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.

Abstract 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

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

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 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 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

Abstract In discrete time, ℓ-blocks of red lights are separated by ℓ-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≤ℓ≤3 and numerically for ℓ≥4.

Abstract 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 observations of the network population vector. These parameters relate to the arrival, routing and departure processes, but also to the (unobservable) Markovian background process. When opting for the classical

Abstract 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

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

Abstract 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 on the linear combination of matrix geometric series in Laplace transform domain, and builds on the availability of an efficient numerical inverse Laplace transformation method.

Abstract 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 (MMFF) processes to analyze the queueing model. First, we define three age processes for the queueing system and convert them into a multi-layer MMFF process. Then we analyze the multi-layer MMFF

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 (Xk,Yk)(k=1,…,n+1) 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

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

Abstract 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

We consider two different continuous-time Markov chain models recently studied in Göbel et al.[8 Göbel, J.; Keeler, H. P.; Krzesinski, A. E.; Taylor, P. G. Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay. Perform. Eval. 2016, 104, 23–41. DOI: 10.1016/j.peva.2016.07.001.[Crossref], [Web of Science ®] , [Google Scholar]], which were created to model the interactions

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

(2020). Preface. Stochastic Models: Vol. 36, 10th International Conference on Matrix-Analytic Methods for Stochastic Models, pp. 173-175.

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 n→∞, Tn has the same asymptotic distribution and moments

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

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  (n−1) for the first time given the process starts in level n. The construction of G and its z-transform G(z) was motivated by the work on stochastic fluid models (SFMs). In this methodology, we first write

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. A

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 data-flows

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 (and clockwise

In this paper, we consider the number of departures seen by a tagged customer while in service in a classical M/M/1 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

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 1−ϵ and heavy-tailed with small

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 Poisson

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 Models

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.

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 inventory

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 of

In this article, we study the asymptotic behavior of the realized quadratic variation of a process ∫0tusdYs(1), where u is a β-Hölder continuous process with β>1−H and Yt(1)=∫0te−sdBasH, where at=HetH and BH is a fractional Brownian motion with Hurst index H∈(0,1). By exploiting the concentration phenomena, we prove almost sure convergence of the quadratic variation, that holds uniformly in time and

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 expressing

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 gene family

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

(2019). IV Workshop on Branching Processes and their Applications (WBPA 2018) – Part II. Stochastic Models: Vol. 35, IV Workshop on Branching Processes and their Applications (WBPA 2018) – Part II, pp. 235-237.

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 operator-self-similar

(2019). IV Workshop on Branching Processes and their Applications (WBPA 2018) – Part I. Stochastic Models: Vol. 35, IV Workshop on Branching Processes and their Applications (WBPA 2018) – Part I, pp. 105-107.

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 Vasicek

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 d=(d1,d2,…,dN), where dk∈Z0+ 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

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 optimal

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 perfect

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

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 are important

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 and

We consider Bienaymé-Galton-Watson and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, in fixed and random environment. In one hand, in the fixed environment case, we give new proofs and derive necessary and sufficient conditions for diffusion approximation to get hold of Feller-Jiřina and Jagers theorems. In the other hand, we propose

We consider critical Sevastyanov branching processes with immigration at random time-points generated by a Poisson random measure with a local intensity r(t)=L(t)t−1 for some slowly varying function L(t). The asymptotic behavior of the probability of non-extinction is studied and conditional limiting distributions of the processes with proper normalization are obtained.

Metastasis, the spread of cancer cells from a primary tumor to secondary location(s) in the human organism, is the ultimate cause of death for the majority of cancer patients. Although studied for more than 180 years, increasing efforts in recent years have significantly contributed to a better understanding of this aspect of tumor development. Adding to this understanding, our current paper proposes

Consider a (2, 1) random walk in an i.i.d. random environment, whose environment involves certain parameter. We construct an M-estimator for the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment (BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant

We consider a device that is designed to perform missions consisting of a random sequence of phases or stages with random durations. The mission process is described by a Markov renewal process and the system is a complex one consisting of a number of components whose lifetimes depend on the phases of the mission. We discuss models and tools to compute system, mission, and phase reliabilities using

We give sufficient conditions on the offspring, the initial and the immigration distributions under which a second-order Galton–Watson process with immigration is regularly varying.

A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations. This flexibility makes the process more appropriate to model real populations due to the fact that the stability in the reproductive capacity and in the immigration

In this paper, we consider a variant of a discrete time Galton–Watson Branching Process in which an individual is allowed to survive for more than one (but finite) number of generations and may also give birth to offsprings more than once. We model the process using multitype branching process and derive conditions on the mean matrix that determines the long-run behavior of the process. Next, we analyze