bluuebunny/crossref_metadata_2025 huggingface dataset:
A limit theorem for a class of supercritical branching processes
Authors: Doney R. A.
Published: July 2016

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn
} form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ
1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn
/mn
to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].
Galton-Watson processes in varying environments
Authors: Jagers Peter
Published: July 2016

Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.
Shock processes by aftereffects and multivariate lack of memory
Authors: Ghurye S. G. , Marshall Albert W.
Published: July 2016

If the survival function satisfies the functional equation

where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.
A multiple-threshold AR(1) model
Authors: Chan K. S. , Petruccelli Joseph D. , Tong H. , Woolford Samuel W.
Published: July 2016

We consider the model Zt
= φ (0, k)+ φ(1, k)Zt

–1 + at
(k) whenever r

k−1<Z

t−1≦r

k
, 1≦k≦l, with r
0 = –∞ and rl
=∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at
(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at
(k), t≧1} independent of {at
(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.
On the exact order of normal approximation in multivariate renewal theory
Authors: Niculescu Ştefan P. , Omey Edward
Published: July 2016

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).
R&D projects analyzed by semimartingale methods
Authors: Aase Knut K.
Published: July 2016

In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.
Age-dependent minimal repair
Authors: Block Henry W. , Borges Wagner S. , Savits Thomas H.
Published: July 2016

A stochastic model is developed to describe the operation in time of the following maintained system setting. A piece of equipment is put in operation at time 0. Each time it fails, a maintenance action is taken which, with probability p(t), is a complete repair or, with probability q(t)=1– p(t), is a minimal repair, where t is the age of the equipment in use at the failure time. It is assumed that complete repair restores the equipment to its good as new condition, that minimal repair restores the equipment to its condition just prior to failure and that both maintenance actions take negligible time.

If the equipment's life distribution F is a continuous function, the successive complete repair times are shown to be a renewal process with interarrival distribution

for t ≧ 0. Preservation and monotone properties of the model extending the results of Brown and Proschan (1983) are obtained.
The proportion of the population never hearing a rumour
Authors: Sudbury Aidan
Published: July 2016

For a model of a rumour, first given a non-rigorous treatment by Maki and Thompson, it is shown that the proportion of the population never hearing the rumour converges in probability to 0.203 as the population size tends to∞.
The association in time of a Markov process with application to multistate reliability theory
Authors: Hjort Nils Lid , Natvig Bent , Funnemark Espen
Published: July 2016

A series of bounds for the availability and unavailability in a fixed time interval, I, for a system of maintained, interdependent components are given in Natvig (1980) in the traditional binary case, and in Funnemark and Natvig (1985) in the multistate case. For the special case of independent components the only assumption needed to arrive at these bounds is that the marginal performance process of each component is associated in I. When these processes are Markovian and binary, a sufficient condition for this to hold is given by Esary and Proschan (1970). In the present paper we generalize this condition to the multistate case, and give an equivalent and much more convenient condition in terms of the transition intensities.
On weak lumpability in Markov chains
Authors: Rubino Gerardo
Published: July 2016

We analyse the conditions under which the aggregated process constructed from an homogeneous Markov chain over a given partition of the state space is also Markov homogeneous. The past work on the subject is revised and new properties are obtained.
Some further results on random splitting of an interval
Authors: Lloyd Christopher J.
Published: July 2016

The random splitting model of Lloyd and Williams (1988) is generalised, to allow first beta splitting distributions and secondly, discrete splitting distributions. Analogues of some earlier results are developed. In particular, the equality of the average value of the largest split and the probability that this is achieved at the first split continues to hold.
Further results for dynamic scheduling of multiclass G/G/1 queues
Authors: Hirayama Tetsuji , Kijima Masaaki , Nishimura Shoichi
Published: July 2016

We consider discrete-time dynamic scheduling problems of the following three types ofG/G/1 queue withKdifferent customer classes: (i) aG/DFR/1 queue withKclasses under preemptive resume service discipline, (ii) aG/IFR/1 queue with two classes under preemptive resume service discipline, and (iii) aG/G/1 queue with two classes under non-preemptive service discipline. Interchange arguments are used to show that simple index policies of different type minimize the total holding cost of customers in a finite-horizon scheduling period for the three cases. Our results extend the result for aG/M/1 queue by Buyukkoc et al. (1985) to general queues.
Realization probability and throughput sensitivity in a closed jackson network
Authors: Cao Xi-Ren
Published: July 2016

Realization probability is a new concept pertaining to perturbation analysis of closed queuing networks. The sensitivities of throughputs in a closed single-class Jackson network can be expressed in terms of realization probabilities. In this paper, based on a discussion of perturbation analysis for networks with state-dependent service rates, we derive some new formulas for sensitivities of throughputs using realization probability.
A functional central limit theorem for the Ewens sampling formula
Authors: Hansen Jennie C.
Published: July 2016

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.
On the two-boundary first-crossing-time problem for diffusion processes
Authors: Buonocore A. , Giorno V. , Nobile A. G. , Ricciardi L. M.
Published: July 2016

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.
On uniform conditional stochastic order conditioned on planar regions
Authors: Shanthikumar J. George , Koo Hui-Wen
Published: July 2016

Sufficient conditions under which two random vectors are ordered in the sense of uniform conditional stochastic order (Whitt (1980), (1982)) with respect to planar regions are given. A natural classification of distributions based on this notion of stochastic order is defined and studied. A negative dependence property of Block et al. (1985) is shown to hold for this class of distributions. An application of these results in statistics is also presented.
Non-negative time series models for dry river flow
Authors: Hutton Jane L.
Published: July 2016

Non-negative time series which are first-order autoregressive with a mixed exponential innovation process are studied. Properties and approximate marginal distributions for such series are found. Modifications to include exact zeroes and to increase variability so that the time series is a more realistic model of rivers which are dry for part of the year are discussed.
Saddlepoint approximations for bivariate distributions
Authors: Wang Suojin
Published: July 2016

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean of n independent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.
Maintenance comparisons: block policies
Authors: Block Henry W. , Langberg Naftali A. , Savits Thomas H.
Published: July 2016

Complete repair and minimal repair models with a block maintenance policy are considered. Each of these models gives rise to a counting process, and these processes are compared stochastically. This contrasts with most previous work on maintenance policies where only univariate marginal comparisons were made. Also a more general block schedule is considered than is customary.
On the matrix occurring in a linear search problem
Authors: Phatarfod R. M.
Published: July 2016

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj
, ···, where pi
is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi
where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.
Some generalized variability orderings among life distributions with reliability applications
Authors: Bhattacharjee M. C.
Published: July 2016

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.
The origin and growth of the Applied Probability Trust
Authors: Hitchcock Mavis
Published: July 2016

An account is given of the foundation of the Applied Probability Trust and the development of its office at the University of Sheffield. The publishing achievements of its first quarter-century are reviewed.
Stochastic models and field trials
Authors: Bartlett M. S.
Published: July 2016

A survey of the development of statistical techniques of field experimentation refers to the unified approach of R. A. Fisher, in which analysis and design were closely linked. Randomization of treatment positions bypassed the need for a precise stochastic model, but sometimes at the cost of diminished accuracy.

Alternative covariance methods associated with the name of J. S. Papadakis, taking note of the yields of neighbouring plots, have stimulated much research in recent years. The results to date suggest that, while improved accuracy can often be achieved, the assessment of accuracy is more dependent on the stochastic model. Robust methods for such assessment still need to be finalized; with these alternative methods a closer link between analysis and design is also advocated.
Typical polymorphisms maintained by selection at a single locus
Authors: Kingman J. F. C.
Published: July 2016

It is known that several different alleles can be maintained at a locus by selection, but only when the various genotypic fitnesses satisfy very special conditions. It is shown in this paper that a population with many possible mutations will evolve in such a way that these conditions arise naturally for most fitness regimes. The first steps are taken towards the assessment of the likely size and shape of the resulting stable polymorphisms.
Stein's method and poisson process convergence
Authors: Barbour A. D.
Published: July 2016

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.
Conditions for strong ergodicity using intensity matrices
Authors: Johnson Jean , Isaacson Dean
Published: July 2016

Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψ
n
of Pn
(ψ nPn
= ψ n
) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.
The transient behaviour of the simple random walk in the plane
Authors: Downham D. Y. , Fotopoulos S. B.
Published: July 2016

For the simple two-dimensional random walk on the vertices of a rectangular lattice, the asymptotic forms of several properties are well known, but their forms can be insufficiently accurate to describe the transient process. Inequalities with the correct asymptotic form are derived for six such properties. The rates of approach to the asymptotic form are derived. The accuracy of the bounds and some practical implications of the results are discussed.
A characterization of the gamma distribution from a random difference equation
Authors: Tollar Eric S.
Published: July 2016

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).
The distribution of evolutionarily stable strategies
Authors: Haigh John
Published: July 2016

Suppose the n × n matrix A gives the payoffs for some evolutionary game, and its entries are the values of independent, identically distributed, continuous random variables. The distribution of the pattern of evolutionarily stable strategies for A will depend, if n ≧ 3, on this underlying distribution. A fairly complete picture for n = 3 is found, and some results are obtained for n ≧ 4.
Pareto processes
Authors: Yeh Hsiaw-Chan , Arnold Barry C. , Robertson Christopher A.
Published: July 2016

An autoregressive process ARP(1) with Pareto-distributed inputs, analogous to those of Lawrance and Lewis (1977), (1980), is defined and its properties developed. It is shown that the stationary distributions are Pareto. Further, the maximum and minimum processes are asymptotically Weibull, and the ARP(1) process is shown to be closed under maximization or minimization when the number of terms is geometrically distributed. The ARP(1) process leads naturally to an extremal process in the sense of Lamperti (1964). Statistical inference for the ARP(1) process is developed. An absolutely continuous variant of the Pareto process is described.
A non-Markov model for the optimum replacement of self-repairing systems subject to shocks
Authors: Rangan A. , Grace R. Esther
Published: July 2016

A system is subject to shocks; each shock at time t increases the cumulative damage λ (t) by a constant amount, while the system is subject to repair in between the shocks which brings down λ (t) at a constant rate. The shock arrival process is an inhomogeneous Poisson process with intensity function λ (t) and each shock weakens the system making it more expensive to run. The long-run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.
On the first-passage times of pure jump processes
Authors: Shared Moshe , Shanthikumar J. George
Published: July 2016

Let Tx
be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx
has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx
to the IFR or PF2 are not available in the literature.
The basic equations for a supplemented GSMP and its applications to queues
Authors: Miyazawa Masakiyo , Yamazaki Genji
Published: July 2016

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.
A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks
Authors: McKenna James
Published: July 2016

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L)
l
) = λ lE((W)
l
) are also true, where (L)
l
= L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji
and Sji, where Nji
is the total number of class j jobs at center i and Sji
is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji
and Wji
are related this way, where Qji
is the number of class j jobs queued, but not in service at center i and Wji
is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.
How large is the support of an ESS?
Authors: Haigh John
Published: July 2016

Suppose the payoffs (aij) in then×nmatrixAare drawn independently from some continuous probability distribution. The number of tactics used in an ESS is investigated. Asymptotic results on the size of the ESS with largest support are given, using the work of Karlin and Kingman on the size of polymorphisms in one-locus multi-allele diploid selection models.
On the null recurrence and transience of a first-order SETAR model
Authors: Guo Meihui , Petruccelli Joseph D.
Published: July 2016

We consider the SETAR(l; 1, ···, 1) model:

where – ∞= r
0 < r
1, < · ·· < rl
= ∞ and for each j {ε t
(j)} forms an i.i.d. zero-mean error sequence independent of {ε t
(i)} for i ≠ j and having a density positive on the real line. Chan et al. (1985) obtained the region of the parameter space on which the process is ergodic, and showed the process to be transient on a subset of the remainder. They conjectured that the process was null recurrent everywhere else. In this paper we show that conjecture to be incorrect and under the assumption of finite variance of the error distributions we resolve the remaining questions of transience or null recurrence for this process.
Time-changing and truncating K-capacity queues from one K to another
Authors: Glasserman Paul , Gong Wei-Bo
Published: July 2016

For , we obtain a K′- capacity queue from a K- capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of an M/G/1/K queue, the time change erases service intervals that begin with more than K′ customers in the systems. This construction yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities in M/G/1/K queues. The same approach proves a strengthened result for ‘detailed' state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A ‘dual' construction yields a different kind of proportionality for the G/M/1/K queue.
Expected frequencies of DNA patterns using whittle's formula
Authors: Cowan Richard
Published: July 2016

Given a realisation of a Markov chain, one can count the numbers of state transitions of each type. One can ask how many realisations are there with these transition counts and the same initial state. Whittle (1955) has answered this question, by finding an explicit though complicated formula, and has also shown that each realisation is equally likely. In the analysis of DNA sequences which comprise letters from the set {A, C, G, T}, it is often useful to count the frequency of a pattern, say ACGCT, in a long sequence and compare this with the expected frequency for all sequences having the same start letter and the same transition counts (or ‘dinucleotide counts' as they are called in the molecular biology literature). To date, no exact method exists; this paper rectifies that deficiency.
Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes
Authors: Lam C. Y. Teresa
Published: July 2016

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.
The abscissa of convergence of the Laplace transform
Authors: Hall Peter , Teugels Jozef L. , Vanmarcke Ann
Published: July 2016

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn
= O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.
On the serial correlation for number in system in the stationary GI/M/1 bulk arrival and GI/Em
/1 queues
Authors: Stanford D. A. , Pagurek B.
Published: July 2016

The generating functions for the serial covariances for number in system in the stationary GI/M/1 bulk arrival queue with fixed bulk sizes, and the GI/Em
/1 queue, are derived. Expressions for the infinite sum of the serial correlation coefficients are also presented, as well as the first serial correlation coefficient in the case of the bulk arrival queue. Several numerical examples are considered.
Pathwise recurrence orders and simulated annealing
Authors: Borkar Vivek S.
Published: July 2016

The Connors–Kumar notion of recurrence orders for state and state transitions of time-inhomogeneous Markov chains is redefined ‘pathwise' and applied to the sample path analysis of the simulated annealing algorithm.
Work-modulated queues with applications to storage processes
Authors: Browne Sid , Sigman Karl
Published: July 2016

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn
, depends upon their delay, Dn
, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.
Stabilities and instabilities in population dynamics
Authors: Jagers Peter
Published: July 2016

Stability in population size is illusory: populations left to themselves either grow beyond all bounds or die out. But if they do not die out their composition stabilizes. These problems are discussed in terms of general abstract, multitype branching processes. The life and descent of a typical individual is described.
Coefficients of ergodicity for stochastically monotone Markov chains
Authors: Pflug G. Ch. , Schachermayer W.
Published: July 2016

In this paper we show that to each distance d defined on the finite state space S of a strongly ergodic Markov chain there corresponds a coefficient ρd
of ergodicity based on the Wasserstein metric. For a class of stochastically monotone transition matrices P, the infimum over all such coefficients is given by the spectral radius of P – R, where R = lim
k

Pk
and is attained. This result has a probabilistic interpretation of a control of the speed of convergence of by the metric d and is linked to the second eigenvalue of P.
A Swiss Army formula of Palm calculus
Authors: Brémaud P.
Published: July 2016

We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.
Slepian models for non-stationary Gaussian processes
Authors: Gadrich Tamar , Adler Robert J.
Published: July 2016

We give explicit expressions for the Slepian model process of non-stationary Gaussian processes following level crossings and local maxima. We also include a detailed analysis of the high-level case.
Some characterizations of the Poisson process and geometric renewal process
Authors: Huang Wen-Jang , Li Shun-Hwa , Su Jyh-Cherng
Published: July 2016

Let γ t
and δ t
denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t
and δ t.
On renewal processes relating to counter models: the case of phase-type interarrival times
Authors: Kao Edward P. C. , Smith Marion Spokony
Published: July 2016

The Type I and Type II counter models of Pyke (1958) have many applications in applied probability: in reliability, queueing and inventory models, for example. In this paper, we study the case in which the interarrival time distribution is of phase type. For the two counter models, we derive the renewal functions of the related renewal processes and propose approaches for their computations.
Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time
Authors: Kijima Masaaki
Published: July 2016

A continuous-time Markov chain on the non-negative integers is called skip-free to the left (right) if the governing infinitesimal generator A = (aij
) has the property that aij
= 0 for j ≦ i ‒ 2 (i ≦ j – 2). If a Markov chain is skip-free both to the left and to the right, it is called a birth-death process. Quasi-limiting distributions of birth–death processes have been studied in detail in their own right and from the standpoint of finite approximations. In this paper, we generalize, to some extent, results for birth-death processes to Markov chains that are skip-free to the left in continuous time. In particular the decay parameter of skip-free Markov chains is shown to have a similar representation to the birth-death case and a result on convergence of finite quasi-limiting distributions is obtained.
On the exact distribution of observed open times in single ion channel models
Authors: Ball Frank , Milne Robin K. , Yeo Geoffrey F.
Published: July 2016

Continuous-time Markov chain models have been widely considered for the gating behaviour of a single ion channel. In such models the state space is usually partitioned into two classes, designated ‘open' and ‘closed', and there is ‘aggregation' in that it is possible to observe only which class the process is in at any given time. Hawkes et al. (1990) have derived an expression for the density function of the exact distribution of an observed open time in such an aggregated Markov model, where brief sojourns in either the open or the closed class are unobservable. This paper extends their result to single ion channel models based on aggregated semi-Markov processes, giving a more direct derivation which is probabilistic and exhibits clearly the combinatorial content.
Yule process approximation for the skeleton of a branching process
Authors: O'Connell Neil
Published: July 2016

The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered. When r = 0 it is trivial. If r > 0 is small and the time unit is taken as α /r generations (α > 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rate α. This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.
On estimating the diffusion coefficient from discrete observations
Authors: Florens-Zmirou Danielle
Published: July 2016

This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.
Sums and maxima of discrete stationary processes
Authors: McCormick William P. , Sun Jiayang
Published: July 2016

This paper considers the joint limiting behavior of sums and maxima of stationary discrete-valued processes. The asymptotic behavior is a cross between a central limit theorem and asymptotic bounds for the distribution of the maxima. Some applications and simulations are also included.
Some characterizations of the exponential and geometric distributions arising in a queueing system with an unreliable server
Authors: Huang Wen-Jang , Shoung Jyh-Ming
Published: July 2016

In this article, we generalize results by Dimitrov and Khalil (1990), Khalil et al. (1991), and van Harn and Steutel (1991) and obtain some characterizations of the exponential and geometric laws.
Optimality of the round-robin routing policy
Authors: Liu Zhen , Towsley Don
Published: July 2016

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.
On maintained systems operating in a random environment
Authors: Wortman Martin A. , Klutke Georgia-Ann
Published: July 2016

This paper examines the availability of a maintained system where the rate of deterioration is governed by an exogenous random environment. We provide a qualitative result that exposes the relationship between remaining lifetime, environment, and repairs. This result leads to simple bounds that can be used to choose inspection rates that guarantee a specified level of availability. The principal result requires no specific distributional assumptions, is intuitively appealing and can be directly applied by practitioners. Our development employs techniques from stochastic calculus.
Subexponential distributions and dominated-variation tails
Authors: Goldie Charles M.
Published: July 2016

For a distribution functionFon (0,∞), regular variation of its tailis known to imply thatFis subexponential. Letbe merely of dominated variation. This note shows thatFneed not be subexponential, and investigates which of the known necessary conditions for subexponentiality become sufficient when insisted upon for such anF.
Plate and line segment processes
Authors: Fava N. A. , Santaló L. A.
Published: July 2016

Random processes of convex plates and line segments imbedded in R
3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.
Scheduled maxima sequences
Authors: Deken Joseph G.
Published: July 2016

We define a vector-valued scheduled maxima sequence
M
by considering simultaneously the maxima of several i.i.d. sequences, with the number of observations considered from each sequence at any time determined by a random scheduling sequence
J.
It is shown that the max-min (vector) sequence derived from i.i.d. can be represented as a mixture of scheduled maxima sequences, giving results for this sequence and the range A functional limit theorem for the scheduled maxima sequence shows convergence to independent extremal processes. Embedding in a scheduled extremal process gives strong laws, central limit theorems, and laws of the iterated logarithm for the record time of the scheduled maxima sequence, and hence for the max-min sequence and the range.
Estimating the law of randomly moving particles by counting
Authors: McDunnough Philip
Published: July 2016

The fluctuation of the counts of particles inmregions are used to yield an estimate of a fundamental motion parameter. The effect of varyingmon the precision of the estimation method is considered. An exact analysis and simulation of the Fürth pedestrian vector process is presented.
Minimum contrast estimation in diffusion processes
Authors: Lánska Vĕra
Published: July 2016

This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.
The rate and variance functions arising from the interaction of two stationary point processes
Authors: Lawrance A. J.
Published: July 2016

The paper gives the mean-time and variance-time functions of responses arising from a general type of interaction between two stationary point processes which are independent of each other. The mean-time function is obtained exactly and explicitly, thereby much improving on earlier work. A Laplace-transformed version of the variance-time function is derived under the general assumptions, and new results for the asymptotic slope and intercept are obtained when one of the point processes is renewal. Attention is drawn to the difficulties of dealing with non-renewal and non-regenerative point processes.
A remark on the central limit question for dependent random variables
Authors: Bradley Richard C.
Published: July 2016

Given a strictly stationary sequence {Xk, k= …, −1,0,1, …} of r.v.'s one defines forn= 1, 2, 3 …,. Here an example of {Xk} is given with finite second moments, for which Var(X1+ … +Xn)→∞ andρn→ 0 asn→∞, but (X1+ … +Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.
Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals
Authors: Ramaswami V. , Neuts Marcel F.
Published: July 2016

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.
Computation of the stationary distribution of the queue size in an M/G/1 queueing system with variable service rate
Authors: Federgruen A. , Tijms H. C.
Published: July 2016

This paper presents a simple and computationally tractable method which recursively computes the stationary probabilities of the queue size in an M/G/1 queueing system with variable service rate. For each service two possible service types are available and the service rule is characterized by two switch-over levels. The computational approach discussed in this paper can be applied to a variety of queueing problems.
Further results on single-lane traffic flow
Authors: Cowan Richard
Published: July 2016

A number of stochastic properties for a model of traffic flowing in a single lane are derived. The arrival process used is one which models (a) the bunching phenomenon of traffic and (b) the fact that vehicles have a spatial dimension and travel at minimum temporal spacing. Weak assumptions about the desired speeds of vehicles are made.
On random simplices in product distributions
Authors: Maehara Hiroshi
Published: July 2016

It is proved that the r-dimensional volume Vn
of the r-simplex spanned by r + 1 sample points taken from the n-fold product distribution of a distribution F is asymptotically normal as n →∞.
Asymptotic likelihood theory for diffusion processes
Authors: Brown B. M. , Hewitt J. I.
Published: July 2016

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.
Autoregression for discrete processes mod 2
Authors: Kanter Marek
Published: July 2016

Given a ‘black box’ of logical elements, a sequence of independent random pulses is fed in and the output from the black box is observed. A procedure is given for estimating the parameters of the black box, in the case when the black box is constructed in a manner analogous to the classical autoregressive scheme.
Orderliness, intensities and Palm-Khinchin equations for multivariate point processes
Authors: Daley D. J. , Milne R. K.
Published: July 2016

Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.
Determination of the order of a Markov chain by Akaike's information criterion
Authors: Tong H.
Published: July 2016

Using Akaike's information criterion, we have presented an objective procedure for the determination of the order of an ergodic Markov chain with a finite number of states. The procedure exploits the asymptotic properties of the maximum likelihood ratio statistics and Kullback and Leibler's mean information for the discrimination between two distributions. Numerical illustrations are given, using data from Bartlett (1966), Good and Gover (1967) and some weather records.
On the relationship between the distribution of maximal queue length in the M/G/1 queue and the mean busy period in the M/G/1/n queue
Authors: Cooper Robert B. , Tilt Borge
Published: July 2016

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk

–i
/Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q
0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = bk

–i
/bk, where bn
is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take Q
0 equal to the mean service time, then Qn
=bn.
On single- and multi-type general age-dependent branching processes
Authors: Doney R. A.
Published: July 2016

We consider a p-type general age-dependent branching process (g.a.d.b.p) in which each individual lives a random length of time and at random times during its life and possibly after its death gives birth to offspring. If this process starts with one individual of type i then it turns out that the number of individuals of type i alive at time t forms a one-type g.a.d.b.p. This fact is exploited to establish a NASC for a properly normalized, supercritical p-type g.a.d.b.p. to have a limiting distribution which is not degenerate at zero.
The existence of moments for stationary Markov chains
Authors: Tweedie R. L.
Published: July 2016

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr
and f(x) = esx
. In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).
Weak moment conditions for time coordinates in first-passage percolation models
Authors: Wierman John C.
Published: July 2016

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.