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The large-time behavior of a nonlinearly coupled pair of measure-valued transport equations with discontinuous boundary conditions, parameterized by a positive real-valued parameter λ, is considered. These equations describe the hydrodynamic or fluid limit of many-server queues with reneging (with traffic intensity λ), which model phenomena in diverse disciplines, including biology and operations research. For a broad class of reneging distributions with finite mean, and service distributions with finite mean and hazard rate function that is either nonincreasing or bounded away from zero and infinity, it is shown that if the fluid equations have a unique invariant state, then the Dirac measure at this invariant state is the unique invariant distribution of the fluid equations. In particular, this implies that the stationary distributions of scaled N-server systems converge to the unique invariant state of the corresponding fluid equations. Moreover, when the mean arrival rate is not equal to the mean service rate, that is, when \lambda \not= 1, it is shown that the solution to the fluid equation starting from any initial condition converges to this unique invariant state in the large-time limit. The proof techniques are different under the two sets of assumptions on the service distribution, as well as under the two regimes λ < 1 and λ > 1. When the hazard rate function is nonincreasing, a reformulation of the dynamics in terms of a certain renewal equation is used, in conjunction with recursive asymptotic estimates. When the hazard rate function is bounded away from zero and infinity, the proof uses an extended relative entropy functional as a Lyapunov function. Analogous large-time convergence results are also established for a system of coupled measure-valued equations modeling a multiclass queue. [ABSTRACT FROM AUTHOR]

Long-term behaviors of biochemical systems are described by steady states in deterministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not hold for the majority of cases. Thus, translating networks to have WR and ZD while preserving the original dynamics was proposed. Yet, this approach is limited because manually obtaining the desired network translation among the large number of candidates is challenging. Here, we prove necessary conditions for having WR and ZD after translation, and based on these conditions, we develop a user-friendly computational package, TOWARDZ, that automatically and efficiently identifies translated networks with WR and ZD. This allows us to quantitatively examine how likely it is to obtain WR and ZD after translation depending on the number of species and reactions. Importantly, we also describe how our package can be used to analytically derive steady states of deterministic models and stationary distributions of stochastic models. TOWARDZ provides an effective tool to analyze biochemical systems. [ABSTRACT FROM AUTHOR]

We consider a queueing system M|G|1 with possible vacations in server operations for principal customers (for example, if a server is leased). A cost optimization problem is solved. As control parameters, we use the probability a of the vacation and its duration. Under fairly general assumptions about the system behavior during vacations, we show that the optimal value of the probability a is either 0 or 1. We also give necessary and sufficient conditions for a vacation to be carried out, i.e., a = 1. With constant vacation durations, we find conditions such that a = 1, and the vacation duration is optimal. Two examples are considered. In the first example, the revenue from the vacation is a linear function of its duration, and, in the second example, the revenue is a quadratic function. [ABSTRACT FROM AUTHOR]

In a general stochastic multistate promoter model of dynamic messenger ribonucleic acid (mRNA)/protein interactions, we identify the stationary joint distribution of the promoter state, mRNA, and protein levels through an explicit "stick-breaking" construction perhaps of interest in itself. This derivation is a constructive advance over previous work where the stationary distribution is solved only in restricted cases. Moreover, the stick-breaking construction allows us to sample directly from the stationary distribution, permitting inference procedures and model selection. In this context, we discuss numerical Bayesian experiments to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper develops a finite approximation approach to find the solution F(x) of integral equations in the form F(x) = ξ(x) + R R κ(x, u)dF(u) ∀x ∈ R, where the unknown distributional solution F(x) defines the measure associated with the integration, and F(x) may not be continuous. To the best of our knowledge, the equation solutions in this framework have never been studied before. However, such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available. [ABSTRACT FROM AUTHOR]

Single-channel systems where service vacation is triggered when the system is free from customers are studied. Vacations in service can mean either a complete shutdown of the server or a transition to a different regime. If, during a vacation, the number of customers in the system reaches a certain fixed threshold, the vacation terminates and the unit resumes standard operations. A delay begins if the system is free from customers after a vacation; the delay interrupts at the moment of arrival of the first customer. Then a new vacation begins if no customers have arrived. The input of the queue outside vacations is a Poisson flow, and the remaining variables, which control the performance of the system (the service time, the durations of vacations and delays) have arbitrary distributions. A stationary distribution for the number of customers in the system is determined under the assumption that the process controlling the number of customers in the system during the vacation and the planned duration of the vacation are independent. We also study the asymptotic behavior for the number of completed vacations. [ABSTRACT FROM AUTHOR]

Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson-Watanabe superprocesses without or with immigration. The strong Feller property in the total variation distance is derived as a by-product. The key of the approach is a set of estimates for the variations of the transition probabilities. The estimates in Wasserstein distance are derived from an upper bound of the kernels induced by the first moment of the superprocess. Those in total variation distance are based on a comparison of the cumulant semigroup of the superprocess with that of a continuous-state branching process. The results improve and extend considerably those of Friesen [Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration, arXiv:1903.05546, 2019] and Stannat [J. Funct. Anal., 201 (2003), pp. 185-227; Ann. Probab., 31 (2003), pp. 1377-1412]. We also show a connection between the ergodicities of the immigration superprocesses and decomposable distributions. [ABSTRACT FROM AUTHOR]

This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing, a real-valued threshold. We show that if < 0, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value > 0, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all t 2 [0;1), approximation in a long but finite interval [0; T] can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any vertex by the rotor walk is at most equal to the expected number of visits by the simple random walk. In particular, this implies that this walk is transient. When these two numbers coincide, we show that the rotor configuration at the end of the process also has the law of OWUSF. Furthermore, if the graph is vertex-transitive, we show that the average number of visits by n consecutive rotor walks converges to the Green's function of the simple random walk as n tends to infinity. This answers a question posed by Florescu et al. (2014). [ABSTRACT FROM AUTHOR]

We propose two algorithms for simulating continuous time Markov chains in the presence of metastability. We show that the algorithms correctly estimate, under the ergodicity assumption, stationary averages of the process. Both algorithms, based on the idea of the parallelreplica method, use parallel computing in order to explore metastable sets more efficiently. The algorithms require no assumptions on the Markov chains beyond ergodicity and the presence of identifiable metastability. In particular, there is no assumption on reversibility. For simpler illustration of the algorithms, we assume that a synchronous architecture is used throughout the paper. We present error analyses, as well as numerical simulations on multiscale stochastic reaction network models in order to demonstrate consistency of the method and its efficiency. [ABSTRACT FROM AUTHOR]

Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides suficient conditions that are very close to the necessary ones for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree. [ABSTRACT FROM AUTHOR]

We give a mathematical framework for exact milestoning, a recently introduced algorithm for mapping a continuous time stochastic process into a Markov chain or semi-Markov process that can be efficiently simulated and analyzed. We generalize the setting of exact milestoning and give explicit error bounds for the error in the milestoning equation for mean first passage times. [ABSTRACT FROM AUTHOR]

Stochastic reaction networks are dynamical models of biochemical reaction systems and form a particular class of continuous-time Markov chains on Nn. Here we provide a fundamental characterization that connects structural properties of a network to its dynamical features. Specifically, we define the notion of "stochastically complex balanced systems" in terms of the network's stationary distribution and provide a characterization of stochastically complex balanced systems, parallel to that established in the 1970s and 1980s for deterministic reaction networks. Additionally, we establish that a network is stochastically complex balanced if and only if an associated deterministic network is complex balanced (in the deterministic sense), thereby proving a strong link between the theory of stochastic and deterministic networks. Further, we prove a stochastic version of the "deficiency zero theorem" and show that any (not only complex balanced) deficiency zero reaction network has a product-form Poisson-like stationary distribution on all irreducible components. Finally, we provide sufficient conditions for when a product-form Poisson-like distribution on a single (or all) component(s) implies the network is complex balanced, and we explore the possibility to characterize complex balanced systems in terms of product-form Poisson-like stationary distributions. [ABSTRACT FROM AUTHOR]

We prove that the stationary distribution of a system of reacting species with a weakly reversible reaction network of zero deficiency in the sense of Feinberg and separable propensity functions admits the tensor-structured approximation in the quantized tensor train (QTT) format. The complexity of the approximation scales linearly with respect to the number of species and logarithmically in the maximum copy numbers as well as in the desired accuracy. Our result covers the classical mass-action and Michaelis--Menten kinetics which correspond to two widely used classes of propensity functions and also to arbitrary combinations of those. New rank bounds for tensor-structured approximations of the probability density function (PDF) of a truncated one-dimensional Poisson distribution are an auxiliary result of the present paper which might be of independent interest. The present work complements recent results obtained by the authors jointly with Khammash and Nip on the tensor-structured numerical simulation of the evolution of system states distributions, driven by the Kolmogorov forward equation of the system, known also as the chemical master equation (CME). For the two kinetics mentioned above and a general reaction network, we also quantify the error of the low-rank tensor-structured approximation of the CME operator in the QTT format. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating the perturbation bounds for the stationary distributions for discrete-time or continuous-time Markov chains on a countable state space. For discrete-time Markov chains, two new normwise bounds are obtained. The first bound is rather easy to obtain since the needed condition, equivalent to uniform ergodicity, is imposed on the transition matrix directly. The second bound, which holds for a general (possibly periodic) Markov chain, involves finding a drift function. This drift function is closely related to the mean first hitting times. Some V -normwise bounds are also derived based on the results in [N. V. Kartashov, J. Soviet Math., 34 (1986), pp. 1493-1498]. Moreover, we show how the bounds developed in this paper and one bound given in [E. Seneta, Adv. Appl. Probab., 20 (1988), pp. 228-230] can be extended to continuous-time Markov chains. Several examples are shown to illustrate our results or to compare our bounds with the known ones in the literature. [ABSTRACT FROM AUTHOR]

This work presents an approach for reducing the number of arithmetic operations involved in the computation of a stationary distribution for a finite Markov chain. The proposed method relies on a particular decomposition of a transition-probability matrix called stochastic factorization. The idea is simple: when a transition matrix is represented as the product of two stochastic matrices, one can swap the factors of the multiplication to obtain another transition matrix, potentially much smaller than the original. We show in the paper that the stationary distributions of both Markov chains are related through a linear transformation, which opens up the possibility of using the smaller chain to compute the stationary distribution of the original model. In order to support the application of stochastic factorization, we prove that the model derived from it retains all the properties of the original chain which are relevant to the stationary distribution computation. Specifically, we show that (i) for each recurrent class in the original Markov chain there is a corresponding class in the derived model with the same period and, given some simple assumptions about the factorization, (ii) the original chain is irreducible if and only if the derived chain is irreducible and (iii) the original chain is regular if and only if the derived chain is regular. The paper also addresses some issues associated with the application of the proposed approach in practice and briefly discusses how stochastic factorization can be used to reduce the number of operations needed to compute the fundamental matrix of an absorbing Markov chain. [ABSTRACT FROM AUTHOR]

A sequence of real numbers (xn) is Benford if the significands, i.e., the fraction parts in the floating-point representation of (xn), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with transition probability matrix P and linfiting matrix P* is Benford if every component of both sequences of matrices (Pn - P*) and (Pn+1 - Pn) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron--Frobenius, this paper derives a simple sufficient condition ("nonresonance") guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications. [ABSTRACT FROM AUTHOR]

We consider generalized maximal branching processes on R+, which were earlier introduced by the author under assumptions providing the ergodicity of these processes. We study a tail asymptotics of stationary distributions dependent on properties of distributions of a number of direct offsprings. It is shown that the asymptotics is essentially different in cases of finite and infinite mean. [ABSTRACT FROM AUTHOR]

We consider the first exit time of a nonnegative Harris-recurrent Markov process from the interval [0, A] as A → ∞. We provide an alternative method of proof of asymptotic exponentiality of the first exit time (suitably standardized) that does not rely on embedding in a regeneration process. We show that under certain conditions the moment generating function of a suitably standardized version of the first exit time converges to that of Exponential(1), and we connect between the standardizing constant and the quasi-stationary distribution (assuming it exists). The results are applied to the evaluation of a distribution of run length to false alarm in change-point detection problems. [ABSTRACT FROM AUTHOR]

For diffusion processes in ℝd with locally unbounded drift coefficients we obtain a sufficient condition for the strict positivity of transition probabilities. To this end, we consider parabolic equations of the form L*μ = 0 with respect to measures on ℝd x (0, 1) with the operator Lu ≔ ∂tμ + ∂xj (ai∂xj u) + bi∂xi u. It is shown that if the diffusion coefficient A = (aij) is sufficiently regular and the drift coefficient b = (bi) satisfies the condition exp(κ∣b∣²) ϵ Lloc¹(μ), where the measure μ is nonnegative, then μ has a continuous density ϱ(x, t) which is strictly positive for t > τ provided that it is not identically zero for t ≦ r. Applications are obtained to finite-dimensional projections of stationary distributions and transition probabilities of infinite-dimensional diffusions. [ABSTRACT FROM AUTHOR]

We devise explicit partitioned numerical methods for second-order-in-time scalar stochastic differential equations, using one Gaussian random variable per timestep. The construction proceeds by analysis of the stationary density in the case of constant-coefficient linear equations, imposing exact stationary statistics in the position variable and absence of correlation between position and velocity; the remaining error is in the velocity variable. A new two-stage "reverse leapfrog" method has good properties in the position variable and is symplectic in the limit of zero damping. Explicit new "Runge-Kutta leapfrog" methods are constructed, sharing the property that qn+1 = qn + 1/2 (pn+pn+1)Δt, whose mean-square velocity order increases with the number of stages. [ABSTRACT FROM AUTHOR]

We obtain lower bounds for solutions to second order elliptic and parabolic equations on the whole space. Our method is based on the study of the dependence of a constant in Harnack's inequality on the coefficients of the equation. As an application we obtain lower bounds for densities of stationary distributions and transition probabilities of diffusion processes with unbounded drift coefficients. [ABSTRACT FROM AUTHOR]

We seek numerical methods for second-order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second-order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous-time and discrete-time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge--Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case. [ABSTRACT FROM AUTHOR]

This paper gives a description of stationary random partitions of positive integers (equivalently, stationary coherent sequences of random permutations) under the action of the infinite symmetric group. Equivalently, all stationary coherent sequences of random permutations are described. This result gives a new characterization of the Poisson--Dirichlet distribution PD(1) with the unit parameter, which turns out to be the unique invariant distribution for a family of Markovian operators on the infinite-dimensional simplex. This result also provides a new characterization of the Haar measure on the projective limit of finite symmetric groups. [ABSTRACT FROM AUTHOR]

We consider the random walk {S[SUBn]}, generated by a sequence {X[SUBk]} of independent identically distributed random variables with EX[SUB1] &epsis; (-∞, 0). The influence of the roots of the characteristic equation 1-E exp([SUBs]X[SUB1]) = 0 in the analyticity strip of the Laplace transform E exp([SUBs]X[SUB1]) on the distribution of the supremum sup[SUBn≥0] S[SUBn] is studied. An analogous problem is investigated for the stationary distribution of an oscillating random walk. [ABSTRACT FROM AUTHOR]

We first derive from abstract results on Feller transition kernels that, under some mild assumptions, a Markov stochastic algorithm with constant step size &epsiv; usually has a tight family of invariant distributions &b.nu;[sup &epsiv;], &epsiv; ∈ (0, &epsiv;o], whose weak limiting distributions as &epsiv; ↓ 0 are all flowinvariant for its ODE. Then the main part of the paper deals with a kind of converse: what are the possible limiting distributions among all flow-invariant distributions of the ODE? We first show that no repulsive invariant (thin) set can belong to their supports. When the ODE is a stochastic pseudogradient descent, these supports cannot contain saddle or spurious equilibrium points either, so that they are eventually supported by the set of local minima of their potential. Such results require only the random perturbation to lie in L[sup 2]. Various examples are treated, showing that these results yield some weak convergence results for the &b.nu; [sup &epsiv;]'s sometimes toward a saddle point when the algorithm is not a pseudogradient. [ABSTRACT FROM AUTHOR]

This paper is concerned with the relative sensitivity of the individual stationary probabilities of an irreducible Markov chain to small relative blockwise perturbations of the transition matrix. It is shown that this sensitivity depends on some quantities related to the stochastic complements of diagonal blocks of the transition matrix. For nearly uncoupled and nearly transient Markov chains, the stationary probabilities are sensitive to general perturbations in the relative sense. However, they are insensitive to small blockwise relative perturbations.

The article offers information on the publication of the monograph by Andrey Nikolaevich Kolmogorov, one of the developers of probability theory. It states that the monograph contains new concepts and notions, which are universally used in probability theory, conditional probabilities as random variables, and probabilities on infinite-dimensional spaces. It also mentions systems of axiomatic construction of probability theory and the construction of general and purely mathematical theory.