Uniprocessor Scheduling under Precedence Constraints PARADES - - PDF document

PARADES

Uniprocessor Scheduling under Precedence Constraints

PARADES G.E.I.E.

Massimo Baleani, Alberto Ferrari, Leonardo Mangeruca, Alberto Sangiovanni-Vincentelli

Outline

• The meaning of precedence constraints
• Motivating example
• Conditions for scheduling under precedence

constraints

– General necessary and sufficient conditions – Sufficient conditions for relevant scheduling techniques

• Optimization problems
• Conclusions

3

• Emerging solution to embedded

Emerging solution to embedded software design issues software design issues

• from manual coding and informal

from manual coding and informal specifications specifications … …

• to capturing functional and non

to capturing functional and non-

• functional requirements by

functional requirements by mathematical models mathematical models

Model Based Design Model Based Design

Motivation

τ τi

i

τ τj

j

Tj Ti Tj

τ τi

i

τ τj

j

Tj Ti Tj

Abstract: “zero computation time” Implementation: non-zero computation time

Ti Ti

Producer Consumer

Tj Ti

τ τj

j

τ τi

i

New Precedence Constraints

Producer Consumer

Tj Ti

τ τj

j

τ τi

i

• τj precedes τi: if the Tj event arrives before the Ti event, the

corresponding job of τj completes before the corresponding job of τi starts

• Tasks subjected to precedence constraints may have different periods
• Deterministic communication in both data-flow and event driven systems

τ τi

i

τ τj

j

Tj Ti Tj Ti

Def.: τj(k) precedes τi(h), denoted τj(k) ≤ τi(h), if fj(k) ≤ si(h) aj(k) ≤ ai(h) => τj(k) ≤ τi(h)

τ τi

i

τi(h)

ai(h) aj(k+1)

τ τj

j

τi(h+1) τj(k+1) τj(k) τj(k+2)

ai(h) ≤ aj(k+1) => τi(h) ≤ τj(k+Blen)

Motivating example

Data is still in the buffer when Consumer executes

Producer Consumer

Tj Ti

τ τj

j

τ τi

i

BUFFER

BUFFER

General Precedence Constraints

ai(h) ≤ aj(k) => τi(h-di,j) ≤ τj(k+ei,j) aj(k) ≤ ai(h) => τj(k-dj,i) ≤ τi(h+ej,i)

τ τi

i

τi(h)

ai(h)

τ τj

j

τj(k+1) τi(h+1) τj(k) τj(k+2)

ai(h) ≤ aj(k) => τi(h) ≤ τj(k+1) aj(k) ≤ ai(h) => τj(k) ≤ τi(h)

aj(k)

Deriving the Precedence Deriving the Precedence Conditions Conditions

Notation: offset, idle, release inter-release time

Oi,j(h,k) ri(h) rj(k)

τ τj

j

τ τi

i

τi(h) τi(h+1) τj(k) τj(k+1)

si(h) Ii(h) fi(h) Ri(h) Ti(h,h+1) ri(h+1)

Precedence Condition: An Example

τ τi

i

τi(h)

τ τj

j

τj(k) τj(k+2) τj(k+1)

Ri Tj

τi(h) ≤ τj(k+2) if 2*Tj + Oi,j ≥ Ri

Ri is an upper bound

• n the response time

Tj is a lower bound

• n the period

Oi,j Oi,j is a lower bound

• n the offset

Tj

Sufficient condition

τ τj

j

τ τi

i

Oi,j(h,k) Ij(k+1) Ri(h-1) Ti(h-1,h) sj(k+1) fi(h-1) Tj(k,k+1)

Ti(h-1,h) - Ri(h-1) + Oi,j(h,k) + Tj(k,k+1) + Ij(k+1) ≥ 0

Precedence Relation

Def: τi(h-1) ≤ τj(k+1) fi(h-1) ≤ sj(k+1) fi(h-1) ≤ sj(k+1)

• τi(h-1)

τj(k+1)

Modeling the Scheduling Modeling the Scheduling Policy Policy

The priority relation

• Priority function: Pi(h,t)

– priority of job τi(h) at time t

• Priority relation: pi,j(h,k)

– pi,j(h,k)=1 if Pi(h,t) > Pj(k,t) for all t – pi,j(h,k)=0 otherwise pi,j(h,k)=1 implies that τj(k) cannot preempt τi(h) Formally: pi,j(h,k)=1 and ri(h) ≤ sj(k) => τi(h) ≤ τj(k)

Precedence Constraint with Priorities

ri(h-1) ≤ sj(k+1)

• τi(h-1)

pi,j(h-1,k+1)=1 and ri(h-1) ≤ sj(k+1) => τi(h-1) ≤ τj(k+1)

τ τj

j

τ τi

i

Oi,j(h,k) Ij(k+1) Ti(h-1,h) sj(k+1) ri(h-1) Tj(k,k+1)

τj(k+1)

Ti(h-1,h) + Oi,j(h,k) + Tj(k,k+1) + Ij(k+1) ≥ 0

Precedence Constraints: Necessary and Sufficient Condition

Ti(h-di,j,h) - Ri(h-di,j) + Oi,j(h,k) + Tj(k,k+ei,j) + Ij(k+ei,j) ≥ 0 pi,j(h-di,j,k+ei,j)=1 and Ti(h-di,j,h) + Oi,j(h,k) + Tj(k,k+ei,j) + Ij(k+ei,j) ≥ 0

with priorities: without priorities:

Ti(h-di,j,h) - [1-pi,j(h-di,j,k+ei,j)]*Ri(h-di,j) + Oi,j(h,k) + Tj(k,k+ei,j) + Ij(k+ei,j) ≥ 0

joining the two conditions together:

Application to Relevant Application to Relevant Scheduling Techniques Scheduling Techniques

Fixed Priority Scheduling: Sufficient Condition

di,j*Ti – (1-pi,j)*Ri + Oi,j+ ei,j*Tj + Ij ≥ 0 Ti(h-di,j,h) - [1-pi,j(h-di,j,k+ei,j)]*Ri(h-di,j) + Oi,j(h,k) + Tj(k,k+ei,j) + Ij(k+ei,j) ≥ 0

General necessary and sufficient condition: Sufficient condition:

• Ri ≥ Ri(h-di,j): upper bound on the response time
• Ti ≤ Ti(h-1,h): lower bound on the inter-release time
• Ij ≤ Ij(k): lower bound on idle time
• Oi,j ≤ Oi,j(h,k): lower bound on offset
• pi,j = pi,j(h,k): fixed priority relation

EDF Scheduling: Sufficient Condition

Di < Dj + di,j*Ti + Oi,j+ ei,j*Tj and di,j*Ti + Oi,j + ei,j*Tj + Ij ≥ 0

• Di: relative deadline
• di(h) = ri(h) + Di: absolute deadline
• Ti: lower bound on the inter-arrival time
• Ij: lower bound on the idle time
• Oi,j: lower bound on the offset

pi,j(h-di,j,k+ei,j)=1 di(h-di,j) < dj(k+ei,j) pi,j(h-di,j,k+ei,j)=1 and Ti(h-di,j,h) + Oi,j(h,k) + Tj(k,k+ei,j) + Ij(k+ei,j) ≥ 0 di(h-di,j) < dj(k+ei,j)

Optimization Problems Optimization Problems

Fixed Priority Scheduling

di,j*Ti – (1-pi,j)*Ri + Oi,j+ ei,j*Tj + Ij ≥ 0 Ri ≤ Di di,j ≤ Di,j and ei,j ≤ Ei,j GIVEN: Ti, Oi,j, Ij, Di FIND: pi,j, di,j and ei,j MINIMIZE COST: ∑i,j(Ai,j * di,j + Bi,j * ei,j) SUCH THAT: NOTE: Ri is an implicit function of pi,j

precedence schedulability functional

EDF Scheduling

GIVEN: Ti, Di, Oi,j, Ij, Dj FIND: ai, aj, di,j and ei,j LET: b be a “small” number Di – b*ai < Dj – b*ai + di,j*Ti + Oi,j+ ei,j*Tj di,j*Ti + Oi,j + ei,j*Tj + Ij ≥ 0 di,j ≤ Di,j and ei,j ≤ Ei,j SUCH THAT: MINIMIZE COST: ∑i,j(Ai,j * di,j + Bi,j * ei,j)

Conclusions

• Generalization of precedence constraints

– Arising from embedded systems design problems

• General necessary and sufficient conditions

– Scheduling algorithm accounted for by priority relation function

• Sufficient conditions for relevant scheduling techniques

– Fixed priority, EDF as special cases of general conditions

• Optimization problems for scheduling embedded systems

under precedence and schedulability constraints

– Minimization of buffers for deterministic communications

• Future works

– Efficient algorithms for optimization problems – Extension to multiprocessor and distributed computation

Thanks for Thanks for Your Your Attention Attention

Notation

ri(h) si(h) fi(h)

τ τj

j

τ τi

i

ai(h)

τi(h) τi(h+1) τj(k) τj(k+1)

Precedence Relation

si(h)

τ τj

j

τ τi

i

τi(h) τj(k)

Def.: τj(k) precedes τi(h), denoted τj(k) ≤ τi(h), if fj(k) ≤ si(h)

fj(k)

aj(k) ≤ ai(h) => τj(k) ≤ τi(h)

τ τi

i

τi(h)

ai(h) aj(k+1)

τ τj

j

τi(h+1)

Motivating example

Data is still in the buffer when Consumer executes

Producer Consumer

Tj Ti

τ τj

j

τ τi

i

BUFFER

ai(h) ≤ aj(k) => τi(h) ≤ τj(k+Blen-1)

τj(k) τj(k-1) τj(k+1)

BUFFER

Bounding the Inter-release Time

τ τi

i

Ti(h-2,h-1) Ti(h-1,h)

τ τi

i(h

(h-

• 2)

2)

τ τi

i(h

(h-

• 1)

1)

τ τi

i(h)

(h) 2*Ti ≤ Ti(h-2,h)

• Ti ≤ Ti(h-1,h): lower bound on the inter-release time
• Ti ≤ Ti(h-1,h) => di,j*Ti ≤ Ti(h-di,j,h)