Quantitative timed simulation functions and refinement metrics for real-time systems

We introduce quantatitive timed refinement metrics and quantitative timed simulation functions, incorporating zenoness checks, for timed systems. These functions assign positive real numbers between zero and infinity which quantify the timing mismatches between two timed systems, amongst non-zeno runs. We quantify timing mismatches in three ways: (1) the maximum timing mismatch that can arise, (2) the "steady-state" maximum timing mismatches, where initial transient timing mismatches are ignored; and (3) the (long-run) average timing mismatches amongst two systems. These three kinds of mismatches constitute three important types of timing differences. Our event times are the global times, measured from the start of the system execution, not just the time durations of individual steps. We present algorithms over timed automata for computing the three quantitative simulation functions to within any desired degree of accuracy. In order to compute the values of the quantitative simulation functions, we use a game theoretic formulation. We introduce two new kinds of objectives for two player games on finite state game graphs: (1) eventual debit-sum level objectives, and (2) average debit-sum level objectives. We present algorithms for computing the optimal values for these objectives for player 1, and then use these algorithms to compute the values of the quantitative timed simulation functions.