Time multidimensional Markov Renewal chains -- An algebraic approach

In this study, a new extension of the Markov Renewal theory is introduced by allowing time to evolve in multiple dimensions. The resulting chains are referred to as multi-time Markov Renewal chains and since this extension is new, the state space is assumed to be finite to cover the theoretical framework of applications, where the possible number of states of a physical system is finite. The flexibility of Markov renewal theory is still present in multiple time dimensions by allowing the sojourn times in the different states of the system to be arbitrarily selected from a multidimensional distribution. The convolution product of multidimensional matrix sequences plays a particular role in the development of the theory and some of its algebraic properties are given and explored, paying particular attention to the existence, the representation and the computation of the convolutional inverse. Some basic definitions and properties of this new class are given as well as the multi-time Markov renewal equations associated to the resulting processes. The practical implementation is achieved very efficiently through a novel adaptation of the Gauss-Jordan algorithm for multidimensional matrix sequences.