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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]

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 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]

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]