the special case of ring networks with indistinguishable items and stationary and ergodic service times. It is shown that as the number of jobs in the network increases towards infinity, the average circulation time converges to the maximum of the expectations of the service times.
Joint exceedances of high levels under a local dependence condition
Authors: Ferreira Helena
Published: July 2016

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.
AB percolation on bond-decorated graphs
Authors: Appel Martin J. B. , Wierman John C.
Published: July 2016

It is known [8] that a certain class of bond-decorated graphs exhibits multiple AB percolation phase transitions. Sufficient conditions are given under which the corresponding AB percolation critical probabilities may be identified as points of intersection of the graph of a certain polynomial with the boundary of the percolative region of an associated two-parameter bond-site percolation model on the underlying undecorated graph. The main result of the article is used to prove that the graphs in [8] exhibit multiple AB percolation critical probabilities. The possibility of identifying AB percolation critical exponents with corresponding limits for the bond-site model is discussed.
Factorial moments for random mappings by means of indicator variables
Authors: English Brian J.
Published: July 2016

A simple identity for the incomplete factorial of sums of zero-one variables is exploited to provide the factorial moments of the number of components and the number of cyclical elements of the random mapping (T, {pi
}) considered by Ross (1981).
The stationary work in system of a G/G/1 gradual input queue
Authors: Kino Issei , Miyazawa Masakiyo
Published: July 2016

This paper is devoted to the study of a stationary G/G/1 queue in which work input is gradually injected into the system and the work load is processed at unit rate. First, assuming that the input process is stationary, the key formula for the stationary distribution of work in system is derived by appeal to the waiting time of the associated regular G/G/1 queue. The main contribution of this paper is the derivation of Laplace-Stieltjes transforms (LSTs) for the stationary distributions of work in system and related random variables in terms of integrations with respect to a waiting time distribution of the associated regular queue. The results are exemplified by giving explicit formulas for the LST of total work for M/Ek
/1 and M/H
2/1. The results generalize the results of Pan et al. (1991) for the M/M/1 gradual input queue.
A coupling of finite particle systems
Authors: Mountford T. S.
Published: July 2016

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.
Critical behavior of percolation and Markov fields on branching planes
Authors: Wu C. Chris
Published: July 2016

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point p
c, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.
Minimizing the expected rank with full information
Authors: Bruss F. Thomas , Ferguson Thomas S.
Published: July 2016

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.
Utilization of the method of linear matrix equations to solve a quasi-birth-death problem
Authors: Feldman Richard M. , Deuermeyer Bryan L. , Valdez-Flores Ciriaco
Published: July 2016

The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.
On zero-avoiding transition probabilities of an r-node tandem queue: a combinatorial approach
Authors: Böhm Walter , Jain J. L. , Mohanty S. G.
Published: July 2016

In this paper we present a simple combinatorial approach for the derivation of zero-avoiding transition probabilities in a Markovian r-node series Jackson network. The method we propose offers two advantages: first, it is conceptually simple because it is based on transition counts between the nodes and does not require a tensor representation of the network. Second, the method provides us with a very efficient technique for numerical computation of zero-avoiding transition probabilities.
Markov processes for modeling and analyzing a new genetic mapping method
Authors: Feingold Eleanor
Published: July 2016

This paper describes a set of stochastic processes that is useful for modeling and analyzing a new genetic mapping method. Some of the processes are Markov chains, and some are best described as functions of Markov chains. The central issue is boundary-crossing probabilities, which correspond top-values for the existence of genes for particular traits. The methods elaborated by Aldous (1989) provide very accurate approximatep-values, as spot-checked against simulations.
An asymptotically exact decomposition of coupled Brownian systems
Authors: Fendick Kerry W.
Published: July 2016

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.
A note on exceedances and rare events of non-stationary sequences
Authors: Hüsler J.
Published: July 2016

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.