Markov Chain States Classification in a Tandem Model with a Cyclic Service Algorithm with Prolongation

Abstract

There is a limited list of papers about crossroads tandems. Usually the following service algorithms are under consideration: a cyclic algorithm with fixed duration, a cyclic algorithm with a loop a cyclic algorithm with regime changes etc. To construct a formal mathematical model of queuing systems nets and crossroads tandems in particular a descriptive approach is usually used. Using this approach input flows and service algorithms are set at the level of content, service duration distribution is known and set via a particular customer service distribution function. However with this approach one can not find nodes output flows distribution, as well as investigate customers’ noninstanteneous transfering between systems and with dependent, different service time distributions. In this paper a new approach is utilized to construct probability models of tandems for conflict queuing systems with different service algorithms in subsystems. Within this approach one can solve a problem of choosing the description for ω elementary outcomes of the stochastic experiment and mathematically correctly define the stochastic process, which describes the entire system, as well as solve the above mentioned problems. Based on a constructively given probabilistic space one can strictly justify the reachability of one state from another the other which in turn gives a full description of the entire essential state space.

Keywords

• stationary distribution
• cybernetic control system
• cyclic algorithm with prolongations
• conflict flows
• multidimensional denumerable discrete-time markov chain
• essential state