A note on the analysis of systems with time dependent transition rates

The behavior of systems in the time domain is of essential importance in many applications of System Engineering such as risk assessment, reliability, logistics and maintenance. The behavior of systems, with constant transitions rates between states, is well known. Such systems are described by the Markov equations. In most realistic systems the state transition rates are time dependent and the Markov equations are inadequate. Such systems are characterized by non exponential distributions which govern the events in the systems. These systems are non markovian i.e future events depend not only on the 'present' state of the system but also on past events. No general state equations are available for the analysis of such systems. The main purpose of this work is to suggest such equations. The concept of 'event density' of the system is introduced and shown to be of fundamental importance. In particular, the 'event density' fulfills a set of integral equations suitable for the analysis of systems with time dependent transition rates. These equations are shown to unify some aspects of system engineering currently treated as independent categories, such as reliability analysis and renewal theory. The equations may support the development of Monte-Carlo methods which are increasingly recognized to be of considerable value in approaching realistic systems. Analytic and numerical aspects of these equations are discussed in some detail.