bluuebunny/crossref_metadata_2025 huggingface dataset:
A spatial model of range-dependent succession
Authors: Krone Stephen M. , Neuhauser Claudia
Published: July 2016

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F
1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F
2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.
Convergence of Markov chains in the relative supremum norm
Authors: Holden Lars
Published: July 2016

It is proved that the strong Doeblin condition (i.e., p

s
(x,y) ≥ a

s
π(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - a

s
)1/s
. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.
Optimal strategy in a dice game
Authors: Haigh John , Roters Markus
Published: July 2016

Computer simulations had suggested that the strategy that maximises the score on each turn in the dice game described by Roters (1998) may not be the optimal way to reach a given target in the shortest time. We give an analytical treatment, backed by numerical calculations, that finds the optimal strategy to reach such a target.
A use of the Stein-Chen method in time series analysis
Authors: Kim Sun-Tsung
Published: July 2016

In this paper, a statistic that has been introduced to test for space-time correlation is considered in a time series context. The null hypothesis is white noise; the alternative is any kind of continuous functional dependence. For an autoregressive process close to the null hypothesis, a bound on the distance between the distribution of the statistic and a Poisson distribution is proved, using the Stein-Chen method. The main difficulty in the proof is that the dependence in the time series is not locally restricted. The result implies asymptotically certain discrimination for a reasonable choice of the thresholds.
Optimal stopping and embedding
Authors: Lamberton Damien , Rogers L. C. G.
Published: July 2016

We use embedding techniques to analyse the error of approximation of an optimal stopping problem along Brownian paths when Brownian motion is approximated by a random walk.
New improved bounds for reliability of consecutive-k-out-of-n:F systems
Authors: Muselli Marco
Published: July 2016

New bounds are found for the reliability of consecutive-k-out-of-n:F systems with equal component failure probabilities. The expressions involved are simple, thus allowing a direct use in the derivation of theoretical properties.

These bounds can also be employed in numerical computations when the value of n or k is so large that the exact calculation of the reliability is not achievable. Comparisons show that the approximation errors exhibited by these new formulas are lower than those of other widely used bounds.
Evaluating first-passage probabilities for spectrally one-sided Lévy processes
Authors: Rogers L. C. G.
Published: July 2016

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.
Hawkes branching point processes without ancestors
Authors: Brémaud Pierre , Massoulié Laurent
Published: July 2016

In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.
The annihilating process
Authors: O'Hely Martin , Sudbury Aidan
Published: July 2016

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. It is shown that with initial configuration ℤ the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all time tends to 1/e. Time-dependent behaviour is also calculated for the tree 𝕋
r
.
Inequalities for the GI/M/1/n loss system
Authors: Abramov Vyacheslav M.
Published: July 2016

The present paper provides simple inequalities for the number of lost customers during a busy period of a GI/M/1/n queueing system.
Optimal sequential allocation with imperfect feedback information
Authors: Mo Charles T. C. , Wu Samuel S. , Chen Robert , Yang Mark C. K.
Published: July 2016

A given number of bullets will be fired sequentially in an attempt to destroy as many targets as possible from a fixed number of targets. The probability of destroying a target at each shot is known. After each shot, there is a report on the state for the target; destroyed or intact. The reports are subject to the usual two types of errors and the probabilities of making these errors are also known. This paper shows that the myopic decision strategy that picks the next target to be the one with the highest intact posterior probability is the optimal strategy.
On some distributional properties of a first-order nonnegative bilinear time series model
Authors: Zhang Zhiqiang , Tong Howell
Published: July 2016

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.
Time dependent analysis of multivariate marked renewal processes
Authors: Dshalalow Jewgeni H.
Published: July 2016

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.
Combinatorial techniques for M/G/1-type queues
Authors: Mercankosk G. , Nair G. M. , Soet W. J.
Published: July 2016

The application of the generalised ballot theorem to queueing theory leads to elegant results for the simple M/G/1 queue. It is thought that such results are not possible for more general M/G/1-type queues. We, however, derive a batch ballot theorem which can be applied to derive the first passage distribution matrix,
G
, for the general M/G/1-type queue.
The coupon subset collection problem
Authors: Adler Ilan , Ross Sheldon M.
Published: July 2016

The coupon subset collection problem is a generalization of the classical coupon collecting problem, in that rather than collecting individual coupons we obtain, at each time point, a random subset of coupons. The problem of interest is to determine the expected number of subsets needed until each coupon is contained in at least one of these subsets. We provide bounds on this number, give efficient simulation procedures for estimating it, and then apply our results to a reliability problem.
On the stationary workload distribution of work-conserving single-server queues: a general formula via stochastic intensity
Authors: Miyoshi Naoto
Published: July 2016

It is well known that a simple closed-form formula exists for the stationary
distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.
On the infinite cluster of Bernoulli bond percolation in Scherk's graph
Authors: Chen Dayue
Published: July 2016

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient for p > ½ and is recurrent for p < ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice for p > ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.
Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence
Authors: Macci Claudio
Published: July 2016

We consider a continuous-time Markov additive process (J

t
,S

t
) with (J

t
) an irreducible Markov chain on E = {1,…,s}; it is known that (S

t
/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (J

t
,S

t
): the averaged parameters model (J

t
,S

t

(A)) and the fluid model (J

t
,S

t

(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (J

t
,S

t

(A)) and (J

t
,S

t

(F)) are faster than the corresponding convergences for (J

t
,S

t
).
Exact asymptotics for a queue with fractional Brownian input and applications in ATM networks
Authors: Duncan Tyrone E. , Yan Yi , Yan Peng
Published: July 2016

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.
Further approaches to computing fundamental characteristics of birth-death processes
Authors: Ball Frank , Stefanov Valeri T.
Published: July 2016

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.
On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials
Authors: Fu James C. , Wang Liqun , Lou W. Y. Wendy
Published: July 2016

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.
Probabilité de ruine éventuelle dans un modèle de risque à temps discret
Authors: Picard Philippe , Lefèvre Claude
Published: July 2016

We continue the study of the discrete-time risk model introduced by Picard et al. (2003). The cumulative loss process (S

t
)
t∊ℕ has independent and stationary increments, the increments per unit of time having nonnegative integer values with distribution {a

i
, i ∊ ℕ and mean ā. The premium receipt process (c

k
)
k∊ℕ is deterministic, nonnegative and nonuniform; in addition, we assume it to be regular in order for there to exist a constant c > ā such that the deviation is bounded as the time t varies. We are interested in whether or not ruin occurs within a finite time. If T is the time of ruin, we obtain P(T = ∞) as the limit of P(T > t) as t → ∞, firstly in the particular case where c = 1/d for some positive d ∊ ℕ, and then in the general case for positive c under the condition that a
0 > ½.
On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials
Authors: Fu James C. , Chang Yung-Ming
Published: July 2016

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.
One-sided variations on interval trees
Authors: Itoh Yoshiaki , Mahmoud Hosam M.
Published: July 2016

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.
Ladder variables, internal structure of Galton–Watson trees and finite branching random walks
Authors: Marckert Jean-François , Mokkadem Abdelkader
Published: July 2016

In this paper, we consider Galton–Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.
Perpetual American put options in a level-dependent volatility model
Authors: Ekström Erik
Published: July 2016

We find the explicit value of perpetual American put options in the constant elasticity of variance model using the concept of smooth fit. We show that the price is increasing in the volatility and convex in the underlying stock price. Moreover, as the model converges to the standard Black and Scholes model, the value of the put is shown to approach the ‘correct’ limit.
Characterizations using record moments in a random record model and applications
Authors: Nagaraja H. N. , Barlevy Gadi
Published: July 2016

We consider a random record model from a continuous parent X with cumulative distribution function F, where the number of available observations is geometrically distributed. We show that, if E(|X|) is finite, then so is E(|R

n
|) whenever R

n
, the nth upper record value, exists. We prove that appropriately chosen subsequences of E(R

n
) characterize F and subsequences of E(R

n
− R

n−1) identify F up to a location shift. We discuss some applications to the identification of wage-offer distributions in job search models.
On the convergence of evolution processes with time-varying mutations and local interaction
Authors: Chen Hsiao-Chi , Chow Yunshyoung
Published: July 2016

This paper analyzes players’ long-run behavior in an evolutionary model with time-varying mutations under both uniform and local interaction rules. It is shown that a risk-dominant Nash equilibrium in a 2 × 2 coordination game would emerge as the long-run equilibrium if and only if mutation rates do not decrease to zero too fast under both interaction methods. The convergence rates of the dynamic system under both interaction rules are also derived. We find that the dynamic system with local matching may not converge faster than that with uniform matching.
The size order of the state vector of discrete-time homogeneous Markov systems
Authors: Kipouridis I. , Tsaklidis G.
Published: July 2016

The size order problem of the probability state vector elements of a homogeneous Markov system is examined. The time t
0 is evaluated, after which the order of the state vector probabilities remains unchanged, and a formula to quickly find a lower bound for t
0 is given. A formula for approximating the mode of the state sizes n

i
(t) as a function of the means En

i
(t), and a relation to evaluate P(n

i
(t) = x+1) by means of certain terms which constitute P(n

i
(t) = x) are derived.
A weak derivative approach to optimization of threshold parameters in a multicomponent maintenance system
Authors: Heidergott Bernd
Published: July 2016

We consider a multicomponent maintenance system controlled by an age replacement policy: when one of the components fails, it is immediately replaced; all components older than a threshold age θ are preventively replaced. Costs are associated with each maintenance action, such as replacement after failure or preventive replacement. We derive a weak derivative estimator for the derivative of the cost performance with respect to θ. The technique is quite general and can be applied to many other threshold optimization problems in maintenance. The estimator is easy to implement and considerably increases the efficiency of a Robbins-Monro type of stochastic approximation algorithm. The paper is self-contained in the sense that it includes a proof of the correctness of the weak derivative estimation algorithm.
Shocks, runs and random sums
Authors: Mallor F. , Omey E.
Published: July 2016

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.
Burn-in procedures for a generalized model
Authors: Cha Ji Hwan
Published: July 2016

In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.
Sufficient conditions for long-range count dependence of stationary point processes on the real line
Authors: Kulik Rafał , Szekli Ryszard
Published: July 2016

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.
Geometric bounds on certain sublinear functionals of geometric Brownian motion
Authors: Hörfelt Per
Published: July 2016

Suppose that {X

s
, 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ
m
→ [0,∞) is a (weighted) l

q
(ℝ
m
)-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L

p
(μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X

s
), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.
Generalized Lorenz curves and convexifications of stochastic processes
Authors: Davydov Youri , Zitikis Ričardas
Published: July 2016

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.
Improved bounds for the large-time behaviour of simulated annealing
Authors: Fontenas Eric , François Olivier
Published: July 2016

We improve on previous finite time estimates for the simulated annealing algorithm which were obtained from a Cheeger-like approach. Our approach is based on a Poincaré inequality.
Stability and exponential convergence of continuous-time Markov chains
Authors: Mitrophanov A. Yu.
Published: July 2016

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.
Transient behaviour of a Galton–Watson process with a large number of types
Authors: Reder Christine
Published: July 2016

Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different modelling assumptions, for instance different values of the number of types.
A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour
Authors: Klüppelberg Claudia , Lindner Alexander , Maller Ross
Published: July 2016

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.
Limit theorems for continuous-time random walks with infinite mean waiting times
Authors: Meerschaert Mark M. , Scheffler Hans-Peter
Published: July 2016

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.
Space filling and depletion
Authors: Baryshnikov Yuliy , Coffman E. G. , Jelenković Predrag
Published: July 2016

For a given k ≥ 1, subintervals of a given interval [0, X] arrive at random and are accepted (allocated) so long as they overlap fewer than k subintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications, including communication networks, surface adsorption-desorption processes, and reservation systems. Our primary interest is in steady-state estimates of the vacant space, i.e. the total length of available subintervals kX - ∑ℓ
i
, where the ℓ
i
are the lengths of the subintervals currently allocated. We obtain explicit results for k = 1 and for general k with all subinterval lengths equal to 2, the classical dimer case of chemical applications. Our focus is on the asymptotic regime of large retention times.
An optimal trade-off between content freshness and refresh cost
Authors: Ling Yibei , Mi Jie
Published: July 2016

Caching is an effective mechanism for reducing bandwidth usage and alleviating server load. However, the use of caching entails a compromise between content freshness and refresh cost. Constant refreshing allows a high degree of content freshness at a greater cost of system resource. Conversely, too little refreshing reduces content freshness but saves the cost of resource usage. To address the freshness-cost problem, we formulate the refresh scheduling problem with a generic cost model and use this cost model to determine an optimal refresh frequency that gives the best trade-off between refresh cost and content freshness. We prove the existence and uniqueness of an optimal refresh frequency under the assumptions that the arrival of updated content is Poisson and the age-related cost monotonically increases with decreasing freshness. In addition, we provide an analytic comparison of system performance under fixed refresh scheduling and random refresh scheduling, showing that two refresh schedulings with the same average refresh frequency are mathematically equivalent in terms of the long-run average cost.
A second-order Markov-modulated fluid queue with linear service rate
Authors: Rabehasaina Landy , Sericola Bruno
Published: July 2016

We consider an infinite-capacity second-order fluid queue governed by a continuous-time Markov chain and with linear service rate. The variability of the traffic is modeled by a Brownian motion and a local variance function modulated by the Markov chain and proportional to the fluid level in the queue. The behavior of this second-order fluid-flow model is described by a linear stochastic differential equation, satisfied by the transient queue level. We study the transient level's convergence in distribution under weak assumptions and we obtain an expression for the stationary queue level. For the first-order case, we give a simple expression of all its moments as well as of its Laplace transform. For the second-order model we compute its first two moments.
Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/n queues
Authors: Hou Zhenting , Liu Yuanyuan
Published: July 2016

This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.
Random dynamical systems with jumps
Authors: Horbacz Katarzyna
Published: July 2016

We consider random dynamical systems with randomly chosen jumps on infinite-dimensional spaces. The choice of deterministic dynamical systems and jumps depends on a position. The system generalizes dynamical systems corresponding to learning systems, Poisson driven stochastic differential equations, iterated function system with infinite family of transformations and random evolutions. We will show that distributions which describe the dynamics of this system converge to an invariant distribution. We use recent results concerning asymptotic stability of Markov operators on infinite-dimensional spaces obtained by T. Szarek.
Stopping the maximum of a correlated random walk, with cost for observation
Authors: Allaart Pieter
Published: July 2016

Let (S

n
)
n≥0 be a correlated random walk on the integers, let M
0 ≥ S
0 be an arbitrary integer, and let M

n
= max{M
0, S
1,…, S

n
}. An optimal stopping rule is derived for the sequence M

n
- nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that M

n
- S

n
≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.
Exponential ergodicity for single-birth processes
Authors: Mao Yong-Hua , Zhang Yu-Hui
Published: July 2016

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.
Numerical integration using V-uniformly ergodic Markov chains
Authors: Mathé Peter
Published: July 2016

We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.
The behavior of multivariate maxima of moving maxima processes
Authors: Zhang Zhengjun , Smith Richard L.
Published: July 2016

In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.
Server advantage in tennis matches
Authors: MacPhee I. M. , Rougier Jonathan , Pollard G. H.
Published: July 2016

We show that the advantage that can accrue to the server in tennis does not necessarily imply that serving first changes the probability of winning the match. We demonstrate that the outcome of tie-breaks, sets and matches can be independent of who serves first. These are corollaries of a more general invariance result that we prove for n-point win-by-2 games. Our proofs are non-algebraic and self-contained.
Infinite divisibility of infinite sums of lower records: a simple proof
Authors: Iksanov Aleksander M.
Published: July 2016

Let {Xn,n= 0, 1,…} be the sequence of the lower records for an arbitrary underlying distributionμon [0, ∞). We show thatis equal in distribution towhere {τi,i= 1, 2,…} is a Poisson flow of unit intensity andgis a right-continuous and nonincreasing function defined byμ. This observation allows us to extend results of Boseet al.and simplify their proofs.
From uniform distributions to Benford's law
Authors: Janvresse Élise , de la Rue Thierry
Published: July 2016

We provide a new probabilistic explanation for the appearance of Benford's law in everyday-life numbers, by showing that it arises naturally when we consider mixtures of uniform distributions. Then we connect our result to a result of Flehinger, for which we provide a shorter proof and the speed of convergence.
The spectral gap and perturbation bounds for reversible continuous-time Markov chains
Authors: Mitrophanov A. Yu.
Published: July 2016

We show that, for reversible continuous-time Markov chains, the closeness of the nonzero eigenvalues of the generator to zero provides complete information about the sensitivity of the distribution vector to perturbations of the generator. Our results hold for both the transient and the stationary states.
The Kelly system maximizes median fortune
Authors: Ethier S. N.
Published: July 2016

It is well known that the Kelly system of proportional betting, which maximizes the long-term geometric rate of growth of the gambler's fortune, minimizes the expected time required to reach a specified goal. Less well known is the fact that it maximizes the median of the gambler's fortune. This was pointed out by the author in a 1988 paper, but only under asymptotic assumptions that might cause one to question its applicability. Here we show that the result is true more generally, and argue that this is a desirable property of the Kelly system.
Multifractal spectra of branching measure on a Galton-Watson tree
Authors: Shieh Narn-Rueih , Taylor S. James
Published: July 2016

IfZis the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|nat each generationn. We use the natural metric ρ(ξ,η) = e−n, wheren= max{k: ξ|k= η|k}, and observe that the local dimension index isd(μ,ξ) = limn→∞log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) whenp1= P{Z= 1} > 0 andZhas finite moments of all positive orders. Because ̱d(μ,ξ) =afor all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.
Conditional law of a branching process observing a subpopulation
Authors: Ceci Claudia , Gerardi Anna
Published: July 2016

The paper is concerned with filtering the cardinality of a branching process observing the cardinality of a subpopulation. In this model, both the processes, state and observation are pure jump processes and may have common jump times. Preliminary properties are studied in the tree framework. A recursive structure for the filtering equation is proved in the supercritical case.
Stochastic aspects of Lanchester's theory of warfare
Authors: Kingman J. F. C.
Published: July 2016

A Markov chain model for a battle between two opposing forces is formulated, which is a stochastic version of one studied by F. W. Lanchester. Solutions of the backward equations for the final state yield martingales and stopping identities, but a more powerful technique is a time-reversal analogue of a known method for studying urn models. A general version of a remarkable result of Williams and McIlroy (1998) is proved.
Inclusion-exclusion methods for treating annihilating and deposition processes
Authors: Sudbury Aidan
Published: July 2016

We consider one-dimensional processes in which particles annihilate their neighbours, grow until they meet their neighbours or are deposited onto surfaces. All of the models considered have the property that they are connected to exponential series often by an inclusion-exclusion argument.
Bisexual Galton-Watson branching process with population-size-dependent mating
Authors: Molina M. , Mota M. , Ramos A.
Published: July 2016

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.
Discrete time analysis of a repairable machine
Authors: Alfa Attahiru Sule , Castro I. T.
Published: July 2016

We consider, in discrete time, a single machine system that operates for a period of time represented by a general distribution. This machine is subject to failures during operations and the occurrence of these failures depends on how many times the machine has previously failed. Some failures are repairable and the repair times may or may not depend on the number of times the machine was previously repaired. Repair times also have a general distribution. The operating times of the machine depend on how many times it has failed and was subjected to repairs. Secondly, when the machine experiences a nonrepairable failure, it is replaced by another machine. The replacement machine may be new or a refurbished one. After the Nth failure, the machine is automatically replaced with a new one. We present a detailed analysis of special cases of this system, and we obtain the stationary distribution of the system and the optimal time for replacing the machine with a new one.
Some bivariate notions of IFR and DMRL and related properties
Authors: Bassan Bruno , Kochar Subhash , Spizzichino Fabio
Published: July 2016

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariate case. Then we introduce certain concepts of bivariate DMRL ageing and we develop a treatment that parallels those developed for bivariate IFR. For both the IFR and DMRL concepts, we analyse a weak and a strong version, and discuss some of the differences between them.
On the moment determinacy of the distributions of compound geometric sums
Authors: Lin Gwo Dong , Stoyanov Jordan
Published: July 2016

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.
Stochastic orders in partition and random testing of software
Authors: Boland Philip J. , Singh Harshinder , Cukic Bojan
Published: July 2016

Testing in order to produce software of high reliability is an area of major concern in software engineering. In an effort to find efficient methods of testing, the comparison of partition and random sampling testing methods has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing, based on their expected efficacy in program debugging, is the probability of detecting at least one failure causing input in the program's domain. However, the goal in software testing is usually to find as many faults as possible in a reasonable period of time, and therefore stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use. We establish various conditions which guarantee that the number of faults discovered in partition testing is stochastically greater than the number discovered in random testing (using a fixed total sample size) for many of the well-established stochastic orders (including the usual stochastic order, the hazard rate order, the likelihood ratio order, and the variability order). The results established also allow us to obtain both upper and lower bounds with these stochastic orders for the sum of n independent Bernoulli random trials (with varying probability of success) in terms of the binomial distribution with parameters n and p.
The house-selling problem with reward rate criterion
Authors: Mazalov Vladimir V. , Saario Vesa
Published: July 2016

We consider a variant of the house-selling problem in which the seller has n objects for sale and the buyers come sequentially one by one. The objective of this paper is to find the seller's optimal strategy for choosing his selling price so as to maximize the average reward per selling attempt until the nth house is sold.
From matchbox to bottle: a storage problem
Authors: Baringhaus Ludwig , Grübel Rudolf
Published: July 2016

We generalize Banach's matchbox problem: demands of random size are made on one of two containers, both initially with content t, where the container is selected at random in the successive steps. Let Z

t
be the content of the other container at the moment when the selected container is found to be insufficient. We obtain the asymptotic distribution of Z

t
as t → ∞ under quite general conditions. The case of exponentially distributed demands is considered in more detail.
Availability of periodically inspected systems subject to Markovian degradation
Authors: Kiessler Peter C. , Klutke Georgia-Ann , Yang Yoonjung
Published: July 2016

This paper studies inspected systems with non-self-announcing failures where the rate of deterioration is governed by a Markov chain. We compute the lifetime distribution and availability when the system is inspected according to a periodic inspection policy. In doing so, we expose the role of certain transient distributions of the environment.
A generic property of phase-type representations
Authors: Commault Christian , Mocanu Stéphane
Published: July 2016

In this paper we introduce the notion of structured phase-type representations. A structured representation corresponds to a class of phase-type representations having the same graph but free parameters. We define a generic property for such a representation. Then we prove that, generically, an irreducible structured representation of a given order corresponds to a distribution whose degree is equal to this order.
Ergodicity and stability of generalised Markov branching processes with resurrection
Authors: Chen Anyue
Published: July 2016

This paper concentrates on investigating ergodicity and stability for generalised Markov branching processes with resurrection. Easy checking criteria including several clear-cut corollaries are established for ordinary and strong ergodicity of such processes. The equilibrium distribution is given in an elegant closed form for the ergodic case. The probabilistic interpretation of the results is clear and thus explained.