Rainfall-runoff Analysis Method Considering the Effect of High-rise Buildings

In this study, a rainfall runoff process analysis method considering the effect of high-rise buildings was proposed. The proposed method was applied to the Yeoksam urban basin in Seoul. For rainfall-runoff analysis, a shot noise process based model was used to independently analyze the runoff from the wall and roof of a high-rise building. Thus, the Yeoksam urban basin was divided into 155 sub-basins for analysis. It was observed that the peak runoff increased by 22.0% in the 9-2 sub-basin. However, in a sub-basin in which the peak runoff increased by 10.0% or more due to high-rise buildings, there was no case where the increase rate of peak runoff was maintained greater than 5.0% until the next sub-basin outlet. Finally, by deriving the runoff hydrograph for the entire Yeoksam urban basin, it was observed that there was no significant difference in rainfall-runoff process, regardless of whether the building was considered. Therefore, it was concluded that the phenomenon of increase in peak runoff due to high-rise buildings occurs only in sub-basin units.

A study on some infinite server queues in discrete-time

This work analysis some discrete-time queueing mechanisms with infinitely many servers.By using a shot noise process, general results on the system size in discrete-time are given both in transient state and in steady state. For this we use the classical differentiation formula of F´a di Bruno. First two moments of the system size and distribution of the busy period of the system are also computed.

On missions’ quality of performance for systems with partially or completely observable degradation

At some instances, it is better to terminate operation of a system than to wait for its failure or completion. However, in this article, we are mostly interested in missions that are cost-effective during the whole mission time and, therefore, do not require termination. Moreover, some requirements for parameters of the considered models that guarantee this cost-effectiveness are analyzed. We consider two failure models for systems executing missions of the fixed duration. In the first model, degradation is partially observed via the number of shocks experienced by a system. Shocks act directly on the failure rate forming the shot-noise process. In the second model, degradation is completely observed and is modeled by the Poisson process. Thus, the number of shocks or the number of failed components is the degradation parameters in our models, respectively. The detailed numerical examples illustrate our findings. Specifically, the bounds for the number of events (shocks or component’s failures) observed at each instant of time that guarantee cost-effectiveness of a mission are obtained.

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Weak convergence of random processes with immigration at random times

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

A functional limit theorem for general shot noise processes

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

A hybrid preventive maintenance model for systems with partially observable degradation

A new model of hybrid preventive maintenance of systems with partially observable degradation is developed. This model combines condition-based maintenance with age replacement maintenance in the proposed, specific way. A system, subject to a shock process, is replaced on failure or at some time ${T}_S$ if the number of shocks experienced by this time is greater than or equal to m or at time $T>{T}_S$ otherwise, whichever occurs first. Each shock increases the failure rate of the system at the random time of its occurrence, thus forming a corresponding shot-noise process. The real deterioration of the system is partially observed via observation of the shock process at time ${T}_S$. The corresponding optimization problem is solved and a detailed numerical example demonstrates that the long-run cost rate for the proposed optimal hybrid strategy is smaller than that for the standard optimal age replacement policy.

Estimation of probability of exceeding a curve by a strictly φ-sub-Gaussian quasi shot noise process

In this paper, we continue to study the properties of a separable strictly φ-sub-Gaussian quasi shot noise process $X(t) = \int_{-\infty}^{+\infty} g(t,u) d\xi(u), t\in\R$, generated by the response function g and the strictly φ-sub-Gaussian process ξ = (ξ(t), t ∈ R) with uncorrelated increments, such that E(ξ(t)−ξ(s))^2 = t−s, t>s ∈ R. We consider the problem of estimating the probability of exceeding some level by such a process on the interval [a;b], a,b ∈ R. The level is given by a continuous function f = {f(t), t ∈ [a;b]}, which satisfies some given conditions. In order to solve this problem, we apply the theorems obtained for random processes from a class V (φ, ψ), which generalizes the class of φ-sub-Gaussian processes. As a result, several estimates for probability of exceeding the curve f by sample pathes of a separable strictly φ-sub-Gaussian quasi shot noise process are obtained. Such estimates can be used in the study of shot noise processes that arise in the problems of financial mathematics, telecommunication networks theory, and other applications.

Lifetime of a structure evolving by cluster aggregation and particle loss, and application to postsynaptic scaffold domains

The dynamics of several mesoscopic biological structures depend on the interplay of growth through the incorporation of components of different sizes laterally diffusing along the cell membrane, and loss by component turnover. In particular, a model of such an out-of-equilibrium dynamics has recently been proposed for postsynaptic scaffold domains which are key structures of neuronal synapses. It is of interest to estimate the lifetime of these mesoscopic structures, especially in the context of synapses where this time is related to memory retention. The lifetime of a structure can be very long as compared to the turnover time of its components and it can be difficult to estimate it by direct numerical simulations. Here, in the context of the model proposed for postsynaptic scaffold domains, we approximate the aggregation-turnover dynamics by a shot-noise process. This enables us to analytically compute the quasi-stationary distribution describing the sizes of the surviving structures as well as their characteristic lifetime. We show that our analytical estimate agrees with numerical simulations of a full spatial model, in a regime of parameters where a direct assessment is computationally feasible. We then use our approach to estimate the lifetime of mesoscopic structures in parameter regimes where computer simulations would be prohibitively long. For gephyrin, the scaffolding protein specific to inhibitory synapses, we estimate a lifetime longer than several months for a scaffold domain when the single gephyrin protein turnover time is about half an hour, as experimentally measured. While our focus is on postsynaptic domains, our formalism and techniques should be applicable to other biological structures that are also formed by a balance of condensation and turnover.