資工系網媒所 NEWS 實驗室 Paper Discussion C. L. Liu and James W. Layland, "Scheduling Algorithms for Multiprogramming in a Hard-Real- Time Environment", JACM, Vol

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Paper Discussion

C. L. Liu and James W. Layland, "Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment", JACM, Vol 20, No. 1, pp. 46--61, 1973

Chihg-Chih Han, Kwei-Jay Lin and Chao-Ju Hou, "Distance-Constrained Scheduling and Its Applications to Real-Time Systems", IEEE Trans. on Computers, Vol 45, No. 7, pp. 814-826 1996

http://www.cs.ccu.edu.tw/~chsueh/courses/4105750_RTS/papers/p46-liu.pdf




Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment

No non-time-critical jobs

Pure Process Control

C. L. Liu and James W. Layland, "Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment", JACM, Vol 20, No. 1, pp. 46--61, 1973


Assumptions

(A1) The requests for all tasks for which hard deadlines exist are periodic, with constant interval between requests.

(A2) Deadlines consist of run-ability constraints only--i.e, each task must be completed before the next request for it occurs.

(A3) The tasks are independent in that requests for a certain task do not depend on the initiation or the completion of requests for other tasks.

(A4) Run-time for each task is constant for that task and does not vary with time. Run-time here refers to the time which is taken by a processor to execute the task without interruption.

(A5) Any nonperiodic tasks in the system are special; they are initialization or failure-recovery routines; they displace periodic tasks while they themselves are being run, and do not themselves have hard, critical deadlines.


Real-Time Task Model

Periodic Task Model

parameters are known a priori

worst-case execution time

period

request time ready time deadline start finish preempt resume

Predictability vs. Schedulability

high predictability and schedulability

easy-to-check schedulability condition


Time-Driven Scheduling Approachwidely used

inflexible, efficient

large space needede.g. cyclic executive

Priority-Driven Scheduling Approachdynamic-priority

e.g. earliest deadline first

fixed-priority e.g. rate monotonic

0 3 6

0 3 6

Scheduling Approaches


Terms in Scheduling

job: an instance of a task

deadline: next request time of the same task

overflow at time t: t is a deadline of unfulfilled request

feasible: scheduled with no overflow (schedulable)

response time: time span from request to finish

critical instance: an instance when a request will have the largest response time

critical time zone: time interval from a critical instance to its finish time


A Fixed Priority Scheduling Algorithm

THEOREM 1. A critical instant for any task occurs whenever the task is requested simultaneously with requests for all higher priority tasks.

A scheduling algorithm is feasible if all tasks can be fulfilled at their critical instants.


A Fixed Priority Scheduling Algorithm (cont.)

THEOREM 2. If a feasible priority assignment exists for some task set, the rate-monotonic priority assignment is feasible for that task set.


Optimality

THEOREM 2. If a feasible priority assignment exists for some task set, the rate-monotonic priority assignment is feasible for that task set.



THEOREM 3. For a set of two tasks with fixed priority assignment, the least upper bound to the processor utilization factor is U = 2 (2 ½ - 1).

case 1

case 2

T1

T2

T1

T2



The minimum U occurs at the boundary of two cases

T1

T2

T1

T2


Least Upper Bound

An easy-to-check schedulability condition

1

)12( /1 mm

n=2, 0.83

n=3, 0.78

n=, 0.69

)/(1

i

m

ii TCU



THEOREM 4. For a set of m tasks with fixed priority order, and the restriction that the ratio between any two request periods is less than 2, the least upper bound to the processor utilization factor is U = m(2 1/m - 1).find optimal conditions

when optimality exists or if it is optimal and condition A does not hold

conflicts, so A must be true.

solve equations at optimal conditionderive bound

relax assumption


T1

T2

T1

T2

T1

T2

T1

T2




THEOREM 5. For a set of m tasks with fixed priority order, the least upper bound to processor utilization is U = m(2 1/m - 1).

The bound of special case is tighter than general case.Rate Monotonic Priority Assignment

The smaller the period, the higher the priority.Rate Monotonic Scheduling Algorithm

generate the whole scheduleon-lineoff-line

priority queue100%?


A Deadline Driven Scheduling Algorithm

THEOREM 6. When the deadline driven scheduling algorithm is used to schedule a set of tasks on a processor, there is no processor idle time prior to an overflow.


A Deadline-Driven Scheduling Algorithm (cont.)

THEOREM 7. For a given set of m tasks, the deadline driven scheduling algorithm is feasible if and only if U 1.

necessary condition (only if)if feasible, the computation demand processor time

sufficient conditionif U 1,

and not feasible

case 10 T T1 T2…Tm

all bi beyond T


Sufficient Condition (cont.)

case 2some bi were carried out before T


A mixed Scheduling Algorithm?

資工系網媒所 NEWS實驗室

21

video serverinter-frame distance must be less than 33ms

robot arm movementneeds steady and smooth operations

satellite/cellular phone trackingservice in constant time interval

more predictable inter-execution time than the periodic task(PT) model

worst inter-execution time for PT is 2xperiod-ei

Distance-Constrained Task Model

Chihg-Chih Han, Kwei-Jay Lin and Chao-Ju Hou, "Distance-Constrained Scheduling and Its Applications to Real-Time Systems", IEEE Trans. on Computers, Vol 45, No. 7, pp. 814-826, 1996


Pinwheel Problem

Given a multiset A={a1, a2, ... ,an}, ai Î N

Find an infinite sequence of symbols from {1, 2, ... , n} such that at least one symbol i within any interval of ai symbols

A={2,3} ® 1 2 1 2 1 2 ...® 1 1 2 1 1 2 ...

A unit-time schedule


Pinwheel schedule 0 1 2 3 4 5 6 7 8 9 10 11 12

01

4

56

7

8

9

10

11

3

2

12

T1 (1,6)

T2 (3,6)

T4 (2,12)

T3 (2,12)

6

T2

T4T1

T1

T3

T2

6 6 6 6 6 6

Pinwheel Schedule


Pinwheel Transformation

Pinwheel

any numbers ® multiples

jitterless, predictable

Signalanalog ® digital

noiseless, ease of control

e.g. CD music

2E.g. A={3,4,6,13} ® {3} B ={3,3,6,12} ={3,3,3×21, 3×22}• single-number reduction

30 6 9

A

B

12

(RM)

(PW)


Pinwheel Scheduling Properties

Good jitter control in schedules

Easy-to-check schedulability

Predictable resource accesses

Easy extension for aperiodic tasks

Predictable end-to-end delay

Network scheduling (INFOCOM’95 by Han & Shin)

Flexible time-driven approach


Pinwheel Scheduler

Given a multi-set of distance constraints of n tasks

A={a1, a2, ... ,an}, density r( ) = A Si (1/ai )

Find another B={b1, b2, ... ,bn}, "i, bi £ ai ,

A is schedulable if B is schedulable.

If bi’s are multiples (bi|bj) and r( ) B £ 1,

B is schedulable.

E.g. A={3,4,6,7} ®(specialized) B ={3,3,6,6}

schedule is “1 2 3 1 2 4 1 2 3 1 2 4 ...”

The goal is to minimize density increase

資工系網媒所 NEWS實驗室27

Previous Works

Sa:x=a1 , dSa ³ 0.5

Sx: a1 /2 < x £ a1 , dSx ³ 0.65

S23:use 2 and 3, dS23 7/12 ³ @ 0.58

Sbc: b=a1 , b £ c < 2b

dSbc 2/3 ³ @ 0.67

CX: the set of all instances schedulable by Scheduler x

CS23

CSx

CSa

CSbc

dX: the schedulable density bound for Scheduler x


Single-node Pinwheel Scheduling

Earlier work: use a base of 2 , find x in (a1/2 , a1]

Sa, Sx, Sbc, Sby, Sxy: 0.5, 0.65, 2/3, …distance constraints of integer, unit execution time

Sr: use a base of 2 , find r in (a1/2 , a1]

real-number distance constraints and execution times

Our contributions:Srg: use a base of g

Srb: try all bases (optimal for single-number reduction)

HSr: near-optimal polynomial-time heuristic

Integer Pinwheel Scheduling: Sxg, Sxb , HSx


Various Pinwheel Schedulers

distance constraint

g=2 optimal g near optimal

heuristic

x is an integer

Sx Sxb HSx

r is a real number

Sr Srb HSr


Multiple-Node Scheduling

All nodes are specialized with the same base using DSr (Distributed Sr Scheduling)

Pinwheel phase alignment

Minimize the total end-to-end delay

Minimize the maximum end-to-end delay

Pinwheelj Pinwheel1 Pinwheel2 Pinwheelm ...

Nodej N1 N2 Nm ...

...


a = {(1,2),(1,3)}

b = {(1,2.1),(2.1,6)}

c = {(1,1.9),(1,4)}

d = {(1,2),(2.1,6)}

e = {(1,1.9),(2.1,6)}

f = {(1,2),(1.1,3)}

: Jitterless

Relationships of Pinwheel Schedulers

CX: the set of all instances schedulable by Scheduler x

CSrb

CSxCSr CSxb

a.

b.

e.

d.c.

f.


Synchronous Scheduling

Intranode: pinwheel scheduling makes task executions regular and predictable, i.e. no jitter.

Internode: in all nodes, transform the periods of all tasks to harmonic numbers so that tasks of the same period are synchronized.

The worst case delay for a task between its arrival and the beginning of its execution on a node is one period.

Nodej N1 N2 Nm ...

...




Simplification


Study of Pinwheel Scheduling

n tasks are schedulable on a node (using HSr)

if the total utilization is £ n (21/n

- 1).

n tasks are schedulable on all nodes (using DSr)

if the utilization on any node is £ 21/n-1

.

Algorithm of O(n log n + nl) to minimize the total or maximum task delay between two nodes with l different periods.

Algorithm of O(mn log n + mnl) to minimize the total end-to-end delay on m nodes.

Polynomial heuristics to reduce the maximum end-to-end delay on m nodes.


Time-driven Scheduling Approach

Off-lineconstruct complete schedule

exponential time and space

generate effective time parametersstart time, resume time, finish time

optimization

On-lineexecute tasks according to off-line generated schedule or parameters, inflexible

0 3 6


SSr: Pinwheel Schedule Constructor

Generate start times and finish times in O(n2)

task5

5.30 10.6 15.9

task1

21.2

task2

task3

task4


Generate effective pinwheel schedule.

Use an on-line scheduler to schedule tasks according to the effect schedule.

Time-Driven Pinwheel Scheduling

Generate start times and finish times using SSr

Next resume time can be found in O(n) the latest finish time of the preempting tasks

task1

task2

task3

task4


TOPS:Time-driven On-line Pinwheel Scheduler

schedulerTOPS-c

(constant)TOPS-l(linear)

TOPS-f(flexible)

TOPS-m(mix)

off-linegenerates

S*,F*,R* S,FS,F,

priorityon-line

generatesR S,F,R

on-line timecomplexity

O(1) O(n) O(n)** O(log n)

spacecomplexity

O(bn/b1) O(n) O(n) O(n)

*S:start time, F:finish time, R:resume time; * * amortized


Pinwheel Transformation

Sr: d ={(1,2),(2.1,6)} ®{(1,2),(2.1,4)}

20 4 6

overflow

Sxb: d ={(1,2),(2.1,6)} ®{(1,2),(2.1,6)}

8


PX: the set of all periodic task sets schedulable by Scheduler xCX: the set of all DCTS’s schedulable by Scheduler x

a = {(1,2),(1,3)}

b = {(1,2),(0.5,3),(1,7)}

c = {(1,2),(1,3),(1,7)}

d = {(1,2),(1.1,3)}

e = {(1,2),(2,3)}

PEDF

PRM

a.

b.

c.d.

e.

CSrb

PRM = CSrb

n £ 2

Relationship between CSrb, PRM , and PEDF


a = {(1,2),(1,3)}b = {(1,2.1),(2.1,6)}c = {(1,1.9),(1,4)}d = {(1,2),(2.1,6)}e = {(1,1.9),(2.1,6)}f = {(1,2),(1,3),(1,7)}g = {(1,2),(1.1,3)} : Jitterless

Relationships of SchedulersCX: the set of all instances schedulable by Scheduler x

CRM

CSrb

CSxCSr CSxb

a.

b.

e.

d.c.

f.g.

資工系網媒所 NEWS實驗室44

Examples of Schedulers

CRM

CSrb

CSx

CSr CSxba.

b. d.c.

a = {(1,2),(1,3)}, 0.83r @

[Sx] = {(1,2),(1,2)}, = 1Fb = {(1,2.1),(2.1,6)}, 0.83r @

[Sx] = {(1,2),(2.1,4)}, 1.03F @[Sr] = {(1,2.1),(2.1,4.2)}, F @

0.98[Sxb] = {(1,2),(2.1,6)} , = 0.85F

c = {(1,1.9),(1,4)}, 0.78r @

[Sr] = {(1,1.9),(1,3.8)} , 0.79F @[Sxb] = {(1,1),(1,4)} , = 1.25F

d = {(1,2),(2.1,6)}, = 0.85r

[Sr] = {(1,1.5),(2.1,6)}, F @ 1.02[Sxb] = {(1,2),(2.1,6)}, = 0.85F


More Examples of Schedulers

e = {(1,1.9),(2.1,6)}, 0.88r @[Sr] ={(1,1.5),(2.1,6)}, 1.02F @

[Sxb] ={(1,1),(2.1,6)}, = 1.35F

[Srb] ={(1,1.9),(2.1,5.7)}, 0.89F @

f = {(1,2),(1,3),(1,7)}, r @0.97[Srb] = {(1,2),(1,2),

(1,6)}, 1.17F @

g = {(1,2),(1.1,3)}, 0.87r @

CRM

CSrb

CSx

CSrCSxb

e.

f.g.


Comparison of Schedulers

Scheduler EDF RM Srb

schedulabilitycondition

1 n (21/n - 1) n (21/n - 1)

jitter high low* none

end-to-end delay long short shortest

resource reservation difficult easy* easy

worst caseblocking time

large large small

systemoverload

unstable stable stable

*for higher priority tasks


Priority Inversion


Priority Inheritance


Priority Inheritance (cont.)

m lower priority task access k distinct semaphoresA job can be blocked at most min(m,k) critical sections

Might have deadlocks

Chain of blockingJ1 access S1, S2 held by J2, J3


Priority Ceiling Protocol

Inherit the Priority Ceiling of a semaphore Priority Ceiling of a semaphore S

The highest priority of the jobs may lock S.

Zi,j,k is the kth critical section in Ji guarded by Sj

Idea came from:When J1 preempts Z2,1, the priority of J1 will be strictly higher than the priorities of all the preempted critical sections, otherwise J1 is blocked and J2 inherits J1’s priority.

A job can be blocked for at most one duration of one critical section.

i, 1 i n, C1/T1+C2/T2+…+Ci/Ti+Bi/Ti i(21/i-1)


Example 2

J1

J0

J20 1 2 3 4 5 6 7 8 9 1011121314 15


Multiprocessor PCP

Static binding vs. dynamic binding

Global semaphore vs. local semaphore

Global critical section vs. local semaphore

Remote blocking

11

12 ,)1(m



Documents

資工系網媒所 NEWS 實驗室 Paper Discussion C. L. Liu and James W. Layland, "Scheduling Algorithms for Multiprogramming in a Hard-Real- Time Environment", JACM, Vol