Download Real-Time Systems: Deriving Utilization Bound for Rate Monotonic Scheduling - Prof. Tarek and more Study notes Computer Science in PDF only on Docsity!
Real-Time Systems
Periodic Tasks
Tarek Abdelzaher
Exercise:
Know Your Worst Case Scenario
Consider a periodic system of two tasks
Let U
i
= C
i
/P
i
(for i = 1,2)
What is the maximum value of:
Πi(1-Ui)
for a schedulable system?
Deriving the Utilization Bound
for Rate Monotonic Scheduling
The minimum utilization case:
C 1
P 1
P 2
Task 1
Task 2
1 1
2 1 2 P P
P C P
= −
−
= 1 + − 1 1
2
1
2
2
1 P
P
P
P
P
C U
−
−
= 1 + − 1
1
2
1
2
1
2
1
2
2
1
P
P
P
P
P
P
P
P
P
P U
1 1
(^2) =
⇒ P
P
= − 1
2 1
1 12 P
P P
P C P
= − 1
2 2 1 1 P
P C P C
Deriving the Utilization Bound
for Rate Monotonic Scheduling
The minimum utilization case:
C 1
P 1
P 2
Task 1
Task 2
1 1
2 1 2 P P
P C P
= −
−
= 1 + − 1 1
2
1
2
2
1 P
P
P
P
P
C U
−
−
= 1 + − 1
1
2
1
2
1
2
1
2
2
1
P
P
P
P
P
P
P
P
P
P U
1 1
(^2) =
⇒ P
P
= − 1
2 1
1 12 P
P P
P CP
P (^1) C 1
P 2
Task 1
Task 2
C 2
Deriving the Utilization Bound
for Rate Monotonic Scheduling
The minimum utilization case:
C 1
P 1
P 2
Task 1
Task 2
1 1
2 1 2 P P
P C P
= −
−
= 1 + − 1 1
2
1
2
2
1 P
P
P
P
P
C U
−
−
= 1 + − 1 1
2
1
2
1
2
1
2
2
1
P
P
P
P
P
P
P
P
P
P U
1 1
(^2) =
⇒ P
P
= − 1
2 1
2 (^1 1) P
P P
C PP
C 1
P 1
P 2
Task 1
Task 2
C 2
2 1 1
1 2 1
P P C
C C P
− =
Solutions
( ) ∏
i i
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P C P P
C P P
2
1
2
2 2
2
2 2
1
2
1
1 1
1
1 1
2 1 1 1 2
1 2 1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
Solutions
∏ (^ +)=
i i
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P C P P
C P P
2
1
2
2 2
2
2 2
1
2
1
1 1
1
1 1
2 1 1 1 2
1 2 1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
Solutions
∏ (^ +)=
i i
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P C P P
C P P
2
1
2
2 2
2
2 2
1
2
1
1 1
1
1 1
2 1 1 1 2
1 2 1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
Solutions
∏ (^ +)=
i i
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P C P P
C P P
2
1
2
2 2
2
2 2
1
2
1
1 1
1
1 1
2 1 1 1 2
1 2 1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
The General Case
∏ (^ +)= =
−
i n n
n i
n n
n n
n
n n
i
i
i
i i
i
i i
n n
i i i
P
P
P
P
P
P
P
P
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P P
C P P
1
2 1
3
1
2
1
1
1
1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
The General Case
∏ (^ +)= =
−
i n n
n i
n n
n n
n
n n
i
i
i
i i
i
i i
n n
i i i
P
P
P
P
P
P
P
P
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P P
C P P
1
2 1
3
1
2
1
1
1
1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
The General Case
∏ (^ +)= =
−
i n n
n i
n n
n n
n
n n
i
i
i
i i
i
i i
n n
i i i
P
P
P
P
P
P
P
P
U
P
P
P
C P
P
C
U
P
P
P
C P
P
C
U
C P P
C P P
1
2 1
3
1
2
1
1
1
1
Critically
Schedulable
Schedulable ∏ i (U^ i+^1 )≤^2
Scheduling Taxonomy
Periodic Task Scheduling
Rate Monotonic EDF
Bound Optimality Bound Optimality
Hyperbolic Bound
Scheduling Taxonomy
Periodic Task Scheduling
Rate Monotonic EDF
Bound Optimality Bound Optimality
Hyperbolic Bound
With
Period=Deadline
Scheduling Taxonomy
Periodic Task Scheduling
Rate Monotonic EDF
With
Deadline < Period
Deadline
Deadline Monotonic Scheduling
Consider a set of periodic tasks where each
task, i, has a computation time, Ci, a period, Pi,
and a relative deadline Di < Pi.
Pi
Di
Deadline Monotonic Scheduling
Consider a set of periodic tasks where each
task, i, has a computation time, C
i
, a period, P
i
and a relative deadline D
i
< P
i
What is the schedulability condition?
Pi
Di
Deadline Monotonic Scheduling
Consider a set of periodic tasks where each
task, i, has a computation time, C
i
, a period, P
i
and a relative deadline D
i
< P
i
Schedulability can’t be worse than if Pi was
reduced to Di. Thus:
i
n
i
i
n
D
C
1 /
Pi
Di
Deadline Monotonic Scheduling
Consider a set of periodic tasks where each
task, i, has a computation time, Ci, a period, Pi,
and a relative deadline Di < Pi.
Schedulability can’t be worse than if P
i
was
reduced to D
i
. Thus:
∑ ≤^ (^ −)
i
n
i
i
n
D
C
1 /
Pi
Di
Problem?
A Better Condition
Worst case interference from a higher priority
task, j?
Pi
Di
Pj Cj
A Better Condition
Worst case interference from a higher priority
task, j?
Pi
Di
Pj Cj
j j
i
C
P
D
A Better Condition
Worst case interference from a higher priority
task, j?
Schedulability condition:
Pj Cj
j j
i
C
P
D
i j
j j
i
i C D
P
D
C ≤
Pi
Di
A Better Condition
Worst case interference from a higher priority
task, j?
Schedulability condition:
Pj Cj
j j
i
C
P
D
i j
j j
i
i C D
P
D
C ≤
My exec. time
Worst case interference,
From higher priority tasks (^) I ,
My deadline
Pi
Di
A Better Condition
Worst case interference from a higher priority
task, j?
Schedulability condition:
Pj Cj
j j
i
C
P
D
i j
j j
i
i C D
P
D
C ≤
My exec. time
Worst case interference,
From higher priority tasks (^) I ,
My deadline
Pi
Di
Problem?
Example
Pj Cj
Pi
Di
Ri Consider a system of two tasks:
Task 1: P 1 =1.7, D 1 =0.5, C 1 =0.
Task 2: P 2 =8, D 2 =3.2, C 2 =
3
- 5 1
- 7
3
3
5 1
7
5
5
5
2
( 2 ) ( 2 ) 2
1 1
( 1 ) ( 2 ) 2
2
( 1 ) ( 1 ) 2
1 1
( 0 ) ( 1 ) 2
2
( 0 ) ( 0 ) 2
1
( 0 )
= + =
=
^ =
= + =
=
^ =
= + =
= =
R I C
C P
R I
R I C
C P
R I
R I C
I C
i i
j j
i
j
R I C
C
P
R
I
= (^) ∑
Example
Pj Cj
Pi
Di
Ri Consider a system of two tasks:
Task 1: P 1 =1.7, D 1 =0.5, C 1 =0.
Task 2: P 2 =8, D 2 =3.2, C 2 =
3
- 5 1
- 7
3
3
5 1
7
5
5
5
2
( 2 ) ( 2 ) 2
1 1
( 1 ) ( 2 ) 2
2
( 1 ) ( 1 ) 2
1 1
( 0 ) ( 1 ) 2
2
( 0 ) ( 0 ) 2
1
( 0 )
= + =
=
^ =
= + =
=
^ =
= + =
= =
R I C
C P
R I
R I C
C P
R I
R I C
I C
i i
j j
i
j
R I C
C
P
R
I
= (^) ∑
3 < 3.2 Ok!
EDF and Processor Demand
Interference is due to only those tasks with
earlier deadlines
Pj Cj
Pi
Di
Ri
Dj
EDF and Processor Demand
Consider demand on the processor due to tasks whose
deadlines have passed
Within any time interval, L, the demand must be less
than L.
Pj Cj
Pi
Di D j C L P
L D
i
n
i i
i ≤
∑ = 1
L
The Processor Demand
Schedulability Test for EDF
Checking the schedulability conditions for all L is not
possible.
Pj Cj
Pi
Di D j C L P
L D
i
n
i (^) i
i (^) ≤
∑ = 1
L
The Processor Demand
Schedulability Test for EDF
Checking the schedulability conditions for all L is not
possible.
Observation 1: Check only within a hyper-period (schedule repeats itself)
Pj Cj
Pi
Di D j C L P
L D
i
n
i (^) i
i (^) ≤
∑ = 1
L
The Processor Demand
Schedulability Test for EDF
Checking the schedulability conditions for all L is not
possible.
Observation 1: Check only within a hyper-period (schedule
repeats itself)
Pj Cj
Pi
Di D j C L P
L D
i
n
i i
i ≤
∑ = 1
L
Least common multiple of
all task periods
The Processor Demand
Schedulability Test for EDF
Checking the schedulability conditions for all L is not
possible.
Observation 1: Check only within a hyper-period (schedule
repeats itself afterwards)
Observation 2: Check only on absolute deadlines
Pj Cj
Pi
Di D j C L P
L D
i
n
i i
i ≤
∑ = 1
L
The Processor Demand
Schedulability Test for EDF
Checking the schedulability conditions for all L is not
possible.
Observation 1: Check only within a hyper-period (schedule repeats itself afterwards)
Observation 2: Check only on absolute deadlines
Observation 3: If U < 1, Demand is trivially satisfied after some point, L*
C L
P
L D
i
n
i i
i ≤
∑ = 1
Time
Demand
Demand=time
L
=?
The Processor Demand
Schedulability Test for EDF
Deriving L
* C L
P
L D
i
n
i (^) i
i ≤
∑ = 1
Time
Demand
Demand=time
L
=?
∑ ∑ ∑ = = =
n
i
i i i
n
i
n
i
i i
i i i i
i
i
i
i
i
C tU P D U P
t D P C P
t D
P
t D
P
t D
1 1 1
The Processor Demand
Schedulability Test for EDF
Deriving L
Time
Demand
Demand=time
L*
∑ ∑ ∑ = = =
n
i
i i i
n
i
n
i
i i
i i i i
i
i
i
i
i
C tU P D U P
t D P C P
t D
P
t D
P
t D
1 1 1
C L
P
L D
i
n
i (^) i
i ≤
∑ = 1
The Processor Demand
Schedulability Test for EDF
Deriving L
Time
Demand
Demand=time
∑ ∑ ∑ = = =
n
i
i i i
n
i
n
i
i i
i i i i
i
i
i
i
i
C tU P D U P
t D P C P
t D
P
t D
P
t D
1 1 1
U
P DU
LU P DU L L
n
i
n i i i
i
i i i −
−
∑
∑
=
= 1
( )
( ) ,
1
L*
C L
P
L D
i
n
i (^) i
i ≤
∑ = 1
