A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rate matrix # of the associated Markov chain in partitions, such that each partition contains rates of the same order of magnitude. An approximate solution to the original problem is obtained by solving each non sti# partition in isolation [17] This technique is incorporated in the package ESP [18]. Example 6 In the folded PN of Figure 18 (the mutual exclusion problem with identical resources) transition t 2 has only the function of regulating the access to the shared resource C s and thus can be modelled by an immediate transition (neglecting, in this case, the access time) The ....

....to the performance and reliability features of the system [7, 20] In order to exploit this peculiarity, the input language must be structured for providing a friendly environment for the specification of the output measures. In the sequel we refer in particular to the language of the ESP package [18]. The stochastic behaviour of a SPN is determined by calculating the occurrence probabilities over the states of the reachability set 1 ) Therefore, the output measures are 26 2l l M4 = 1 0 1 0 ) M5 = 0 1 1 0 ) Figure 22: Markov chain defined over the tangible states of the PN of ....

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A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phasetype distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

.... for numerical truncation and roundoff, exact results are obtained, but the approach has two limitations: the number of SPN markings increases combinatorially, rendering unfeasible the solution of large models, and generally distributed activities must be modeled using phase type (PH) expansion [15]. PH distributions can approximate any distribution arbitrarily well, but it is difficult to exploit this fact in practice because the expansion exacerbates the state space size problem. Discrete distributions for the timing of SPNs have received less attention. This is unfortunate, since ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.

.... More complex definitions of SPNs resulting nevertheless in an underlying CTMC are in use, such as the GSPNs, where immediate transitions having zero firing time can also exist under certain conditions [2] and the PH SPNs, where the firing time has arbitrary continuous time phasetype distributions [20]. However, these destroy the one to one mapping between states of the PN and states of the CTMC, so we do not consider them further. The results we mention can be easily extended to these classes. In the Gauss Seidel method, a single vector is needed, and its entries must be updated ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In 1985.

.... a system further led to several de nitions of related, but not exactly equivalent, stochastic Petri net (SPN) formalisms [14, 4, 5, 11, 12] Most of the work has focussed on SPNs having an underlying continuous time Markov chain (CTMC) This is achieved by associating an exponentially or PHtype [13] distributed ring time to each transition. The continuous support interval [0; 1) of these distributions implies that the modeler does not have to worry about contemporary events in the model: they have null probability. However, this property implies that it is actually dicult to model systems ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.

....several ways by researchers in the performance or reliability eld. These extensions embed the model into a timed environment, usually by associating a time to the transitions (sometimes to the places) of the PN. The most general extensions, allowing the usage of stochastic times and probabilities [1, 2, 3, 4, 5, 6, 7, 8], can generally be classi ed as Stochastic PN models (SPN) It is certainly interesting to compare di erent extensions, either to evaluate them individually, or to decide the best suited to model a speci c system. The measuring stick for this purpose is the modeling power of the model, a ....

.... 1 j jp 2 j jp 3 j Enabling function p 2 f 3 f 1 = minfjp 1 j; 4g; f 2 = jp 1 j jp 4 j; f 3 = 1 jp 1 j = i jp 3 j = k (k 1) jp 1 j = maxfi 4; 0g jp 2 j = j i l jp 3 j = k 1 Variable i o bag Figure 1: Some Petri net constructs multiple arc [3, 4, 5, 6, 7, 8] A multiplicity can be associated to the arcs input or output arcs. This extension is almost unanimously used, so it is often included into the standard de nition of PN. An input arc from p to t with multiplicity k requires k tokens in p to enable t, and it causes their removal upon ring. An ....

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A. Cumani, \ESP - A package for the evaluation of stochastic Petri nets with phasetype distributed transitions times," in Proc. of the Int. Workshop on Timed Petri Nets, (Torino, Italy), July 1985.

.... is sometimes of greater interest: instantaneous availability, interval availability, and reliability (for a fault tolerant system) response time distribution of a program (for performance evaluation of software) computational availability (for a degradable system) Except for a few instances [6, 10, 11, 40], transient analysis of (G)SPN models has not received much attention in the past. Analytical models are often used to evaluate alternatives during the design phase of a system, when exact values for all the input parameters may not be known yet. It is then useful to estimate how the output ....

....speci cation. This check is turned o by setting a identically equal to TRUE. 2.2 Run time speci cation of the GSPN An important feature of CSPL is the ability to exploit C language constructs to represent a large class of GSPNs within a single CSPL le. In contrast to other (G)SPN packages [6, 10] that allow the run time speci cation of scalar parameters such as the initial number of tokens in a place or the value of a parameter used in the de nition of a rate or probability, SPNP provides a function to input a value at run time, before reading the speci cation of the GSPN. This input ....

[Article contains additional citation context not shown here]

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proceedings of the International Workshop on Timed Petri Nets, Torino, Italy, July 1985.

.... for numerical truncation and roundo , exact results are obtained, but the approach has two limitations: the number of SPN markings increases combinatorially, rendering unfeasible the solution of large models, and generally distributed activities must be modeled using phase type (PH) expansion [15]. PH distributions can approximate any distribution arbitrarily well, but it is dicult to exploit this fact in practice because the expansion exacerbates the state space size problem. Discrete distributions for the timing of SPNs have received less attention. This is unfortunate, since ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.

....stored 23. 33. Structured GSPN: CTMC (DTMC) JOR, diagonal recomputed 24. Structured GSPN: CTMC UNIFORM, diagonal stored 25. Structured GSPN: CTMC UNIFORM, diagonal recomputed 40. Approximation: CTMC Cross aggregation 50. Linear Algebra: Meschach Table 1: SNS menu from the package ESP [10]. We omit this discussion here, since numerical analysis of the kanban systems in our case study would become even more time and memory consuming due to the additional state space growth. 3 SNS implementation 3.1 Execution and textual output The command to run SNS is sns [ h] m] n] s] ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.

....is evidenced by a grow ing body of literature concerning the development of such models and their application in model based performance and or dependability evaluation. Included here is the work of Behr et al. 14] Evaluation Nets) Chiola ( 15] Generalized Stochastic Petri Nets) Cumani ([16]; Stochastic Petri nets with phase type distributions) Dugan et al. 17] Ex tended Stochastic Petri Nets) Godbersen and Meyer ( 18] Function Nets) Marsan et al. 19] Generanzed Stochastic Petri Nets) Molloy ( 13, 20] Timed Petri Nets) Natkin ( 12] Timed Petri Nets) and Tarn ( 21] ....

A. Cumani, "ESP - a package of the evaluation of stochastic Petri nets with phase-type distributed transition times," in Proc. International Workshop on Timed Petri Nets, Torino, Italy, July 1985, pp. 144 151.

....(prd) preemptive resume (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 ....

.... nd the proper way to analyze PN with concurrently active general transitions [16, 17] A possible approach for the analysis of SPN models, with concurrently active prs and prd general transitions, is through the continuous time Phase type (CPH) approximation of generally distributed ring times [11]. With this technique, the marking process of the NMSPN is approximated by a continuous time Markov chain with an expanded state space [11] In this paper, we discuss a modeling technique for the analysis of NMSPNs that relaxes some of the restrictions present in currently available SPN analysis ....

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A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....devoted to define models with generally distributed timings and to merge in the same model random variables and deterministic duration. Analytical solutions are possible in special cases, and the approximation of the original problems by means of CPH distributions is a rather well known technique [7]. This paper is aimed at emphasizing that DPH approximation may provide a more convenient alternative with respect to CPH approximation, and also to provide a way to quantitatively support this choice. Furthermore, the use of DPH approximation can be extended from stochastic models to functional ....

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985.

.... More complex definitions of SPNs resulting nevertheless in an underlying CTMC are in use, such as the GSPNs, where immediate transitions having zero firing time can also exist under certain conditions [2] and the PH SPNs, where the firing time has arbitrary continuous time phasetype distributions [20]. However, these destroy the one to one mapping between states of the PN and states of the CTMC, so we do not consider them further. The results we mention can be easily extended to these classes. In the Gauss Seidel method, a single vector is needed, and its entries must be updated ....

A. Cumani. ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy, July 1985.

....Two general directions have been taken in this area: the rst one [5, 9] introduces general distributions of ring time with specic conditions, which allows one to extract so called subordinated Markov chains and compute steady state and transient probabilities of the states. The second approach [16, 12, 7] introduces Phase type (PH) distribution ring time without further restrictions, and compute the resulting embedded Continous Time Markov Chain (CTMC) This needs to precisely dene the stochastic semantics of these new transitions and to cope with the bi dimensional complexity induced by the ....

....of the net, exponential under specic hypotheses with respect to the number of stages of the Coxian distributions. In the framework of this last approach, previous works have taken into account the stochastic semantics problem either at the net level [7] or at the underlying CTMC level directly [12]. Unfortunately, the complexity problem has not been tackled. Our approach deals not only with the stochastic semantics allowing the most general situations but also with the complexity of the Markov chain. We exploit the concurrency structure LAMSADE URA CNRS 825, Universit# Paris Dauphine, ....

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A. Cumani. ESP - a package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proc. of the International Workshop on Timed Petri Nets, pages 144151, Torino, Italy, July 1-3 1985. IEEE Computer Society Press. 21

....GDT SPN is converted into a CTMC de ned over an expanded state space. The cardinality of the expanded state space is of the order of the cross product of the timed reachability set of the basic PN times the state spaces of the PH distributions of the k random variables. The program package ESP [23] realizes the GDT SPN model with PH distributions. According to De nition 1, the program allows the user to assign a speci c memory policy to each PN transition so that the di erent execution policies can be put to work. The important point about the ESP package is that the expanded CTMC is ....

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phasetype distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....and any number of concurrent or conflicting transitions with generally distributed firing times. Moreover, the expansion technique can be easily implemented by a computer program, starting from the basic specification at the PN level, so that all the solution steps can be hidden from the modeler [48]. The drawback of this approach is, of course, the explosion of the state space that can be alleviated by resorting to the use Kronecker operators for matrices [63] 21 A very recent and interesting modification of the expansion technique, resorts to the use of discrete PH type random variables ....

.... An overview of the methods and tools available to estimate the parameters of a PH distribution from a given cdf can be found in [12] The expansion algorithm can be performed automatically by a computer program, and is driven by the execution policy associated with different transitions [48]. The result of the expansion algorithm is that each marking of the original PN is blown into a macrostate in the new state space. When the firing times of the original PN are already PH distributed, this approach provides exact results. Otherwise, a preliminary step is needed in order to ....

[Article contains additional citation context not shown here]

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....variable and the structure state process is a CTMC, the completion time problem can be alternatively evaluated by expanding the state space taking into account all the possible stages of the PH distribution. A completely automated tool that implements the state space expansion technique is in [4]. In this paper we devote our attention to the case when the structure state process is a SMP, and the method of the state space expansion can not be applied. 3 Example: Series System with Repair In this section we illustrate the application of Theorem 1 through a numerical example of a series ....

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....of all the other transitions. A semimarkov SPN more suited for applications has been discussed in [23] In this de nition, the transitions are partitioned into three classes: exclusive, competitive and concurrent. Only exclusive or competitive transitions are allowed to be non exponential. Cumani [22] has realized a package in which each PN transition can be assigned a PH [39] distributed ring time. We refer to this model as PHSPN. The peculiar feature of the PHSPN model is that it can support any combination of the memory policies de ned in [1] and that the related analytical solution can be ....

....kinds of preemptive service policies, is developed in length in order to put in evidence the limits and the potentialities of the di erent approaches. The GDT SPN is formally de ned in Section 2. Section 3 focuses on two main realizations of the GDT SPN, namely: the PHSPN implemented by Cumani in [22] and the DSPN introduced by Ajmone and Chiola in [3] and further generalized by di erent authors [15, 16, 10] into the MRSPN model. A comparative discussion of the modeling power of the considered models is performed in Section 4 through an example. Starting from a simple queuing system, more ....

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A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phasetype distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....time to b. 4 Numerical examples We compare three computational methods: namely, the numerical inverse transformation of C (s; F (s; IT) the numerical inverse transformation of the shifted function C a (s; F b (s; ITSF) and the PH approximation technique [1] (PH) In the latter case we approximate the constant work reqirement a and the downtime constraint b by an Erlang distribution of assigned order and with the same expected value (a or b, respectively) Figure 1 depicts the defective distribution of the completion time for the UdDs system, where ....

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....and any number of concurrent or con icting transitions with generally distributed ring times. Moreover, the expansion technique can be easily implemented by a computer program, starting from the basic speci cation at the PN level, so that all the solution steps can be hidden from the modeler [53]. The drawback of this approach is, of course, the explosion of the state space that can be alleviated by resorting to the use Kronecker operators for matrices [69] A very recent and interesting modi cation of the expansion technique, resorts to the use of discrete PH type random variables [41, ....

.... An overview of the methods and tools available to estimate the parameters of a PH distribution from a given cdf can be found in [16] The expansion algorithm can be performed automatically by a computer program, and is driven by the execution policy associated with di erent transitions [53]. The result of the expansion algorithm is that each marking of the original PN is blown into a macrostate in the new state space. When the ring times of the original PN are already PH distributed, this approach provides exact results. Otherwise, a preliminary step is needed in order to ....

[Article contains additional citation context not shown here]

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

....several classes of SPN s have been developed to deal with non exponentially distributed events [7, 3] The main reason for this is the observation that it is possible to identify a wide number of real situations in which deterministic or generally distributed events occur. Phase type distributions [9] can be used to approximate the occurrence time of a generally distributed event, but the main drawback of this approach is in the often prohibitive state space size of the expanded Continuous Time Markov Chain (CTMC) The analysis of stochastic systems with non exponential timing requires the ....

....proposed method through a a MATHEMATICA program. Columns 4 and 5 ,instead, provide the results obtained by approximating the deterministic service time tr 2 by an Erlang distribution of order 10 and 1000 respectively, and evaluating the resulting PHSPN model [3] utilizing the method described in [9]. Columns from 6 to 8 give the results of the same example when an enabling memory policy is assumed. More speci cally, column 6 contains the results obtained through a MATHEMATICA implementation, while columns 7 and 8 provide the results obtained a through a PHSPN model. Two are the main ....

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phasetype distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144-151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

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A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985.

No context found.

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985.

No context found.

A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

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Aldo Cumani, "ESP - a package for the evaluation of stochastic Petri nets with phase-type distributed transition times," in Proceedings of International Workshop on Timed Petri Nets, Torino, Italy, July 1--3, 1985, pp. 144--151.

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A. Cumani. Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, pages 144--151, Torino (Italy), 1985. IEEE Computer Society Press no. 674.

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