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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]