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07. Using a GSPN for Performance Evaluation

07. Using a GSPN for Performance Evaluation - CS4 Modelling...

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CS4 Modelling and Simulation LN-7 7 Using a GSPN for Performance Evaluation In this note we will consider two aspects of using a GSPN model of a system once it has been constructed: generating and solving a corresponding Markov process, and deriving performance measures. At the end of the note we will summarise the assumptions we are making when we use GSPN models. 7.1 Generating and solving the corresponding Markov process As stated in Section 5.2 generating the Markov process underlying an SPN model is very straightforward. We take advantage of the isomorphism between the reachability graph of the SPN and the state transition diagram of the Markov process. If the markings of the SPN are { M 0 , M 1 , . . . , M N } (where M 0 is the initial marking), then the states of the Markov process will be { x 0 , x 1 , . . . , x N } generated as follows: we associate a state, x i , in the Markov process with every marking, M i , in the reachability graph of the SPN; the transition rate from state x i (corresponding to marking M i ) to state x j ( M j ), is obtained as the sum of the firing rates of the transitions that are enabled in M i and whose firings generate marking M j . If we consider the very simple SPN model shown below, M 0 = (1 , 0) and M 1 = (0 , 1), are the only possible markings. Suppose that the firing rates of transitions T 1 , T 2 and T 3 are λ 1 , λ 2 and λ 3 respectively. T 1 T 2 T 3 P 1 P 2 λ 2 λ 1 + λ 3 x 0 x 1 Q = λ 2 λ 2 λ 1 + λ 2 ( λ 1 + λ 2 ) We associate states x 0 and x 1 with the markings M 0 and M 1 . The transition rate from x 0 to x 1 is λ 2 because T 2 is the only transition enabled in M 0 whose firing results in M 1 . The transition rate from x 1 to x 0 is λ 1 + λ 3 since both T 1 and T 3 are enabled in M 1 , and the firing of either of them will result in the marking M 0 . Once the Markov process corresponding to an SPN model has been generated, it is solved in exactly the same way as Markov processes which are constructed directly during modelling. The steady state probability distribution is found by solving the global balance equations, together with the normalisation condition. Two additional features were added to SPN notation to give GSPN notation: inhibitor arcs and immediate transitions. The effect of inhibitor arcs in a GSPN model is to alter the reachability graph of model: some markings and transition firings which would have been possible in the absence of the inhibitor arcs may no longer be possible. However, the 46
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CS4 Modelling and Simulation LN-7 effect of this new type of arcs only impacts on the generation of the reachability graph, not on the subsequent generation of the underlying Markov process. Once the reachability graph has been constructed obeying the constraints of the inhibitor arcs, they may be forgotten as far as generating the Markov process is concerned. Unfortunately, the same is not true for immediate transitions.
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  • Spring '10
  • MohammadAbdolahiAzgomiPh.D
  • Markov process, Markov chain, Continuous-time Markov process, Arc, M4 motorway, reachability graph

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