Generalized stochastic Petri nets
GSPN
Xiaolan XIE
Ecole Nationale Supérieure des Mines de Saint Etienne
1
PLAN
1. Introduction to stochastic Petri nets by examples
2. Petri nets with inhibitors and priority
4. Generalized stochastic Petri nets
5. Applications
6. Extensions
2
2.
Introduction to GSPN by examples
3
Example 1 :
Consider an unreliable machine with:
1/ : mean time between failures (MTTF)
1/ : mean time to repair (MTTR)
1/p : mean processing time
state-transition diagram of CTMC :
Stochastic Petri net model :
p1
p3
p2
t2 (p)
t1
t3
immediate tr.

1
0
Legend:
p
t4 ( )
t5 ( )

p4
4
timed tr.
Reachability graph :
p1
p3
p2
t4
0001
1000
t1
0100
t5
t2 (p)
t1
t4 ( )
t3
t5 ( )
t2
p4
0010
Removing transient states leads to an isomorphic CTMC :

0100
0001

5
p
t3
Results :
• 
• Failure rate (frequence of t4) : 
• Production rate (frequence of t2) : pp
Remark :
The conflict between t2 and t4 is solved by competition of clocks
(failure models)
p1
p3
p2

0100
0001
t2 (p)
t1
p
t4 ( )
t5 ( )

p4
6
t3
Example 2 :
• Two types of products P1 and P2 are produced.
• The production of each product requires two operations with
- the first operation on a shared flexible machine
- and the second operation on a dedicated machine.
• P1 has priority to the shared machine but cannot preempt on-going
operations
• There is at most one product of each type in the system at all moment
• When the production of a product is finished, a new one of the same
type is dispatched into the system
7
Stochastic Petri net model :
p1
t1
p4
p7
t4
p2
p5
t2
t5
p3
p6
t3
t6
P1 has priority to the shared machine
8
Reachability graph :
Markov chain :
1001001 M1
t1
0101000 M2
t2
t3 0011001
t4
M3
t6
t6

0010100
1000100 M5
t5
M6
1000011
t1
0010011
M8
M4
t1

 1000100 M5

0100010

t3
p1

0100010 M7
t2
t6
0101000 M2
t5
M4 0010100
t3

p7
p2
p5
t2
t5
p3
p6
t3
t6
M8
Convention : Immediate transitions have priority over timed transitions
9
t4
M7

0010011
p4
Results for the case i = :
•
• Utilization ratio of the shared machine:

• Production rate of P1 (frequence of t3) :

• Production rate of P1 (frequence oft6) :

0101000 M2

0010100
p1
M4
t1


 1000100 M5

0100010


p4
p7
t4
p2
p5
t2
t5
p3
p6
t3
t6
M7

0010011
M8
10
Remarks :
- It is not obvious to build directly the Markov chain
- Stochastic Petri nets offer an elegant tool for generation of correct
Markov chain models of complex systems
- Immediate transitions seem a good way for representation of conflicts
and priorities
11
3.
Petri nets with inhibitors and
priority
12
Definition
PN = (P, T, I, O, M0, H, )
where :
• P : set of places
• T : set of transitions
• I : P x T  IN : PRE function, i.e. arcs from places to transitions and
their weights
• O : P x T  IN : POST function, i.e. arcs from transitions and places
their weights
• M0 : P  IN : initial marking
• H : P x T  IN : generalized inhibitor arcs
•  : T  IN : priority degrees of transitions
13
Example :
t2
p3

t1
K
p1
t4
p6

t6

2
p2


t3

p4
t5

p7
t7
Effect of inhibitor :
Transition t5 is not firable if M(p6) ≥ 2 .
Effect of priority :
Transitions t1, t6 and t7 are not allowed to fire if at least one transition
of higher priority is firable.
14
Firing rules:
- Let :
•t : set of input places of transition t
t• : set of output places of t
°t : set of places linked to t by inhibitor arcs
R1 : A transition t is said to have concession in marking M if M(p) ≥
I(p, t), p •t and M(p) < H(p), t), p °t.
Let (M) the set of transitions that have concession in M
R2 : A transition tj is said firable if
tj  (M) and j ≥ k, tk  (M)
R3 : Firing a transition t leads to the following marking :
M' = M + O(t) - I(t)
15
Structures of conflicts
• (Without priority) : A transition tm is said in effective conflict with tl in
marking M if
- tm has concession in M
- tl is firable
- tm no longer has concession in M' such that M(tl > M'.
Example : t2, t3 and M1 obtained after firing t1.
t2
p3

t1
K
p1
t4
p6

t6

2
p2


t3

p4
16
t5

p7
t7
• (with priority) : A transition tk is said in indirect effective conflict with tl
having the same degree of priority in M if
- tk is firable in M,
- tl is firable in M,
- firing tl leads to a sequence  of higher priority transitions such that,
after , no transition of priority higher than k is firable, and
- tk is not firable in M' such that M(tl > M'.
Example :
p3
p1
t1
t3


p2
t2

p4
17
p5
Structural properies of PN
If the Markov chain of a stochastic Petri net has a finite state space and is
ergodic, then
- it is consistent :
 Y  0, CY  0
- and it is conservative :
 X  0, X T C  0
where C = O - I.
18
4.
Generalizd Stochastic Petri nets
GSPN
19
Definition : A generalized stochastic Petri net is an 8-tuples:
GSPN = (P, T, I, O, M0, H, , W)
where
• PN = (P, T, I, O, M0, H, ), the underlying Petri net, is a Petri net with
inhibitors and priority composed of:
- transitions of priority 0 that are timed transitions with exponentially
distributed transition firing times (timed transitions)
- transitions of priority n ≥ 1 that are immediate transitions (nimmediates transitions)
• W : T  IR+ : is a function such that
- W(t) is the rate of the exponential distribution, i.e. inverse of the
mean firing time, if t is timed
- W(t) is a weight associated with t if t is immediate
20
Tangibles and intangibles markings
- A marking M is said tangible if no immediate transition is firable
- It is said intangible if at least one immediate transition is firable.
21
Dynamics of a GSPN
• To each timed transition is associated a clock initiated with the
corresponding exponential distribution
• For a tangible marking,
- the clocks of all firable transitions tick down at the same speed
- the next transition to fire is that whose clock reaches zero first
22
• For an intangible marking,
- we choose one firable transition ti to fire next with probability :
Pt i M
W t i 

tE M Wt 
where E(M) is the set of firable transitions. Remind that all transitions
in E(M) are immediate transitions of the same degree.
- firing ti leads to a new marking
• If the new marking is till intangible, the above is repeated.
Otherwise, clocks of new transitions are generated and the system
evolves in the resulting new tangible marking
23
Results :
• The sequence of tangible markings is a CTMC.
• the average sojourn time in a tangible marking M is exponentially
distributed with mean


tE  M W t 
1
where E(M) is the set of transition firable at M, and W(t) is the
parameter of the exponential distribution of t
• the probability of firing a firable transition tk in a tangible marking M
is:
Pt k M 
24
Wt k 
tE M Wt 
Generation of the isomorphic CTMC
The CTMC can be derived from the reachability graph (RG) :
- by establish a one-to-one correspondence betwwen the states of the
CTMC and tangible markings
RG
CTMC
M
t
M*
25
- by introducing an arc (or a state-transition) u for all arc t of RG
connecting two tangible markings. The rate of u is W(t)
RG
CTMC
M
W(t)
M
t
M*
M*
26
- by introducing an arc v for all path t1t2…tk of RG connecting two
tangible markings M and M* via intangible markings M1,M2, …, Mk-1.
The rate of v is :
W(t1) P(t2 / M1) .P(t3 / M2) … P(tk / Mk-1)
where
W t i 
tE M Wt 
Pt i M 
RG
CTMC
M
t
M*
27
Performance evaluation
• Steady state distribution :
j : probablity of being in tangible marking Mj
•Firing frequency of a timed transition tk
k 

 
j : t k E M j
  j
k Mj
where k(Mj) is the firing rate of tk in Mj
• Average number of tokens in place p :
EM(p)   M j p  j
j
• Average sojourn time of tokens in a place p (Little's law):
EM(p)
E dp 
 tp•  t
 
28
Methodology
1. Construction of the GSPN model
2. Validation of the model. For example, the verification of the
consistency and conservativeness.
3. Definition of performance measures
4. Generation of the reachability graph
5. Derive the isomorphic CTMC
6. Verification of the ergodicity of the CTMC
7. Compute the steady state distribution of the CTMC
8. Computation of the performance measures
Remark : Softwares are available for automatic generation and
solution of CTMC (GreatSPN, …)
29
5. Applications
1. A failure-prone production line
M1
B
Assumption :
idle machines cannot fail
Parameters :
i : failure rate
i : repair rate
pi : production rate
H : buffer capacity
30
M2
GSPN :
p1
p3
p2
t2
t1
t3
p9
t7
t6
t9
t10
p4
M1
B
p7
p6
t4
t5
p5
H+1 p10
p8
M2
31
t8
Structural properties :
• Conservative as the GSPN is covered by three p-invariants :
{p1, p2, p3, p4}
{p2, p3, p9, p10}
{p5, p6, p7, p8}
• Consistent as the GSPN is covered by three t-invariants:
{t1, t2, t3, t6, t7, t8}
{t4, t5}
{t9, t10}
32
Performance :
- Production rate (frequency of t7) :

TH 
t 7  j
p
1
j / M j (p6)0
- Average WIP :
t1
WIP   M j p9 j
p
4
- Utilization ratios of the machines :

j
UM2 
j / M j (p2)0

j

j
j / M j (p6)0
- Idle rate of the machines :
IM1 

 j et IM2 
j / M j (p3)0
j / M j (p5)0
33
p
3
t2
t4
t5
j
UM1 
p
2
p
5
H+ p1
1 0
t3
p9
p
6
t6
t1
0
p
7
t7
t9
p
8
t8
2. An NC machine
• The operation cycle of the machine is as follows :
Phase 1 : Read informations of the product to process
Phase 2 : Simultaneous execution of two commands: preparation of
machining tools and preparation of NC program
Phase 3 : Process the product
Phase 4 : Test of the quality. If the quality is insufficient, then the
machine repeats phases 2 and 3 for other treatments. Otherwise, the
product is unloaded.
34
• GSPN :
t9
p9
r
p2
p1
t1
t2
p3
t3
p4
t4
p5
p6
t7
p8
t5
p7
t6
q
t8
with r + q = 1.
Phase 1 : Read the product informations
Phase 2 : Simultaneous execution of two commands: tool preparation and NC program
loading
Phase 3 : Process the product
Phase 4 : Qualit test. With probability q, the quality is insufficient, then the machine repeats
phases 2 and 3 for other treatments. Otherwise, the product is unloaded.
35
Structural properties :
•Conservative as covered by p-invariants :
{p1, p2, p3, p5, p7, p8, p9}
{p1, p2, p4, p6, p7, p8, p9}
• Consistent as covered by t-invariants :
{t1, t2, t3, t4, t5, t6, t7, t9}
{t2, t3, t4, t5, t6, t8}
t9
p9
r
p2
p1
t1
t2
p3
t3
p4
t4
p5
p6
p8
t5
p7
t6
q
36
t7
t8
Reachability graph :
M1
100000000
t1
M2
010000000
t2
M3
001100000
t3
000110000
t4
M4
000011000
t5
M7
000000100
t6
M8
000000010
t7
M9
M1 100000000
1
M3
t4
001001000
t3
M5
M6
000000001
Markov Chain :
001100000

000110000
M4

M7
001001000
M5

000000100
t8
t9

q 6
r 6
M9

000000001
t9
p9
r
p2
p1
t1
37
p3
t2
p4
t3
t4
p5
p6
t7
p8
t5
p7
t6
q
t8
• The state-transition diagam is strongly connected and the CTMC is
ergodic.
• Steady state distribution is solution of the system:

q


r

M1 100000000
1
M3
001100000

000110000
M4

M7

001001000
M5

000000100
q 6
r 6
M9
38
000000001

• Performance measures :
- Production rate (frequency of t9) :
M1 100000000
1
TH = 9
M3
- mean production cycle :
001100000

000110000
M4

T = 1/TH

001001000
M5

- mean time of phase 2 (Little's law):
M7
Nb of tokens in p3 and p5  3   4   5
d2 

frequency of arrivals in p3 11   7 6 q
000000100
q 6
r 6
- Utilization ratio of the machine :
M9
D = 7

000000001
t9
p9
r
p2
p1
39
t1
t2
p3
t3
p4
t4
p5
p6
t7
p8
t5
p7
t6
q
t8
6.
Extensions:
40
1. Choice of immediate transitions with random switches
• Each random switch is associated with a set S of transition and it
indicates, for each marking, the probability of firing a transition of thi
set
• Switches which are functions of markings are particularly useful
• Example :
if M(p2) < M(p3) , P(t1) = 1, P(t2) =0
if M(p2) > M(p3), P(t1) = 0, P(t2) =1
if M(p2) = M(p3), P(t1) = 0.5, P(t2) = 0.5
t1
p2
p1
t2
p3
41
2. Marking dependent firing times
Example :t(M) =t * Min{M(p), p  •t}
42
3. Approximation of general distribution
• Consider a random variable D of mean D and standard deviation D.
p1
D
p2
• Case : D > D. The delay of the subnetwork N1 and D have the same
mean and same standard deviation if :
1

 2 
 D
n   D    1  



 D  
,
n(n  1) 2D  n 2D 


n 1

n  n(n  1) 2  n 2
D
D
D
 2  

n 1

1



…
N1 :
p1



n
43
p2
• Case : D < D. The delay of the subnetwork N2 and D have the same
mean and same standard deviation if :
2
2

 h  2  D D   D ,
2
2
2
r1  2  D  D   D

1 - r1
N2 :
p1
h
r1
44
p2