12/11/2015 periodic tasks aperiodic tasks · 12/11/2015 1 periodic tasks + aperiodic tasks 2...
TRANSCRIPT
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
1
Periodic tasks+
Aperiodic tasks
2
Handling aperiodic tasks
Aperiodic tasks are typically activated by thearrival of external events (notified by interrupts).
From one hand, one objective of the kernel is toreduce the response time of aperiodic tasks(interrupt latency).
On the other hand, aperiodic task executionshould not jeopardize schedulability.
3
Aperiodic Scheduling
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Consider a simple example with 2 periodic tasks (scheduled by RM) and a single aperiodic job with Ca = 2 arriving at time t = 2:
4
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
If aperiodic jobs are scheduled in background (i.e., during idle times left by periodic tasks) their response times are too long:
5
Immediate service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
deadline miss
Response Time = 2
On the other hand, if interrupts service routines are scheduled at the highest priority, the other tasks can miss their deadlines:
6
Aperiodic tasks with HARD deadlines must beguaranteed under worst-case conditions.
Off-line guarantee is only possible if we canbound interarrival times (sporadic tasks).
Hence sporadic tasks can be guaranteed asperiodic tasks with Ci = WCETi and Ti = MITi
WCET = Worst-Case Execution TimeMIT = Minimum Interarrival Time
HARD aperiodic tasks
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
2
7
SOFT aperiodic tasks
Aperiodic tasks with SOFT deadlines should beexecuted as soon as possible, but withoutjeopardizing HARD tasks.
We may be interested in
minimizing the response time of each aperiodic request
performing an on-line guarantee
How can we achieve these goals?
8
Aperiodic Servers
A server is a kernel activity aimed at controlling theexecution of aperiodic tasks.
Normally, a server is a periodic task having twoparameters:
Cs capacity (or budget)Ts server period
To preserve periodic tasks, no more than Csunits must be executed every period Ts
9
Aperiodic service queue
Service queue
Server
READY queue
periodic/sporadicHARD tasks
aperiodicSOFT tasks
CPU
The server is scheduled as any periodic task.
Priority ties are broken in favor of the server.
Aperiodic tasks can be selected using an arbitraryqueueing discipline.
10
Fixed-priority Servers
Polling Server
Deferrable Server
Sporadic Server
Slack Stealer
11
Dynamic Polling Server
Dynamic Sporadic Server
Total Bandwidth Server
Tunable Bandwidth Server
Constant Bandwidth Server
Dynamic-priority Servers
12
Polling Server (PS)
At the beginning of each period, the budget isrecharged at its maximum value.
Budget is consumed during job execution.
When the server becomes active and there areno pending jobs, Cs is discharged to zero.
When the server becomes active and there arepending jobs, they are served until Cs > 0.
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
3
13
Background service
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
Response Time = 9
Let’s take the previous example:
14
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 1
C2 = 3
RM + Polling Server
0
PSCs = 1Ts = 4
Response Time = 7
4 8 12
15
Computing Ulub for RM + PST1 < Ti < 2T1
Cs = T1 – Ts
1
2
Cs Cs
C1Ts 2Ts
C2
n-1
0
Cn–1
nCn
T1
T2
Tn–1
Tn
•••
C1 = T2 – T1
C2 = T3 – T2
Cn–1 = Tn – Tn–1
Ts
1n
1kkssn CCTC
PS
= 2Ts – Tn
16
n
ns
n
nnsub T
TT
T
TT
T
TTUU
2
1
1
1
12
nT
T
T
T
T
TUU
n
s
n
nsub
2
11
2
Definingi
ii T
TR 1 we can write:
nT
TRUU
n
sn
iisub
21
1
Computing Ulub for RM + PS
17
Note that
nP
KRUU
n
iisub
1
1
1
1
1 T
TRP n
n
ii
hence:
nT
TRUU
n
sn
iisub
21
1
Cs = T1 – Ts 11 ss
UT
T
PUT
T
T
T
T
T
sn
s
n
s 1
1
222 1
1
1
2
sUK
Computing Ulub for RM + PS
18
nP
KRUU
n
iisub
1
1
Minimizing with respect to Ri we have:
01
PR
K
R
U
ii
ub for ni KR /1
Hence: 1/1lub n
sPSRM KnUU
Computing Ulub for RM + PS
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
4
19
PS properties
In the worst-case, the PS behaves as a periodictask with utilization Us = Cs/Ts
Aperiodic tasks execute at the highest priority ifTs = min(T1, … , Tn).
Liu & Layland analysis gives that:
11
2)(
/1
lub
n
ss
PSRM
UnUnU
20
RM + PS schedulability
Us
0 1
1
ln2
1
2ln)(lub
ss
PSRM
UUnU
)1()( /1lub n
sPSRM KnUnU
1
2
sUK
21
1
2)1(
1
s
n
ii U
U
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Polling Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12max P
U s )( ssaaaa CTCR 22
Response time under PS
Consider a PS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
initial delay # full service periods final chunk
Ca
(Cs, Ts)
ra
a = Ts ra
ra
Ts
1s
aa C
C saaa CC
23
Deferrable Server (DS)
Is similar to the PS, but the budget is notdischarged if there are no pending requests.
Keeping the budget improves responsiveness,since jobs can be served within a period.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
1224
RM + Deferrable Server
1
2
ape2
4
0
8
126
0 4 8 1262 10
C1 = 2
C2 = 1
0 4 12
DSCs = 1Ts = 5
Response Time = 3
8
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
5
25
However, DS does not behaves like a periodic task and it ismore invasive than PS.
Keeping the budget decreases the utilization bound.
ape
0 4 8 1262 10
0 4 8
DSCs = 1Ts = 4
Deferrable Server (DS)
There can be two serverexecutions close to each other
26
0 4 8
DSCs = 2Ts = 4
0 4 8 1262 10
0 5 10
1
2
0 4 8 1262 10
0 5 10
ape
2
deadlinemiss
Deferrable Server (DS)
27
Analysis of RM + DS
112
2)(
1
lub
n
s
ss
DSRM
U
UnUnU
DS
1
Cs Cs
C1Ts +Cs 2Ts+Cs
2C2
T1
Cs
T2
2Cs = T1 – Ts
C1 = T2 – T1
C2 = T3 – T2
C1
C2
28
Us
0 1
1
ln2
)(lub nU DSRM
PS
DS
)1()( /1lub n
sPSRM KnUnU
1
2
sPS U
K12
2
s
sDS U
UK
RM + DS schedulability
29
12
2)1(
1
s
sn
ii U
UU
Analysis with Hyperbolic Bound
A set of periodic tasks is schedulable by Rate Monotonic in the presence of a Deferrable Server with utilization Us if
n
iiUP
1
)1(Defining
the maximum server utilization that guarantees theschedulability of the periodic task set is
12
2max
P
PU s
30
Response time under DS
Consider a DS running at the highest priority and anaperiodic job arriving when the server is idle:
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
saa C
CC sasaa CCC
initial delay
ra
12/11/2015
6
31
Response time under DS
If a task arrives close to the next server period, a value ea < Cs
is executed.
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
aaa C
eC saaaa CeC
initial delay
ra
ea = min(a, Cs)In general:
32
Designing server parameters
1. Determine Us using
2. Define Us Us
3. Define Ts = min (T1, …, Tn)
4. Compute Cs = UsTs
max
max
server
n
ii KU
1
)1(
33
Sporadic Server (SS)
It preserves the budget like DS, but it is lessaggressive than DS, since the budget is replenishedonly Ts units after its consumption.
SS is not activated periodically, but from the analysispoint of view it behaves like a period task withcomputation time Cs and period Ts.
ape
0 4 8 1262 10
0 4 8
SSCs = 1Ts = 4
12
2
Ts Ts+1 +1
34
Sporadic Server rules
Rule 1At time tA, at which the following event occurs:
(qs > 0) AND ( pending aperiodic requests)
set the replenishment time in the future at time RT = tA + Ts
Letqs the current server budget
ds the current server deadline
Rule 2At time tI, at which the following event occurs:
(qs 0) OR ( pending aperiodic requests)
set the replenishment amount equal to the budget Cape(tA, tI)consumed in the interval [tA, tI].
36
Total Bandwidth Server (TBS)
It is a dynamic priority server, used along withEDF.
Each aperiodic request is assigned a deadline sothat the server demand does not exceed a givenbandwidth Us .
Aperiodic jobs are inserted in the ready queueand scheduled together with the HARD tasks.
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Response time under DS
If a task arrives close to the next server period, a value ea < Cs
is executed.
ape
full service periods final chunk
Ca
(Cs, Ts)
asaaa TR
a = Ts ra
ra
Ts
1
s
aaa C
eC saaaa CeC
initial delay
ra
ea = min(a, Cs)In general:
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Designing server parameters
1. Determine Us using
2. Define Us Us
3. Define Ts = min (T1, …, Tn)
4. Compute Cs = UsTs
max
max
server
n
ii KU
1
)1(
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Sporadic Server (SS)
It preserves the budget like DS, but it is lessaggressive than DS, since the budget is replenishedonly Ts units after its consumption.
SS is not activated periodically, but from the analysispoint of view it behaves like a period task withcomputation time Cs and period Ts.
ape
0 4 8 1262 10
0 4 8
SSCs = 1Ts = 4
12
2
Ts Ts+1 +1
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Sporadic Server rules
Rule 1At time tA, at which the following event occurs:
(qs > 0) AND ( pending aperiodic requests)
set the replenishment time in the future at time RT = tA + Ts
Letqs the current server budget
ds the current server deadline
Rule 2At time tI, at which the following event occurs:
(qs 0) OR ( pending aperiodic requests)
set the replenishment amount equal to the budget Cape(tA, tI)consumed in the interval [tA, tI].
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Total Bandwidth Server (TBS)
It is a dynamic priority server, used along withEDF.
Each aperiodic request is assigned a deadline sothat the server demand does not exceed a givenbandwidth Us .
Aperiodic jobs are inserted in the ready queueand scheduled together with the HARD tasks.