Abstract
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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
