Analysis of computational systems: Discrete Markov analysis of computer programs

A PROGRAM with a number of subroutines can be represented by a flow diagram; Figure 1. The nodes represent the subroutines and the directed branches indicate the allowed transitions between them. Given a program consisting of n subroutines R<subscrpt>1</subscrpt>, R<subscrpt>2</subscrpt> .....R<subscrpt>n</subscrpt>, two matrices are also assumed to be known, viz., an n × 1 matrix of execution times of each subroutine and a n × n matrix <underline>P</underline>, such that its ij-th element P<subscrpt>ij</subscrpt> is the branching probability that the program will branch to subroutine j from subroutine i. We shall assume P<subscrpt>ij's</subscrpt> are statistically independent so that the model of the computer program is that of a discrete Markov process. The expected time to complete a program is then a summation of all possible statistically weighted paths that begin at an initial or starting subroutine and end at the terminal subroutine.