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AbstractWe consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process {(M1(t),M2(t)):t≥0}, which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process {(M1(t),M2(t)):t≥0} to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of {(M1(t),M2(t)):t≥0}. Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process is formulated and analysed, which is used to obtain a better understanding of the population dynamics. At first, we validate that the stochastic system has a unique global solution with any initial value. Then we analyse the stochastic dynamics of the model in detail, including pth moment boundedness, asymptotic pathwise estimation in turn. After that, we obtain sufficient conditions for the existence of a stationary distribution of the system by adopting stochastic Lyapunov function methods. In addition, under some mild conditions, we derive the specific form of covariance matrix in the probability density near the quasi-positive equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

Abstract The classical compartment model is often used to study the spread of an epidemic with one virus. However, there are few types of research on epidemic models with multiple viruses. The article aims to propose two new deterministic and stochastic SIIIRS models with multiple viruses and saturation incidences. We obtain asymptotic properties of disease-free and several endemic equilibria for the deterministic model. In the stochastic case, we prove the existence and uniqueness of positive global solutions. The extinction and persistence of diseases are obtained under different threshold conditions. We analyze the existence of stationary distribution through a suitable Lyapunov function. The results indicate that the extinction or persistence of the two viruses is closely related to the intensity of white noise interference. Specifically, considerable white noise is beneficial for the extinction of diseases, while slight one can lead to long-term epidemics of diseases. Finally, numerical simulations illustrate our theoretical results and the effect of essential parameters. [ABSTRACT FROM AUTHOR]

This paper proposes a stochastic reaction–diffusion predator–prey system with fear under the interference of space-time white noise by using the related theories of stochastic partial differential equations. The motivation of this paper is twofold: (i) mathematically, to try to investigate the effects of environmental noise and diffusion on population dynamics; (ii) biologically, to study how environmental noise affects the permanence of species. First, we analyse the well-posedness of solutions. Then sufficient conditions for extinction and permanence of prey and predator populations are derived. Moreover, we present the existence and uniqueness of stationary distribution of this system. Finally, based on numerical simulations and theoretical analysis, it is revealed that (i) high-intensity white noise can lead to the extinction of predator populations; (ii) low-intensity white noise may prolong the period of periodic solutions. This study provides new theoretical guidance for exploring ecological issues in the face of environmental disturbance and spatial diffusion. [ABSTRACT FROM AUTHOR]

We formulated a Coronavirus (COVID-19) delay epidemic model with random perturbations, consisting of three different classes, namely the susceptible population, the infectious population, and the quarantine population. We studied the proposed problem to derive at least one unique solution in the positive feasweible region of the non-local solution. Sufficient conditions for the extinction and persistence of the proposed model are established. Our results show that the influence of Brownian motion and noise on the transmission of the epidemic is very large. We use the first-order stochastic Milstein scheme, taking into account the required delay of infected individuals. [ABSTRACT FROM AUTHOR]

A stochastic epidemic model with random noise transmission is taken into account, describing the dynamics of the measles viral infection. The basic reproductive number is calculated corresponding to the stochastic model. It is determined that, given initial positive data, the model has bounded, unique, and positive solution. Additionally, utilizing stochastic Lyapunov functional theory, we study the extinction of the disease. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. We employed optimal control principles and examined stochastic control systems to regulate the transmission of the virus using environmental factors. Graphical representations have been offered for simplicity of comprehending in order to further verify the acquired analytical findings. [ABSTRACT FROM AUTHOR]

In this article, a stochastic differential equation model is proposed to investigate how the environmental noise affects the transmission dynamics of the interaction between crime, criminality and victimization with the influence of case reporting through the short message service (SMS). We prove that there is a unique global positive solution of the stochastic system with any positive initial value which is fundamental in the dynamics of population modeling. Moreover, we get sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that both criminal and victim will be persistent and coexistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we make up adequate conditions for complete eradication and wiping out of the diseases. In a biological viewpoint, the existence of a stationary distribution shows that the diseases will be prevalent and persistent in the long term. [ABSTRACT FROM AUTHOR]

Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that the diseases will be prevalent and persistent in the long term. In addition, we make up adequate conditions for complete eradication and wiping out of the diseases. Some numerical simulations are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic general epidemic model with relapse, graded cure and nonlinear incidence rate. We show the existence and uniqueness of a global positive solution for the model under some mild conditions. We also give some sufficient conditions for ensuring the existence of an ergodic stationary distribution for the model. Moreover, we analyse the extinction of the disease described by the model. Finally, we provide some numerical simulations to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate conditions for complete eradication and wiping out of the infectious disease. [ABSTRACT FROM AUTHOR]

In this article, based on the existing literature, we extend a deterministic delayed SIS epidemic model with vaccination to the stochastic version. First, we show that the solution of system is globally unique, positive and stochastically bounded. Then, constructing suitable Lyapunov function, we study the stochastic stability of system at the disease-free equilibrium. Moreover, the threshold R ̂ 0 which determines the extinction or persistence in time mean of the disease is proposed. When R ̂ 0 is less than 1, the disease will disappear; while R ̂ 0 is greater than 1, the disease will persist. Moreover, we study the existence of the ergodicity and the stationary distribution of the model, which can help us better understand the dynamic behavior and statistical characteristics of stochastic delayed biological models. Finally, we apply the results to hepatitis B in Chinese mainland. Using the nonlinear Least-Square method, we estimate the parameter of the model and obtain the threshold R ̂ 0 = 0.0556. We forecast that hepatitis B infections in Chinese mainland will drop to about 20 million in 2100 from 90 million in 2015, and give the probability densities of Hepatitis B infections each year. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long term. Some numerical simulations are introduced to illustrate our theoretical result. [ABSTRACT FROM AUTHOR]

Recently, Zhang et al. show that if ∑ i = 1 N π i β (i) ≠ 0 , then the Cox-Ingersoll-Ross (CIR) model with Markov switching (see below, the SDE (1.2)) is ergodic in the Wasserstein distance if and only if ∑ i = 1 N π i β (i) > 0. In this article, we will show that if ∑ i = 1 N π i β (i) = 0 , the Cox-Ingersoll-Ross (CIR) model with Markov switching is non-ergodic. Explicit expressions for the mean and variance of the CIR model with Markov switching are obtained. As a byproduct, the explicit expressions for mean of stationary distribution and second-order moments for such model are presented. Besides, we find the necessary and sufficient conditions of weak stationarity for CIR model with Markov switching. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a stochastic Holling type-II predator–prey model with prey refuge and fear effect. We first prove the existence and uniqueness of the global positive solution. Then we perform the survival analysis of the model, including the existence of a unique ergodic stationary distribution and the extinction of the model. Numerical simulations are carried out to validate our analytical results. Our findings indicate that the white noise is adverse to the growth of predator and prey populations, and the increase of fear effect will lead to the decrease of predator density, but with no obvious effect on prey density. [ABSTRACT FROM AUTHOR]

A stochastic susceptible-infected-recovered-susceptible model with vaccination includes stochastic variation in its parameters. Sufficient conditions for the extinction and the existence of the stationary distribution of the population are proved. [ABSTRACT FROM AUTHOR]

In this paper, we focus on ergodicity and transience of the Cox–Ingersoll–Ross interest rate model driven by stable processes with random switching. Under some assumptions, we prove that the interest rate process and switching process has a unique stationary distribution. In addition, some sufficient conditions for transience of such interest rate model are given. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic susceptible-infective-removed-infective (SIRI) epidemic model with relapse. We show that the densities of the distributions of the solutions can converge in L 1 to an invariant density or can converge weakly to a singular measure under certain condition. We also find the support of the invariant density. Moreover, we establish sharp sufficient criteria for the extinction of the disease in two cases. The results show that the smaller white noise can assure the existence of a stationary distribution which implies the persistence of the disease while the larger white noise can lead to the extinction of the disease. [ABSTRACT FROM AUTHOR]

This paper studies a class of stochastic SIRS epidemic models with general nonlinear incidence f (S , I , R) and varying population sizes. A new threshold value is obtained. The sufficient conditions for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution in probability meaning are established. As applications of the main results, the stochastic SIRS epidemic models with standard incidence, Beddington–DeAngelis incidence and nonlinear incidence rate h (S) g (I) are discussed, and a series of new criteria for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution are established. [ABSTRACT FROM AUTHOR]

We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations. [ABSTRACT FROM AUTHOR]

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider ergodicity and transience for Lotka–Volterra model driven by α-stable processes () with Markovian switching. Sufficient conditions for ergodicity and extinction of such model are given. Two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations. [ABSTRACT FROM AUTHOR]

We consider solutions to so-called stochastic fixed point equation , where Ψ is a random Lipschitz function and R is a random variable independent of Ψ. Under the assumption that Ψ can be approximated by the function , we show that the tail of R is comparable with the one of A, provided that the distribution of is tail equivalent. In particular, we obtain new results for the random difference equation. [ABSTRACT FROM AUTHOR]

This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L1 to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

This paper considers two S-DI-A models in random environments. Firstly, using Has’minskii theory of periodic solution, we show that stochastic periodic S-DI-A model has a nontrivial positive periodic solution if. Then, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to S-DI-A model perturbed by white and telephone noises. Finally, examples are introduced to illustrate the results developed. [ABSTRACT FROM AUTHOR]

In this paper, we initially propose a stochastic SIRI epidemic model. The existence, uniqueness, and boundedness of the positive solution of the stochastic system are derived. By using some novel Lyapunov functions and stochastic analysis tools, we mainly investigate asymptotical behaviors of the SIRI model under stochastic perturbations according to the basic reproduction number. That is, when(or) and some conditions on white noise are satisfied, the solution of stochastic model will oscillate around the disease-free equilibrium (or endemic equilibrium) of the deterministic model and the estimate for the oscillation amplitude is obtained. Furthermore, the existence of a stationary distribution and the ergodicity of solutions are studied for the stochastic system. Finally, some numerical examples are provided to illustrate the effectiveness of the obtained results. [ABSTRACT FROM PUBLISHER]

We study the backwards Markov chain for the Bak–Sneppen model of biological evolution and derive its corresponding reversibility equations. We show that, in contrast to the forwards Markov chain, the dynamics of the backwards chain explicitly involve the stationary distribution of the model, and from this we derive a functional equation that the stationary distribution must satisfy. We use this functional equation to derive differential equations for the stationary distribution of Bak–Sneppen models in which all but one or all but two of the fitnesses are replaced at each step, subject to certain conditions on the relative locations of the replaced species. This gives a unified way of deriving Schlemm’s expressions for the stationary distributions of the isotropic four-species model, the isotropic five-species model, and the anisotropic three-species model. [ABSTRACT FROM AUTHOR]

In this paper, we consider sufficient conditions for the stationary distribution of generalized Ait-Sahalia-type interest rate model. We show that ifr⩾ 2ρ − 1, then the generalized Ait-Sahalia interest rate model has a unique stationary distribution and ifr< 2ρ − 1, then the generalized Ait-Sahalia interest rate model is recurrent relative to the domain (0, ϵ), for any ϵ > 0. Besides, some recursion formulas for the stationary distribution are presented. Finally, some numerical simulations are used to illustrate our results. [ABSTRACT FROM AUTHOR]

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results. [ABSTRACT FROM AUTHOR]

This paper focuses on retarded stochastic differential equations (SDEs), which include SDEs of neutral type and SDEs driven by Lévy processes without finite second moments. Treating ‘pure delay’ systems in which both the drift and the diffusion coefficients depend only on the delayed state but not on the current state, most of the existing methods become not applicable. Using the variation-of-constants formulae, we overcome the difficulties due to the lack of the information of the state at the current time. We obtain existence and uniqueness of stationary distributions for retarded SDEs that need not satisfy dissipative conditions. In addition, we provide a partial answer to a problem posed in Reiß et al. [ABSTRACT FROM AUTHOR]

Consider a Markov chain with finite state {0, 1, …,d}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations: (1) the absorbing time of statedwhen the chain starts from any stateiand absorbing at stated; (2) the passage time of any stateiwhen the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly. [ABSTRACT FROM PUBLISHER]

We illustrate practical uses of logistic regression and Markov chains by applying these concepts to the problem of developing optimal strategy in the popular basketball game of H-O-R-S-E. Based on data collected by the authors, we estimate model parameters for each author, describe strategies of optimizing each author’s probability of winning, and calculate the stationary distribution of a Markov chain that arises from the game. [ABSTRACT FROM AUTHOR]

We compute the stationary distribution of a continuous-time Markov chain that is constructed by gluing together two finite, irreducible Markov chains by identifying a pair of states of one chain with a pair of states of the other and keeping all transition rates from either chain. The result expresses the stationary distribution of the glued chain in terms of quantities of the two original chains. Special emphasis is given to the cases when the stationary distribution of the glued chain is a multiple of the equilibria of the original chains, and when not, for which bounds are derived. [ABSTRACT FROM AUTHOR]

We first introduce fuzzy finite Markov chains and present some of their fundamental properties based on possibility theory. We also bring in a way to convert fuzzy Markov chains to classic Markov chains. In addition, we simulate fuzzy Markov chain using different sizes. It is observed that the most of fuzzy Markov chains not only do have an ergodic behavior, but also they are periodic. Finally, using Halton quasi-random sequence we generate some fuzzy Markov chains, which are compared with the ones generated by the RAND function of MATLAB. Therefore, we improve the periodicity behavior of fuzzy Markov chains. [ABSTRACT FROM PUBLISHER]

In many situations, serious damage and considerable financial losses are caused by non-repairable failures of a system. Redundant systems and maintenance policies are commonly employed to improve reliability. This paper is focused on the modelling of a complex cold standby system by analysing the effectiveness and costs of preventive maintenance, always in an algorithmic form. The online unit of the system is subject to wear failures and external shocks. The online unit can go through an indeterminate number of degradation levels before failure. This one is observed when inspections occur. Inspections are performed at random intervals, and when one takes place, the unit is taken to the preventive maintenance facility if it is necessary. The preventive maintenance time and cost is different depending on the degradation level observed. If only one unit is performing, a minimal maintenance policy is adopted in order to optimise system behaviour. Reliability measures such as the conditional probability of failure are worked out in a well-structured and algebraic form in transient and stationary regimes by using algorithmic methods. The stationary distribution is calculated using matrix analytic methods, and rewards are included in the model. An optimisation example shows the versatility of the model presented. [ABSTRACT FROM AUTHOR]

In this article, we consider the longest-queue-served-first queueing system, in which two types of customers arrive, independently and respectively according to two Poisson processes with possibly unequal arrival rates, to two separate queues served, with possibly unequal service rates, by a single exponential server. This system finds important applications and has been studied before. Our focus is on the tail asymptotic behavior for the stationary distribution when the system is stable. We model this system into a reflected random walk in the half plane. This random walk reveals two different properties, respectively, along the direction of the minimum of the two queues and the direction of the difference between the two queues. Specifically, the one-dimensional process for the minimum is reducible, which makes the tail analysis along the difference direction easier, but the process for the difference is irreducible, for which the analysis is more challenging (similar to the analysis for a non-singular genus one reflected random walk). We extend the kernel method for random walks in the quarter plane to this system to obtain exact tail asymptotic expressions along both directions, which are new contributions to the literature. [ABSTRACT FROM AUTHOR]

In this article, we consider a discrete-time semi-Markov process with finite state space and an observation censored at an arbitrary fixed time. Some intermediate results concerning the empirical estimation of the mean recurrence times of the embedded Markov process and the mean sojourn times of the semi-Markov process are given. We study two nonparametric estimators for the stationary distribution of the semi-Markov process and examine their asymptotic properties, such as strong consistency and asymptotic normality, as the length of the observation tends to infinity. Finally, a numerical application is presented to illustrate the comparison of the two estimators. [ABSTRACT FROM PUBLISHER]

In recent years, reflected diffusion processes have played an important role in mathematical finance because of the existence of the constraint policies to special economic regime, such as a storage system, the exchange rate target-zone dynamics, barrier options, and so on. We are to propose a class of Cox–Ingersoll–Ross interest rate processes which has double sided reflected barriers so that the central bank can constrain the interest rate to fluctuate within a pre-specified zone. In this paper, we are mainly interested in the stationary distribution and first hitting times about the reflected Cox–Ingersoll–Ross interest rate processes. Therefore, we will supply an explicit expression of the stationary distribution and some related analytic formulas of first passage times. [ABSTRACT FROM PUBLISHER]

In the last decade, mobile phones and mobile devices using mobile cellular telecommunication network connections have become ubiquitous. In several developed countries, the penetration of such devices has surpassed 100 percent. They facilitate communication and access to large quantities of data without the requirement of a fixed location or connection. Assuming mobile phones usually are in close proximity with the user, their cellular activities and locations are indicative of the user's activities and movements. As such, those cellular devices may be considered as a large scale distributed human activity sensing platform. This paper uses mobile operator telephony data to visualize the regional flows of people across the Republic of Ireland. In addition, the use of modified Markov chains for the ranking of significant regions of interest to mobile subscribers is investigated. Methodology is then presented which demonstrates how the ranking of significant regions of interest may be used to estimate national population, results of which are found to have strong correlation with census data. [ABSTRACT FROM PUBLISHER]

Wireless multi-hop networks are an important type of networks, which have been studied in the literature by several researchers. For example, in a recent study, Guillemin et al. (2013) obtained exact tail asymptotics for the two stationary marginal distributions for wireless 3-hop networks with stealing based on boundary value problems. Efficient algorithms and good approximations for system performance could be derived based on this asymptotic property. However, it has been pointed out (see Fayolle and Iasnogorodski, 2014) that the key boundary value problem proposed in Guillemin et al. (2013) is incorrect, and an erratum by the authors has been published, Guillemin et al. (2014), with further details to appear in the near future. In this paper, we revisit this wireless 3-hop network with stealing. Using a different approach, the kernel method, we obtain exact tail asymptotics not only for the marginal distributions, but also for the joint distributions, which matches exactly with the results in Guillemin et al. (2013) (and Guillemin et al., 2011). Based on this result, impact of stealing can be clearly revealed. [ABSTRACT FROM AUTHOR]

We study tail asymptotics of the stationary distribution for the [ABSTRACT FROM AUTHOR]

This article considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For case (i), we present a weaker sufficient condition for the subexponential asymptotics than those given in the literature. As for case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in case (ii) is different from that in case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii). [ABSTRACT FROM PUBLISHER]

We consider a multi-dimensional diffusion process (Xt)t≥0with drift vectorband diffusion matrix Σ. This process is observed atn+1 discrete times with regular sampling interval Δ. We review sufficient conditions for the existence and uniqueness of an invariant density. In a second step, we assume that the process is stationary, and estimate the drift functionbon a compact setKin a nonparametric way. For this purpose, we consider a family of finite dimensional linear subspaces ofL2(K), and compute a collection of drift estimators on every subspace by a penalized least-squares approach. We introduce a penalty function and select the best drift estimator. We obtain a bound for the risk of the resulting adaptive estimator. Our method fits for any dimensiond, but, for safe of clarity, we focus on the cased=2. We also provide several examples of two-dimensional diffusions satisfying our assumptions, and realize various simulations. Our results illustrate the theoretical properties of our estimators. [ABSTRACT FROM PUBLISHER]

This article is concerned with the long-time behavior of neutral genetic population models with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter. [ABSTRACT FROM AUTHOR]

We consider a two-server queueing system with the special feature of cost-conscious and jockeying customers. It is assumed that a new arriving customer joins one of the two queues only if the associated costs for him do not exceed a given upper threshold c. These costs are given by a linear combination of the current queue length at the chosen server and a service fee. Thus, if Ni are the current queue lengths, βi > 0 are the service fees and αi > 0 are certain prespecified weights for the two servers (i ∈ {1, 2}), the customer enters the system if and only if min [α1N1 + β1, α2N2 + β2] ≤ c, and in this case he goes to server 1 or 2 depending on which of the two terms is smaller; if equality occurs, he selects each server with probability 1/2. A customer waiting in line jockeys if, due to a departure from the other line during his waiting time, his residual costs become larger than the costs he will have after switching to the other server. In this model, new customers do not necessarily join the shortest queue (if they enter the system at all) and customers may jockey to a longer queue (if such a move reduces residual costs). Under the assumption that the arrival times form a Poisson process and the service times are exponential we derive the stationary distribution of the two-dimensional Markov process of queue lengths. [ABSTRACT FROM AUTHOR]