bluuebunny/crossref_metadata_2025 huggingface dataset:
An infinite-server queue subject to an extraneous phase process and related models
Authors: Purdue P. , Linton D.
Published: July 2016

We consider an infinite-server queueing system in an extraneous environment. Initially it is shown that the systems of interest can be decomposed into a two-stage system. The primary system is an infinite-server queue with many customer types subject to a clearing mechanism. The secondary system is a special type of bulk-arrival, infinite-server queue. We derive results for the primary and secondary systems separately and combine the results to find the mean steady-state behavior of the original system.
The correlated random walk
Authors: Renshaw Eric , Henderson Robin
Published: July 2016

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.
A machine interference model with threshold effect
Authors: Karmeshu  , Jaiswal N. K.
Published: July 2016

A non-linear stochastic model for the single-server machine interference problem is investigated. The service rate is a rational function of the number of failed machines, the numerator as well as the denominator of the function being linear inn.Depending on the relative values of the parameters involved, the service rate may increase or decrease withn, exhibiting the incentive or disincentive effect. The analysis is carried out by the usual diffusion approximation technique. A notable feature of this model is the threshold effect exhibited in the mean queue size.
A fixed-point property for Galton–Watson processes with generation dependence
Authors: Fearn Dean H.
Published: July 2016

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.
Lognormal genesis
Authors: Brown Gavin , Sanders J. W.
Published: July 2016

Models have been proposed in many diverse areas to generate a lognormal distribution and the underlying idea has always been some form of the law of proportionate effect. In a sense any model must resemble this recipe: take logs and apply the central limit theorem. Our model is no exception. However our formulation is designed to encompass the previous models and demonstrate that the fundamental concept is one of classification. This supersedes a multitude of models which incorporate a mechanism peculiar to a specific application in order to use the law of proportionate effect; we illustrate with applications of the model to ecology, econometrics and geostatistics.
On the first-order bilinear time series model
Authors: Pham Tuan Dinh , Tran Lanh Tat
Published: July 2016

The paper investigates some properties of the first-order bilinear time series model: stationarity and invertibility. Estimates of the parameters are obtained by a modified least squares method and shown to be strongly consistent.
Diffusion approximations for the cooperative service of voice and data messages
Authors: Lehoczky J. P. , Gaver D. P.
Published: July 2016

A diffusion approximation is developed for a voice and data communication system. Voice traffic operates as an M/M/v/v loss system, while data is a Markovian queueing system with a random number of channels assigned. The data process is shown to be approximated by a certain Wiener process with reflecting barrier at 0. System behavior is described using this approximation. The methodology is based on the work of Burman (1979).
Another look at the two-lift problem
Authors: Kaplan Norman L.
Published: July 2016

How often will a store with two elevators be without service assuming that repairs are quick? In this note it is shown that, under very mild assumptions on the operating- and repair-time distributions, the times when the building is without service are asymptotically Poisson.
Waveforms and velocities for models of spatial infection
Authors: Renshaw Eric
Published: July 2016

A new technique is developed for deriving velocities of propagation and shapes of travelling wave profiles in non-linear situations. Two models are considered for the spread of infection through a linear community. In one the infected individuals move, in the other the phenomenon of infection.
On the classical Bonferroni inequalities and the corresponding Galambos inequalities
Authors: Walker A. M.
Published: July 2016

Let (A
1
A
2, · ··, An
) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn
the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn
= m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.

Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.
Exit times for a class of random walks exact distribution results
Authors: Jacobsen Martin
Published: July 2016

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also determined and the paper concludes with exact distribution results concerning exits from bounded intervals. The proofs use simple martingale techniques together with some classical expansions of polynomials and Rouché's theorem from complex function theory.
Fisher information and statistical inference for phase-type distributions
Authors: Bladt Mogens , Esparza Luz Judith R. , Nielsen Bo Friis
Published: July 2016

This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.
Duality results for Markov-modulated fluid flow models
Authors: Ahn Soohan , Ramaswami Vaidyanathan
Published: July 2016

We establish some interesting duality results for Markov-modulated fluid flow models. Though fluid flow models are continuous-state analogues of quasi-birth-and-death processes, some duality results do differ by the inclusion of a scaling factor.
Computing the invariant law of a fluid model
Authors: Rogers L. C. G. , Shi Z.
Published: July 2016

In this paper, we discuss a variety of methods for computing the Wiener-Hopf factorization of a finite Markov chain associated to a fluctuating additive functional. The importance of this is that the equilibrium law of a fluid model can be expressed in terms of these Wiener–Hopf factors. The diagonalization methods considered are actually quite efficient, and provide an effective solution to the problem.
On the identification of continuous-time Markov chains with a given invariant measure
Authors: Pollett P. K.
Published: July 2016

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.
Quasi-stationary distributions for a Brownian motion with drift and associated limit laws
Authors: Martinez Servet , Martin Jaime San
Published: July 2016

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution lim
t→∞
P

x
(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.
On redundancy allocations in systems
Authors: Singh Harshinder , Misra Neeraj
Published: July 2016

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C
1 and C
2), having independent random lifetimes X
1 and X
2, and suppose a component C having random lifetime X (independent of X
1 and X
2) is available for active redundancy with one of the components. Let U
1 = min(max(X
1, X), X
2) and U
2 = min(X
1, max(X
2, X)), so that U
1 (U
2) denote the lifetime of a system obtained by allocating C to C
1 (C
2). We consider the criterion where C
1 is preferred to C
2 for redundancy allocation if . Here we investigate the problem of allocating C to C
1 or C
2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.
Inspection times for stand-by units
Authors: Parmigiani Giovanni
Published: July 2016

This paper studies optimal timing of inspections for units in cold stand-by. The problem is approached in continuous time. The policies developed adapt optimally to information provided by failures of the main unit and by ageing of both the standby and the main unit.

Results include a general form for the optimal policy, consisting of a recursive relation between successive inspection times, and of a renewal-type integral equation to determine the value of the objective function for a given policy. Further results are derived for the case in which the failure densities describing the process are logconcave. Then the stopping rule is easily characterized. It is also shown that the interval between inspections decreases with time, and that the optimal schedule is unique. A binary search algorithm for finding the solution is outlined.
Servers in tandem with k-stage blocking and communications-type flow
Authors: Avi-Itzhak Benjamin
Published: July 2016

Modeling of manufacturing lines and data communications makes frequent use of tandem queues with blocking. Here we present and study such a system with a k-stage blocking scheme in which processing in each station requires attendance of the server of that station together with the servers of the next k – 1 stations. This scheme describes the conventional manufacturing and communications blocking schemes but is also representative of a wider range of applications. Explicit expressions for residence times, throughput, queueing times, waiting times and other measures of performance are obtained for the case of communications type flow and just-in-time input. Use of the results in modeling and analyzing a highway traffic flow situation is presented.
Approximations and boundary conditions for continuous-time threshold autoregressive processes
Authors: Hyndman Rob J.
Published: July 2016

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.
Classes of life distributions and renewal counting process
Authors: Chen Yi-Hau
Published: July 2016

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.
On the input-output map of a G/G/1 queue
Authors: Chang Cheng-Shang
Published: July 2016

In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.
Simple random walk statistics. Part II: Continuous time results
Authors: Böhm W. , Panny W.
Published: July 2016

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.
An invariant of representations of phase-type distributions and some applications
Authors: Commault C. , Chemla J. P.
Published: July 2016

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.
Poissonian approximation for the tagged particle in asymmetric simple exclusion
Authors: Ferrari P. A. , Fontes L. R. G.
Published: July 2016

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt
= Nt
− Bt
+ B
0
, where {N
t} is a Poisson process of parameter (p – q)(1– λ) and {Bt
} is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt
} converges to Brownian motion. A previous result says that in the scale t
1/2, the position Xt
is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.
A note on disjoint-occurrence inequalities for marked Poisson point processes
Authors: van den Berg J.
Published: July 2016

For (marked) Poisson point processes we give, for increasing events, a new proof of the analog of the BK inequality. In contrast to other proofs, which use weak-convergence arguments, our proof is ‘direct' and requires no extra topological conditions on the events. Apart from some well-known properties of Poisson point processes, the proof is self-contained.
Strong approximations for some open population epidemic models
Authors: O'Neill Philip
Published: July 2016

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.
Stability of feed-forward fluid networks with Lévy input
Authors: Kaspi Haya , Kella Offer
Published: July 2016

We consider a stochastic fluid network with independent subordinator inputs to the various stations and deterministic internal flow which is of feed-forward type. We show that under suitable conditions the process of fluid contents in the station has a limiting distribution, where the limit holds in total variation and is independent of the initial condition. We also show that this limiting distribution is of product form only for trivial setups.
Rate modulation in dams and ruin problems
Authors: Asmussen Søren , Kella Offer
Published: July 2016

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.
Optimal policies for scheduling repairs and allocating heterogeneous servers
Authors: Righter Rhonda
Published: July 2016

We consider the machine repairman, or resource utilization, model in which there is a finite source of jobs with non-identically distributed exponential return times and a single server with job dependent service times. We also consider a related problem of scheduling jobs at heterogeneous servers. We construct a coupling framework that provides a simple unified proof that strengthens many of the results in the literature, and generalizes easily to prove several new results.
Une généralisation du modèle de diffusion de Bernoulli–Laplace
Authors: Taïbi Djaouad
Published: July 2016

A generalization of the Bernoulli–Laplace diffusion model is proposed. We consider the case where the number of balls exchanged is greater than one. We show that the stationary distribution is the same as in the classical scheme and we give the mean and the variance of the process. In a second stage, we study the asymptotic approximation based on the diffusion process. A solution of transition density is given using Legendre polynomials.
A general formula for the downtime distribution of a parallel system
Authors: Haukås Harald , Aven Terje
Published: July 2016

In this paper we study the problem of computing the downtime distribution of a parallel system comprising stochastically identical components. It is assumed that the components are independent, with an exponential life-time distribution and an arbitrary repair time distribution. An exact formula is established for the distribution of the system downtime given a specific type of system failure scenario. It is shown by performing a Monte Carlo simulation that the portion of the system failures that occur as described by this scenario is close to one when we consider a system with quite available components, the most common situation in practice. Thus we can use the established formula as an approximation of the downtime distribution given system failure. The formula is compared with standard Markov expressions. Some possible extensions of the formula are presented.
The stability of storage models with shot noise input
Authors: Lund Robert B.
Published: July 2016

We examine the existence of limiting behavior, or stability, for storage models with shot noise input and general release rules. The shot noise feature of the input process allows the individual inputs to gradually enter the store.

We first show that a store under the unit release rule is stable if and only if the traffic intensity is less than one; this extends the classic result of Prabhu (1980) to the case of shot noise input. The stability of the unit release rule store and various stochastic orderings are then used to derive a sufficient condition for a store with a general release rule to be stable. Finally, we show that when restricted to a compact state space, our storage model is always stable.

An important component of the paper is the methodology employed: coupling and stochastic monotonicity play a key role in analyzing the non-Markov processes encountered.
Superprophet inequalities for independent random variables
Authors: Wittmann Rainer
Published: July 2016

As well as having complete knowledge of the future, a superprophet can also alter the order of observation as it is presented to a player without foresight, whose strategy is known to the prophet. It is shown that a superprophet can only do twice as well as his counterpart, if the underlying random sequence is independent.
Value iteration in a class of average controlled Markov chains with unbounded costs: necessary and sufficient conditions for pointwise convergence
Authors: Cavazos-Cadena Rolando , Fernández-Gaucherand Emmanuel
Published: July 2016

This work concerns controlled Markov chains with denumerable state space, (possibly)unboundedcost function, and an expectedaveragecost criterion. Under aLyapunov function condition, together with mild continuity-compactness assumptions, a simplenecessary and sufficientcriterion is given so that the relative value functions and differential costs produced by thevalue iterationscheme converge pointwise to the solution of the optimality equation; this criterion is applied to obtain convergence results when the cost function is bounded below or bounded above.
Sample quantiles of additive renewal reward processes
Authors: Dassios Angelos
Published: July 2016

The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.
Properties of moments for s-order equilibrium distributions
Authors: Nanda Asok K. , Jain Kanchan , Singh Harshinder
Published: July 2016

In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.
Stochastic models of damped vibrations
Authors: Elshamy Maged
Published: July 2016

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε
(t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε
(t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.
A Pollaczek–Khintchine formula for M/G/1 queues with disasters
Authors: Jain Gautam , Sigman Karl
Published: July 2016

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.
Braess's paradox in a loss network
Authors: Bean N. G. , Kelly F. P. , Taylor P. G.
Published: July 2016

Braess's paradox is said to occur in a network if the addition of an extra link leads to worse performance. It has been shown to occur in transportation networks (such as road networks) and also in queueing networks. Here, we show that it can occur in loss networks.
A correspondence between product-form batch-movement queueing networks and single-movement networks
Authors: Coleman J. L. , Henderson W. , Pearce C. E. M. , Taylor P. G.
Published: July 2016

A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.
Stochastic orders based on ratios of Laplace transforms
Authors: Shaked Moshe , Wong Tityik
Published: July 2016

The purpose of this paper is to study two notions of stochastic comparisons of non-negative random variables via ratios that are determined by their Laplace transforms. Some interpretations of the new orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory are described.
Strong convergence of proportions in a multicolor Pólya urn
Authors: Gouet Raúl
Published: July 2016

We prove strong convergence of the proportions Un
/Tn
of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.
Dimensioning a multiple hashing scheme
Authors: Barbour A. D. , Phatarfod R. M.
Published: July 2016

The number of items of data which are irretrievable without additional effort after hashing can be greatly reduced if several hash tables are used simultaneously. Here we show that, in a multiple hashing scheme, this number has a distribution very close to Poisson. Thus choosing the number and sizes of the tables to minimize the expected number of irretrievable items is the right way to dimension a scheme.
Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations
Authors: Li Wei , Shi Dinghua , Chao Xiuli
Published: July 2016

This note introduces reliability issues to the analysis of queueing systems. We consider an M/G/1 queue with Bernoulli vacations and server breakdowns. The server uptimes are assumed to be exponential, and the server repair times are arbitrarily distributed. Using a supplementary variable method we obtain a transient solution for both queueing and reliability measures of interest. These results provide insight into the effect of server breakdowns and repairs on system performance.
A proof of the persistence criterion for a class of superprocesses
Authors: Wang Yong-Jin
Published: July 2016

Gorostiza and Wakolbinger (1991), and Dawson and Perkins (1991) established the same persistence criterion for a class of critical branching particle systems and for a class of superprocesses respectively. In this note we take another approach to the criterion and present a simpler proof of it.
Cost comparison of a spectrum of self-organizing rules
Authors: Chong K. S. , Lam K.
Published: July 2016

Consider the following self-organizing rule called POS(i): after a book in the jth position of a shelf is borrowed, it is moved up one position if j

i, and is moved to the ith position if j > i. This is a family of move-forward rules, with POS(l) being the move-to-front rule and POS(n − 1) being the transposition rule where n is the number of books to be organized. We derive explicitly the stationary distribution under the POS(i) rule and show that its search cost compares favorably with that of move-to-front rule under any book access probabilities p
1, p
1, ···, pn
.
On the inverse of the first hitting time problem for bidimensional processes
Authors: Lefebvre Mario
Published: July 2016

Bidimensional processes defined by dx(t) = ρ (x, y)dt and dy(t) = m(x, y)dt + [2v(x, y)]1/2dW(t),

where W(t) is a Wiener process, are considered. Let T(x, y) be the first time the process (x(t), y(t)), starting from (x, y), hits the boundary of a given region in . A theorem is proved that gives necessary and sufficient conditions for a given complex function to be considered as the moment generating function of T(x, y) for some bidimensional diffusion process. Examples are given where the theorem is used to construct explicit solutions to first hitting time problems and to compute the infinitesimal moments that correspond to the chosen moment generating function.
Partial balances in batch arrival batch service and assemble-transfer queueing networks
Authors: Chao Xiuli
Published: July 2016

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.
An M/G/1 queue with multiple types of feedback and gated vacations
Authors: Boxma Onno J. , Yechiali Uri
Published: July 2016

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.
A note on stochastic ordering of order statistics
Authors: Ma Chunsheng
Published: July 2016

A necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.
Mechanical models for cell movement – locomotion, translocation, migration
Authors: Watkins Joseph C.
Published: July 2016

This paper provides a detailed stochastic analysis of leucocyte cell movement based on the dynamics of a rigid body. The cell's behavior is studied in two relevant anisotropic environments displaying adhesion mediated movement (haptotaxis) and stimulus mediated movement (chemotaxis). This behavior is modeled by diffusion processes on three successively longer time scales, termed locomotion, translocation, and migration.
Stochastic comparison of point random fields
Authors: Georgii Hans-Otto , Küneth Torsten
Published: July 2016

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.
Reliability analysis for systems in a random environment
Authors: Petakos Kyriakos , Tsapelas Theothosios
Published: July 2016

We consider some important systems in reliability theory situated in a random environment, where shocks occur and cause component failure in a specific way. We study some appropriate coefficients, which play an important role in the reduction of our systems to a linear combination of parallel subsystems.
A product-form ‘loss network' with a form of queueing
Authors: Berezner S. A. , Krzesinski A. E. , Taylor P. G.
Published: July 2016

We show that a form of queueing can be introduced into the standard fixed-routing loss network model while retaining a product-form invariant measure.
The moments of FIND
Authors: Paulsen Volkert
Published: July 2016

To study the limiting behaviour of the random running-time of the FIND algorithm, the so-called FIND process was introduced by Grübel and Rösler [1]. In this paper an approach for determining the nth moment function is presented. Applied to the second moment this provides an explicit expression for the variance.
On the asymptotic size and duration of a class of epidemic models
Authors: Svensson Åke
Published: July 2016

Models for epidemic spread of infections are formulated by defining intensities for relevant counting processes. It is assumed that an infected individual passes through k stages of infectivity. The times spent in the different stages are random. Many well-known models for the spread of infections can be described in this way. The models can also be applied to describe other processes of epidemic character (such as models for rumour spreading). Asymptotic results are derived both for the size and for the duration of the epidemic.
A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains
Authors: Martínez Servet , Vares Maria Eulália
Published: July 2016

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.
On the local limit theorem for non-uniformly ergodic Markov chains
Authors: Séva Marc
Published: July 2016

Using an approach similar to that of Guivarc'h and Hardy (1988), we show that the local limit theorem holds for a Markov chain on a countable state space, with non-uniform ergodicity, when the recurrence is fast enough. We present a detailed study of a typical example, the reflected random walk on the positive half-line with negative mean and finite exponential moment. The results can be extended to some random walks on ℕ.
Stochastic differential equations for ruin probabilities
Authors: Møller Christian Max
Published: July 2016

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.

Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.
Equality of critical densities in continuum percolation
Authors: Meester Ronald
Published: July 2016

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.
On transforming an index for generalised bandit problems
Authors: Glazebrook K. D. , Greatrix S.
Published: July 2016

Nash (1980) demonstrated that index policies are optimal for a class of generalised bandit problem. A transform of the index concerned has many of the attributes of the Gittins index. The transformed index is positive-valued, with maximal values yielding optimal actions. It may be characterised as the value of a restart problem and is hence computable via dynamic programming methodologies. The transformed index can also be used in procedures for policy evaluation.
An L 2 convergence theorem for random affine mappings
Authors: Burton Robert M. , Rösler Uwe
Published: July 2016

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of thenth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.
A linear birth-and-death predator–prey process
Authors: Coffey John
Published: July 2016

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ
1
X and death rate μ
1
X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ
2
Y and the death rate is . It is proven that and iff
Characterization of a general class of life-testing models
Authors: Ghitany M. E. , El-saidi M. A. , Khalil Z.
Published: July 2016

In this paper we establish a characterization theorem for a general class of life-testing models based on a relationship between conditional expectation and the failure rate function. As a simple application of the theorem, we characterize the gamma, Weibull, and Gompertz distributions, since they have many probabilistic and statistical properties useful in both biometry and engineering reliability.
On unbounded hazard rates for smoothed perturbation analysis
Authors: Fu Michael C. , Hu Jian-Qiang
Published: July 2016

Many applications of smoothed perturbation analysis lead to estimators with hazard rate functions of underlying distributions. A key assumption used in proving unbiasedness of the resulting estimator is that the hazard rate function be bounded, a restrictive assumption which excludes all distributions with finite support. Here, we prove through a simple example that this assumption can in fact be removed.
Sur la Lp
convergence des martingales liées au recouvrement
Authors: Al Hua Fan
Published: July 2016

In a coverage problem introduced by Dvoretzky (1956) the circle is covered with lengths The associated indexed martingale can be considered as a set of random densities with respect to Lebesgue measure, of which the limit is a random measure. The total masses of the set constitute a new martingale. From a theorem of Shepp (1972), this martingale does not converge in a space if . If 0 < α < 1 it converges in all spaces , and in this paper we demonstrate that it converges in all spaces Lp
for p > 2. We also obtain quite precise estimates for the moments of the total mass of the limit measure, showing that the characteristic function of this total mass is an entire function of order 1/(1 − α). Thus in some sense the mass is distributed in a bounded domain.
Some general characteristics of two-state reliability models for maintained systems
Authors: Marlow Norman A. , Tortorella Michael
Published: July 2016

Formulas for the asymptotic failure rate, long-term average availability, and the limiting distribution of the number of long ‘outages' are obtained for a general class of two-state reliability models for maintained systems. The results extend known formulas for alternating renewal processes to a wider class of point processes that includes sequences of dependent or non-identically distributed operating and repair times.
Tail events of simulated annealing Markov chains
Authors: Niemiro Wojciech
Published: July 2016

We consider non-homogeneous Markov chains generated by the simulated annealing algorithm. We classify states according to asymptotic properties of trajectories. We identify recurrent and transient states. The set of recurrent states is partitioned into disjoint classes of asymptotically communicating states. These classes correspond to atoms of the tail sigma-field. The results are valid under the weak reversibility assumption of Hajek.
Some asymptotic results for exponential functionals of Brownian motion
Authors: Gruet J.-C. , Shi Z.
Published: July 2016

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.
Probability bounds on the finite sum of the binary sequence of order k
Authors: Dhar Sunil K. , Jiang Xulun
Published: July 2016

The cumulative distribution of the finite sum of the binary sequence of order k is studied and some of its applications discussed. Certain properties of this sequence are investigated and uniformly superior bounds for the cumulative distribution under minimal information on the ‘success' probabilities are derived.
On weak convergence within a Lorenz ordering family of distributions
Authors: Lin Gwo Dong
Published: July 2016

We first introduce a Lorenz ordering family of distributions which are related to the gamma distribution, and then prove that the weak convergence within this family is equivalent to the convergence of each moment sequence of positive orders to the corresponding moment of the limiting distribution.
A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process
Authors: Du Qing
Published: July 2016

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα
i
. The arrival rate at level i is λ i
and the service time is exponentially distributed with rate μ i
. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.
A Girsanov transformation for birth and death on a Brownian flow
Authors: Phelan Michael J.
Published: July 2016

We consider a system of particles in birth and death on a Brownian flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of counting measures. The system depends on a handful of parameters including a rate of drift, diffusion, birth, and killing. We exhibit a Girsanov formula for the absolutely continuous change of particle system by way of a change of drift, birth, and killing rates. We can only allow for a restrictive change of drift on the flow, but for fairly unrestrictive change of birth and death rates. The result is therefore of some interest in problems of statistical inference from passive transport of transient tracers.
Stochastic comparisons of order statistics under multivariate imperfect repair
Authors: Hu Taizhong
Published: July 2016

A monotone coupling of order statistics from two sets of independent non-negative random variables Xi, i = 1, ···, n, and Yi, i = 1, ···, n, means that there exist random variables X′i, Y′i, i = 1, ···, n, on a common probability space such that , and a.s. j = 1, ···, n, where X(1) ≦ X(2) ≦ ·· ·≦ X(n) are the order statistics of Xi
, i = 1, ···, n (with the corresponding notations for the X′, Y, Y′ sample). In this paper, we study the monotone coupling of order statistics of lifetimes in two multi-unit systems under multivariate imperfect repair. Similar results for a special model due to Ross are also given.
A non-exponential generalization of an inequality arising in queueing and insurance risk
Authors: Willmot Gordon E.
Published: July 2016

An exponential inequality is generalized to one involving the tail of a decreasing failure rate distribution. The results are then applied in various situations, notably when the exponential inequality does not apply.
A heterogeneous blocking system in a random environment
Authors: Falin G.
Published: July 2016

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.