Asymptotic analysis of closed queueing network with time-dependent service parameters and priority applications
The object of the research is a closed queuing network with the single-type applications of two classes such as priority and non-priority. It is assumed that the service parameters in each of the network systems, as well as the transition probability of applications between systems, depend on time. The route of transportation of applications of each class is defined by an arbitrary stochastic matrix of transition probabilities. Queuing time each of line systems distributed exponentially. Applications are selected for the service in accordance with the discipline FIFO. The random vector that defines the state of the network at any given time and forming a Markov random process was learned. The goal of this study is the asymptotic analysis of the Markov process describing the state of the queuing network, in case of large number of applications, and the determination of the mean relative number of applications in network systems at any given time. A system of difference-differential Kolmogorov equations for the state probabilities of Markov process with continuous time and finite number of states was obtained in the main part of the paper. The case of a large number of serviced applications is considered, transition to a vector having a continuous distribution was done. A differential equation in partial derivatives of the second order, which is the Fokker-Planck-Kolmogorov probability density function of the investigated process, was derived. A system of linear ordinary differential equations of first order for the average values of the components of the network status was received. The solution to this system allows predicting the mean relative number of applications in each of queuing systems in the interested time instance. Software procedure with calculation examples was developed in computer mathematical system of Maple. One example of a calculation of the average number of applications in network systems was proposed. The obtained results can be used in the mathematical modeling of various economic, technical and other systems using closed queuing networks of defined structure. Described network can be used as a mathematical model of the customer requests processing in insurance or logistics companies.