Abstract In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions

Abstract The article is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviations and the entropy minimization problem. Both of them appear when considering a dynamical system

Abstract We study a general k dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index α∈(0,1). When the arrival rate is sped up by a factor nγ, the transition probabilities of the underlying Markov chain are divided by nγ and the service times are divided by n, we identify two regimes (”fast arrivals”, when γ>α, and” equilibrium”

Abstract In this article, consider a new continuous-time bidimensional renewal risk model with constant force of interest, in which every kind of business is assumed to pay two classes of claims called the first and second ones, respectively. Suppose that the first class of claim vectors form a sequence of independent and identically distributed random vectors following a general dependence structure

Abstract The Omega model assumes that the company continues to do business even when the surplus is negative. However, when the surplus is negative, the bankruptcy probability increases when the deficit grows. We study the Omega model with two state dependent bankruptcy rates, where bankruptcy occurs with probability 1 when the surplus is below a given negative level. Both the classical Cramér-Lundberg

Abstract The Lévy-driven CARMA(2,1) process is a popular one with which to model stochastic volatility. However, there has been little development in statistical tools to verify this model assumption and assess the goodness-of-fit using high frequency realized volatility. When a Lévy-driven CARMA(2,1) is observed at high frequencies, the unobserved driving process can be approximated from the observed

Abstract In this article, we consider residual lives of series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components with the joint distribution of their lifetimes being modeled by an Archimedean copula. By considering two such systems with different numbers of components within each subsystem, we establish various stochastic orderings and aging

Abstract In this paper, we analyze the sojourn time of an entire batch in a M[X]/M/1 processor sharing queue, where geometrically distributed batches arrive according to a Poisson process and individual jobs require exponentially distributed service times. By conditioning on the number of jobs in the queue and the number of jobs in a tagged batch, we establish recurrence relations for conditional sojourn

Abstract The number of common friends (or connections) in a graph is a commonly used measure of proximity between two nodes. Such measures are used in link prediction algorithms and recommendation systems in large online social networks. We obtain the rate of growth of the number of common friends in a linear preferential attachment model. We apply our result to develop an estimate for the number of

Abstract Hawkes processes have been widely studied in both theory and applications because of their self-exciting effects, but there are still some questions and room left for exploring the self-exciting point processes. In this paper, an ephemeral self-exciting Hawkes process is developed, which contains 1-memory and 0-memory self-exciting point processes. After giving the definition of the ephemeral

Abstract In this paper, we investigate optimal power management in parallel processing systems composed of one queue and several identical processing stations. Power consumption is controlled by setting some of the stations into an inactive state, where they consume less power but are unable to provide service. This way, we are faced with the conflicting objective of minimizing power consumption while

Abstract In the stochastic delay differential neoclassical growth system, the white noises stochastically perturb some parameters. we show that the stochastic system has the positive solutions, which will not explode to infinity in a finite time and, in fact, will be ultimately bounded. A numerical test shows the effectiveness of the theoretical results.

(2021). Fetschrift in honour of Prof. Peter Taylor. Stochastic Models: Vol. 37, A Festschrift in Honour of Prof. Peter Taylor, pp. 1-4.

Abstract We consider a virtual network (VN) embedding problem on a large-scale substrate physical network. We permit varying capacities and cost rates, and reservation of resources (physical links and nodes) for more profitable later VN requests. Our aim is to maximize the long-run average revenue by controlling the allocation of physical components to arriving VN requests. We propose an index policy

Abstract The expression given in an earlier paper for the ladder height distribution is erroneous. After deriving the correct version, the expression for ruin probability is provided.

In this paper, we study a single-server polling model with two queues. Customers arrive at the queues according to two independent Poisson processes. The server spends random amounts of time in each queue, regardless of the amounts of work present at the queues. The service speed is not constant; it is assumed that the server works at speed rixi at queue i when its current workload equals xi, i = 1

Abstract Motivated by communication networks, we study a single-server queue with varying service rates and multiple customers’ types. The difference between types of customers is defined by the profits contributed to the system. We show that the optimal trunk reservation policy exists and we compare the optimal control levels with different parameter values. Furthermore, when the optimal trunk reservation

This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. \ We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics algebraically for $2\leq\ell\leq3$ and numerically for $\ell\geq4$.

The setting considered in this paper concerns a discrete-time multivariate population process under Markov modulation. Our objective is to estimate the model parameters, based on periodic observati...

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process,

Abstract In this paper, we investigate the ruin probabilities of non-homogeneous risk models. By employing martingale method, the Lundberg-type inequalities of ruin probabilities of non-homogeneous renewal risk models are obtained under weak assumptions. In addition, for the periodic and quasi-periodic risk models the adjustment coefficients of the Lundberg-type inequalities are obtained. Finally,

Abstract In this paper, we propose a relative recursive regression estimator for censored data defined by the stochastic approximation algorithm to deal with the presence of outliers or when the response is usually positive. We give the central limit theorem and the strong pointwise convergence rate for our proposed nonparametric relative recursive estimators under some mild conditions. We finally

Abstract It is well known that the numerical stability of many finite difference time-stepping algorithms for solving the Kolmogorov differential equations for Markov jump processes depends on the magnitude of the spectral radius of the rate matrix. In this paper, we develop bounds on the spectral radius that rigorously establish that the spectral radius typically scales in proportion to the maximal

The article presents a numerical analysis approach for the transient solution of a piecewise homogeneous quasi-birth-death process. The proposed approach computes the transient probabilities based ...

In this paper, we study a double-sided queueing model with marked Markovian arrival processes and finite discrete abandonment times. We apply the theory of multi-layer Markov modulated fluid flow (...

Abstract In the original Bitcoin paper and other research papers since, it is assumed that Bitcoin blocks are mined at the instants of a homogeneous Poisson process. Based on blockchain block arrival data and stochastic analysis of the block arrival process, we demonstrate that this is not the case. We present a framework for studying the block arrival process, including two refined mathematical models

Abstract This paper investigates a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poisson process. Drift estimates are based on Kalman filter techniques and described by the

Abstract Consider a random sample from a bivariate random variable. If the sample is ordered according to the Y-variates, then the X-variate paired with the r-th order statistic is called the concomitant of the r-th order statistic (or the r-th concomitant). Given a sample of independent, identically distributed and bivariate phase-type random variables with continuous density function, we provide

Abstract This article studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends

We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochastic decomposition. Each one can be seen as an independent

Abstract Current evolutionary biology models usually assume that a phenotype undergoes gradual change. This is in stark contrast to biological intuition, which indicates that change can also be punctuated—the phenotype can jump. Such a jump could especially occur at speciation, i.e., dramatic change occurs that drives the species apart. Here we derive a Central Limit Theorem for punctuated equilibrium

Abstract A man has a house with n doors. Initially he places k pairs of walking shoes at each door. For each walk, he chooses one door at random, and puts on a pair of shoes, returns after the walk to a randomly chosen door and takes off the shoes at the door. Let Tn be the first time a door is chosen to walk out but with no shoes available. We show that as Tn has the same asymptotic distribution and

Abstract In this paper, we are concerned with strong convergence rate of Euler scheme for time-inhomogeneous one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process at point zero. We use a space transform in order to remove the local time from this class of stochastic differential equations. We provide the approximation of Euler for the stochastic differential

Abstract We consider two different continuous-time Markov chain models recently studied in Göbel et al.[8], which were created to model the interactions between a small pool of miners, and a larger collection of miners, within the Bitcoin network. The first model we discuss represents the case where all miners behave honestly and follow the Bitcoin protocol, while the second model represents the case

In 1953, Grothendieck [G] characterized locally convex Hausdorff spaces which have the Dunford-Pettis property and used this property to characterize weakly compact operators u : C(K) → F , where K is a compact Hausdorff space and F is a locally convex Hausdorff space (briefly, lcHs) which is complete. Among other results, he also showed that there is a bijective correspondence between the family of

Abstract We apply physical interpretations to construct algorithms for the key matrix G of discrete-time quasi-birth-and-death (dtQBD) processes which records the probability of the process reaching level for the first time given the process starts in level n. The construction of G and its z-transform was motivated by the work on stochastic fluid models (SFMs). In this methodology, we first write a

Abstract The focus of this work is the numerical application of a stochastic decision support model for the patient admission scheduling problem with random arrivals and departures. Here, we discuss the methodology for applying our model to real-world problems. We outline a solution approach for efficient computation, provide a numerical analysis of the model, and illustrate the methodology with examples

Abstract We study a future Internet networking paradigm where instead of congestion control an open loop traffic control is applied. We aim to give theoretical foundations for data transfer controlled only by the access points of the network. The key characteristics of networks without congestion control are stability and efficiency addressed in this paper. We consider the queue length processes of

Abstract We prove a counterintuitive result concerning the expected hitting/absorption time for a class of Markov chains. The “paradox” already shows itself in the following elementary example that is suitable for undergraduate teaching: Batman and the Joker perform independent discrete-time random walks on the vertices of a square until they meet, starting from opposite vertices. Batman always moves

Abstract In this paper, we consider the number of departures seen by a tagged customer while in service in a classical processor sharing queue. By exploiting the underlying orthogonal structure of this queuing system revealed in an earlier study, we compute the probability mass function of the random variable under consideration. We moreover derive the asymptotic behavior of this probability mass function

Abstract We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability and heavy-tailed with small

Abstract In the stochastic network model of Britton and Lindholm, the number of individuals evolves according to a supercritical linear birth and death process, and a random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. We derive a rate for the convergence of the degree distribution in this model towards the mixed

Abstract A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic

Abstract This paper presents matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Currently, there is no symbolic construction available to obtain the most concentrated ME distributions, and the numerical optimization-based approaches to construct them have many pitfalls. We present a numerical optimization-based procedure which avoids numerical issues.

Abstract We investigate a new class of separable systems which exhibit a product-form stationary distribution. These systems consist of parallel production systems (servers) at several locations, each with a local inventory under base stock policy, connected with a common supplier network. Demand of customers arrives at each production system according to a Poisson process and is lost if the local

Abstract This article provides a computational formula for the the stability of the probability of ruin of the compound Poisson risk process. This stability is the infinitesimal standardized variation of the probability of ruin resulting from pointwise perturbation of the distribution of the individual claim amounts. The proposed formula is obtained from the saddlepoint approximation to the distribution

Abstract In this article, we study the asymptotic behavior of the realized quadratic variation of a process where u is a β-Hölder continuous process with and where and BH is a fractional Brownian motion with Hurst index By exploiting the concentration phenomena, we prove almost sure convergence of the quadratic variation, that holds uniformly in time and on the full range As an application, we construct

Abstract In this article, we develop a new methodology for integrating epistemic uncertainties into the computation of performance measures of Markov chain models. We developed a power series algorithm that allows for combining perturbation analysis and uncertainty analysis in a joint framework. We characterize statistically several performance measures, given that distribution of the model parameter

Abstract A gene family is a set of evolutionarily related genes formed by duplication. Genes within a gene family can perform a range of different but possibly overlapping functions. The process of duplication produces a gene that has identical functions to the gene it was duplicated from with subsequent divergence over time. In this paper, we explore different models for the ongoing evolution of a

(2019). Correction. Stochastic Models: Vol. 35, No. 4, pp. 523-523.

The Workshop on Branching Processes and their Applications (WBPA) was held during 10–13 April 2018 in Badajoz, Spain. This conference gave continuity to such important previous meetings organized with the aim to facilitate the exchange of research ideas in this field and related processes. The First World Congress on Branching Processes was held in Varna (Bulgaria) in 1993 to celebrate the first 150

Abstract In this paper we construct vector-valued multi operator-stable random measures that behave locally like operator-stable random measures. The space of integrable functions is characterized in terms of a certain quasi-norm. Moreover, a multi operator-stable moving-average representation of a random field is presented which behaves locally like an operator-stable random field which is also o

Abstract This article studies Ornstein-Uhlenbeck (OU) diffusions with sticky reflection at zero. Sticky reflecting OU diffusions are defined as weak solutions of a system of SDEs involving the local time at the boundary at zero. The transition semigroup and the distribution of the first hitting time up are characterized analytically via their spectral representations. The results are applied to the

Abstract We deal with a general preferential attachment graph model with multiple type edges. The types are chosen randomly, in a way that depends on the evolution of the graph. In the N-type case, we define the (generalized) degree of a given vertex as where is the number of type k edges connected to it. We prove the existence of an a.s. asymptotic degree distribution for a general family of preferential

Abstract In this article, we consider an optimal execution problem with fixed time horizon and bounded transaction rate, which is more natural in practice. We show that, different from traditional stochastic control or singular control problems, this problem is of the stochastic bang-bang control type. Under some parameter settings we show that the optimal control does not involve buy action, and the

Abstract In Jacob et al. [A General Class of Population-Dependent Two-Sex Processes with Random Mating. Bernoulli 2017, 23, 1737–1758], a new class of two-sex branching processes in discrete time was introduced. These processes present the novelty that, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behavior with only

Abstract We describe two classes of Gaussian multi-self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions

Abstract A subordinate diffusion is a Markovian jump-diffusion or pure jump process obtained by time changing a diffusion process with an independent Lévy or additive subordinator. This class of processes has found many applications in financial modeling. In this paper, we develop sufficient conditions and necessary conditions for the distributions of two subordinate diffusions to be equivalent, which

Abstract Branching processes in random environments arise in a variety of applications such as biology, finance, and other contemporary scientific areas. Motivated by these applications, this article investigates the problem of ancestral inference. Specifically, the article develops point and interval estimates for the mean number of ancestors initiating a branching process in i.i.d. random environments