Wyniki wyszukiwania - BazTech

1

The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model

Yavuz Mehmet, Boulaasair Lahcen, Bouzahir Hassane, Diop Mamadou Abdoul, Benaid Brahim

Journal of Applied Mathematics and Computational Mechanics

|

2024

|

Vol. 23, nr 1

121-134

EN

The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results.

2

Distribution Tails for Solutions of Sde Driven By An Asymmetric Stable Lévy Process

Eon Richard, Gradinaru Mihai

Probability and Mathematical Statistics

|

2020

|

Vol. 40, Fasc. 2

317--330

EN

The behaviour of the tails of the invariant distribution for stochastic differential equations driven by an asymmetric stable Lévy process is obtained. We generalize a result by Samorodnitsky and Grigoriu [8] where the stable driving noise was supposed to be symmetric.

3

Martingale approach for tail asymptotic problems in the generalized Jackson network

Miyazawa M.

Probability and Mathematical Statistics

|

2017

|

Vol. 37, Fasc. 2

395--430

EN

We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

4

Analysis of an MAP/PH/1 queue with flexible group service

Brugno A., D’Apice C., Dudin A., Manzo R.

International Journal of Applied Mathematics and Computer Science

|

2017

|

Vol. 27, no. 1

119--131

EN

A novel customer batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at the service completion moment is less than this size, the server does not start the next service until the number of customers in the system reaches this size or a random limitation of the idle time of the server expires, whichever occurs first. Customers arrive according to a Markovian arrival process. An individual customer’s service time has a phase-type distribution. The service time of a batch is defined as the maximum of the individual service times of the customers which form the batch. The dynamics of such a system are described by a multi-dimensional Markov chain. An ergodicity condition for this Markov chain is derived, a stationary probability distribution of the states is computed, and formulas for the main performance measures of the system are provided. The Laplace–Stieltjes transform of the waiting time is obtained. Results are numerically illustrated.

5

Asymptotic behavior of ultimately contractive iterated random Lipschitz functions

Alsmeyer G., Hölker G.

Probability and Mathematical Statistics

|

2009

|

Vol. 29, Fasc. 2

321--336

EN

Let (Fn)n≥0 be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space (X; d) with Lipschit constants L1; L2; : : : For n ≥0, denote by Mx n = Fn○ : : : ○ F1(x) and ^Mx n = Fn○ : : : ○ F1(x) the associated sequences of forward and backward iterations, respectively. If E log+ L1 < 0 (mean contraction) and E log+ d ( F1(x0); x0) is finite for some x0ЄX, then it is known (see [9]) that, for each x Є X, the Markov chain Mx n converges weakly to its unique stationary distribution π, while ^M xn is a.s. convergent to a random variable ^M∞ which does not depend on x and has distribution π. In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for ^M x n, which then also lead to corresponding assertions for Mx n via Mx n d= ^M x n for all n and x, where d= means equality in law. Here our purpose is to demonstrate how these methods are extended to the more general situation where only ultimate contraction, i.e. an a.s. negative Lyapunov exponent limn→∞ n−1 log l(Fn○ : : : ○ F1) is assumed (here l(F) denotes the Lipschitz constant of F). This not only leads to an extension of the results from [2] but in fact also to improvements of the obtained convergence rate.

6

Asymptotic behavior of some random splitting schemes

Białkowski M., Wesołowski J.

Probability and Mathematical Statistics

|

2002

|

Vol. 22, Fasc. 1

181--191

EN

We consider three new schemes of random splitting of a unit interval.These schemes are related to settings considered earlier in literature. Essentially we are concerned with asymptotic behavior of sequences of subdivisions. In all three cases almost sure or weak limits are obtained for a sequence of points of divisions. The two of the schemes considered are dual to each other in the sense of the contraction principle of Chamayou and Letac [2].