## DISCRETE-TIME MARKOVIAN STOCHASTIC PETRI NETS (1995)

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Citations: | 21 - 8 self |

### BibTeX

@MISC{Ciardo95discrete-timemarkovian,

author = {Gianfranco Ciardo},

title = { DISCRETE-TIME MARKOVIAN STOCHASTIC PETRI NETS },

year = {1995}

}

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### Abstract

We revisit and extend the original definition of discrete-time stochastic Petri nets, by allowing the firing times to have a “defective discrete phase distribution”. We show that this formalism still corresponds to an underlying discrete-time Markov chain. The structure of the state for this process describes both the marking of the Petri net and the phase of the firing time for of each transition, resulting in a large state space. We then modify the well-known power method to perform a transient analysis even when the state space is infinite, subject to the condition that only a finite number of states can be reached in a finite amount of time. Since the memory requirements might still be excessive, we suggest a bounding technique based on truncation.

### Citations

782 |
Finite Markov Chains
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(Show Context)
Citation Context ...ffl" whenever t is not enabled and either it cannot become enabled again or its phase is going to be reset upon becoming enabled). The difference between the size of the two DTMCs is due to a lum=-=ping [22]-=- of the states, and it would be even greater if t 3 had a more complex distribution. By postponing the probabilistic decision as much as possible, the second DTMC lumps states (011; ffl12), (011; ffl2... |

259 | A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems
- Marsan, Conte, et al.
- 1984
(Show Context)
Citation Context ...firing time. State-space-based analysis has been mostly applied to SPNs whose underlying process is a continuous-time Markov chain (CTMC), that is, to SPNs with exponentially distributed firing times =-=[3, 12, 25, 26]-=-. Except 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 solutio... |

245 |
Introduction to Stochastic Processes
- Çinlar
- 1975
(Show Context)
Citation Context ...y-state probability vector of the actual process is then obtained by weighting ~saccording to the holding times, a well known result applicable to the steady-state solution of any semi-Markov process =-=[8]-=-:ss = ~ss h s ( P u2S ~su h u ) \Gamma1 . For transient analysis, the same idea can be applied, but in a much more restricted way. If, at step n, every state in S [n] is such that the minimum time bef... |

84 |
Automated Generation and Analysis of Markov Reward Models using Stochastic Reward Nets
- Ciardo, Blakemore, et al.
- 1992
(Show Context)
Citation Context ...firing time. State-space-based analysis has been mostly applied to SPNs whose underlying process is a continuous-time Markov chain (CTMC), that is, to SPNs with exponentially distributed firing times =-=[3, 12, 25, 26]-=-. Except 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 solutio... |

73 | A characterization of the stochastic process underlying a stochastic Petri net
- Ciardo, German, et al.
- 1993
(Show Context)
Citation Context ... basic unit step, possibly with parameter equal one, that is, constants. [29] restates these results in more detail, and uses the concept of weight to break the ties, following [3] and, more closely, =-=[13]-=-. Generalized Timed Petri Nets (GTPN) have also been proposed [19], where the steps of the geometric firing times for each transition can be arbitrary, unrelated, real numbers. A DTMC can be obtained ... |

65 |
Numerical Transient Analysis of Markov models
- Reibman, Trivedi
- 1988
(Show Context)
Citation Context ...ver, acyclic state spaces arise only in special cases, such as reliability models of non-repairable systems. For transient analysis of a general CTMC, Jensen's method [21], also called uniformization =-=[17, 27]-=-, is widely adopted. [18] outlines a dynamic implementation of the algorithm, where the state space is explored as the computation of the transient probability vector proceeds, not in advance, as norm... |

50 |
A generalized timed petri net model for performance analysis
- Holliday, Vernon
- 1987
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Citation Context ...tants. [29] restates these results in more detail, and uses the concept of weight to break the ties, following [3] and, more closely, [13]. Generalized Timed Petri Nets (GTPN) have also been proposed =-=[19]-=-, where the steps of the geometric firing times for each transition can be arbitrary, unrelated, real numbers. A DTMC can be obtained by embedding, but the analysis is restricted to steady-state behav... |

44 |
On the Integration of Delay and Throughput Measures in Distributed Processing Models
- Molloy
- 1981
(Show Context)
Citation Context ...firing time. State-space-based analysis has been mostly applied to SPNs whose underlying process is a continuous-time Markov chain (CTMC), that is, to SPNs with exponentially distributed firing times =-=[3, 12, 25, 26]-=-. Except 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 solutio... |

43 |
Esp - A package for the evaluation of stochastic Petri nets with phase-type distributed transition times
- Cumani
- 1985
(Show Context)
Citation Context ...ations: 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) expans=-=ion" [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 dist... |

40 |
Markoff Chains as an Aid in the Study of Markoff Processes
- Jensen
- 1953
(Show Context)
Citation Context ...tional numerical approaches. However, acyclic state spaces arise only in special cases, such as reliability models of non-repairable systems. For transient analysis of a general CTMC, Jensen's method =-=[21]-=-, also called uniformization [17, 27], is widely adopted. [18] outlines a dynamic implementation of the algorithm, where the state space is explored as the computation of the transient probability vec... |

35 |
Finding Transient Solutions in Markovian Event Systems Through Randomization
- Grassmann
- 1991
(Show Context)
Citation Context ...se only in special cases, such as reliability models of non-repairable systems. For transient analysis of a general CTMC, Jensen's method [21], also called uniformization [17, 27], is widely adopted. =-=[18]-=- outlines a dynamic implementation of the algorithm, where the state space is explored as the computation of the transient probability vector proceeds, not in advance, as normally done. This allows to... |

31 | The influence of random delays on parallel execution times, Performance Evaluation 21
- Adve, Vernon
- 1993
(Show Context)
Citation Context ...roximate it arbitrarily well as the size of the step decreases. Furthermore, there is evidence supporting the use of deterministic instead of exponential distributions when modeling parallel programs =-=[1]-=-. If all the firing distributions are geometric with the same step, the underlying process is a discrete-time Markov chain (DTMC) [25]. Such SPNs can model synchronous behavior, as well as the main as... |

28 | Numerical analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions
- Lindemann, Shedler
- 1996
(Show Context)
Citation Context ... and solving its underlying stochastic process, a DTMC. Section 6, examines approaches to cope with large state spaces. 342 Chapter 20 2 THE PN FORMALISM We recall the (extended) PN formalism used in =-=[12, 14]-=-. A PN is a tuple \Gamma P; T ; D \Gamma ; D + ; D ffi ; ; g;s[0] \Delta where: P is a finite set of places, which can contain tokens. A markings2 IN jP j defines the number of tokens in each place p ... |

23 |
Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes
- Melamed, Yadin
- 1984
(Show Context)
Citation Context .... If the CTMC contains widely different rates, the number of matrix-vector multiplications required by uniformization can be excessive. Proposals to alleviate this problem are selective randomization =-=[24]-=- and adaptive uniformization [28], both based on the idea of allowing different uniformization rates, according to the set of states that can be reached at each step. The latter, in addition, can be u... |

23 |
Introduction to Stochastic Processes
- inlar
- 1975
(Show Context)
Citation Context ...state probability vector of the actual process is then obtained by weighting ~ according to the holding times, a well known result applicable to the steady-state solution of any semi-Markov process [8]: s = ~ s h s P u2S ~ u h u 1 . For transient analysis, the same idea can be applied, but in a much more restricted way. If, at step n, every state in S [n] is such that the minimum time befor... |

21 |
GSPNs versus SPNs: What is the actual role of immediate transitions
- Chiola, Donatelli, et al.
- 1991
(Show Context)
Citation Context ...d geometric distribution for the size of the batch arrivals. Incidentally, we observe that the continuous version of this SPN, where t and y are exponentially distributed, shows that Proposition 1 in =-=[9]-=- does not hold for unbounded systems: there is no SPN with only exponentially distributed firing times equivalent to this GSPN (equivalently, there is no SPN with only geometrically distributed firing... |

17 |
A Complete Model for Representing the Coordination of Asynchronous Processes
- Agerwala
- 1974
(Show Context)
Citation Context ... modeled by place p 2 and transition y. Unfortunately, finiteness of R is an undecidable question for the class of Petri nets we defined, since transition priorities alone make them Turing equivalent =-=[2]. S s;s 0 -=-can be infinite for a particular s 0 . If R is finite, this requires the presence of arbitrarily long paths over a finite set of vanishing states, just as for a "vanishing loop" in a GSPN [1... |

14 |
Approximate Performance Models of Polling Systems using Stochastic Petri Nets
- Choi, Trivedi
- 1992
(Show Context)
Citation Context ...rom researchers in the performance and reliability arena and have been extensively applied to the performance and reliability modeling of computer, communication, manufacturing, and aerospace systems =-=[4, 5, 7, 10, 23]-=-. While there is agreement on the appropriateness of SPNs as a description formalism for a large class of systems, two radically different solution approaches are commonly employed: simulation and sta... |

14 |
Analysis of large stochastic Petri net models
- Ciardo
- 1989
(Show Context)
Citation Context ...ty matrix \Pi and an initial state probability vectors[0] . The state probability vector at step n can be obtained with the iteration (power method):s[n] =s[n\Gamma1] \Pi. This result was extended in =-=[11]-=- to include immediate transitions, which fire in zero time, and geometric firing distributions with steps multiple of a basic unit step, possibly with parameter equal one, that is, constants. [29] res... |

14 |
Means and Variances of Time Averages in Markovian Environments
- Grassmann
- 1987
(Show Context)
Citation Context ...ver, acyclic state spaces arise only in special cases, such as reliability models of non-repairable systems. For transient analysis of a general CTMC, Jensen's method [21], also called uniformization =-=[17, 27]-=-, is widely adopted. [18] outlines a dynamic implementation of the algorithm, where the state space is explored as the computation of the transient probability vector proceeds, not in advance, as norm... |

12 | Stochastic Petri Net Analysis of a Replicated File System
- Dugan, Ciardo
- 1989
(Show Context)
Citation Context ...rom researchers in the performance and reliability arena and have been extensively applied to the performance and reliability modeling of computer, communication, manufacturing, and aerospace systems =-=[4, 5, 7, 10, 23]-=-. While there is agreement on the appropriateness of SPNs as a description formalism for a large class of systems, two radically different solution approaches are commonly employed: simulation and sta... |

11 |
The in uence of random delays on parallel execution times
- Adve, Vernon
- 1993
(Show Context)
Citation Context ...roximate it arbitrarily well as the size of the step decreases. Furthermore, there is evidence supporting the use of deterministic instead of exponential distributions when modeling parallel programs =-=[1]-=-. If all the firing distributions are geometric with the same step, the underlying process is a discrete-time Markov chain (DTMC) [25]. Such SPNs can model synchronous behavior, as well as the main as... |

11 |
inlar. Introduction to Stochastic Processes
- C
- 1975
(Show Context)
Citation Context ...y-state probability vector of the actual process is then obtained by weighting ~saccording to the holding times, a well known result applicable to the steady-state solution of any semi-Markov process =-=[8]-=-:ss = ~ss h s \Gamma P u2S ~su h u \Delta \Gamma1 . For transient analysis, the same idea can be applied, but in a much more restricted way. If, at step n, every state in S [n] is such that the minimu... |

6 |
Amulti-level solution algorithm for steady state Markovchains
- Horton, Leutenegger
- 1994
(Show Context)
Citation Context ...ns=s\Pi, which can be rewritten as the homogeneous linear systems(\Pi \Gamma I) = 0, subject to P s2Sss = 1. Fast iterative methods such as successive over-relaxation (SOR) [12] or multilevel methods =-=[20]-=- can then be employed, although their convergence is not guaranteed. Fig. 5 offers an example of an ergodic DTMC obtained from a DDP-SPN. 6 COPING WITH LARGE STATE SPACES The power method algorithm de... |

6 |
Performabilty modeling of an automated manufacturing system with deterministic and stochastic petri nets
- Lindemann, Ciardo, et al.
- 1993
(Show Context)
Citation Context ...rom researchers in the performance and reliability arena and have been extensively applied to the performance and reliability modeling of computer, communication, manufacturing, and aerospace systems =-=[4, 5, 7, 10, 23]-=-. While there is agreement on the appropriateness of SPNs as a description formalism for a large class of systems, two radically different solution approaches are commonly employed: simulation and sta... |

6 |
Reseaux de Petri stochastiques. These de docteur ingeneur, CNAM-Paris
- Natkin
- 1980
(Show Context)
Citation Context |

4 |
Modeling and analysis of transfer lines and production networks using Generalized Stochastic Petri Nets
- Al-Jaar, Desrochers
- 1988
(Show Context)
Citation Context |

2 |
Adaptive Uniformization", Stochastic Models
- Moorsel, Sanders
- 1994
(Show Context)
Citation Context ...ferent rates, the number of matrix-vector multiplications required by uniformization can be excessive. Proposals to alleviate this problem are selective randomization [24] and adaptive uniformization =-=[28]-=-, both based on the idea of allowing different uniformization rates, according to the set of states that can be reached at each step. The latter, in addition, can be used with infinite state spaces ev... |

2 |
D-timed Petri nets and modelling of timeouts and protocols
- Zuberek
- 1988
(Show Context)
Citation Context ...is restricted to steady-state behavior and the state space of the DTMC can be infinite even when the underlying untimed PN has a finite reachability set. Analogous considerations hold for D-timed PNs =-=[30]-=-. Discrete-time Markovian stochastic Petri nets 341 We generalize and formalize the results in [13] and show how, using phaseexpansion, a DTMC can be obtained even if the firing time distributions are... |

2 |
Stochastic Petri net analysis of a replicated le system
- Dugan, Ciardo
- 1989
(Show Context)
Citation Context ...rom researchers in the performance and reliability arena and have been extensively applied to the performance and reliability modeling of computer, communication, manufacturing, and aerospace systems =-=[4, 5, 7, 10, 23-=-]. While there is agreement on the appropriateness of SPNs as a description formalism for a large class of systems, two radically dierent solution approaches are commonly employed: simulation and stat... |

1 |
Timed Petri nets performance models for fiber optic LAN architectures
- Marsan, Signore
- 1987
(Show Context)
Citation Context |

1 |
Reliability analysis of non repairable systems using stochastic Petri nets
- Barkaoui, Florin, et al.
- 1988
(Show Context)
Citation Context ...ine step, arbitrary steps can be approximated, but this increases the state space. Approaches to solve models with a large state space have been proposed for both steady-state and transient analysis. =-=[6]-=- considers the reliability study of a SPN with exponentially distributed firing times, under the condition that the reachability graph is acyclic. The underlying CTMC is then acyclic as well, and a st... |

1 |
Souza e Silva and P. Mej ' ia Ochoa. State space exploration in Markov models
- de
- 1992
(Show Context)
Citation Context ...res to generate the state space S and \Pi, and then to iterate using Eq. (20.4), hence it assumes a finite S. However, a "dynamic" state space exploration has been proposed to remove this re=-=striction [16, 18]-=-. The general idea is to start from the initial state, or Discrete-time Markovian stochastic Petri nets 351 set of initial states and iteratively compute both the set of reachable states and the proba... |

1 |
A new approach to discrete time stochastic Petri nets
- Zijal, German
- 1994
(Show Context)
Citation Context ... in [11] to include immediate transitions, which fire in zero time, and geometric firing distributions with steps multiple of a basic unit step, possibly with parameter equal one, that is, constants. =-=[29]-=- restates these results in more detail, and uses the concept of weight to break the ties, following [3] and, more closely, [13]. Generalized Timed Petri Nets (GTPN) have also been proposed [19], where... |

1 |
Timed Petri nets performance models for optic LAN architectures
- Marsan, Signore
- 1987
(Show Context)
Citation Context ...rom researchers in the performance and reliability arena and have been extensively applied to the performance and reliability modeling of computer, communication, manufacturing, and aerospace systems =-=[4, 5, 7, 10, 23-=-]. While there is agreement on the appropriateness of SPNs as a description formalism for a large class of systems, two radically dierent solution approaches are commonly employed: simulation and stat... |

1 |
chains as an aid in the study of Marko processes
- Marko
- 1953
(Show Context)
Citation Context ...tional numerical approaches. However, acyclic state spaces arise only in special cases, such as reliability models of non-repairable systems. For transient analysis of a general CTMC, Jensen's method =-=[21]-=-, also called uniformization [17, 27], is widely adopted. [18] outlines a dynamic implementation of the algorithm, where the state space is explored as the computation of the transient probability vec... |

1 | Timed Petri nets performance models for �ber optic LAN architectures - Marsan, Signore - 1987 |

1 | Finite Markov Chains. D.Van Nostrand-Reinhold - Kemeny, Snell - 1960 |

1 | on Analysis and Optimization of Systems, Discrete Event Systems - Conf - 1994 |