Homothecy, bifurcations, continuity and monotonicity in timed continuous Petri nets under infinite server semantics

This work studies how equilibrium markings and throughputs change in Timed Continuous Petri Net (TCPN) systems as transition firing rates vary. In particular, it analyzes the bifurcations of the former, and the discontinuities and non-monotonicities of the latter; specifically, using structural objects of the net, such as P-semiflows, T-semiflows, and configurations, among others, the following properties can be obtained. For Join Free TCPN systems, a sufficient structural condition guaranteeing that the equilibrium markings do not bifurcate when firing rates vary, is derived. A dual result is obtained for Choice Free TCPN systems. For Mono-T-Semiflow TCPN systems, the equilibrium throughput is investigated; using a time-scale (a homothetic) property it is proven that a discontinuity of the equilibrium throughput implies its non-monotonicity, even if not evident at first glance. This is a connection of two timed behavioral properties of the equilibrium throughput. Moreover, a sufficient structural condition, parametrized by the equilibrium markings, ensures its continuity under firing rate variations. It is also proven that the monotonicity of the equilibrium throughput can be characterized by the previous structural condition. The convergence of the marking evolution of TCPN systems to its equilibrium markings is also discussed.