A Kind of Phase Enlargement of Semi-Markov Systems on the Example of Modeling a Synchronous Automated Line

Abstract

The paper deals with the phase enlargement of semi-Markov systems that does not require the definition of the stationary distribution of the embedded Markov chain. Phase enargement is the equivalent replacement of a semi-Markov system with a common phase space of states by a system with a discrete state space. The determination of the stationary distribution of the embedded Markov chain for systems with a continuous phase space of states is a complex and not always solvable problem, since in a number of cases it leads to the solution of integral equations with kernels containing the sum and difference of variables. For such equations only a particular solution is known, but there are no general solutions for the present day. For this purpose, we prove a lemma on the form of the distribution function of the difference of two random variables, provided that the first is greater than the subtrahend. It is shown that the form of the distribution function of the difference RV under this condition depends on one constant, which is determined by the numerical method for solving the equation given in the lemma. The use of this method is demonstrated using the example of simulation of an automated synchronous line with a free cycle of operation. Automated synchronous lines with a free operation cycle are increasingly used in machine-tool construction as having a significant advantage in comparison with synchronous lines operating in a rigid cycle, as well as with automated lines with a reflex control that ensure the transfer of products in the presence of polling of end-of-service sensors installed at each position.