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Rate Monotonic Analysis
Introduction
Periodic tasks
Extending basic theory
Synchronization and priority inversion
Aperiodic servers
Case study: BSY-1 Trainer
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Synchronization & Priority Inversion
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Software Engineering Institute
Sample Problem: Synchronization
Periodics
Servers
Emergency
100 msec
τ1
50 msec
20 msec
Data Server
2 msec
150 msec
τ2
20 msec
5 msec
Deadline 6 msec
after arrival
40 msec
Comm Server
350 msec
τ3
Aperiodics
10 msec
10 msec
Routine
40 msec
2 msec
100 msec
Desired response
20 msec average
τ2’s deadline is 20 msec before the end of each period.
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Synchronization & Priority Inversion
Carnegie Mellon University
Software Engineering Institute
Priority Inversion in Synchronization
Legend
τ1:{...P(S1)...V(S1)...}
τ3:{...P(S1)...V(S1)...}
Critical section
(S1 locked)
Executing
S1 locked
attempts to lock S1 (blocked)
S1 unlocked
Blocked
τ1(H)
B
τ2(M)
S1 locked
S1 unlocked
τ3(L)
Time
3
Synchronization & Priority Inversion
B
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Software Engineering Institute
Priority Inversion
Delay to a task’s execution caused by interference from
lower priority tasks is known as priority inversion.
Priority inversion is modeled by blocking time.
Identifying and evaluating the effect of sources of
priority inversion is important in schedulability
analysis.
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Synchronization & Priority Inversion
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Sources of Priority Inversion
Synchronization and mutual exclusion
Non-preemptable regions of code
FIFO (first-in-first-out) queues
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Synchronization & Priority Inversion
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Software Engineering Institute
Accounting for Priority Inversion
Recall that task schedulability is affected by
• preemption: two types of preemption
- can occur several times per period
- can only occur once per period
• execution: once per period
• blocking: at most once per period for each source
The schedulability formulas are modified to add a
“blocking” or “priority inversion” term to account for
inversion effects.
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Synchronization & Priority Inversion
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UB Test with Blocking
Include blocking while calculating effective utilization
for each tasks:
C j Ci Bi 1
f i = ∑ ------ + ----- + ----- + ----- ∑ C k
T
T i T i T i k ∈ H1
j ∈ Hn j
Hn Preemption
(can hit n times)
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Synchronization & Priority Inversion
Execution
Blocking
H1 Preemption
(can hit once)
Carnegie Mellon University
Software Engineering Institute
RT Test with Blocking
Blocking is also included in the RT test:
i–1
an + 1 = Bi + Ci +
an
----- C j
Tj
∑
j=1
i
where a 0 = B i +
∑ Cj
j=1
Perform test as before, including blocking effect.
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Synchronization & Priority Inversion
Carnegie Mellon University
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Example: Considering Blocking
Consider the following example:
Periodics tasks
τ1
τ2
τ3
100 msec
25 msec
200 msec
50 msec
Data Structure
10 msec
30 msec
300 msec
100 msec
What is the worst-case blocking effect (priority
inversion) experienced by each task?
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Synchronization & Priority Inversion
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Example: Adding Blocking
Task τ2 does not use the data structure. Task τ2
experiences no priority inversion.
Task τ1 shares the data structure with τ3. Task τ1 could
have to wait for τ3 to complete its critical section. But
worse, if τ2 preempts while τ1 is waiting for the data
structure, τ1 could have to wait for τ2’s entire
computation.
This is the resulting table:
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Synchronization & Priority Inversion
Task
Period
Execution
Time
Priority
τ1
τ2
τ3
100
200
300
25
50
100
High
Medium
Low
Blocking Deadline
Delays
30+50
0
0
100
200
300
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Software Engineering Institute
UB Test for Example
Recall UB test with blocking:
C j Ci Bi 1
f i = ∑ ------ + ----- + ----- + ----- ∑ C k
T
T i T i T i k ∈ H1
j ∈ Hn j
f1
C1 B1
25
80
= ------ + ------ = --------- + --------- = 1.05 > 1.00
100 100
T1 T1
f2
C1 C2
25
50
= ------ + ------ = --------- + --------- = 0.50 < U ( 2 )
100 200
T1 T2
f3
C1 C2 C3
25
50 100
= ------ + ------ + ------ = --------- + --------- + --------- = 0.84 > U ( 3 )
100 200 300
T1 T2 T3
RT test shows
Not schedulable
τ3 is schedulable
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Synchronization Protocols
No preemption
Basic priority inheritance
Highest locker’s priority
Priority ceiling
Each protocol prevents unbounded priority inversion.
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Nonpreemption Protocol
τ2:{...P(S1)...V(S1)...}
τ4:{...P(S1)...V(S1)...}
Ready
τ1(H)
Legend
S1 locked
Β
Executing
Blocked
Ready
τ2
Ready
τ3
S1 locked
S1 unlocked
τ4(L)
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Synchronization & Priority Inversion
Time
B
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Software Engineering Institute
Basic Inheritance Protocol (BIP)
τ2:{...P(S1)...V(S1)...}
τ4:{...P(S1)...V(S1)...}
Legend
S1 locked
Executing
τ1(H)
Blocked
S1 locked S1 unlocked
Attempts to lock S1
Ready
τ2
Β
Ready
τ3
S1 locked
S1 unlocked
τ4(L)
Time
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Synchronization & Priority Inversion
B
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Software Engineering Institute
Highest Locker’s Priority Protocol
τ2:{...P(S1)...V(S1)...}
τ4:{...P(S1)...V(S1)...}
Legend
S1 locked
τ1(H)
Executing
Blocked
Ready
τ2
Β
Β
Ready
τ3
S1 locked
S1 unlocked
τ4(L)
Time
15
Synchronization & Priority Inversion
B
Carnegie Mellon University
Software Engineering Institute
Priority Ceiling Protocol (PCP)
τ2:{...P(S1)...V(S1)...}
τ4:{...P(S1)...V(S1)...}
Legend
S1 locked
τ1(H)
Executing
Attempts to lock S1
Blocked
S1 locked S1 unlocked
Ready
τ2
Β
Ready
τ3
S1 locked
S1 unlocked
τ4(L)
Time
16
Synchronization & Priority Inversion
B
Carnegie Mellon University
Software Engineering Institute
Example Of Chained Blocking (BIP)
τ1:{...P(S1)...P(S2)...V(S2)...V(S1)...}
Legend
τ2:{...P(S1)...V(S1)...}
τ3:{...P(S2)...V(S2)...}
S2 locked
Attempts to lock S2 (blocked)
Attempts to lock S1 (blocked)
τ1(H)
S1 locked
Β
S1 locked
Blocked
Β
S1 unlocked
τ2(M)
S2 locked
S2 unlocked
τ3(L)
0
1
2
3
4
5
6
7
8
9
10
11
Time
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Synchronization & Priority Inversion
12
13
B
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Software Engineering Institute
Blocked At Most Once (PCP)
τ1:{...P(S1)...P(S2)...V(S2)...V(S1)...}
Legend
τ2:{...P(S1)...V(S1)...}
τ3:{...P(S2)...V(S2)...}
S2 locked
S1 locked
S2 locked
S2 unlocked
S1 locked
Ceiling
S1 unlocked
Attempts to lock S1 (blocked)
τ1(H)
C
S1 locked
Attempts to lock S1 (blocked)
S1 unlocked
τ2(M)
S2 unlocked
S2 locked
τ3(L)
0
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Synchronization & Priority Inversion
1
2
3
4
5
6
7
8
9
10
11
Time
12
13
C
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Software Engineering Institute
Deadlock: Using BIP
τ1:{...P(S1)...P(S2)...V(S2)...V(S1)...}
Legend
τ2:{...P(S2)...P(S1)...V(S1)...V(S2)...}
S1 locked
S2 locked
attempts to lock S2 (blocked)
Blocked
locks S1
τ1(H)
B
Attempts to lock S1 (blocked)
S2 locked
τ2(M)
0
1
2
3
4
5
6
7
8
9
10
11
Time
19
Synchronization & Priority Inversion
12
13
B
Carnegie Mellon University
Software Engineering Institute
Deadlock Avoidance: Using PCP
Legend
τ1:{...P(S1)...P(S2)...V(S2)...V(S1)...}
S1 locked
τ2:{...P(S2)...P(S1)...V(S1)...V(S2)...}
S2 locked
Ceiling
locks S2
attempts to lock S1 (blocked)
locks S1
τ1(H)
C
locks S2
locks S1
unlocks S1
unlocks S2
τ2(M)
0
1
2
3
4
5
6
7
8
9
10
11
Time
20
Synchronization & Priority Inversion
12
13
C
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Software Engineering Institute
Summary of Synchronization
Protocols
Protocol
Nonpreemptible
critical sections
Highest locker’s
priority
Basic inheritance
Priority ceiling
Bounded
Priority
Inversion
Blocked at
Most Once
Deadlock
Avoidance
Yes
Yes1
Yes1
Yes
Yes1
Yes1
Yes
Yes
No
Yes2
No
Yes
1
Only if tasks do not suspend within critical sections
2
PCP is not affected if tasks suspend within critical sections
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Sample Problem with
Synchronization
When basic priority inheritance protocol is used:
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Synchronization & Priority Inversion
Task
Period
Execution
Time
Priority
τ1
τ2
τ3
100
150
350
20
40
100
High
Medium
Low
Blocking Deadline
Delays
20+10
10
0
100
130
350
Carnegie Mellon University
Software Engineering Institute
UB Test for Sample Problem
This format is sometimes called a schedulability model
for the task set:
f1
C1 B1
20
30
= ------ + ------ = --------- + --------- = 0.500 < U ( 1 )
100 100
T1 T1
f2
C1 C2 B2
20
40
10
= ------ + ------ + ------ = --------- + --------- + --------- = 0.534 < 0.729
100 150 150
T1 T2 T2
U ( 2, .80 ) = 0.729
f3
C1 C2 C3
20
40 100
= ------ + ------ + ------ = --------- + --------- + --------- = 0.753 < U ( 3 )
100 150 350
T1 T2 T3
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Synchronization & Priority Inversion