Quasi-Static Scheduling of
Embedded Software Using
Free-Choice Petri Nets
Marco Sgroi, Alberto Sangiovanni-Vincentelli
University of California at Berkeley
Luciano Lavagno
Cadence Berkeley Labs
Yosinori Watanabe
Cadence European Labs
EE249 - Fall2001
1
Outline
• Motivation
• Scheduling Free-Choice Petri Nets
• Algorithm
2
Embedded Software Synthesis
• System specification: set of concurrent functional
blocks (DF actors, CFSMs, PNs, …)
• Software implementation: set of concurrent
software tasks
• Two sub-problems:
– Generate code for each task (software synthesis)
– Schedule tasks dynamically (dynamic scheduling)
• Goal: minimize real-time scheduling overhead
3
Petri Nets Model
4
Petri Nets Model
t12
t13
t16
Schedule: t12, t13, t16...
a=5
c=a+b
5
Petri Nets Model
Task 1
Task 2
+ RTOS
Shared Processor
Task 3
6
Classes of Scheduling
• Static: schedule completely determined at compile
time
• Dynamic: schedule determined at run-time
• Quasi-Static: most of the schedule computed at
compile time, some scheduling decisions made at
run-time (but only when necessary)
7
Embedded Systems Specifications
Specification
Scheduling
Real-time Control
(preemption, suspension)
Dynamic
Data-dependent Control
(if ..then ..else, while ..do)
Quasi-Static
Data Processing (+, -, *...)
Static
8
Data Processing
IIR 2nd order filter
o(n)=k1 o(n-1) + k2 i(n)
i
*k2
+
o
*k1
Schedule: i, *k2, *k1, +, o
Schedule: i, *k1, *k2, +, o
9
Data computation (Multirate)
Fast Fourier Transform
Multirate Data Flow network
Petri Net
i
256
i
256
o
256
o
FFT
256
Schedule: ii…i FFT oo…. o
Sample rate conversion
A
B
2
3
C
2
7
D
8 7
E
5
F
Schedule: (147A) (147B) (98C) (28D) (32E) (160F)
10
Data-dependent Control
*2
i
o
>0
/2
Schedule: i, if (i>0) then{ /2} else{ *2}, o
• Petri Nets provide a unified model for mixed control
and data processing specifications
• Free-Choice (Equal Conflict) Nets: the outcome of a choice
depends on the value of a token (abstracted non-deterministically)
rather than on its arrival time
11
Existing approaches
• Lee - Messerschmitt ‘86
– Static Data Flow: cannot specify data-dependent control
• Buck - Lee ‘94
– Boolean Data Flow: scheduling problem is undecidable
• Thoen - Goossens - De Man ‘96
– Event graph: no schedulability check, no minimization of
number of tasks
• Lin ‘97
– Safe Petri Net: no schedulability check, no multi-rate
• Thiele - Teich ‘99
– Bounded Petri Net: partial schedulability check, complex
(reachability-based) algorithm
12
PNs and BDF
BDF network
Petri Net
t2 <=0
F
F
T
t1
<=0
>0
T
>0
TFF
t4
>0
t5
t3
Switch/Select vs. choice/merge
PNs: No correlation between different choices
13
PNs and BDF
BDF network
>0
F
F
T
T
F
PNs are notdeterminate
Petri Net
t2
t4
t6
t3
t5
t7
t8
t1
14
PNs and BDF
BDF network
>0
F
F
T
T
Petri Net
t2
t4
t6
t3
t5
t7
t8
t1
TF
PNs are notdeterminate
15
PNs and BDF
BDF network
>0
F
F
T
T
Petri Net
t2
t4
t6
t3
t5
t7
t8
t1
TTF
PNs are notdeterminate
16
Existing approaches
• Lee - Messerschmitt ‘86
– Static Data Flow: cannot specify data-dependent control
• Buck - Lee ‘94
– Boolean Data Flow: scheduling problem is undecidable
• Thoen - Goossens - De Man ‘96
– Event graph: no schedulability check, no minimization of
number of tasks
• Lin ‘97
– Safe Petri Net: no schedulability check, no multi-rate
• Thiele - Teich ‘99
– Bounded Petri Net: partial schedulability check, complex
(reachability-based) algorithm
17
Scheduling Petri Nets
• Petri Nets provide a unified model for mixed control and
dataflow specification
• Most properties are decidable
• A lot of theory available
• Abstract Dataflow networks by representing if-then-else
structures as non-deterministic choices
• Non-deterministic actors (choice and merge) make the
network non-determinate according to Kahn’s definition
• Free-Choice: the outcome of a choice depends on the value of
a token (abstracted non-deterministically) rather than on its
arrival time
18
Bounded scheduling
(Marked Graphs)
• A finite complete cycle is a finite sequence of transition
firings that returns the net to its initial state
• Bounded memory
• Infinite execution
• To find a finite complete cycle solve f(s) D = 0
t1
t2
2
t2
t3
2
1 0
D = -2 1
0 -2
T-invariant f(s) = (4,2,1)
t1
2
t3
f(s) D = 0 has no solution
No schedule
19
Bounded scheduling
(Marked Graphs)
• Existence of a T-invariant is only a necessary condition
• Verify that the net does not deadlock by simulating the
minimal T-invariant [Lee87]
t1
t1
t2
2
t3
3
2
2
3
t3
t2
2
T-invariant f(s) = (4,2,1)
T-invariant f(s) = (3,2,1)
t1t1t1t1t2t2t4 (0,0)
(0,0)
Not enough initial tokens
s = t1t1t1t1t2t2t4
Deadlock
20
Free-Choice Petri Nets (FCPN)
Marked Graph (MG)
Free-Choice
Confusion (not-Free-Choice)
Free-Choice:
choice depends on token value rather than arrival time
easy to analyze (using structural methods)
21
Bounded scheduling
(Free-Choice Petri Nets)
• Can the “adversary” ever force token overflow?
t1
t3
t5
t6
t4
t7
t2
t1
t2
t3
t5
t6
22
Bounded scheduling
(Free-Choice Petri Nets)
• Can the “adversary” ever force token overflow?
t1
t3
t5
t4
t7
t6
t2
t1
t2
t4
t7
23
Bounded scheduling
(Free-Choice Petri Nets)
• Can the “adversary” ever force token overflow?
t3
t1
t5
t2
t7
t4
t6
24
Bounded scheduling
(Free-Choice Petri Nets)
• Can the “adversary” ever force token overflow?
t3
t1
t5
t2
t7
t4
t6
25
Bounded scheduling
(Free-Choice Petri Nets)
• Can the “adversary” ever force token overflow?
t3
t1
t5
t2
t7
t4
t6
26
Schedulability (FCPN)
• Quasi-Static Scheduling
• at compile time find one schedule for every
conditional branch
• at run-time choose one of these schedules
according to the actual value of the data.
S={(t1 t2 t4),(t1 t3 t5)}
27
Bounded scheduling
(Free-Choice Petri Nets)
• Valid schedule S
t1
• is a set of finite firing sequences that return the net
to its initial state
• contains one firing sequence for every combination
of outcomes of the free choices t2
t4
t1
(t1 t2 t4)
t2
t4
t3
t5
S={(t1 t2 t4),(t1 t3 t5)}
t1
t3
t2
t5
t4
(t1 t3 t5)
t3
t5
Schedulable
28
How to check schedulability
• Basic intuition: every resolution of datadependent choices must be schedulable
• Algorithm:
– Decompose (by applying the Reduction Algorithm) the
given Equal Conflict Net into as many Conflict-Free
components as the number of possible resolutions of
the non-deterministic choices.
– Check if every component is statically schedulable
– Derive a valid schedule, i.e. a set of finite complete
cycles one for each conflict-free component
29
Allocatability
(Hack, Teruel)
• An Allocation is a control function that chooses which
transition fires among several conflicting ones ( A: P T).
• A Reduction is the Conflict Free Net generated from one
Allocation by applying the Reduction Algorithm.
• A ECN is allocatable if every Reduction generated from an
allocation is consistent.
• Theorem: A ECN is schedulable iff
– it is allocatable and
– every Reduction is schedulable (following Lee)
30
Reduction Algorithm
t2
t4
t2
t6
t1
t4
t6
t1
t3
t7
t5
t5
t7
T-allocation A1={t1,t2,t4,t5,t6,t7}
t2
t6
t1
t2
t4
t6
t1
t7
t7
t5
t1
t2
t4
t2
t4
t6
t1
t4
t6
t7
31
How to find a valid schedule
t2
t4
t1
Conflict Relation Sets:{t2,t3},{t7,t8}
T-allocations:
t3
t7
t5
t9
A1={t1,t2,t4,t5,t6,t7,t9,t10}
A2={t1,t3,t4,t5,t6,t7,t9,t10}
A3={t1,t2,t4,t5,t6,t8,t9,t10}
A4={t1,t3,t4,t5,t6,t8,t9,t10}
t6
t8
t10
32
Valid schedule
t1
t4
t2
t1
t1
t9
t1
t3
t5
t7
t6
t4
t2
t7
t6
t9
t8
t10
(t1 t2 t4 t6 t7 t9 t5) (t1 t2 t4 t6 t8 t10)
t1
t1

S
t1
t1
t6
t3
t5
t5
t6
t8
t10
(t1 t3 t5 t6 t7 t9 t5) (t1 t3 t5 t6 t8 t10)
t 2 t 4 t 6 t 7 t 9 t 5
t 3 t 5 t 6 t 7 t 9 t 5

t 2 t 4 t 6 t 8 t10 
t 3 t 5 t 6 t 8 t10 
33
C code implementation
t2
t1
p2
t4
2
p1
p4
t3
t6
p3
t5
t7
S={(t1 t2 t1 t2 t4 t6 t7 t5)
(t1 t3 t5 t6 t7 t5)}
Task 1:
Task 2:
{ t1;
{ t6;
if (p1)
t7;
then{ t2;
t5;
count(p2)++;
}
if (count(p2) = 2)
then{ t4;
count(p2) = count(p2) - 2;}
else{ t3;
t5;}
}
}
34
Application example:
ATM Switch
Incoming cells
(non-periodic)
Input cells: accept?
Internal buffer
Output cells: emit?
Outgoing cells
Clock
(periodic)
• No static schedule due to:
– Inputs with independent rates
(need Real-Time dynamic scheduling)
– Data-dependent control
(can use Quasi-Static Scheduling)
35
Petri Nets Model
36
Decomposition with min # of tasks
Input cell processing
2 Tasks
Output cell processing
37
Real-time scheduling of
independent tasks
Task 1
+ RTOS
Task 2
Shared Processor
38
Functional decomposition
Accept/discard cell
4 Tasks
Output time selector
Clock divider
Output cell enabler
39
ATM: experimental results
Functional partitioning (4+1 tasks)
Sw Implementation
Number of tasks
Lines of C code
Clock cycles
QSS (2 tasks)
QSS
Functional partitioning
2
1664
197526
5
2187
249726
40
Conclusion
• Advantages of Quasi-Static Scheduling:
QSS minimizes run-time overhead with respect to Dynamic
Scheduling by
Automatic partitioning of the system functions into a
minimum number of concurrent tasks
The underlying model is FCPN: can check schedulability
before code generation
• Future work
– Larger PN classes
– Code optimizations
41