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We introduce quantatitive timed refinement and timed simulation (directed)metrics, incorporating zenoness check s, for timed systems. These metricsassign positive real numbers between zero and infinity which quantify the\emph{timing mismatches} between two timed systems, amongst non-zeno runs. Wequantify timing mismatches in three ways: (1) the maximal timing mismatch thatcan arise, (2) the "steady-state" maximal timing mismatches, where initialtransient timing mismatches are ignored; and (3) the (long-run) average timingmismatches amongst two systems. These three kinds of mismatches constitutethree important types of timing differences. Our event times are the\emph{global times}, measured from the start of the system execution, not justthe time durations of individual steps. We present algorithms over timedautomata for computing the three quantitative simulation distances to withinany desired degree of accuracy. In order to compute the values of thequantitative simulation distances, we use a game theoretic formulation. Weintroduce two new kinds of objectives for two player games on finite-state gamegraphs: (1) eventual debit-sum level objectives, and (2) average debit-sumlevel objectives. We present algorithms for computing the optimal values forthese objectives in graph games, and then use these algorithms to compute thevalues of the timed simulation distances over timed automata.

Quantitative Timed Simulation Functions and Refinement Metrics for Timed Systems (Full Version)