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Calculating the precondition of a particular partial-ordered set of events is often necessary in software testing, such as for generating test cases. Things become even more complicated when the execution time is added to the picture. If the execution time of two processes along a partial order does not match each other, the precondition of the partial order is false and then the partial order is identified as time unbalanced partial order. We present its formal definition and an algorithm to distinguish it. Then we suggest a method to fill the gap of the execution time of participating processes. This method can also be adopted to simplify the calculation of the minimal and maximal bounds of a time parameter. [ABSTRACT FROM AUTHOR]

When testing a system, it is often necessary to execute a suspicious trace in a realistic environment. Due to nondeterministic choices existing in concurrent systems, such a particular trace may not be scheduled for execution. Thus it is useful to compute the probability of executing the trace. Our probabilistic model of real-time systems requires that for each transition, the period from the time when its enabling condition becomes satisfied to the time when it is fired is bounded and the length of the period obeys a probabilistic distribution. This model is not Markovian if the distribution is not exponential. Therefore it cannot be analyzed by Markov processes. We propose to use integration to calculate the probability for a path. Then we discuss the possibility to optimize the calculation. [ABSTRACT FROM AUTHOR]