Stochastic simulation and analysis of the operation of computing systems with structural redundancy

A random Bernoulli process with continuous time and a finite number of states (random events) is proposed. The process is obtained by two mutually complementary methods - directly from the Poisson process with an intensity parameter that depends on time and methods of queuing theory, from a queuing system with two parameters. In the first case, the process was formalized on a probability space with measure, as a measurable function of time. The intensity of the Poisson process was considered as a measure. The Bernoulli process for each fixed time was obtained as a conditional distribution from a suitable Poisson distribution. The parameter of the Poisson distribution was determined from the differential equation, in the formulation of which the approximation of the Bernoulli formula by the Poisson formula was essentially used. In the second method, standard methods of queuing theory were used. A two- parameter queuing model was formulated in which for all customer flows the time between occurrence of neighboring customers was a random value satisfying the exponential law. The model was formalized by a system of differential equations, whose analytical solution represented the continuous-time Bernoulli process. In finding solutions, the method of generating functions was used. It is of interest to derive the Bernoulli process both from the probability space constructed for the Poisson process and from the queuing theory model. The authors believe that the proposed process can be generalized to a wider class of functions than that used in the work, down to measurable ones. The possibilities for the practical application of the continuous-time Bernoulli process will undoubtedly be expanded, since its discrete analog is well known in many fields of science and technology.

Section: Markov Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Continuous-Time Time Bernulli Process

Pavsky¹,

Pavsky²,

Иванова³

2017

Solution of Stochastic Differential Equations of the Models of Queueing Theory by the Method of Generationg Function

Иванова¹,

Pavskii²,

Pavsky

2016

The object is mathematical models and their formalization by differential equation systems. The aim is to popularize stochastic models and differential equation systems which solution allows an analytical form. A model formulation and a process of finding a solution to equation systems are of interest. In the queueing theory many models are formalized by systems of linear differential equations with one or more parameters in which distribution of states of queueing systems are unknown functions. In such systems Markov processes are often grounding in the theory of differential equations construction; in a special case postulates of Poisson process are used. Analytical solution of equation systems exists but it is hard to find by traditional methods. In our study we offer a method which allows to find an analytical solution not only for probability distribution but also for moments of any order from one equation system. Description of procedure of differential equation generation for moments of random order varieties is presented. The method is based on the usage of generating (characteristic) functions. This method is effective because it allows to find solutions for moments (here it is expectation and variance) without complex probability calculations. It is especially important in empirical researches of systems that consist of many elements. For example, when we analyze function effectiveness of operating and designed scaling computing systems and supercomputers. Three models and their formalization by differential equation systems that correspond to stochastic processes and analytical solutions of diverse complexity are formulated. Connection between stochastic differential equations systems and their solutions with probability distributions that are classical in probability theory is shown.

Probabilistic Space for Calculation of Probabilistic Characteristics of a Three-Parameter Queueing System Model

Pavsky¹,

Pavsky²,

Иванова³

et al. 2017

A model of queueing theory is proposed that describes a queueing system with three parameters, which has important practical applications. The model is based on the continuous time Markov process with a discrete number of states. The model is formalized by a probabilistic space in which the space of elementary events is a set of inconsistent states of the queueing system; and the probabilistic measure is a probability distribution corresponding to a set of elementary events, that is, each elementary event is associated with the probability of the system staying in this state, for each fixed time moment. The model is represented by a system of ordinary differential equations, compiled by methods of queueing theory (Kolmogorov equations). To find the solution of the system of equations, the method of generating functions is used. For the generating function, a partial differential equation is obtained. Finding the generating function completes the construction of a probability space. The latter means that for any random variables and functions defined on the resulting probability space, one can find their probabilistic characteristics. In particular, analytical expressions of the moments (mathematical expectations and variances) of random functions that depend on time are obtained. The peculiarity of finding a solution is that it is obtained not from the probability distribution, but directly from the partial differential equation, which represents a system of ordinary differential equations. For the probability distribution, the solution was found by a combinatorial method, which made it possible to significantly reduce the computations. To apply the formulas in engineering calculations, we consider the stationary case, to which a considerable simplification of the calculations corresponds. A relationship between a system of differential equations and a polynomial distribution known in probability theory is shown. The results are used in the analysis of the reliability of the operation of scalable computing systems; graphical implementation is shown