A Min-Max Problem for the Computation of the Cycle Time Lower Bound in Interval-Based Time Petri Nets

The time Petri net with firing frequency intervals (TPNF) is a modeling formalism used to specify system behavior under timing and frequency constraints. Efficient techniques exist to evaluate the performance of TPNF models based on the computation of bounds of performance metrics (e.g., transition throughput, place marking). In this paper, we propose a min-max problem to compute the cycle time of a transition under optimistic assumptions. That is, we are interested in computing the lower bound. We will demonstrate that such a problem is related to a maximization linear programming problem (LP-max) previously stated in the literature, to compute the throughput upper bound of the transition. The main advantage of the min-max problem compared to the LP-max is that, in addition to the optimal value, the optimal solutions provide useful feedback to the analyst on the system behavior (e.g., performance bottlenecks). We have implemented two solution algorithms, using CPLEX APIs, to solve the min-max problem, and have compared their performance using a benchmark of TPNF models, several of these being case studies. Finally, we have applied the min-max technique for the vulnerability analysis of a critical infrastructure, i.e., the Saudi Arabian crude-oil distribution network.