Methodology and Computing in Applied Probability最新文献

In this paper we obtain some “general” two-sided bounds for the delayed renewal function, in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times. Also, we give a sequence of monotone non-decreasing (non-increasing) lower (upper) general bounds converging to the delayed renewal function. By considering several aging or reliability classes for the distribution of the interarrival times (e.g., (DFR), bounded mean residual lifetime, (NBUE), (NWUE), bounded failure rate, (DMRL), (IMRL)) we give upper and lower bounds for the delayed renewal function, and moreover by assuming the usual stochastic order between the first and the subsequent interarrival times, we give sequences of monotone non-decreasing (non-increasing) lower (upper) bounds converging to the delayed renewal function. Also, some sequences of bounds for the delayed renewal function in terms of the ordinary renewal function are given. Sequences of monotone non-decreasing (non-increasing) lower (upper) bounds for the delayed renewal density are also given. Finally, we obtain upper and lower bounds for the expected number of renewals over a finite interval, and as a result, we get an improvement of the upper bounds obtained by Lorden (Ann Math Statist 41:520–527, 1970) and Losidis and Politis (2022) for the expected number of renewals over a finite interval under the ordinary renewal process.

This paper concerns the dynamics of a stochastic regime-switching four-species food chain model with distributed delays, Lévy jumps and harvesting in a polluted environment. First, under some basic assumptions, criterions about stochastic persistence and extinction of each species are established. Then, sufficient and necessary conditions for the existence of optimal harvesting policy (OHP) are obtained, as well as the expressions for the optimal harvesting effort (OHE) and the maximum of expectation of sustainable yield (MESY). The effects of both environmental noises and delays on the stochastic persistence, extinction and OHP of the model are revealed. Some results on food chain models without telephone noises or distributed delays are extended and improved.

This paper considers some asymptotics of value-at-risk (VaR) and conditional value-at-risk (CVaR) estimates in the cases of extended negatively dependent (END) and widely orthant dependent (WOD) sequences. The Bahadur representation and strong consistency of VaR estimator are obtained, and the strong convergence rate of CVaR estimator is obtained based on END and WOD sequences. In addition, the asymptotic normality of VaR estimator is given based on END sequence. Finally, some simulations to study the numerical performance of the consistency for VaR and CVaR estimators are given in the case of WOD sample. Our results extend and improve some corresponding results.

A modified differentiated vacation queueing system with a close-down period and impatient customers is investigated in this research paper. The proposed model can be applied to study the energy management mechanisms of WiMAX and tethered high altitude platform (HAP) systems. In this system, the WiMAX mobile station (MS) switches to a close-down period for a random duration at the end of the busy period. At the end of the close-down period, if the system is non-empty, the MS switches to a functional state. Otherwise, the MS takes an ordinary vacation. The MS switches to a working vacation (WV) if the system is empty at the end of the ordinary vacation period. Jobs arriving during the WV are served at a slower service rate. Jobs may become impatient in any state of the system except during the regular busy period. Explicit expressions for the transient and steady-state probabilities of this system are obtained using continued fraction and confluent hypergeometric functions. Furthermore, performance indices such as mean, variance, average abandonment rate, and system throughput are computed. Finally, a cost-profit analysis and graphical illustrations are provided.

We offer an alternative proof, using the Stein-Chen method, of Bollobás’ theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.

We investigate the dynamics of customer geometric abandonment within a queueing framework characterized by renewal input batch arrivals and multiple vacations. Customers’ impatience becomes evident when confronted with server vacations, triggering instances of abandonment. This phenomenon reduces the number of customers within the system during abandonment epochs following a geometric distribution. The probability of customers leaving the queue escalates with prolonged waiting times. We derive concise and closed-form expressions for system-length distributions at pre-arrival and arbitrary epochs by harnessing the power of supplementary variable and difference operator methods. Furthermore, we elucidate specific instances of our model, shedding light on its versatility. To substantiate our theoretical framework, we provide a series of illustrative numerical experiments presented through meticulously crafted tables and graphs, thereby showcasing the robustness and applicability of our methodology.

The main aim of this article is to demonstrate the collocation method based on the barycentric rational interpolation function to solve nonlinear stochastic differential equations driven by fractional Brownian motion. First of all, the corresponding integral form of the nonlinear stochastic differential equations driven by fractional Brownian motion is introduced. Then, collocation points followed by the Gauss-quadrature formula and Simpson’s quadrature method are used to reduce them into a system of algebraic equations. Finally, the approximate solution is obtained using Newton’s method. The rigorous convergence and error analysis of the presented method has been discussed in detail. The proposed method has been applied to some well-known stochastic models, such as the stock model and a few other examples, to demonstrate the applicability and plausibility of the discussed method. Also, the numerical results of the collocation method based on the barycentric rational interpolation function and barycentric Lagrange interpolation function get compared with an exact solution.