WebSPN 3.3

• Available measures
• How to define a Reward Function f
• How to define <Event>
• Examples of valid statement

pr[Event]

steady state probability of Event

(return a value)
 

pr(time)[Event]

instantaneous probability of Event

(return a value)
 

cycle(time1, time2 [, step]) pr[Event]

probability of Event in interval [time1, time2]

(return an array of values)

---

av_token[Place Name]

steady state average number of tokens in Place Name

(return a value)
 

av_token(time)[Place Name]

instantaneous average number of tokens in Place Name

(return a value)
 

cycle(time1, time2 [, step]) av_token[PlaceName]

average number of tokens in Place Name in interval [time1, time2]

(return an array of values)

---

pr_enabled[Transition Name]

steady state probability that Transition Name is enabled

(return a value)
 

pr_enabled(time)[Transition Name]

instantaneous probability that Transition Name is enabled

(return a value)
 

cycle(time1, time2 [, step]) pr_enabled[TransitionName]

probability that Transition Name is enabled in interval [time1, time2]

(return an array of values)

---

exp[Reward Function]

steady state expected reward rate of Reward Function

(return a value)
 

exp(time)[Reward Function]

instantaneous expected reward rate of Reward Function

(return a value)
 

cycle(time1, time2 [, step]) exp[Reward Function]

expected reward rate of Reward Function in interval [time1, time2]

(return an array of values)

---

cum_exp[Reward Function]

steady state expected accumulated reward rate of Reward Function

(only if the Petri Net have absorption marking)
(return a value)
 

cum_exp(time)[Reward Function]

instantaneous expected accumulated reward rate of Reward Function

(return a value)
 

cycle(time1, time2 [, step]) cum_exp[RewardFunction]

expected accumulated reward rate of Reward Function in interval [time1,time2]

(return an array of values)

---

mtta

Mean Time To Absorption

(only if the Petri Net have absorption marking)
(return a value)

<Reward Function Expression>:

a numerical expression in which the only functions allowed are:

mark[Place Name]

return the number of tokens in Place Name
(return an array of values)

enabled[Transition Name]

return 1 if Transition Name is enabled, 0 otherwise
(return an array of values)

<Conditional Reward Function Expression>:

an IF THEN ELSE statement in the following form:

if <Event> then <Reward Function Expression> else <Reward Function Expression>

(ELSE clausole is not optional)

The numerical expression consist of an algebric expression with parenthesis and the following operators (the order represents the precedence too):

- : unary minus

^ : power operator; is considered left associative

2^3^4 is evaluated as (2^3)^4, not 2^(3^4). To specified a specific evaluation order use the parenthesis.

*, /

+, -

Is a boolean expression with parenthesis and the following operators (the order represents the precedence too):

! : not

& : and

| : or

• #PlaceName <RelationalOperator> <numerical expression>
• mark[PlaceName] <RelationalOperator> <numerical expression>
• ?TransitionName
• is_enabled[TransitionName]

Note:

# and mark are equivalent

? and is_enabled are equivalent

The relational operators are:

==

<

<=

>

>=

!=

If the expression value is a value, the form of assignment statement is:

variable_name = expression

If the expression value is an array of values, the form of assignment statement is:

variable_array_name := expression

To view the value of one or more variables that represent a value (NOT AN ARRAY) the PRINT keyword must be used, i.e.:

print(a, b, ...)

The arrays obtained by means of CYCLE keyword are automatic visualized in the result dialog.

If an array represents a REWARD FUNCTION only the relative expected reward rate and expected accumulated reward rate can be viewed.

How to calculate and view the steady state average number of tokensin a place:

m1 = av_token[PlaceName]
print(m1)

How to calculate and view the instantaneous average number of tokens in a place:

m1 = av_token(7)[PlaceName]
  or
t = 7
m1 = av_token(t)[PlaceName]

print(m1)

How to calculate and view the Mean Time To Absorption (MTTA):
(mtta is calculated only for Petri Net with absorption marking)

t = mtta
print(t)

How to calculate and view the steady state probability of an event:

p = pr[#place1>0 & #place2=0 | #place3==5]
print(p)

How to calculate and view the instantaneous probability of an event:

p = pr(15)[#place1>0 & #place2=0 | #place3==5]
  or
time = 15
p = pr(time)[#place1>0 & #place2=0 | #place3==5]>

print(p)

How to calculate and view the steady state probability that a transition is enabled:

p = pr_enabled[TransitionName]
  or
p = pr[?TransitionName]

print(p)

How to calculate and view the instantaneous probability that a transition is enabled:

p = pr_enabled(20)[TransitionName]
  or
time = 20
p = pr_enabled(time)[TransitionName]

print(p)

How to define a reward function:

f := mark[PlaceName]
  or
f := enabled[TransitionName]
  or
f := if mark[PlaceName]>0 then 1 else 0
  or
f := if mark[PlaceName]>0 then mark[PlaceName]*10 else mark[PlaceName]/10

How to calculate and view steady state expected reward rate of a reward function:

p = exp[f]
print(p)

How to calculate and view steady state expected accumulated reward rate of a reward function:

p = cum_exp[f]
print(p)

How to calculate and view instantaneous expected reward rate of a reward function:

p = exp(25)[f]
print(p)

How to calculate and view steady instantaneous accumulated reward rate of a reward function:

p = cum_exp(10)[f]
print(p)

How to calculate and view the value of a measure in interval [t1, t2]

v:=cycle(t1, t2 [, step]) pr[#PlaceName >=0]
  or
v:=cycle(t1, t2 [, step]) av_token[PlaceName]
  or
v:=cycle(t1, t2 [, step]) pr_enabled[TransitionName]
  or
v:=cycle(t1, t2 [, step]) exp[RewardFunction]
  or
v:=cycle(t1, t2 [, step]) cum_exp[RewardFunction]

step is optional; if it is not present or is lower then delta it is set to delta

i.e.:

a = pr[ #p_0>0 & is_enabled[t_1] ]

a = pr[ #p_0>0 &
is_enabled[t_1] ]