By by Sorin Manolache.
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Additional info for Analysis and optimisation of real-time systems with stochastic behaviour
ANALYSIS ALGORITHM 39 cessor states of a state si consists of those states that can directly be reached from state si . Let Zi denote the time when state si is entered. State si can be reached from any of its predecessor states sj ∈ Pi . Therefore, the probability P(Zi ≤ t) that state si is entered before time t is a weighted sum over j of probabilities that the transitions sj → si , sj ∈ Pi , occur before time t. The weights are equal to the probability P(sj ) that the system is in state sj prior to the transition.
The system takes the transition s2 → s6 when the attempted completion time of τ2 (running in s2 ) exceeds 5. The completion time is the sum of the starting time of τ2 (whose probability density is given by z2 ) and the execution time of τ2 (whose probability density is given by 2 ). Hence, the probability density of the completion time of τ2 is given by the convolution z2 ∗ 2 of the above mentioned densities. Once this density is computed, the probability of the completion time being larger than 5 is easily computed by integrating the result of the convolution over the interval (5, ∞).
In our example, s8 , s9 , s14 , and s19 are back states of order 0, while s20 , s25 and s30 are back states of order 1. Obviously, there cannot be any transition from a state belonging to a hyperperiod H to a state belonging to a lower hyperperiod than H (s → s , s ∈ Hk , s ∈ Hk ⇒ Hk ≤ Hk ). Consequently, the set S of all states belonging to hyperperiods greater or equal to Hk can be constructed from the back states of an order smaller than k. We say that S is generated by the aforementioned back states.
Analysis and optimisation of real-time systems with stochastic behaviour by by Sorin Manolache.