Uniprocessor Scheduling under Precedence Constraints PARADES - PDF document

Uniprocessor Scheduling under Precedence Constraints PARADES PARADES G.E.I.E. Massimo Baleani, Alberto Ferrari, Leonardo Mangeruca, Alberto Sangiovanni-Vincentelli PARADES 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

Model Based Design Model Based Design PARADES � 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 Key Belt Alarm Generation MEM Process Process Process Belt Generation Process Timer MEM Process 3 PARADES Motivation T j T i τ j τ j Producer Consumer τ i τ τ j τ i τ τ i j i Abstract: “zero T j T j computation time” T i T i τ j τ j τ i τ i Implementation: non-zero T j T j computation time T i T i

PARADES New Precedence Constraints • τ j precedes τ i : if the T j event arrives before the T i event, the corresponding job of τ j completes before the corresponding job of τ i starts • Tasks subjected to precedence constraints may have different periods • Deterministic communication in both data-flow and event driven systems τ j τ T j T i j τ i τ i Producer Consumer τ j τ i τ τ j i T j T j T i T i Def.: τ j (k) precedes τ i (h), denoted τ j (k) ≤ τ i (h), if f j (k) ≤ s i (h) a j (k) ≤ a i (h) => τ j (k) ≤ τ i (h) PARADES Motivating example T j T i Data is still in the buffer when Consumer executes BUFFER Producer Consumer a i (h) ≤ a j (k+1) => τ i (h) ≤ τ j (k+B len ) τ j τ τ i τ j i BUFFER τ j (k) τ j (k+1) τ j (k+2) τ j τ j τ i τ i τ i (h) τ i (h+1) a i (h) a j (k+1)

PARADES General Precedence Constraints a i (h) ≤ a j (k) => τ i (h-d i,j ) ≤ τ j (k+e i,j ) a j (k) ≤ a i (h) => τ j (k-d j,i ) ≤ τ i (h+e j,i ) τ j (k) τ j (k+1) τ j (k+2) τ j τ j τ i τ i τ i (h) τ i (h+1) a j (k) a i (h) a i (h) ≤ a j (k) => τ i (h) ≤ τ j (k+1) a j (k) ≤ a i (h) => τ j (k) ≤ τ i (h) PARADES Deriving the Precedence Deriving the Precedence Conditions Conditions

PARADES Notation: offset, idle, release inter-release time τ j (k) τ j (k+1) r i (h) r j (k) s i (h) f i (h) r i (h+1) τ τ j j τ τ i i O i,j (h,k) τ i (h) τ i (h+1) I i (h) R i (h) T i (h,h+1) PARADES Precedence Condition: An Example τ j (k) τ j (k+1) τ j (k+2) τ j τ j τ i τ i τ i (h) R i O i,j T j T j O i,j is a lower bound R i is an upper bound T j is a lower bound on the offset on the response time on the period τ i (h) ≤ τ j (k+2) if 2* T j + O i,j ≥ R i Sufficient condition

PARADES Precedence Relation Def: τ i (h-1) ≤ τ j (k+1) � f i (h-1) ≤ s j (k+1) f i (h-1) s j (k+1) τ j τ R i (h-1) j τ i τ i τ j (k+1) T i (h-1,h) O i,j (h,k) T j (k,k+1) I j (k+1) τ i (h-1) f i (h-1) ≤ s j (k+1) � T i (h-1,h) - R i (h-1) + O i,j (h,k) + T j (k,k+1) + I j (k+1) ≥ 0 PARADES Modeling the Scheduling Modeling the Scheduling Policy Policy

PARADES The priority relation • Priority function: P i (h,t) – priority of job τ i (h) at time t • Priority relation: p i,j (h,k) – p i,j (h,k)=1 if P i (h,t) > P j (k,t) for all t – p i,j (h,k)=0 otherwise p i,j (h,k)=1 implies that τ j (k) cannot preempt τ i (h) Formally: p i,j (h,k)=1 and r i (h) ≤ s j (k) => τ i (h) ≤ τ j (k) PARADES Precedence Constraint with Priorities p i,j (h-1,k+1)=1 and r i (h-1) ≤ s j (k+1) => τ i (h-1) ≤ τ j (k+1) r i (h-1) s j (k+1) τ j τ j τ i τ i τ j (k+1) T i (h-1,h) O i,j (h,k) T j (k,k+1) I j (k+1) τ i (h-1) r i (h-1) ≤ s j (k+1) � T i (h-1,h) + O i,j (h,k) + T j (k,k+1) + I j (k+1) ≥ 0

PARADES Precedence Constraints: Necessary and Sufficient Condition without priorities: T i (h-d i,j ,h) - R i (h-d i,j ) + O i,j (h,k) + T j (k,k+e i,j ) + I j (k+e i,j ) ≥ 0 with priorities: p i,j (h-d i,j ,k+e i,j )=1 and T i (h-d i,j ,h) + O i,j (h,k) + T j (k,k+e i,j ) + I j (k+e i,j ) ≥ 0 joining the two conditions together: T i (h-d i,j ,h) - [1-p i,j (h-d i,j ,k+e i,j )]*R i (h-d i,j ) + O i,j (h,k) + T j (k,k+e i,j ) + I j (k+e i,j ) ≥ 0 PARADES Application to Relevant Application to Relevant Scheduling Techniques Scheduling Techniques

PARADES Fixed Priority Scheduling: Sufficient Condition General necessary and sufficient condition: T i (h-d i,j ,h) - [1-p i,j (h-d i,j ,k+e i,j )]*R i (h-d i,j ) + O i,j (h,k) + T j (k,k+e i,j ) + I j (k+e i,j ) ≥ 0 • R i ≥ R i (h-d i,j ): upper bound on the response time • T i ≤ T i (h-1,h): lower bound on the inter-release time • I j ≤ I j (k): lower bound on idle time • O i,j ≤ O i,j (h,k): lower bound on offset • p i,j = p i,j (h,k): fixed priority relation Sufficient condition: d i,j *T i – (1-p i,j )*R i + O i,j + e i,j *T j + I j ≥ 0 PARADES EDF Scheduling: Sufficient Condition p i,j (h-d i,j ,k+e i,j )=1 and T i (h-d i,j ,h) + O i,j (h,k) + T j (k,k+e i,j ) + I j (k+e i,j ) ≥ 0 • D i : relative deadline • d i (h) = r i (h) + D i : absolute deadline • T i : lower bound on the inter-arrival time • I j : lower bound on the idle time • O i,j : lower bound on the offset p i,j (h-d i,j ,k+e i,j )=1 � d i (h-d i,j ) < d j (k+e i,j ) d i (h-d i,j ) < d j (k+e i,j ) D i < D j + d i,j *T i + O i,j + e i,j *T j and d i,j *T i + O i,j + e i,j *T j + I j ≥ 0

PARADES Optimization Problems Optimization Problems PARADES Fixed Priority Scheduling GIVEN : T i , O i,j , I j , D i FIND : p i,j , d i,j and e i,j SUCH THAT : d i,j *T i – (1-p i,j )*R i + O i,j + e i,j *T j + I j ≥ 0 precedence schedulability R i ≤ D i functional d i,j ≤ D i,j and e i,j ≤ E i,j ∑ i,j (A i,j * d i,j + B i,j * e i,j ) MINIMIZE COST : NOTE : R i is an implicit function of p i,j

PARADES EDF Scheduling GIVEN : T i , D i , O i,j , I j , D j LET : b be a “small” number FIND : a i , a j , d i,j and e i,j SUCH THAT : D i – b*a i < D j – b*a i + d i,j *T i + O i,j + e i,j *T j d i,j *T i + O i,j + e i,j *T j + I j ≥ 0 d i,j ≤ D i,j and e i,j ≤ E i,j ∑ i,j (A i,j * d i,j + B i,j * e i,j ) MINIMIZE COST : PARADES 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

PARADES Thanks for Thanks for Your Your Attention Attention PARADES Notation τ j (k) τ j (k+1) r i (h) s i (h) f i (h) a i (h) τ j τ j τ i τ i τ i (h) τ i (h+1)

PARADES Precedence Relation τ j (k) s i (h) f j (k) τ τ j j τ τ i i τ i (h) Def.: τ j (k) precedes τ i (h), denoted τ j (k) ≤ τ i (h), if f j (k) ≤ s i (h) a j (k) ≤ a i (h) => τ j (k) ≤ τ i (h) PARADES Motivating example T j T i Data is still in the buffer when Consumer executes BUFFER Producer Consumer a i (h) ≤ a j (k) => τ i (h) ≤ τ j (k+B len -1) τ j τ τ i τ j i BUFFER τ j (k-1) τ j (k) τ j (k+1) τ j τ j τ i τ i τ i (h) τ i (h+1) a i (h) a j (k+1)

PARADES Bounding the Inter-release Time • T i ≤ T i (h-1,h): lower bound on the inter-release time • T i ≤ T i (h-1,h) => d i,j *T i ≤ T i (h-d i,j ,h) τ i τ τ τ i τ i τ i (h (h- -2) 2) i (h (h- -1) 1) i (h) (h) τ i τ i T i (h-2,h-1) T i (h-1,h) 2*T i ≤ T i (h-2,h)