bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on networks of infinite-server queues
Authors: Harrison J. Michael , Lemoine Austin J.
Published: July 2016

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.
Dynamically optimized replacement with a Markovian renewal process
Authors: Siedersleben Johannes
Published: July 2016

We consider a system that gradually deteriorates. At random times Tn
the system is inspected and may or may not be replaced, but it must be if its state is too bad. The process of deterioration is described by a Markovian renewal process. Under some monotonicity conditions we derive a policy for the finite and the infinite horizon which minimizes the total discounted costs. This paper is a generalization of a result of Feldman (1977).
The determination of the age of a clone from characteristics of its geographical distribution
Authors: An Sudbury Aid , Clifford Peter
Published: July 2016

Each point with integer coordinates in d dimensions is occupied by one individual. These individuals produce offspring at a Poisson rate 1, and these offspring migrate and displace other individuals. With probability u (the mutation rate) an offspring is of an entirely new type. A number of points N
0 will be occupied by the same type as the individual at the origin. It is shown that the distribution of N
0 arising from an ancient mutation does not differ greatly from the distribution of N
0 when the mutation is recent. However, the geographical spread is shown to be important, and a central limit theorem is proved for the age of the mutant clone given that a representative is present at a large distance from the origin.
Maintainable regions in a Markov manpower model
Authors: Davies G. S.
Published: July 2016

We examine the geometry of regions of maintainable structures arising in a Markov manpower model. The regions are described in terms of convex hulls, and it is shown that for systems divided into two or three grades these regions form an increasing sequence. It is also shown that the monotonie property fails quite drastically for a four-graded system.

An open problem is discussed for a related model.
Random exchanges of information
Authors: Haigh John
Published: July 2016

Suppose that n persons each know a different piece of information, and that, whenever a pair of them talk on the telephone, each tells the other all the information he knows at that time. If the calls are made at random, we show that the expected number of calls required for everyone to know all n pieces of information is asymptotically 1.5 n log n + O(n). This sharpens an earlier result of D. W. Boyd and J. M. Steele. Some numerical comparisons are given.
Ruin probabilities in a diffusion environment
Authors: Grandell Jan , Schmidli Hanspeter
Published: July 2016

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.
Asymptotics of conditional moments of the summand in Poisson compounds
Authors: Rolski Tomasz , Tomanek Agata
Published: July 2016

Suppose that N is a ℤ+-valued random variable and that X,X
1,X
2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean EN

k
or the conditional variance varN

k
as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.
On exact sampling of stochastic perpetuities
Authors: Blanchet Jose H. , Sigman Karl
Published: July 2016

A stochastic perpetuity takes the formD∞=∑n=0∞exp(Y1+⋯+Yn)Bn, whereYn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively byDn+1=AnDn+Bn,n≥0, whereAn=eYn;D∞then satisfies the stochastic fixed-point equationD∞D̳AD∞+B, whereAandBare independent copies of theAnandBn(and independent ofD∞on the right-hand side). In our framework, the quantityBn, which represents a random reward at timen, is assumed to be positive, unbounded with EBnp<∞ for somep>0, and have a suitably regular continuous positive density. The quantityYnis assumed to be light tailed and represents a discount rate from timenton-1. The RVD∞then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples ofD∞. Our method is a variation ofdominated coupling from the pastand it involves constructing a sequence of dominating processes.
A comparison of cross-entropy and variance minimization strategies
Authors: Chan Joshua C. C. , Glynn Peter W. , Kroese Dirk P.
Published: July 2016

The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.
Exact simulation of the stationary distribution of the FIFO M/G/c queue
Authors: Sigman Karl
Published: July 2016

We present an exact simulation algorithm for the stationary distribution of the customer delayDfor first-in–first-out (FIFO) M/G/cqueues in which ρ=λ/μ<1. We assume that the service time distributionG(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distributionGe(x)=μ∫0xP(S>y)dyare such that samples of them can be simulated. We further assume thatGhas a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which isD. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.
On some tractable growth-collapse processes with renewal collapse epochs
Authors: Boxma Onno , Kella Offer , Perry David
Published: July 2016

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.
Volatility determination in an ambit process setting
Authors: Barndorff-Nielsen Ole E. , Graversen Svend Erik
Published: July 2016

The limit behaviour in probability of realised quadratic variation is discussed under a relatively simple ambit process setting. The relation of this to the underlying volatility/intermittency field is in focus, especially as concerns the question of no volatility/intermittency memory.
The escape probability for integrated Brownian motion with non-zero drift
Authors: Atkinson R. A. , Clifford Peter
Published: July 2016

We consider the two-dimensional process {X(t), V(t)} where {V(t)} is Brownian motion with drift, and {X(t)} is its integral. In this note we derive the joint density function of T and V(T) where T is the time at which the process {X(t)} first returns to its initial value. A series expansion of the marginal density of T is given in the zero-drift case. When V(0) and the drift are both positive there is a positive probability that {Χ (t)} never returns to its initial value. We show how this probability grows for small drift. Finally, using the Kontorovich–Lebedev transform pair we obtain the escape probability explicitly for arbitrary values of the drift parameter.
On a generalization of the ehrenfest urn model
Authors: Dette Holger
Published: July 2016

Krafft and Schaefer [14] considered a two-parameter Ehrenfest urn model and found then-step transition probabilities using representations by Krawtchouk polynomials. For a special case of the model Palacios [17] calculated some of the expected first-passage times. This note investigates a generalization of the two-parameter Ehrenfest urn model where the transition probabilitiespi,i+1andpi,i+1are allowed to be quadratic functions of the current statei. The approach used in this paper is based on the integral representations of Karlin and McGregor [9] and can also be used for Markov chains with an infinite state space.
Detecting optimal and non-optimal actions in average-cost Markov decision processes
Authors: Lasserre Jean B.
Published: July 2016

We present two sufficient conditions for detection of optimal and non-optimal actions in (ergodic) average-cost MDPs. They are easily interpreted and can be implemented as detection tests in both policy iteration and linear programming methods. An efficient implementation of a recent new policy iteration scheme is discussed.
A probabilistic interpretation of the degree of fuzziness
Authors: Handa Kenji , Kalukottege Prasansa , Ogura Yukio
Published: July 2016

In this paper we give a probabilistic interpretation for the function d(f), which has been proposed as a measure of fuzziness of a fuzzy set f. For this we construct random patterns which approximate the fuzzy set f and show that, for large n, the number of possible outcomes of the nth random pattern is approximately 2
nd(f)
. We also give the best possible constant K in a more accurate approximation .
The departure process of the M/G/1 queueing model with server vacation and exhaustive service discipline
Authors: Tang Yinghui
Published: July 2016

In this paper we study the departure process of M/G/1 queueing models with a single server vacation and multiple server vacations. The arguments employed are direct probability decomposition, renewal theory and the Laplace–Stieltjes transform. We discuss the distribution of the interdeparture time and the expected number of departures occurring in the time interval (0, t] from the beginning of the state i (i = 0, 1, 2, ···), and provide a new method for analysis of the departure process of the single-server queue.
A new approach to the G/G/1 queue with generalized setup time and exhaustive service
Authors: Li Huan , Zhu Yixin
Published: July 2016

We consider a class of G/G/1 queueing models with independent generalized setup time and exhaustive service. It is shown that a variety of single-server queueing systems with service interruption are special cases of our model. We give a simple computational scheme for the moments of the stationary waiting time and sojourn time. Our numerical investigations indicate that the algorithm is quite accurate and fast in general. For the M/G/1 case, we are able to derive a recursive formula for the moments of the stationary waiting time, which includes the Takács formula as a special case. It immediately results in the stochastic decompòsition property which can be found in the literature.
Asymptotic distributions for the Ehrenfest urn and related random walks
Authors: Voit Michael
Published: July 2016

The distributions of nearest neighbour random walks on hypercubesin continuous timet0 can be expressed in terms of binomial distributions; their limit behaviour fort, N →∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity forN →∞.Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.
On a Markov chain approach for the study of reliability structures
Authors: Koutras M. V.
Published: July 2016

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.
An infinite variance solidarity theorem for Markov renewal functions
Authors: Sgibnev M. S.
Published: July 2016

Let , be a recurrent Markov renewal process and Mik
(t) be the expected value of Nk
(t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii
are finite, the differences μ ij Mki
(t) – t behave asymptotically the same for all states i and k.
Restrictions on the saddle-point solution in the game of teamball
Authors: Cowan Richard , Yu Philip L. H. , Ferguson Thomas S.
Published: July 2016

Cowan (1992) has addressed the question of how a team should allocate its players to achieve the optimal balance between offence and defence. He posed a simple territorial game called ‘Teamball' and derived a saddle-point solution under conditions which are claimed to be fairly general. In this paper we find that some restrictions are indeed required, leading to an interesting analysis.
Approximate versions of Melamed's theorem
Authors: Barbour A. D. , Brown Timothy C.
Published: July 2016

In 1979, Melamed proved that, in an open migration process, the absence of ‘loops' is necessary and sufficient for the equilibrium flow along a link to be a Poisson process. In this paper, we prove approximation theorems with the same flavour: the difference between the equilibrium flow along a link and a Poisson process with the same rate is bounded in terms of expected numbers of loops. The proofs are based on Stein's method, as adapted for bounds on the distance of the distribution of a point process from a Poisson process in Barbour and Brown (1992b). Three different distances are considered, and illustrated with an example consisting of a system of tandem queues with feedback. The upper bound on the total variation distance of the process grows linearly with time, and a lower bound shows that this can be the correct order of approximation.
The censored Markov chain and the best augmentation
Authors: Zhao Y. Quennel , Liu Danielle
Published: July 2016

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.
On closure of some partial orderings under mixtures
Authors: Nanda Asok K. , Jain Kanchan , Singh Harshinder
Published: July 2016

Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literature. These are more general in the sense that most of the earlier known partial orderings reduce to particular cases of these orderings. Moreover, these orderings have helped in defining a new and useful ageing criterion. In this paper, we discuss preservation properties of these orderings under mixtures. Results are expected to be useful in engineering and reliability, where mixtures of distributions arise naturally in numerous situations. Moreover, existing results, specifically for LR, FR and ST orderings under the operation of mixtures, reduce to special cases.
Stein's method for geometric approximation
Authors: Peköz Erol A.
Published: July 2016

The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.
The minimax bookie
Authors: Barry Daniel , Hartigan John A.
Published: July 2016

A bookmaker makes a book on a horse race: he offers odds against the various horses winning the race, and gamblers accept bets at those odds when they find the odds attractive. The book at a particular time consists of the bookmaker's winnings according to the different outcomes of the race if the race were run at that time. We consider strategies the bookmaker might adopt when deciding how to alter his quoted odds as bets accumulate. The bookmaker is assumed to behave conservatively in the sense that he tries to minimise his expected maximum loss over all possible outcomes of the race.
Shortest common superstrings of random strings
Authors: Alexander Kenneth S.
Published: July 2016

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.
Stability and structural properties of stochastic storage networks
Authors: Kella Offer , Whitt Ward
Published: July 2016

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.
Some majorization orderings of heterogeneity in a class of epidemics
Authors: Tong Y. L.
Published: July 2016

The theory of majorization is applied to investigate the effects of heterogeneity in a class of epidemics. In particular, the heterogeneity of the components of the contact vector of a given susceptible is studied. It is shown that if the heterogeneity can be partially ordered through majorization, then the probability functions of escaping infection can be similarly ordered. The results can be applied to AIDS research on the spread of the HIV virus.
On allocation of spares at component level versus system level
Authors: Singh Harshinder , Singh R. S.
Published: July 2016

Design engineers are well aware that a system where active spare allocation is made at the component level has a lifetime stochastically larger than the corresponding system where active spare allocation is made at the system level. In view of the importance of hazard rate ordering in reliability and survival analysis, Boland and El-Neweihi (1995) recently investigated this principle in hazard rate ordering and demonstrated that it does not hold in general. They showed that for a 2-out-of-n system with independent and identical components and spares, active spare allocation at the component level is superior to active spare allocation at the system level. They conjectured that such a principle holds in general for a k-out-of-n system when components and spares are independent and identical. We prove that for a k-out-of-n system where components and spares have independent and identical life distributions active spare allocation at the component level is superior to active spare allocation at the system level in likelihood ratio ordering. This is stronger than hazard rate ordering, thus establishing the conjecture of Boland and El-Neweihi (1995).
Large deviations of heavy-tailed random sums with applications in insurance and finance
Authors: Klüppelberg C. , Mikosch T.
Published: July 2016

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.
The propagation of chaos of multitype mean field interacting particle systems
Authors: Feng Shui
Published: July 2016

A result for the propagation of chaos is obtained for a class of pure jump particle systems of two species with mean field interaction. This result leads to the corresponding result for particle systems with one species and the argument used is valid for particle systems with more than two species. The model is motivated by the study of the phenomenon of self-organization in biology, chemistry and physics, and the technical difficulty is the unboundedness of the jump rates.
On the second Borel-Cantelli lemma for strongly mixing sequences of events
Authors: Tasche Dirk
Published: July 2016

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.
Laplace transforms and the renewal equation
Authors: Kebir Y.
Published: July 2016

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.
Stochastic comparisons of random minima and maxima
Authors: Shaked Moshe , Wong Tityik
Published: July 2016

Let X
1, X
2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X
1, X
2,…, XN
) and max{X
1, X
2,…, XN
}.
Stochastic comparisons for fork-join queues with exponential processing times
Authors: Frostig Esther , Lehtonen Tapani
Published: July 2016

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.
Stochastic storage networks: stationarity and the feedforward case
Authors: Kella Offer
Published: July 2016

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.
A perpetuity and the M/M/∞ ranked server system
Authors: Preater J.
Published: July 2016

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.
Duration of a secretary problem
Authors: Yeo Geoffrey F.
Published: July 2016

The distribution of the number of items drawn in a secretary problem, with an order s selection role and a success if any of the best s items is selected, is obtained by a probabilistic argument. Moments and asymptotics readily follow.
Extremes for shot noise processes with heavy tailed amplitudes
Authors: McCormick William P.
Published: July 2016

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k
} are the points of a renewal process and {Ak
} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.
A power-law model and other models for long-range dependence
Authors: Martin R. J. , Walker A. M.
Published: July 2016

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.
A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals
Authors: Barnes John A. , Meili Richard
Published: July 2016

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.
On a property of a refusals stream
Authors: Abramov Vyacheslav M.
Published: July 2016

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.
Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains
Authors: Bollabás Béla , Stacey Alan
Published: July 2016

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc
< 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc
≈ 0.6445, this bound is fairly tight.
On the rate of Poisson process approximation to a Bernoulli process
Authors: Xia Aihua
Published: July 2016

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.
Limit motion of an Ornstein–Uhlenbeck particle on the equilibrium manifold of a force field
Authors: Calzolari Antonella , Marchetti Federico
Published: July 2016

In this paper we consider a position–velocity Ornstein-Uhlenbeck process in an external gradient force field pushing it toward a smoothly imbedded submanifold of . The force is chosen so that is asymptotically stable for the associated deterministic flow. We examine the asymptotic behavior of the system when the force intensity diverges together with the diffusion and the damping coefficients, with appropriate speed. We prove that, under some natural conditions on the initial data, the sequence of position processes is relatively compact, any limit process is constrained on , and satisfies an explicit stochastic differential equation which, for compact , has a unique solution.
On defining long-range dependence
Authors: Heyde C. C. , Yang Y.
Published: July 2016

Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.
A prophet inequality for independent random variables with finite variances
Authors: Kennedy D. P. , Kertz R. P.
Published: July 2016

It is demonstrated that for each n ≧ 2 there exists a minimal universal constant, cn, such that, for any sequence of independent random variables {Xr, r ≧ 1} with finite variances, , where the supremum is over all stopping times Τ, 1 ≦ Τ ≦ n. Furthermore, cn
≦ 1/2 and lim inf
n→ ∞
cn
≧ 0.439485 · ··.
Stochastic indicator analysis of contaminated sites
Authors: Christakos George , Hristopulos Dionissios T.
Published: July 2016

We formulate stochastic indicator parameters that characterize pollution levels in geographical regions with heterogeneous contaminant distributions. The indicator parameters are expressed in terms of the random fields representing the contaminant distributions and the critical threshold level specified by health and environmental standards. Certain theoretical results are proven regarding univariate and bivariate indicator parameters. The analytical expressions obtained are general and can be used in practice for various types of contaminant distributions. A test of ergodicity-breaking is suggested for scientific and engineering applications in terms of the indicator parameters. Fractal characteristics of the indicator parameters are discussed. The effects of modelling and observation scale on exceedance contamination analysis are examined. Indicator random field parameters are studied on both continuum and lattice domains using analytical means and numerical simulations.
M/M/∞ transience revisited
Authors: Preater J.
Published: July 2016

We take a fresh look at some transient characteristics of an M/M/∞ queue, studied previously by Guillemin and Simonian using delicate complex analysis. Along the way we obtain the Laplace transform of the joint distribution of the duration, number of arrivals and swept area associated with a busy period of an M/M/1 queue.
State aggregation and discrete-state Markov chains embedded in a class of point processes
Authors: Cao Xi-Ren
Published: July 2016

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.
Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime
Authors: Coyle A. J. , Taylor P. G.
Published: July 2016

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in theGI/M/n/nqueueing system.
Note on generalizations of Mecke's formula and extensions of H = λG
Authors: Miyazawa Masakiyo
Published: July 2016

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group, and relates integrations concerning them to each other. For, we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version ofH = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions ofH = λG.This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.
Mixture models for time series
Authors: Jalali Assad , Pemberton John
Published: July 2016

In this paper we extend the class of zero-order threshold autoregressive models to a much richer class of mixture models. The new class has the important property of duality which, as we show, corresponds to time reversal. We are then able to obtain the time reversals of the zero-order threshold models and to characterise the time-reversible members of this subclass. These turn out to be quite trivial. The time-reversible models of the more general class do not suffer in this way. The complete stationary distributional structure is given, as are various moments, in particular the autocovariance function. This is shown to be of ARMA type. Finally we give two examples, the second of which extends from the finite to the countable mixture case. The general theory for this extension will be given elsewhere.
Phase transitions of some non-linear stochastic models
Authors: Feng Shui
Published: July 2016

A class of non-linear stochastic models is introduced. Phase transitions, critical points and the domain of attraction are discussed in detail for some concrete examples. For one of the examples the explicit expression for the domain of attraction and the rates of convergence are obtained.
An optimal inspection–repair–replacement policy for standby systems
Authors: Yeh Lam
Published: July 2016

In this paper, an optimal maintenance model for standby systems is studied. An inspection–repair–replacement policy is employed. Assume that the state of the system can only be determined through an inspection which may incorrectly identify the system state. After each inspection, if the system is identified as in the down state, a repair action will be taken. It will be replaced some time later by a new and identical one. The problem is to determine an optimal policy so that the availability of the system is high enough at any time and the long-run expected cost per unit time is minimized. An explicit expression for the long-run expected cost per unit time is derived. For a geometric model, a simple algorithm for the determination of an optimal solution is suggested.
The distribution of Brownian quantiles
Authors: Yor Marc
Published: July 2016

The distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Lévy [9], [10] and more recently by Miura [11]. Three proofs are given, two of them involving last-passage times of Brownian motion, before time 1, at a given level.
Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature
Authors: Balabushkin A. N.
Published: July 2016

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.
The final outcome of an epidemic model with several different types of infective in a large population
Authors: Ball Frank , Clancy Damian
Published: July 2016

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.
Robustness of the Ewens sampling formula
Authors: Joyce Paul
Published: July 2016

Under the assumptions of the neutral infinite alleles model,K(the total number of alleles present in a sample) is sufficient for estimatingθ(the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator forθbased onK. It is shown that the same estimator used to estimateθunder neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.
On the estimation of probabilities for birth and death processes
Authors: Zeifman A. I.
Published: July 2016

Let X(t) be a non-homogeneous birth and death process. In this paper we develop a general method of estimating bounds for the state probabilities for X(t), based on inequalities for the solutions of the forward Kolmogorov equations. Specific examples covered include simple estimates of Pr(X(t) < j | X(0) = k) for the M(t)/M(t)/N/0 and M(t)/M(t)/N queue-length processes.
A note on a model for restoring a destroyed region
Authors: Hüsler J.
Published: July 2016

We introduce a model for the healing process of a destroyed region and use some relations to the coverage models. The restoring model depends on the region which is hit n times at random. Each random part of the region which is destroyed is restored at a certain fixed speed, but the restoring process may start after a random delay. We focus mainly on the total healing time until the whole region is restored, and analyse its limit distribution as n tends to ∞. The dependence of this limit distribution on the different ingredients is of interest. We consider two cases with different geometrical influence. One case is restricted to a one-dimensional region, that means the circumference of a circle or the unit interval as region. The results follow from extreme value theory. We discuss also some particular cases and examples.
On the calculation of the reliability of general load sharing systems
Authors: Lee Shiowjen , Durham S. , Lynch J.
Published: July 2016

Harlow et al. (1983) have given a recursive formula which is fundamental for computing the bundle strength distribution under a general class of load sharing rules called monotone load sharing rules. As the bundle size increases, the formula becomes prohibitively complex and, by itself, does not give much insight into the relationship of the assumed load sharing rule to the overall strength distribution. In this paper, an algorithm is given which gives some additional insight into this relationship. Here it is shown how to explicitly compute the bundle strength survival distribution by using a new type of graph called the loading diagram. The graph is parallel in structure and recursive in nature and so would appear to lend itself to large-scale computation. In addition, the graph has an interesting property (which we refer to as the cancellation property) which is related to the asymptotics of the Weibull as a minimum stable law.
Adaptive linear list reorganization under a generalized query system
Authors: Valiveti R. S. , Oommen B. J. , Zgierski J. R.
Published: July 2016

We consider the problem of reorganizing a linear list, when the individual queries consist of accesses to a subset of the elements stored, as opposed to the individual elements themselves. In this paper, which to our knowledge represents the first reported result in this model of query processing, we first propose a simple model for a query generator which emits set queries. Subsequently, we present extensions to the classical move-to-front (MTF) and transposition (TR) rules under this generalized query generation mechanism and analyze their performance.
Analysis of Markov renewal shock models
Authors: Igaki Nobuko , Sumita Ushio , Kowada Masashi
Published: July 2016

A trivariate stochastic process is considered, describing a sequence of random shocks {Xn
} at random intervals {Y
n} with random system state {Jn
}. The triviariate stochastic process satisfies a Markov renewal property in that the magnitude of shocks and the shock intervals are correlated pairwise and the corresponding joint distributions are affected by transitions of the system state which occur after each shock according to a Markov chain. Of interest is a system lifetime terminated whenever a shock magnitude exceeds a prespecified level z. The distribution of system lifetime, its moments and a related exponential limit theorem are derived explicitly. A similar transform analysis is conducted for a second type of system lifetime with system failures caused by the cumulative magnitude of shocks exceeding a fixed level z.
Some EATA properties for marked point processes
Authors: Kofman D. , Korezlioglu H.
Published: July 2016

We derive an ESTA property for marked point processes similar to Wolff's PASTA property for ordinary (non-marked) point processes, via a stochastic integral approach. This new ESTA property allows us to extend a known result on the conditional PASTA property and to derive an ASTA property for batch arrival processes. We also present an application of our results.
The extremal index and clustering of high values for derived stationary sequences
Authors: Weissman Ishay , Cohen Uri
Published: July 2016

Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.
A monotonicity result for a single-server loss system
Authors: Chao Xiuli , Dai Liyi
Published: July 2016

We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt
(c), μt
(c)) = (λct
, μct
), where (λt
, μt
) are governed by an irreducible Markov process with infinitesimal generator Q = (qij
)m × m such that (λt
, μt
) = (λi
, μi
) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c
∗], where c
∗ = 1/maxi Σ
j
≠i
qij
/(λi
+ μi
). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.
General necessary conditions for partially observed optimal stochastic controls
Authors: Li Xunjing , Tang Shanjian
Published: July 2016

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.