bluuebunny/crossref_metadata_2025 huggingface dataset:
Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2
Authors: Franco Tertuliano , Neumann Adriana , Valle Glauco
Published: July 2016

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T

d
. In the discrete torus N
-1
T

d

N
, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is e

j
, then the exchange rate is defined as N
-1 times the absolute value of the inner product between e

j
and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.
A Stochastic Breakdown Model for an Unreliable Web Server System and an Optimal Admission Control Policy
Authors: Cha Ji Hwan , Lee Eui Yong
Published: July 2016

Web servers have to be protected against overload since overload can lead to a server breakdown, which in turn causes high response times and low throughput. In this paper, a stochastic model for breakdowns of server systems due to overload is proposed and an admission control policy which protects Web servers by controlling the amount and rate of work entering the system is studied. Requests from the clients arrive at the server following a nonhomogeneous Poisson process and each requested job takes a random time to be completed. It is assumed that the breakdown rate of the server depends on the number of jobs which are currently being performed by the server. Based on the proposed model, the reliability function and the breakdown rate function of the server system are derived. Furthermore, the long-run expected number of jobs completed per unit time is derived as the efficiency measure, and the optimal admission control policy which maximizes the efficiency will be discussed.
Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions
Authors: Duffy Ken R. , Torrisi Giovanni Luca
Published: July 2016

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.
Stochastic Intensity for Minimal Repairs in Heterogeneous Populations
Authors: Cha Ji Hwan , Finkelstein Maxim
Published: July 2016

In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.
Scaling of High-Quantile Estimators
Authors: Degen Matthias , Embrechts Paul
Published: July 2016

Enhanced by the global financial crisis, the discussion about an accurate estimation of regulatory (risk) capital a financial institution needs to hold in order to safeguard against unexpected losses has become highly relevant again. The presence of heavy tails in combination with small sample sizes turns estimation at such extreme quantile levels into an inherently difficult statistical issue. We discuss some of the problems and pitfalls that may arise. In particular, based on the framework of second-order extended regular variation, we compare different high-quantile estimators and propose methods for the improvement of standard methods by focusing on the concept of penultimate approximations.
A Dynamic Network in a Dynamic Population: Asymptotic Properties
Authors: Britton Tom , Lindholm Mathias , Turova Tatyana
Published: July 2016

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model, we derive a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation ρ (of neighbouring nodes) which shows that ρ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.
On a time deformation reducing nonstationary stochastic processes to local stationarity
Authors: Genton Marc G. , Perrin Olivier
Published: July 2016

A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.
Minimal repair in heterogeneous populations
Authors: Finkelstein M. S.
Published: July 2016

The notion of minimal repair is generalized to the case when the lifetime distribution function is a continuous or a discrete mixture of distributions (heterogeneous population). The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding stochastic intensities are defined and analyzed for the point processes generated by both types of minimal repair. Some generalizations are discussed. Several simple examples are considered.
The spectral representation of Bessel processes with constant drift: applications in queueing and finance
Authors: Linetsky Vadim
Published: July 2016

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of orderν> −1 with constant driftμ≠ 0. Whenν> -½ andμ< 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 <ν< -½ andμ< 0, there is only one eigenvalueλ0= 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.
Phase Transition and Law of Large Numbers for a Non-Symmetric One-Dimensional Random Walk with Self-Interactions
Authors: Vermet Franck
Published: July 2016

We study a not necessarily symmetric random walk with interactions on ℤ, which is an extension of the one-dimensional discrete version of the sausage Wiener path measure. We prove the existence of a repulsion/attraction phase transition for the critical value λc≡ −μ of the repulsion coefficient λ, where μ is a drift parameter. In the self-repellent case, we determine the escape speed, as a function of λ and μ, and we prove a law of large numbers for the end-point.
A Result on Networks of Queues with Customer Coalescence and State-Dependent Signaling
Authors: Chao Xiuli , Zheng Shaohui
Published: July 2016

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Hendersonet al. (1994), as well as those of Chaoet al. (1996).
P
λ
M-policy for a dam with input formed by a compound Poisson process
Authors: Lee Eui Yong , Ahn Sung K.
Published: July 2016

An infinite dam with input formed by a compound Poisson process is considered. As an output policy, we adopt the P
λ

M
-policy. The stationary distribution and expectation of the level of water in the reservoir are obtained.
Absorption sampling and the absorption distribution
Authors: Kemp Adrienne W.
Published: July 2016

The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.
A perturbation method for the structured coalescent with strong migration
Authors: Notohara Morihiro
Published: July 2016

By applying a perturbation method to the structured coalescent process in population genetics theory, we obtain the approximate solutions of the moment generating functions for the total coalescence time, the number of segregating sites among sampled DNA sequences and the number of allele types in a sample in the case of strong migration.
Rates of convergence of stochastically monotone and continuous time Markov models
Authors: Roberts G. O. , Tweedie R. L.
Published: July 2016

In this paper we give bounds on the total variation distance from convergence of a continuous time positive recurrent Markov process on an arbitrary state space, based on Foster-Lyapunov drift and minorisation conditions. Considerably improved bounds are given in the stochastically monotone case, for both discrete and continuous time models, even in the absence of a reachable minimal element. These results are applied to storage models and to diffusion processes.
On the waiting time in a janken game
Authors: Maehara Hiroshi , Ueda Sumie
Published: July 2016

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each round (if any) retire from the rest of the game, and (3) the game ends when only one player (winner) is left. Let W

n
be the number of rounds played through the game. Among other things, it is proved that (2/3)
n

W

n
is asymptotically (as n → ∞) distributed according to the exponential distribution with mean ⅓, provided that each player chooses one of the three strategies (scissors, paper, rock) with equal probability and independently from other players in any round.
On the dynamics and performance of stochastic fluid systems
Authors: Konstantopoulos Takis , Last Günter
Published: July 2016

A (generalized) stochastic fluid system Q is defined as the one-dimensional
Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative ‘integral representation’ for Q. This representation forms the basis for deriving a ‘Little's law’ for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a ‘dual process’ to Q. Examples of models for X, including finite variation Lévy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation.
Weakly approaching sequences of random distributions
Authors: Belyaev Yuri , Sjöstedt-de Luna Sara
Published: July 2016

We introduce the notion of weakly approaching sequences of distributions, which is a generalization of the well-known concept of weak convergence of distributions. The main difference is that the suggested notion does not demand the existence of a limit distribution. A similar definition for conditional (random) distributions is presented. Several properties of weakly approaching sequences are given. The tightness of some of them is essential. The Cramér-Lévy continuity theorem for weak convergence is generalized to weakly approaching sequences of (random) distributions. It has several applications in statistics and probability. A few examples of applications to resampling are given.
Filtering equations for the conditional law of residual lifetimes from a heterogeneous population
Authors: Gerardi A. , Spizzichino F. , Torti B.
Published: July 2016

We consider a probabilistic model of a heterogeneous population P subdivided into homogeneous sub-cohorts. A main assumption is that the frailties give rise to a discrete, exchangeable random vector. We put ourselves in the framework of stochastic filtering to derive the conditional distribution of residual lifetimes of surviving individuals, given an observed history of failures and survivals. As a main feature of our approach, this study is based on the analysis of behaviour of the vector of ‘occupation numbers’.
Truncation approximations of invariant measures for Markov chains
Authors: Tweedie R. L.
Published: July 2016

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n)
P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n)
P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.
The Frobenius–Harper technique in a general recurrence model
Authors: Warren Di
Published: July 2016

We present a general recurrence model which provides a conceptual framework for well-known problems such as ascents, peaks, turning points, Bernstein's urn model, the Eggenberger–Pólya urn model and the hypergeometric distribution. Moreover, we show that the Frobenius-Harper technique, based on real roots of a generating function, can be applied to this general recurrence model (under simple conditions), and so a Berry–Esséen bound and local limit theorems can be found. This provides a simple and unified approach to asymptotic theory for diverse problems hitherto treated separately.
Cell loss asymptotics for buffers fed with a large number of independent stationary sources
Authors: Likhanov Nikolay , Mazumdar Ravi R.
Published: July 2016

In this paper we derive asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources. The technique is based on scaling, measure change and local limit theorems and extends the recent results of Courcoubetis and Weber on buffer overflow asymptotics. We discuss the cases when the buffers are of the same order as the transmission bandwidth as well as the case of small buffers. Moreover we show that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods, which are typical candidates for so-called ‘self-similar’ inputs, showing that the asymptotic cell loss probability behaves in much the same manner for such sources as for the Markovian type of sources, which has important implications for statistical multiplexing. Numerical validation of the results against simulations are also reported.
The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease
Authors: Farrington C. P. , Grant A. D.
Published: July 2016

We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.
Asymptotic distributions for downtimes of monotone systems
Authors: Gåsemyr J. , Aven Terje
Published: July 2016

Consider a monotone system with independent alternating renewal processes as component processes, and assume the component uptimes are exponentially distributed. In this paper we study the asymptotic properties of the distribution of the rth downtime of the system, as the failure rates of the components converge to zero. We show that this distribution converges, and the limiting function has a simple form. Thus we have established an easy computable approximation formula for the downtime distribution of the system for highly available systems. We also show that the steady state downtime distribution, i.e. the downtime distribution of a system failure occurring after an infinite run-in period, converges to the same limiting function as the failure rates converge to zero.
Favorable red and black on the integers with a minimum bet
Authors: Ruth Kevin
Published: July 2016

In a superfair red and black gambling house where the player must bet at least 2 units at each stage, a gambler wishes to maximize the probability of reaching a goal integer N before reaching zero. For win probability p > 1/2, when N is even, an optimal strategy is to bet 3 units when you have 3 units or N − 3 units and to bet 2 units otherwise. When N is odd, there are two strategies which are optimal depending on the value of the win probability p. When p is smaller than a certain value, p
*, the above strategy is optimal, and when p is larger than this value, the timid strategy is optimal.
Large buffer asymptotics for the queue with fractional Brownian input
Authors: Massoulie Laurent , Simonian Alain
Published: July 2016

In this paper, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H ∊ [1/2,1[is studied. By applying general results on suprema of centred Gaussian processes, in particular, we show that

for large x. Explicit formulae for constants κ, γ and L are given in terms of H and system parameters.
Branching and tree indexed random walks on fractals
Authors: Telcs András , Wormald Nicholas C.
Published: July 2016

This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.
Approximations of boundary crossing probabilities for a Brownian motion
Authors: Novikov Alex , Frishling Volf , Kordzakhia Nino
Published: July 2016

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.
Sequential selection of an increasing subsequence from a sample of random size
Authors: Gnedin Alexander V.
Published: July 2016

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence.

We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.
The busy cycle of the reflected superposition of Brownian motion and a compound Poisson process
Authors: Perry David , Stadje Wolfgang
Published: July 2016

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.
A stochastic covariate failure model for assessing system reliability
Authors: Ebrahimi Nader
Published: July 2016

Many failure mechanisms can be traced to an underlying deterioration process, and stochastically changing covariates may influence this process. In this paper we propose an alternative model for assessing a system's reliability. The proposed model expresses the failure time of a system in terms of a deterioration process and covariates. When it is possible to measure deterioration as well as covariates, our model provides more information than failure time for the purpose of assessing and improving system reliability. We give several properties of our proposed model and also provide an example.
On the total time spent in records by a discrete uniform sequence
Authors: Grübel Rudolf , Reimers Anke
Published: July 2016

We consider the sum S

d
of record values in a sequence of independent
random variables that are uniformly distributed on 1,…,d. This sum can be interpreted as the total amount of time spent in record lifetimes in the standard renewal theoretic setup. We investigate the distributional limit of S

d
and some related quantities as d→∞. Some explicit values are given for d=6, a case that can be interpreted as a simple game of chance.
Distribution of the scan statistic for a sequence of bistate trials
Authors: Fu James C.
Published: July 2016

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic S

n
(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of S

n
(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of S

n
(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.
Random walk in a random environment with correlated sites
Authors: Komorowski T. , Krupa G.
Published: July 2016

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z

d
. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.
Polling systems with periodic server routeing in heavy traffic: distribution of the delay
Authors: Olsen Tava Lennon , van der Mei R. D.
Published: July 2016

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.
On the martingale property of stochastic exponentials
Authors: Wong Bernard , Heyde C. C.
Published: July 2016

We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a ‘candidate measure’ the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical finance are also given.
A symmetrized Euler scheme for an efficient approximation of reflected diffusions
Authors: Bossy Mireille , Gobet Emmanuel , Talay Denis
Published: July 2016

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step.
A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options
Authors: Barrieu P. , Rouault A. , Yor M.
Published: July 2016

One approach to the computation of the price of an Asian option involves the Hartman–Watson distribution. However, numerical problems for its density occur for small values. This motivates the asymptotic study of its distribution function.
The first rendezvous time of Brownian motion and compound Poisson-type processes
Authors: Perry D. , Stadje W. , Zacks S.
Published: July 2016

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.
The eigentime identity for continuous-time ergodic Markov chains
Authors: Mao Yong-Hua
Published: July 2016

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generatorL. When the essential spectrum is empty, let {0 = λ0< λ1≤ λ2≤ ···} be the whole spectrum ofLin L2. Then ∑n≥1λn-1< ∞ implies the existence of the spectral gapαofLin L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1λn-1< ∞ if and only ifα> 0.
On pair and tuple formation under independent Poisson or renewal arrival processes
Authors: Borovkov K.
Published: July 2016

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).
M/GI/1 queues with services of both positive and negative customers
Authors: Zhu Yijun , Zhang Zhe George
Published: July 2016

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.
Existence of phase transition of percolation on Sierpiński carpet lattices
Authors: Shinoda Masato
Published: July 2016

We study Bernoulli bond percolation on Sierpiński carpet lattices, which is a class of graphs corresponding to generalized Sierpiński carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.
On a new approach to calculating expectations for option pricing
Authors: Borovkov K. , Novikov A.
Published: July 2016

We discuss a simple new approach to calculating expectations of a specific form used for the pricing of derivative assets in financial mathematics. We show that in the ‘vanilla case’, the expectations can be found by simply integrating the respective moment generating function with a certain weight. In situations corresponding to barrier-type options, we just need to carry out one more integration. The suggested approach appears to be the first (and, apart from Monte Carlo simulation, the only) one to allow the pricing of discretely monitored exotic options when the underlying asset is modelled by a general Lévy process. We illustrate the method numerically by calculating the price of a discretely monitored lookback call option in the cases when the underlying follows the geometric Brownian and variance-gamma processes.
A new theorem on the existence of invariant distributions with applications to ARCH processes
Authors: Kazakevičius Vytautas , Leipus Remigijus
Published: July 2016

A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.
Distribution of the number of customers served in an M/G/1 retrial queue
Authors: Lopez-Herrero M. J.
Published: July 2016

We present a recursive method of computation for the probability that at most k customers were served during the busy period of an M/G/1 retrial queue.
On the behaviour of some new ageing properties based upon the residual life of k-out-of-n systems
Authors: Li Xiaohu , Zuo Ming J.
Published: July 2016

In this paper, we investigatek-out-of-nsystems with independent and identically distributed components. Some characterizations of the IFR(2), DMRL, NBU(2) and NBUC classes of life distributions are obtained in terms of the monotonicity of the residual life given that the (n-k)th failure has occurred at timet≥ 0. These results complement those reported by Belzunce, Franco and Ruiz (1999). Similar conclusions based on the residual life of a parallel system conditioned by the (n-k)th failure time are presented as well.
Multivariate Poisson flows on Markov step processes
Authors: Beutler Frederick J. , Melamed Benjamin
Published: July 2016

A Markov step process Z equipped with a possibly non-denumerable state space X can model a variety of queueing, communication and computer networks. The analysis of such networks can be facilitated if certain traffic flows consist of mutually independent Poisson processes with respective deterministic intensities λ
i
(t). Accordingly, we define the multivariate counting process N = (N
1, N
2, · ·· N
c) induced by Z; a count in Ni
occurs whenever Z jumps from x (χ into a (possibly empty) target set . We study N through the infinitesimal operator Ã of the augmented Markov process W = (Z, N), and the integral relationship connecting Ã with the transition operator T

t
of W. It is then shown that Ni
depends on a non-negative function ri
defined on χ; ri
(x) may be interpreted as the expected rate of increase in Ni
, given that Z is in state x.

A multivariate N is Poisson (i.e., composed of mutually independent Poisson streams N

i
) if and only if simultaneous jumps are impossible in a certain sense, and if the conditional expectation E[r

i
(Z(t) | 𝒩

i
] = E[r

i
(Z(t))] for i = 1, 2, ···, c and each t ≧ 0, where 𝒩

i
is the σ-algebra σ{N(s), s ≦ t}. Necessary and sufficient conditions are also specified that, for each s ≦ t, the variates [N

i
(t)-N

i
(s)] are mutually independent Poisson distributed; this involves a weakened version of E[r

i
(Z(v)) | N(v) – N(u)] = E [r

i
(Z(v))] for i = 1, 2, ···, c and all 0 ≦ u ≦ v.

It is shown that the above criteria are automatically met by the more stringent classical requirement that N(t) and Z(t) be independent for each t ≧ 0.
A note describing the marginal benefits of spares on availability
Authors: Stulman A. , Mehrez A.
Published: July 2016

In this note we consider the marginal effect of adding an additional spare to a system containing operating units and their spares. We shall see that under the circumstance that the number of spares is greater than the number of repair facilities, the marginal effect will assume a simple interesting structure.
General shock models associated with correlated renewal sequences
Authors: Shanthikumar J. G. , Sumita U.
Published: July 2016

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn
) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn
is correlated to the length Yn
of the interval since the last shock, or to the length Yn
of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.
Measurement of a wandering signal amid noise
Authors: Hannan E. J.
Published: July 2016

A formula is given for the response function of a filter which extracts a signal generated by a non-stationary process from amid noise. The non-stationarity is due to the presence of zeros on the unit circle of the function characterising the difference equation generating the signal. The signal is broken into a trend component and a stochastic integral in which t occurs only through the factor exp it λ and it is shown that the filter which optimally extracts the latter perfectly represents the former. The considerations cover the case of a vector series. Applications to problems in seasonal variation measurement are indicated.
Asymptotic properties of the periodogram of a discrete stationary process
Authors: Olshen Richard A.
Published: July 2016

Suppose x
1,…, xN
are indefinitely many observations on a stochastic process which is weakly stationary with spectral density f(λ), – π ≦ λ ≦ π. An asymptotically unbiased, and to that extent plausible, estimate of 4rf(λ)is the periodogram

Yet the periodograms of processes which possess spectral densities are notoriously subject to erratic behavior.
Branching and clustering models associated with the ‘lost-games distribution’
Authors: Kemp Adrienne W. , Kemp C. D.
Published: July 2016

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi
with probability generating functions gi
(s), i = 1, 2, 3, if g
3(s) = g
1[g
2(s)], we say that X
3 is X
1 generalized by X
2 and write X
3~ X
1
V
X
2.
The departure process from the GI/G/1 Queue
Authors: Vlach Thomas L. , Disney Ralph L.
Published: July 2016

The departure process from the GI/G/1 queue is shown to be a semi-Markov process imbedded at departure points with a two-dimensional state space. Transition probabilities for this process are defined and derived from the distributions of the arrival and service processes. The one step transition probabilities and a stationary distribution are obtained for the imbedded two-dimensional Markov chain.
An LCFS finite buffer model with finite source batch input
Authors: Van Dijk Nico M.
Published: July 2016

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.
Obituary: Michel Métivier 1931–1988
Authors: Joffe A.
Published: July 2016

The sudden death of Michel Métivier in Paris on 10 October 1988 in his fifty-seventh year came as a great shock to the scientific community. Although he was known to be seriously ill with cancer, everyone had hoped for the best, including Michel himself whose deep religious faith and immense joie de vivre sustained him to the end.
A population process with offspring distribution depending on both population size and generation
Authors: Santana Pedro Saavedra , Garcia Miguel Sanchez
Published: July 2016

A population process {Zn
} is defined, with offspring distribution depending on both population size and generation number. As is well known, in models where the offspring distribution depends only on the size, under most conditions extinction surely results. The model we now introduce would mean that if history has an influence on evolution of the population, it is possible to obtain models for which a probability as high as desired exists that the population size balances between two given values.
Characterizations of exponential distributions by independent non-stationary record increments
Authors: Pfeifer Dietmar
Published: July 2016

A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.
Some notes on the total service time in certain loss systems
Authors: Parthasarathy P. R. , Sharafali M.
Published: July 2016

In this paper we discuss the total service rendered by all the servers in certain loss systems.
OBITUARY: Dr Norma Ruth McArthur
Authors: Gani J.
Published: July 2016

Norma McArthur, one of the four founding trustees of the Applied Probability Trust, died at her home in Canberra, Australia, on 17 January 1984. She had fought a long battle with cancer, exhibiting great resilience, clear-sightedness and courage to the last. When we took leave of each other in Canberra in August 1983, she was making plans for our next meeting in July 1984.
The slow server problem
Authors: Rubinovitch Michael
Published: July 2016

The problem is what to do with a slow server in a service facility which has fast and slow servers. Should the slow server be used to render service, or is it better not to use it at all? Simple models for answering this question are formulated and studied.
The busy period of the queueing system M/G/∞
Authors: Stadje Wolfgang
Published: July 2016

For the queueing system M/G/∞ some distributions connected with the associated busy periods are derived.
Inequalities and bounds for the scheduling of stochastic jobs on parallel machines
Authors: Pinedo Michael , Schechner Zvi
Published: July 2016

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj
, a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.
Symmetric sampling procedures, general epidemic processes and their threshold limit theorems
Authors: Martin-Löf Anders
Published: July 2016

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.
A genealogical approach to variable-population-size models in population genetics
Authors: Donnelly Peter
Published: July 2016

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.
The variance of a Poisson process of domains
Authors: Kellerer A. M.
Published: July 2016

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair-distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region.
Multivariate distributions for the life lengths of components of a system sharing a common environment
Authors: Lindley Dennis V. , Singpurwalla Nozer D.
Published: July 2016

In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.
On the concavity of the waiting-time distribution in some GI/G/1 queues
Authors: Szekli R.
Published: July 2016

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN
is established under the assumption that Xi
are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.
Optimal consumption with stochastic prices in continuous time
Authors: Björk T. , Myhrman J. , Persson M.
Published: July 2016

In this paper we consider an economic agent (the consumer) having initial wealth x(0), which he is allowed to spend during the time interval [0, T]. There are N consumer goods, the prices of which are described by a system of stochastic differential equations, and the consumer can spend his money as he pleases provided that he does not get into debt. The consumer's problem is to maximize his expected utility over [0, T]. Given a separable utility function we show that the optimal consumption policy is linear in the wealth variable regardless of the structure of the price system, and if we assume that the price processes are geometric Brownian motions we can give the optimal consumption policy explicitly.