multiprocessor scheduling real-time systems · eda222/dit161 – real-time systems, chalmers/gu,...

EDA222/DIT161 – Real-Time Systems, Chalmers/GU, 2010/2011 Lecture #15 Updated February 27, 2011

1

Implementation

Specification

•  Dynamic scheduling -- Earliest-deadline-first scheduling •  Processor-demand analysis

Task Ci Di Ti

1 1 1

1 2 3

2 4 8

Simulate an execution of the tasks:

The tasks are not schedulable even though

U =

12

+14

+18

=78

= 0.875 < 1

misses its deadline!

t 0 6 8 4 2


2

•  Let represent the number of instances of that must complete execution before .

•  The total processor demand up to is

CP (0, L) = Ni

LCii=1

n

∑

0, L − Di⎡⎣ ⎤⎦

•  We can ignore instances of the task that arrived during the interval since for these instances. Di > L

L − Di , L⎡⎣ ⎤⎦

t 0 L

Instance with Di > L

•  The total processor demand is thus

Ni

L =L − Di

Ti

⎢⎣⎢

⎥⎦⎥

+ 1

CP (0, L) =

L − Di

Ti

⎢⎣⎢

⎥⎦⎥

+ 1⎛⎝⎜

⎞⎠⎟

Cii=1

n

∑

Di ≤ Ti

∀L : CP (0, L) ≤ L

CP (0, L)


3

∀L ∈K : CP (0, L) ≤ L

K = Dik | Di

k = kTi + Di , Dik ≤ LCM T1,…,Tn{ }, 1 ≤ i ≤ n, k ≥ 0{ }

Task Ci Di Ti

2 2 3

3 7 12

4 8 16

t1

H blocked

t2 t

t H

t M

normal execution

critical region

L

H and L share resource R

H misses its deadline


4

The tasks are assigned preemption levels, the properties of which are: –  The preemption level of task is denoted –  Task is not allowed to preempt another task unless –  If has higher priority than and arrives later, then must

have a higher preemption level than .

τ i π i

τ i τ j π i > π jτ i τ j τ iτ j

π i > π j ⇔ Di < Dj

system-wide ceiling is set to R’s resource ceiling (= H’s preemption level)

system-wide ceiling is restored H blocked

H

M

normal execution

critical region

preemption level (H) > preemption level (M) > preemption level (L)

L

H and L share resource R

t

t

t

Again: note the similarity to the behavior of ICPP CP

i =L − Dk

Tk

⎢⎣⎢

⎥⎦⎥

+ 1⎛⎝⎜

⎞⎠⎟

Ckk =1

i

∑ +L − Di

Ti

⎢⎣⎢

⎥⎦⎥

+ 1⎛⎝⎜

⎞⎠⎟

Bi

π i > π j ⇔ i < j

∀L ∈K ,∀i ∈ 1,n[ ] : CPi (0, L) ≤ L


5

τ i

1. Determine the worst-case resource ceiling for each critical region, that is, assume the run-time situation where the corresponding resource is unavailable.

2. Identify the tasks that have a preemption level lower than and that calls critical regions with a worst-case resource ceiling equal to or higher than the preemption level of . τ i

τ i

3. Consider the times that these tasks lock the actual critical regions. The longest of those times constitutes the blocking factor .

R1 R2

Task Ci Di Ti

6 10

50 7 17 50 10 25

50 HR1 HR2

- 2 3 2

1 2

HR3

2 2 2

- 1 1 3

2 1 1

3 1

μR1 μR2 μR3 Task Ci Di Ti

R3

τ 2

τ 2

Examine the schedulability of the tasks when the SRP (Stack Resource Policy) protocol is used.

a) Derive the ceilings (dynamic and worst-case) of the resources. b) Derive the blocking factors for the tasks. c) Show whether the tasks are schedulable or not.

U ≤ n(21/ n − 1)

U ≤ 1

∀i : Ri = Ci +

Ri

Tj

⎡

⎢⎢

⎤

⎥⎥ C j ≤ Di

∀j∈hp( i)∑

∀L :

L − Di

Ti

⎢⎣⎢

⎥⎦⎥

+ 1⎛⎝⎜

⎞⎠⎟i=1

n

∑ Ci ≤ L

Di = Ti Di ≤ Ti

multiprocessor scheduling real-time systems · eda222/dit161 – real-time systems, chalmers/gu,...

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