Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
A Petri Nets Graphic Method of Reduction Using Birth-Death Processes R. Zemouri, D. Racoceanu, N. Zerhouni Laboratoire d’Automatique de Besançon (LAB), UMR CNRS 6596 25, rue Alain Savary, 25000 Besançon (France)
[email protected]
Abstract
2 Overview of stochastic Petri nets simplification methods 2.1 Graphical methods of simplification
Stochastic Petri nets are powerful tool for performance evaluation of concurrent systems like parallel computing, communication network and production systems. In many practical applications, performance evaluation using this model is very difficult because of the great dimension of the marking space. In this paper, we present a new graphical method for the reduction of stochastic Petri nets, applied for safe production system modeling. The approach is based on the principle of places interactivity in the model, function of transition firing rates. The reduction of the Petri net is applied directly on the graphical model after a simple analysis of places efficiency by using mathematical techniques of birth-death processes. Thus, the problem of the model dimension is solved since our method is independent of the marking graph.
These methods of simplification allow a transformation of SPN into a more simple one, by keeping some important properties of the initial model, like liveness and boundedness, but it is not always possible to give a physical interpretation to this reduction [3], [4], [6]: -
2.2 Analytic simplification methods One of these methods is based on a development proposed by Racoceanu [9], concerning the application of the singular perturbation method for the reduction of Markov chains. An extension of this technique is used afterward for the simplification of the stochastic Petri net [2], [10]. This method consists in decoupling the slow and fast dynamics of the SPN. The meaning of slow and fast models of this category of methods is very interesting for the study of the transient behavior of the marking probabilities. The default of this method is not to take into account the steady state behavior of the system. Besides, the method uses the Markov process associated to the SPN marking. This graph has frequently a great number of states and the results of the study can diverge and become quickly illegible. Racoceanu [8], [10] initialized the study of such a random model simplification in steady state. It seems interesting to study this problem in the case of the SPN by introducing a graphic reduction method.
Key Words Stochastic Petri Nets, Markov Process, Birth-Death Process, Production System Safety, Maintenance Process, Singular Perturbations.
1
Introduction
A complete and effective performances analysis of any production system requires the use of random parameters. Stochastic models like stochastic Petri nets (SPN) find also a great applicability for modeling and simulation [1],[5],[11]. However, real industrial systems are often complex and the associate algorithms become complicate and can diverge. There are many simplification methods to reduce the dimension of such kind of model. Every place of the model represents a state of the studied system, so most existing reduction methods do not conserve physical meaning of the model and imply an important loss of information. Some researches try to keep the meaning of the reduced model, by eliminating insignificant states. Firstly, we present a brief survey of such existing methods of SPN simplification.
0-7803-6475-9/01/$10.00© 2001 IEEE
Fusion of places; Transition fusion; Elimination of places and transitions in loop.
3 New graphic method of SPN simplification 3.1 Studied systems
46
Our study concerns vivacious, bounded stochastic Petri ) defined by nets, M(Pi) ≤ N ( i ∈{1,…, n}, N ∈ such as:
-
`
N, the upper bound of number of tokens, which can contain the place P.
Ρ = {P1, …, Pn} is the finite set of places, Γ = {T1, …, Tm} the finite set of transitions, Λ = {µ 1, …, µ m} the set of transition firing rates, Pre, Post : ΡxΓ→{0,1} incidence applications M0 : Ρ → {0,1, …., N} the initial marking.
Pr Figure 2. Generic SPN
Having applied several series of rate using the method of Amodeo, we observe that token attraction or rejection depend on the value of the firing rates. To estimate their impact on the marking in steady-state regime, we study the interactivity of every place with the rest of the model, according to its input and output rates. We will be also able to know if the studied place tends to attract or to reject tokens. This is illustrated in Figure 1, where µs and µe are respectively the sum of output and input firing rates of the place P1. The place Pr is a generic place that represents the rest of the model. So the study of the interactions between the place P1 and the other places is reduced to the study of its behavior reporting to the generic place Pr. P1
After a transient period, during which each place is going to attract or to reject tokens according to the values of the firing rates, the system evolves towards one of markings Mi represented in Figure 3: N. µs
P3
P2
µ4
µ3
N –1
0
1
µs
2µs
2
M1
0
N -1
N
(N-1) µe
3µe M2
1
…………
Nµe MN-1
MN
Figure 3. Generic SPN markings graph Leaving the state Mi, the system can evolve only towards two Mi-1 or Mi+1 states with the respective firing rates (i)µe and (N-i)µs. This particular characteristic is that of birth-death processes. These processes are a special case of a Markov chain in which the process makes transitions only to the neighboring states of its current position [7]. Thus, we can use all the analytical power of these processes to obtain a mathematical expression of places communication.
Pr
µs = µ1 + µ2 µe = µ3 + µ4
N. µs
M0
Figure 1. Interaction of the place P1 with the rest of the model
(N-1) µs
(N-2) µs
2µe
µs
2µs
M2
M1 µe
MN
MN-1
3µe
(N-1) µe
N µe
Figure 4. Birth-death process associated to the generic SPN
3.3 Use of birth-death process theory We can generalize the idea of the study of a place with regard to the rest of the model. We obtain thus a reduced two places stochastic Petri net, called generic Petri net (Figure 2), with: -
(N-2) µs N –2
2µe
M0
µs
µe
(N-1) µs
N
µe
P1
µ1
µs
µe
3.2 Places interactivity
µ2
P1
N
Birth and death rates of our N+1 states system are the following: λn = (N - n) µs µn = n µe , with n = 0,…,N
the studied place P, the place Pr representing the rest of the model, µs and µe the sums of all input and output firing rates of the place P
(1)
In such a system, the probability πn being in a state n has the next form:
47
ρ n .(1 − ρ ) πn = 1 − ρ N +1
1
ρ = µS/µe
, with
(2)
0.9
N1=100
0.8 0.7
This result allows us to deduct the steady-state average marking M(Pi) of the place Pi: N
M(Pi) =
∑M n =0
n
Strong state
0.6
R(P) 0.5 1
0.4
( Pi ).π n
(3)
0.3
s
0.2
N2 =10
with Mn(Pi), the marking of the place Pi when the system is in the n state.
0
0
1
2
f1 f2
If we apply this formula for the place P of the model of figure 2, we obtain: M(P) = N -
n (1 − ρ ) N n.ρ . N +1 ∑ 1− ρ n =0
(
)
µ ρ = µS
7
8
9
ρ = µS → ∞ µe and consequently,
µs>>µe ⇒
limρ→∞ M(P) = limρ→∞ ( N -
10
(6)
1−ρ N n =0 N +1 ∑nρ ) 1−ρ n =0
(7)
limρ→∞ R(P) = 0 In this case, the place P does not contain tokens in steady-state regime, so the system converges to the state MN, corresponding to the marking [0 N]t. The place P tends to reject tokens rather than keeping them. The action associated to P appears rarely. We can qualify these states as weak or unstable states. Their elimination can only simplify our network without a consequent loss of information.
From the expression of the average marking of the place P, we obtain the steady-state efficiency of this place: N n (1−ρ) . ∑ n.ρ N +1 1−ρ N n =0
f3
6
1st case :
3.4 Use of the results for twoweighting scale decoupling of the SPN
M(P) N = 1-
5
All the curves pass by the point (1, 0.5), corresponding to an efficiency R=50% and an output/input rate ratio ρ = 1. According to ρ=µs/µe, R(P) can evolve in two manners:
place Pi tends to attract tokens: M(Pi) >> 0 and so Pi can not be eliminated because the corresponding state of the system is significant. place Pi tends to reject the tokens (M(Pi) ≈ 0), so the state corresponding to this place is in no way significant and consequently, the place Pi can be neglected.
R(P) =
4
Figure 5. Evolution of the efficiency R of the place P, function of the rate ρ and the capacity N
(4)
This result shows that the relation between the input and output firing rates is very important and has a strong influence on the steady-state behavior of the system. According to the rate ratio ρ, places can have two different possible evolutions:
-
3
Weak state
e
Thus, we find an expression of the steady-state mean marking, according to the capacity N of the place P and to its input and output firing rates.
-
N
0.1
2nd case: µe >> µs
(5)
⇒
ρ=
µS → 0 µe
(8)
and by next:
Figure 5 illustrate the evolution of the efficiency R(P) function of the rate ratio ρ between output and input firing rate of the place P. The relation R = F(ρ) depends on the upper marking bound N of the place P:
limρ→0 M(P) = limρ→0 ( N so: limρ→0 R(P) = 1
48
1−ρ N n N +1 ∑nρ ) = N 1−ρ n =0
(9)
corresponds to the real system. The places eliminated are negligible in the system evolution.
In this case, we remark that the place P has an efficiency of 100%, corresponding to a steady-state marking of N tokens. The state M0 corresponding to the marking [N 0]t has a strong occurrence probability. The physical state or action corresponding to this place is very significant (strong state) and must be considered.
3.5 Firing rates calculus for the reduced SPN Having listed and eliminated weak places of the system with our method, we obtain a reduced model with new firing rates. In Figure 6, for example, P5 represents a weak place and P1 ,P2 ,P3 and P4 strong ones, the token flow between places Pi (i-th input place of P5) and Pj (j-th output place of P5) depends on the smallest rate between these places. After the simplification (Fig.6.b), we obtain the new firing rate µij:
On the other hand, the efficiency R(P) tends quickly to zero for great capacities (N) when ρ is greater than 1, and evolves slowly to attempt 10% for a value ρ = 10 for the 1-bounded places (N=1).
Conclusion: According to the interaction between the places, we conclude that the Petri net performances depend on the firing rates values. We calculate the average marking of places. This expression allows us to estimate the dynamics of every place in steady-state regime by estimating its efficiency, and so to see the effect of every place in the distribution of tokens in the model: some will tend to attract tokens towards them, and the others will reject them. Figure 5 give the place efficiency evolution function of output/input firing rates ratio ρ, for various values of N. In order to evaluate the weight of a place, we define an efficiency threshold s, s ∈ [0,1] that shares places in two corresponding groups called strong and weak. Corresponding to this threshold, we have a bound f of rate ratio ρ (f∈R+) such as: s = 1−
(1 − f ) (1 − f N +1 ) N
µij = min (µi , µj ) P1
µ24 µ23
µ4 P3
P4
µ14
P3
-a-
P4 -b-
Figure 6. Calculus of firing rates of the reduced SPN in the case of multiple inputs/outputs
n=0
with: µij , the firing rate between the places Pi and Pj (Fig.6.b), µi , the firing rate between Pi and P5 (Fig.6.a), µj , the firing rate between P5 and Pj (Fig.6.a).
This bound depends essentially on the capacity N of the place. It is close to 1, for places having an important capacity, and grows with the decrease of N.
(10)
4
Industrial application
To estimate the performances of our method, we apply those results to a maintenance model elaborate in the frame of a global project with our automotive industrial partner. The studied model corresponds to the maintenance flow of spot-welding tongs between the robotized (flow-shop) production line and the repair shop. The various phases of this circuit are the following: - Waiting zone after breakdown in a special place of the production line, - Clips transit to the repair shop, - Waiting zone for reparation, - Test bed, - Transit of clips to the different sectors of the flowshop.
The efficiency of the place P is important. The state or the action associated to this place is significant and so it cannot be eliminated. 2nd case: ρ >f ⇒ µs > f.µe ⇒ R(P) < s ⇒ M(P) < s.N
P2
µ13
P5
µ3
n
We obtain thus for the 1st case: ρ s ⇒ M(P) >s.N
P1
µ2
µ1
N
∑ n. f
P2
(11)
The place P has a weak efficiency. The state or the action associated to this place is insignificant, so it can be eliminated without important loss of information in the model. This method allows a simplification of the SPN by operating directly on the graphic model. The only necessary calculation is that of the efficiency on every place. After the reduction, the reduced model still
49
P1
At first, we apply the singular perturbation method [2], [9] to the model and we compare the results with those given by our graphic method. The column R(P) gives the efficiency of the corresponding place according to N and ρ (the rate ratio between output and input sum of firing rates). For an efficiency threshold of 20%, we obtain a bound f =4 for the coefficient ρ. Hachured cells (tab 1) indicate the places with efficiency lower than 20%, and the surrounded cells represent the places, which are eliminated by the singular perturbation method. We remark that all places eliminated by the singular perturbation techniques have values of efficiency R(P) lower than 20%. The graphical method seems more complete than the singular perturbation method, thanks to the exhaustive elimination of all weak states. In this sense, we see (tab 1) that some weak states escaped from the elimination by singular perturbations.
Collecting point Transit of the clip (µ1)
P2
Arrival zone of the repair shop Check-in (µ2)
P3
Reparation End of the reparation (µ3)
P6 Repairman availability
P4
Wait for test Enter to the test bed (µ4)
P5
Test Validation of the reparation (µ5)
P7
To illustrate the meaning of the reduced model in our application, we take for example, the second column data of Table 1. The places eliminated by our method are:
Wait for return to the production line Transit to the production line (µ6)
Figure 7. Maintenance flow modeling of a spot-welding tong.
P2 : waiting in the flow–shop, P3 : reparation, P5 : the test bed.
A random transition period is necessary for every stage, so every transition can be characterized by a firing rate. Thus, we obtain the stochastic Petri net of Figure 7, with µi the different firing rates. This algorithm was programmed with the Matlab software and can be programmed with any other mathematical software. The table 1 gives the results of the SPN simplification using two methods: singular perturbations, and graphical method for different management cases (so different firing rates).
µ1 µ2 µ3 µ4 µ5 µ6
Thus the reduced stochastic Petri net so obtained is the following: P1
Transit, check-in, and reparation of the spot-welding clip ( µ123) P6 Repairman availability
P4
Table 1. Results using singular perturbations and graphic reduction methods
µH µV ρ
5 3
3
3
3
3
P5 5
4
3
3
µH µV ρ
µH µV ρ
5 3
P7
100 5
Wait for return to the production line Transit to the production line ( µ6)
µH µV ρ
Wait for passage on the test bed Test bed and validation ( µ45)
Figure 8. Reduced SPN
5 3
Collecting point
5 3
The reparation delay and the test duration of spotwelding clips are relatively small with a very short waiting time in the repair shop. The major part of the cycle is distributed among:
-
place eliminated by singular perturbations R(P) < 0,2 - elimination by graphical method
50
Waiting in the collecting points of the production (assembly) line, ( P1) Waiting for passage on the test bed, ( P4) Waiting for return ( P7).
The firing rates have the following values:
5
µ123 = min (µ1 , µ2 , µ3 ) = 1 µ45 = min (µ4 , µ5 ) = 5 µ6 = 4 Solutions for this type of maintenance strategy will be directed to - an optimization of the transit times, especially since the mechanic is often available (P6), - a control of the passage of spot-welding clips in the test bed, because the time of the test is very short, the operator is often available, but clips waiting time is too long (P4).
6
This technique allows us to watch the performances of the repair shop by establishing a dashboard having the efficiency R of each place as a performance indicator. The optimum to reach is the equilibrium between the flow of the spot-welding tongs in all the places of the model. This equilibrium corresponds to an equality between the input and output rates, which gives a communication coefficient of ρ = 0.5. Thus we can localize the part of the circuit to be optimized. We can also test this model to evaluate the impact of any amelioration operation on the performances and the flow of the system.
µs ≈ 0
References
[1] Alain J., Stochastic petri nets. Belgium French Netherland’Summer School on Discret Event Systems, Spa – Belgium, 1993. [2] Amodeo L., Contribution à la simplification et à la commande des réseaux de Petri stochastiques. Application aux systèmes de production, PhD, Belfort, France, 1999. [3] Brams G., Réseaux de Petri, théorie e pratique Tome 1 :Théorie et analyse – Tome 2 : Modélisation et applications, Edition Masson, 1983. [4] David R., et Alla H., Du Grafcet aux réseaux de Petri, Edition HERMES (2e édition) , 1992. [5] Florin G., Stochastic Petri nets : Properties, applications and tools, Microelectronics and Reliability, 31(4), 1991, pp.669-697. [6] Murata T., Petri nets Properties, analysis and applications, IEEE, 77(4), 1989. [7] Ng Chee Hock, Queueing Modelling Fundamentals, JOHN WILEY & SONS 1996. [8] Racoceanu D., A. El Moudni, M. Ferney, N. Zerhouni., On a New Method of Markov Chain Reduction. Mathematical Modelling of Systems, vol. 1, no 3, pp. 83-101, 1995. [9] Racoceanu D., Contribution à la modélisation et à l’analyse des chaînes de Markov à échelles de temps et échelles de pondération multiples. Application à la gestion d’un système hydro-énergétique, PhD, Belfort, France, 1997. [10] Racoceanu D. and Zerhouni N., Use of Singular Perturbations for the Reduction of Manufacturing System Models, IFAC Congress MCPL-2000, Grenoble, France, 2000. [11] Ruegg A., Processus stochastiques, Presses polytechniques romandes, 1989. [12] Zhou, M.C. and J. Ma, Reduction and Approximation of Stochastic Petri Nets with Multiple Input Multiple-Output Modules, Preprints of 12th IFAC World Congress, vol. 4, 159-163, Sydney, Australia, 1993.
Let us note that the method of singular perturbation becomes impracticable with an important number of places or tokens, because of the enormous number of states of the marking graph, which gives an untreatable Markov generator. This problem does not occur in our method, because we don’t need to establish the marking graph; a simple calculation of R (P) = M (p) /N is enough. The application of this method can be extended to the simplification of other more complex systems (with divergent or convergent component) being able to be represented by a stochastic Petri net. We can study for example a rare but catastrophic situation (example of a fatal breakdown) as shown in Figure 9: µ e
