System In Random Environment Research Articles

Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phaseiwith probabilityqi,i= 1, 2, …,d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

Dynamics of multivariate default system in random environment

We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting of enlargement of filtrations and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. Finally, we present a martingale characterization result with respect to the observable information filtration on the market.

Modeling and Analysis of Switching Diffusion Systems: Past-Dependent Switching with a Countable State Space

Motivated by networked systems in random environment and controlled hybrid stochastic dynamic systems, this work focuses on modeling and analysis of a class of switching diffusions consisting of continuous and discrete components. Novel features of the models include the discrete component taking values in a countably infinite set, and the switching depending on the value of the continuous component involving past history. In this work, the existence and uniqueness of solutions of the associated stochastic differential equations are obtained. In addition, Markov and Feller properties of a function-valued stochastic process associated with the hybrid diffusion are also proved. In particular, when the switching rates depend only on the current state, strong Feller properties are obtained. These properties will pave a way for future study of control design and optimization of such dynamic systems.

Reliability of dynamic systems in random environment by extreme value theory

A practical method is developed for estimating the performance of highly reliable dynamic systems in random environment. The method uses concepts of univariate extreme value theory and a relatively small set of simulated samples of system states. Generalized extreme value distributions are fitted to state observations and used to extrapolate Monte Carlo estimates of reliability and failure probability beyond data. There is no need to postulate functional forms of extreme value distributions since they are selected by the estimation procedure. Our approach can be viewed as an alternative implementation of the method in [7] for estimating system reliability. Numerical examples involving Gaussian and non-Gaussian system states are used to illustrate the implementation of the proposed method and assess its accuracy.

Euler hydrodynamics for attractive particle systems in random environment

Nous obtenons la limite hydrodynamique trempée, sous un changement d’échelle hyperbolique, pour un système de particules attractif sur $\mathbb{Z}$ en milieu aléatoire ergodique, avec un nombre borné de particules par site. Notre résultat est une loi forte des grands nombres. Nous l’illustrons sur différents exemples.

Stochastic comparisons for the number of working components of a system in random environment

In terms of stochastic orders, the purpose of this paper is to show how the random environment can affect the number of working components of a system with heterogeneous components sharing a common random environment. Applications to a class of semiparametric mixture models, stress-strength model and warm standby system are presented.

The stochastisation hypothesis and the spacing of planetary systems

We introduce the stochastisation hypothesis which aims to provide a framework to deal with physical systems in random environment. We apply the stochastisation hypothesis in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al. [“A stochastic model for the orbits of planets and satellites: an interpretation of Titius-Bode law,” Expo. Math. 4, 363–373 (1983)], Nottale [“The quantization of the solar system,” Astron. Astrophys. 315, L9 (1996)], and Laskar [“On the spacing of planetary systems,” Phys. Rev. Lett. 84(15), 3240–3243 (2000)10.1103/PhysRevLett.84.3240]. These results give both a particular law for the distribution of planetary orbits.

Stationary Distributions of Discrete-Time Markov Chains in Random Environment: Exact Computations and Bounds

A general model for the description of a discrete-time Markov chain in random environment is introduced. We study the stationary distribution of this model and its relationship with the Palm (embedded) distributions at certain environmental transition epochs. We also study the Events See Time Averages (ESTA) property for this model and we show that it is related to certain partial balance equations and a product-form stationary distribution. Next, we study certain forms of interaction between the environmental process and the process of interest that lead to tractable stationary distributions. More specifically, the stationary distributions of a large class of Markov chains in random environment can be computed efficiently using α-potentials of smaller Markov chains. This decomposability result is also the basis for developing approximations for the stationary distributions of intractable models that evolve in a slowly changing random environment. Moreover, explicit error bounds for these approximations are given. The model can be used for the performance evaluation of discrete time queueing systems in random environment.

Comparing single-server loss systems in random environment

Considers a single-server loss system in a random environment. The environment is determined by a finite-state Markov process. When the environment is in state i, the arrival process is Poisson with rate /spl lambda//sub i/, and the service time is exponential with rate /spl mu//sub i/=1,/spl middot//spl middot//spl middot/m. We show that the blocking probability of this system is bounded from below (above) by that of the same system with a more (less) regular arrival and service pattern. This result supports Ross's conjecture (1978) that the blocking probability is smaller when the arrival process is more regular and suggests its validity in scenarios with dependent arrival and service processes. Such structural properties are useful in obtaining bounds and approximations for system performance. We then fix the marginal service (arrival) process and search for an arrival (service) process with the same long-run rate that minimizes the blocking probability. It is shown that the optimal solution is the one such that the arrival and service rates are proportional. This result is in contrary with the case of independent arrival and service processes, where the system performance reaches its minimum when the arrival is replaced by Poisson, and it provides insight into the understanding of the effect of nonstationarity on system performance.

A consecutive k-out-of- n reliability system in random environment

Linear and circular consecutive k-out-of-n reliability system are assumed to be situated in a random environment such that shocks generated by the environment will cause failure with probability p to one randomly chosen component. Explicit formulas for k = 2 and 3 are derived for generating functions and expectations and variances for the system time to failure. An equivalene between linear and circular systems is derived.

Invariant probabilities for systems in a random environment — With applications to the brusselator

In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator.

Further results of replacement problem of a parallel system in random environment

We have recently discussed the replacement problem of a parallel system in a random environment. This paper extends the same replacement problem for the following two cases which are more plausible: (i) The probability that an operating component fails by the j th shock depends on the number j. (ii) The replacement cost is a linear function of failed components. The expected cost of the above model is obtained. A numerical example is finally presented when the probability of failure time of a component is a negative binomial distribution.

Further results of replacement problem of a parallel system in random environment

We have recently discussed the replacement problem of a parallel system in a random environment. This paper extends the same replacement problem for the following two cases which are more plausible: (i) The probability that an operating component fails by the j th shock depends on the number j. (ii) The replacement cost is a linear function of failed components. The expected cost of the above model is obtained. A numerical example is finally presented when the probability of failure time of a component is a negative binomial distribution.

Replacement problem of a parallel system in random environment

Råde considered a parallel system with n components in a random environment. The shocks cause the components to fail with certain probabilities. This paper considers the replacement policy for the above system in which it is exchanged before failure if the total number of failed components is more than k, and it is replaced if all of n components have failed. An optimum number of k which will minimize the expected cost is obtained. A numerical example is finally presented.

Replacement problem of a parallel system in random environment

Råde considered a parallel system with n components in a random environment. The shocks cause the components to fail with certain probabilities. This paper considers the replacement policy for the above system in which it is exchanged before failure if the total number of failed components is more than k, and it is replaced if all of n components have failed. An optimum number of k which will minimize the expected cost is obtained. A numerical example is finally presented.

Discrete Extreme Distributions

Discrete extreme distributions can be defined to correspond with extreme distributions in continuous time. The application to a n-unit parallel system in random environment is shown.

Reliability systems in random environment

A parallel reliability system is subject to shocks which are generated by a renewal point process. The shocks cause the components to fail with certain probabilities. First we study the case when the components fail independently of each other with the same probability. For this case simple closed expressions for the Laplace-Stieltjes transform and expectation of time to system failure are derived. Then a case of dependent components is studied: the probability of failure of a component depends on the number of functioning components. Some extensions are mentioned.

Reliability systems in random environment

A parallel reliability system is subject to shocks which are generated by a renewal point process. The shocks cause the components to fail with certain probabilities. First we study the case when the components fail independently of each other with the same probability. For this case simple closed expressions for the Laplace-Stieltjes transform and expectation of time to system failure are derived. Then a case of dependent components is studied: the probability of failure of a component depends on the number of functioning components. Some extensions are mentioned.