Before revision:  We use this representation to develop a modified version of the Next Reaction Method . We then demonstrate how this modified Next Reaction Method is the appropriate method to use to simulate systems in which the rate constants are functions of time (that is, systems that are time dependent Markov Processes). Finally, we extend our modified Next Reaction Method to systems with delays and compare its efficiency with those of the recent algorithms designed for systems with delays.
After revision:  We use this representation to develop a modified version of the Next Reaction Method and, in a way that achieves efficiency over existing approaches for exact simulation, extend it to systems with delays and compare its efficiency with those of the recent algorithms designed for systems with delays.