G. Ciardo and R. Zijal. Discrete deterministic and stochastic Petri nets. In ICASE Technical Report No. 96-72, pages 1-25. NASA, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... a new attention has been devoted to discrete models since it has been observed that they can be utilized in the numerical solution of non Markovian processes, or they are more closely related to physical observations [15,16] Moreover, new emphasis has been put on discrete stochastic Petri Nets [5,6,17]. Finally, DPHs may have a wide range of applicability in stochastic models in which random times must be combined with constant durations. In fact, one of the most interesting property of the DPH distributions is that they can represent in an exact way a number of distributions with nite ....

G. Ciardo and R. Zijal. Discrete deterministic and stochastic Petri nets. In ICASE Technical Report No. 96-72, pages 1-25. NASA, 1996.

....(prs) and preemptive repeat identical (pri) respectively. These stochastic extensions have increased the descriptive power of SPNs, as well as the computational e ort required for their solution. Many SPN modeling tools have recently been proposed or developed (e.g. ESP [11] GreatSPN [6] SPNP [9], DSPNExpress [16] TimeNet [13] UltraSAN [10] Some of the above tools have also implemented the possibility of including some non Markovian features thus extending the range of applicability of PNs. Their main limitations regard the kind and number of generally distributed ring time (GEN) ....

....present in currently available SPN analysis packages. This approach is based on a discrete time approximation of the stochastic behavior of the marking process, hence it can be considered as a discrete time version of the phase type expansion technique. A similar approach can be found in [9], where Discrete Deterministic and Stochastic PNs (DDSPNs) are presented and race policies equivalent to our prd and prs policies are considered. The main di erences with our approach consist in the intrinsic assumption of a discrete time model, the lack of the pri policy and the absence of a full ....

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G. Ciardo and R. Zijal. Discrete deterministic and stochastic petri nets. Technical report, NASA, 1996.

....can be assigned to transitions of Petri nets. Usually geometric distributions or mixtures of geometric distributions are used. First approaches have been published about 15 years ago [16] but also more recent extensions of the basic class of nets with discrete time steps have been proposed [23, 24]. To distinguish continuous and discrete time SPNs, we denote the former as CTSPNs and the latter as DTSPNs. DTSPNs describe an underlying Discrete Time Markov Chain (DTMC) The major problem with this model class is that transitions re concurrently such that steps instead of interleavings have ....

.... . An external observer who can only see visible transitions labeled with some action from Act cannot notice such a step. At the level of the DTMC, transition steps can no longer be distinguished, and we observe the stochastic process as usual for discrete time models like SPNs in discrete time [16, 23, 24]. If we assume that an observer does not know when a step takes place, s)he cannot see ring of a set of internal transitions resulting in an empty multiset of transition labels. This behavior can be described by transforming the reachability graph by skipping unobservable transitions. The ....

R. Zijal, G. Ciardo, and G. Hommel. Discrete deterministic and stochastic Petri nets. In K. Irmscher, Ch. Mittasch, and K. Richter, editors, MMB'97, Aktuelle Probleme der Informatik: Band 1. VDE Verlag, 1997.

....case of time extended Petri nets with exponentially, deterministically, and expolynomially timed transitions without restrictions is referred to as SPNs, for convenience. ffl Discrete Time Approach: The temporal behavior of a discrete deterministic and stochastic Petri net (DDSPN) [14, 6] is characterized by its underlying discrete time scale. An exponentially distributed firing time is approximated by its discrete analog the geometric distribution (cf. Sec. 3.8) Deterministic firing times are represented by a special case of the geometric distribution. Immediate transitions ....

R. Zijal and G. Ciardo. Discrete Deterministic and Stochastic Petri Nets. ICASE Technical Report 96-72, Institute for Computer Applications in Science and Engineering, NASA/Langley Research Center, Hampton, VA, 1996. submitted for MMB'97.

....their values; in case of pri a g is resetted whereas fl g does not change. The algorithm for the analysis of non markovian PN we have developed is based on a time discretization approach which allows to deal with the prs, prd and pri preemption policies [11] A similar approach can be found in [7], where Discrete Deterministic and Stochastic PNs (DDSPNs) are presented and race policies equivalent to our prd and prs policies are considered. The main differences with our approach consist in the intrinsic assumption of a discrete time model, the lack of the pri policy and the absence of a ....

G. Ciardo and R. Zijal. Discrete deterministic and stochastic petri nets. Technical report, NASA, 1996.

....when the ring probability is set to 1 (e.g. Const(4) Geom(1; 4) in Fig. 1) An immediate transition ( ring in zero time) can be modeled by a DTP containing a single absorbing phase 0. A more detailed presentation of DTPs and of corresponding discrete ring time distributions can be found in [4, 14]. Consequently, a state s of a DDSPN consists of two discrete components, the marking and the vector containing the phase for each transition: s = 2 IN Each entry t of represents the current phase of the DTP associated to transition t. De nition 3.1 Formally, a DDSPN is a ....

.... t where t = t :G t ( t ; t ) 0 t g: 8 2 S; 8t 2 E( 8u 2 T; 8i; j 2 u ; F t;u ( i; j) is the probability that the phase of transition u changes from i to j when transition t res in marking . F is used for the speci cation of race policies [1] for transitions. See [14] for a more detailed discussion of race policies in DDSPNs. Again, all combinations of possible new phases for all transitions need to be considered when is changed by the ring of t in leading to the construction of the set F(t; such that 8 2 F(t; is a possible ....

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R. Zijal and G. Ciardo. Discrete Deterministic and Stochastic Petri Nets. ICASE Technical Report 96-72, Institute for Computer Applications in Science and Engineering, NASA/Langley Research Center, Hampton, VA, 1996.

....firing probability ff is set to 1 (e.g. Const(4ffi) Geom(1; 4ffi) in Fig. 1) An immediate transition (firing in zero time) can be modeled by a DTP containing a single absorbing phase 0. A more detailed presentation of DTPs and of corresponding discrete firing time distributions can be found in [4, 14]. Consequently, a state s of a DDSPN consists of two discrete components, the marking and the vector OE containing the phase for each transition: s = OE) 2 IN jP j Theta IN jT j : Each entry OE t of OE represents the current phase of the DTP associated to transition t. Definition ....

....[ OE 0 t :G t ( OE t ;OE 0 t ) 0 fOE 0 t g: ffl 8 2 S; 8t 2 E( 8u 2 T; 8i; j 2 Phi u ; F t;u ( i; j) is the probability that the phase of transition u changes from i to j when transition t fires in marking . F is used for the specification of race policies [1] for transitions. See [14] for a more detailed discussion of race policies in DDSPNs. Again, all combinations of possible new phases for all transitions need to be considered when OE is changed by the firing of t in leading to the construction of the set F(t; OE) such that 8OE 0 2 F(t; OE) OE 0 is a possible ....

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R. Zijal and G. Ciardo. Discrete Deterministic and Stochastic Petri Nets. ICASE Technical Report 96-72, Institute for Computer Applications in Science and Engineering, NASA/Langley Research Center, Hampton, VA, 1996.

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