1 A class of Generalized Stochastic Petri Nets for the performance Evaluation of Mulitprocessor Systems By M. Almone, G. Conte Presented by Yinglei Song

2 Outline Background Background  Modeling concurrent systems with Petri Nets  Stochastic Petri Nets (SPN)  Markov Chains (MC)  Generalized Stochastic Petri Nets (GSPN) The steady state distribution in GSPN The steady state distribution in GSPN Computing the steady state distribution more efficiently. Computing the steady state distribution more efficiently. Examples. Examples. Numerical results. Numerical results.

3 Petri Nets A model that consists of A model that consists of  P, a set of places  T, a set of transitions  A, a set of directed arcs  M, a vector that stands for the number of tokens in each place. (referred to as a marking). The reachability set of a marking. The reachability set of a marking. k-bounded Petri Nets. k-bounded Petri Nets.

4 An Example A Petri Net for modeling bisexual population A Petri Net for modeling bisexual population

5 Stochastic Petri Nets The modeling ability of a PN is limited The modeling ability of a PN is limited  transition occurs with different probabilities in real systems.  New parameter sets are needed for modeling different transition rates or probabilities.  A new parameter set R is thus added to the definition of Petri Nets.  A Stochastic Petri Net (SPN) is defined as a five-tuple (P, T, A, M, R).

6 A Markov Chain A Markov Model (MM) is comprised of A Markov Model (MM) is comprised of A Markov Chain (MC) is a sequence states generated following transitions in an MM. A Markov Chain (MC) is a sequence states generated following transitions in an MM.  S: a set of states  T: a set of transitions  P: a set of probabilities associated with each transition

7 SPN and MC It can be proved that SPN is equivalent to a MC It can be proved that SPN is equivalent to a MC The set of states in MC is equivalent to the set of all possible markings in the corresponding SPN The set of states in MC is equivalent to the set of all possible markings in the corresponding SPN The transition probabilities in the MC can be computed with transition rates in the corresponding SPN The transition probabilities in the MC can be computed with transition rates in the corresponding SPN The transition probability matrix can thus be determined from the transition rates in SPN The transition probability matrix can thus be determined from the transition rates in SPN

8 SPN and MC The sojourn time in each marking is an exponentially distributed random variable with average: The sojourn time in each marking is an exponentially distributed random variable with average:

9 SPN and MC The transition probabilities in the corresponding MC is determined by: The transition probabilities in the corresponding MC is determined by:

10 The transition matrix of MC The transition matrix of a MC is defined as: The transition matrix of a MC is defined as:

11 The dynamics of MC The dynamical equation of a MC can be written as: The dynamical equation of a MC can be written as:

12 The steady state distribution of MC The steady state distribution of the MC is a fixed point of the dynamical equation: The steady state distribution of the MC is a fixed point of the dynamical equation:

13 Generalized Stochastic Petri Nets Neither PN nor SPN is able to perfectly model all the real systems. Neither PN nor SPN is able to perfectly model all the real systems. Transition rates in real systems may span a wide range including a few orders of magnitude. Transition rates in real systems may span a wide range including a few orders of magnitude. GSPN is a model that allows both timed transitions and immediate transitions. GSPN is a model that allows both timed transitions and immediate transitions. GSPN is able to model real systems with an appropriate granularity of time. GSPN is able to model real systems with an appropriate granularity of time.

14 An example of GSPN

15 Switching probabilities of the GSPN

16 The reachability set of the GSPN

17 Timed and intermediate transitions may be correlated

18 Time vs. State in GSPN

19 Outline Background Background  Modeling concurrent systems with Petri Nets  Stochastic Petri Nets (SPN)  Markov Chains (MC)  Generalized Stochastic Petri Nets (GSPN) The steady state distribution in GSPN The steady state distribution in GSPN Computing the steady state distribution more efficiently. Computing the steady state distribution more efficiently. Examples. Examples. Numerical results. Numerical results.

20 GSPN steady state distribution Two types of markings (states) exist in a GSPN: Two types of markings (states) exist in a GSPN:  Tangible states are markings that are associated with only timed transitions.  Vanishing states are markings that are associated with at least on immediate transition.

21 Assumptions The reachability set of GSPN is finite. The reachability set of GSPN is finite. Transition rates remain constant and do not evolve with time. Transition rates remain constant and do not evolve with time. The initial marking is reachable with a nonzero probability from any marking in the reachability set. The initial marking is reachable with a nonzero probability from any marking in the reachability set. No marking exists that “absorbs” the process. No marking exists that “absorbs” the process.

22 Notations Following notations are used to derive the steady state distribution: Following notations are used to derive the steady state distribution: S: the set of states in the SPP. S: the set of states in the SPP. T: the set of tangible states in S. T: the set of tangible states in S. V: the set of vanishing states in S. V: the set of vanishing states in S.

23 The steady state distribution The steady state distribution must satisfy: The steady state distribution must satisfy:

24 Outline Background Background  Modeling concurrent systems with Petri Nets  Stochastic Petri Nets (SPN)  Markov Chains (MC)  Generalized Stochastic Petri Nets (GSPN) The steady state distribution in GSPN The steady state distribution in GSPN Computing the steady state distribution more efficiently. Computing the steady state distribution more efficiently. Examples. Examples. Numerical results. Numerical results.

25 Efficient computation of steady state distribution The inverse of the transition matrix needs to be computed in time The inverse of the transition matrix needs to be computed in time The dimensionality of the transition matrix can become very big. The dimensionality of the transition matrix can become very big. The computation of the inverse of the transition matrix can become very inefficient. The computation of the inverse of the transition matrix can become very inefficient. More efficient approaches are needed for computing the steady state distribution. More efficient approaches are needed for computing the steady state distribution.

26 The approach The dimensionality of the transition matrix can be reduced by observing the figure: The dimensionality of the transition matrix can be reduced by observing the figure: t1 ir j

27 The effective transition matrix If we only consider the tangible states, the transition matrix can be computed with: If we only consider the tangible states, the transition matrix can be computed with:

28 Outline Background Background  Modeling concurrent systems with Petri Nets  Stochastic Petri Nets (SPN)  Markov Chains (MC)  Generalized Stochastic Petri Nets (GSPN) The steady state distribution in GSPN The steady state distribution in GSPN Computing the steady state distribution more efficiently. Computing the steady state distribution more efficiently. Examples. Examples. Numerical results. Numerical results.

29 An example

30 A GSPN for the system

31 A simplified model

32 Another simplified model

33 A third simplified example

34 Interesting questions Can we further simplify the GSPN used such that all resources can be abstracted as tokens? Can we further simplify the GSPN used such that all resources can be abstracted as tokens? If the answer is “no”, what actually determines that, the topology of the system? If the answer is “no”, what actually determines that, the topology of the system? Is a mathematical proof possible? Is a mathematical proof possible?

35 Outline Background Background  Modeling concurrent systems with Petri Nets  Stochastic Petri Nets (SPN)  Markov Chains (MC)  Generalized Stochastic Petri Nets (GSPN) The steady state distribution in GSPN The steady state distribution in GSPN Computing the steady state distribution more efficiently. Computing the steady state distribution more efficiently. Examples. Examples. Numerical results. Numerical results.

36 Numerical results The upper bound (M is infinitely large) The upper bound (M is infinitely large) The lower bound (M is equal to b) The lower bound (M is equal to b) To understand the dependence of the throughput on M, further investigation is needed. To understand the dependence of the throughput on M, further investigation is needed. GSPN provides a convenient way for this purpose. GSPN provides a convenient way for this purpose.

37 Results

38 Conclusion Extended from SPN and PN, the GSPN model can provide a finer description of the real system. Extended from SPN and PN, the GSPN model can provide a finer description of the real system. The GSPN is mathematically equivalent to a MC. The GSPN is mathematically equivalent to a MC. The steady state distribution of GSPN can be efficiently computed. The steady state distribution of GSPN can be efficiently computed. Real system can be analyzed to deeper level if GSPN is adopted. Exact solutions can be obtained for some complicated situations. Real system can be analyzed to deeper level if GSPN is adopted. Exact solutions can be obtained for some complicated situations.