A Theory of Rate-Based Execution A Theory of Rate-Based Execution - - PDF document

1 1

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

http://www.cs.unc.edu/Research/Dirt/

Kevin Jeffay

Department of Computer Science Department of Computer Science University of North Carolina University of North Carolina at Chapel Hill at Chapel Hill jeffay@ jeffay@cs cs. .unc unc. .edu edu

Steve Goddard

Computer Science & Engineering Computer Science & Engineering University of Nebraska Ð Lincoln University of Nebraska Ð Lincoln goddard goddard@ @cse cse. .unl unl. .edu edu

2 2

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

WhatÕs wrong with the Liu & WhatÕs wrong with the Liu & Layland Layland model? model?

¥ ¥ Loosely speaking, Loosely speaking, nothing is periodic or nothing is periodic or sporadic in a sporadic in a distributed system distributed system

Acquire Display

Display Initiation Time

Display Network Reception

¥ ¥ The essential problem The essential problem seems to be the seems to be the requirement that the requirement that the arrival process be arrival process be somehow constrained somehow constrained

3 3

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

Goals Goals

¥ ¥ Extend the Liu and Extend the Liu and Layland Layland theory of real-time theory of real-time processor scheduling to: processor scheduling to: Ð ÐSupport notions of execution rate that are Support notions of execution rate that are more general than periodic or sporadic more general than periodic or sporadic execution execution Ð ÐSupport integrated real-time device and Support integrated real-time device and application processing application processing Ð ÐSupport responsive non-real-time computing Support responsive non-real-time computing

4 4

Rate-Based Execution Rate-Based Execution

Concept Concept ¥ ¥ Schedule tasks at the Schedule tasks at the average average rate rate at which they are at which they are expected to be invoked expected to be invoked Ð Ð Make buffering a first-class concept in the model Make buffering a first-class concept in the model Ð Ð Understand the fundamental relationships between Understand the fundamental relationships between feasibility, response time, and processing rate feasibility, response time, and processing rate ¥ ¥ Develop a model of tasks wherein: Develop a model of tasks wherein: Ð Ð Tasks complete execution before a well-defined deadline Tasks complete execution before a well-defined deadline Ð Ð Tasks make progress at application-specified rates Tasks make progress at application-specified rates Ð Ð No constraints are placed on the external environment No constraints are placed on the external environment

5 5

Rate-Based Execution Rate-Based Execution

Formal model Formal model

¥ ¥ Process make progress at the rate of processing Process make progress at the rate of processing x x events every events every y y time units, each event is time units, each event is processed within processed within d d time units time units

ti,j + di if 1 ≤ j ≤ xi MAX(ti,j + di , D(i, jÐxi)+yi ) if j > xi D(i, j) =

¥ ¥ For task For task i i with rate specification ( with rate specification (x xi

i,

, y yi

i,

, d di

i), the

), the j jth

th event for task

event for task i i, arriving at time , arriving at time t ti

i,j ,j, will be

, will be processed by time processed by time

Ð Ð Deadlines occur at least Deadlines occur at least d d time units after a job is time units after a job is released released Ð Ð Deadlines separated by at least Deadlines separated by at least y y time units time units

6 6

Rate-Based Execution Rate-Based Execution

Example: Periodic arrivals, periodic service Example: Periodic arrivals, periodic service

¥ ¥ Task with rate specification ( Task with rate specification (x x = 1 = 1, , y y = 2 = 2, , d d = 2 = 2) )

J1 J2 J4 J5 J6 J7 J8 J9 J10 J11 J12

0 2 4 6 8 10 12 14 16 18 20 22 24

J3

ti,j + di if 1 ≤ j ≤ xi MAX(ti,j + di , D(i, jÐxi)+yi ) if j > xi D(i, j) =

7 7

Rate-Based Execution Rate-Based Execution

Bursty Bursty arrivals arrivals

¥ ¥ Task with rate specification ( Task with rate specification (x x = 1 = 1, , y y = 2 = 2, , d d = 6 = 6) )

0 2 4 6 8 10 12 14 16 18 20 22 24 26

J1 J2 J3 J4 J5 J6 J7 J8 J9

8 8

Rate-Based Execution Rate-Based Execution

Bursty Bursty arrivals arrivals

¥ ¥ Task with rate specification ( Task with rate specification (x x = 3 = 3, , y y = 6 = 6, , d d = 6 = 6) )

0 2 4 6 8 10 12 14 16 18 20 22 24 26

J1 J2 J3 J4 J5 J6 J7 J8 J9

9 9

Rate-Based Execution Rate-Based Execution

Comparison of different rate specifications Comparison of different rate specifications

Rate specification Rate specification ( (x x = = 1 1, , y y = = 2 2, , d d = = 6 6) ) Rate specification (x = 3, y = 6, d = 6)

J1 J2 J3 J4 J5 J6 J7 J8 J9 0 2 4 6 8 10 12 14 16 18 20 22 24 26 J1 J2 J3 J4 J5 J6 J7 J8 J9 0 2 4 6 8 10 12 14 16 18 20 22 24 26

10 10

Using RBE Tasks Using RBE Tasks

What problems do they solve? What problems do they solve?

¥ ¥ RBE tasks provide a RBE tasks provide a more natural way of more natural way of modeling inbound modeling inbound packet processing of packet processing of fragmented messages fragmented messages

0 2 4 6 8 10 12

Rate specification (x = 3, y = 6, d = 6) J1 J2 J3 J4 J5 J6

Acquire Display

Display Initiation Time

11 11

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

Feasibility under preemption constraints Feasibility under preemption constraints

¥ ¥ Feasibility conditions for periodic and sporadic Feasibility conditions for periodic and sporadic tasks, for all other known execution tasks, for all other known execution environments, also hold for environments, also hold for RBE RBE tasks tasks

Ð Ð Feasibility under non-preemptive scheduling Feasibility under non-preemptive scheduling Ð Ð Feasibility under scheduling with critical sections Feasibility under scheduling with critical sections Ð Ð Feasibility under scheduling with interrupt handlers Feasibility under scheduling with interrupt handlers

¥ ¥ Thus feasibility is not inherently a function of Thus feasibility is not inherently a function of release times release times

Ð Ð Under deadline-driven scheduling, feasibility is a Under deadline-driven scheduling, feasibility is a function of the implementation of a task set function of the implementation of a task set Ð Ð Under static-priority scheduling, feasibility is a Under static-priority scheduling, feasibility is a function of the behavior of the external environment function of the behavior of the external environment

12 12

Feasibility of RBE Tasks Feasibility of RBE Tasks

Feasibility under preemptive scheduling Feasibility under preemptive scheduling

¥ ¥ Feasibility conditions of Feasibility conditions of RBE RBE tasks with rate tasks with rate specifications ( specifications (x x, , y y, , c c, , d d) are precisely the same as ) are precisely the same as for periodic tasks for periodic tasks

L

T3 T2 T1 ∀L, L > 0: L ≥ Σ

i=1 n

xici L Ð di + yi yi

13 13

¥ ¥ But canÕt an RBE task be modeled as But canÕt an RBE task be modeled as x x instances instances

• f a periodic task (with some appropriate
• f a periodic task (with some appropriate

precedence relationship between instances)? precedence relationship between instances)?

L

T = (x, y, d) = (3, 6, 6)

J1 J2 J3 J4 J5 J6

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

On the relationship to periodic tasks On the relationship to periodic tasks

14 14

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

A corollary on static priority scheduling A corollary on static priority scheduling

¥ ¥ Under a static priority scheduling scheme, the Under a static priority scheduling scheme, the processor demand in any interval can be unbounded processor demand in any interval can be unbounded

L J1 J2 J3 J4 J5 J6

L ≥ Σ

j=1 i

cj L pj

Ð Ð Thus event driven, rate-based execution is not possible Thus event driven, rate-based execution is not possible under static priority scheduling schemes under static priority scheduling schemes

15 15

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

Summary Summary

¥ ¥ Traditional Liu & Traditional Liu & Layland Layland theory is not directly theory is not directly applicable to distributed real-time systems applicable to distributed real-time systems ¥ ¥ The theory of scheduling periodic & sporadic The theory of scheduling periodic & sporadic tasks applies verbatim to RBE tasks tasks applies verbatim to RBE tasks

Ð Ð Polynomial & pseudo-polynomial time Polynomial & pseudo-polynomial time schedulability schedulability conditions exist for conditions exist for

È È Preemptive scheduling Preemptive scheduling È È Non-preemptive scheduling Non-preemptive scheduling È È Scheduling with interrupt handlers Scheduling with interrupt handlers È È Scheduling with critical sections Scheduling with critical sections

Ð Ð The The earliest-deadline-first earliest-deadline-first scheduling algorithm is scheduling algorithm is

• ptimal
• ptimal

16 16

A Theory of Rate-Based Execution A Theory of Rate-Based Execution

Summary Summary

¥ ¥ The feasibility of a set of Òperiodic tasksÓ was The feasibility of a set of Òperiodic tasksÓ was never inherently a function of the periodic arrival never inherently a function of the periodic arrival requirement requirement

Ð Ð The only requirement is that exist a minimal separation The only requirement is that exist a minimal separation between deadlines between deadlines

¥ ¥ But if static priority scheduling methods are But if static priority scheduling methods are employed then (in the worst case) periodic employed then (in the worst case) periodic arrivals are required arrivals are required

Ð Ð Static priority methods require a well-behaved Static priority methods require a well-behaved external environment external environment Ð Ð Deadline methods require a well-behaved operating Deadline methods require a well-behaved operating system system