tailieunhanh - Petri nets applications Part 15

Tham khảo tài liệu 'petri nets applications part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Assessing Risks in Critical Systems using Petri Nets 551 impulsive function where S t - t 0 to V t T e S t-T to t T where J . dx 1 the area -w of which is given by rign at firing of g. Racum t J Rinst p drp is an accumulated reward variable at t. 0 Raver t .Racum t is an average reward variable at t. The variable Rinst t has no stochastic behavior due to the presence of impulses S German 2000 . However variables Racum t and Raver t have stochastic behavior and can be characterized by their probability distributions. As mentioned by Muppala et al. 1994 the SRN output measures are expressed in terms of expectation E of the reward variables defined as the amount of interest in the analysis. Therefore considering E Racum t as the expectation of accumulated reward variable Racum t German 2000 13 14 t t E Racum t X rrn .J n x dx X X ri g .J Pn x dx neS 0 geT neS 0 Where nn t P N t n neS is the probability of transient state which represents the probability of the net being in the marking n at time t given that E 1 v t n nn t and t Ịpg x dx E number of fires of geT at marking neS and during the period 0 a t . 0 Considering E Raver t as the expectation of the average reward variable Raver t E Rm. t E 1 Rcum t - .E Ip. t For the instantaneous reward variable Rinst t it is possible to calculate its expectation if Rinst t does not contain impulses S . Therefore obtaining E Rinst t through Racum t we have E R. t E R t rr 7T t rig 09g t 15 13 p Snst t 1. 13 acum t X llnJin t X X -Tn t dt neS geTị neS So as to simplify the equating the definition of the rign impulsive reward vector can be supressed from a net under consideration. Thus making rign 0 VgeT V neS we have t Rinst t X rrn . W t n E Racum t X rrn J x x dx 16 neS neS 0 And the other variables are calculated as a consequence of the definition above. The analysis of a net using reward equations requires the calculation of its probability of state rnn t P N t n neS. Redefining it as the vector n t rci t n2 t . nm t S 1 2 .