Probabilistic Real-Time Analysis Luca Santinelli and Liliana - - PowerPoint PPT Presentation

Probabilistic Real-Time Analysis

Luca Santinelli and Liliana Cucu-Grosjean

Trio Team, INRIA Nancy Grand-Est France

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms Real-Time System: S = (A, R, Γ)

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms Real-Time System: S = (A, R, Γ) Scheduling Algorithms A ◮ Priority-based ◮ Static or Dynamic Timing constraints...

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms Real-Time System: S = (A, R, Γ) Resource R ◮ Computational resource ◮ Communication resource ◮ · · ·

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms Real-Time System: S = (A, R, Γ) Task set Γ Γ = {τ1, τ2, . . . , τn} τi = (Oi, Ci, Ti, Di) Timing constraints → Deadline Di

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Problem Statement

◮ Real-Time Systems: timing constraints to enforce and guarantee in all the conditions ◮ Task Scheduling: task execution through scheduling algorithms Real-Time System: S = (A, R, Γ) WORST-CASE SYSTEM and ANALYSIS

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Outline

1

Problem Statement

2

Real-Time Analysis

3

Motivations to Probabilities

4

Probabilities

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Real-Time Analysis: Abstractions

◮ Workload bound function wbf: the maximum amount of resource required by that task ◮ Demand bound function dbf: the minimum amount of resource demanded by that task in order to execute and meet its timing constraint ◮ supply bound function sbf: the minimum resource provisioning from a resource provisioning system element

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Real-Time Analysis: Abstractions

◮ Workload bound function wbf: the maximum amount of resource required by that task ◮ Demand bound function dbf: the minimum amount of resource demanded by that task in order to execute and meet its timing constraint ◮ supply bound function sbf: the minimum resource provisioning from a resource provisioning system element Functions

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Real-Time Analysis: Approximations

Approximating the resource supply sbf bdf(t) = max{0, α(t − ∆)} α = lim

t→∞

sbf(t) t ∆ = inf{q | α(t − q) ≤ sbf(t) ∀t}

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Real-Time Analysis: Approximations

Approximating the resource supply sbf bdf(t) = max{0, α(t − ∆)} α = lim

t→∞

sbf(t) t ∆ = inf{q | α(t − q) ≤ sbf(t) ∀t} Approximations applicable to workload and resource demand

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Real-Time Analysis: Approximations

Approximating the resource supply sbf bdf(t) = max{0, α(t − ∆)} α = lim

t→∞

sbf(t) t ∆ = inf{q | α(t − q) ≤ sbf(t) ∀t} Approximations applicable to workload and resource demand Approximations

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Real-Time Analysis: Schedulability

Schedulability A real-time system is schedulable if all the tasks composing the system meet their deadline while executing Scheduling conditions are defined by comparing the resource request (workload or resource demand) and the resource provisioning

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Real-Time Analysis: Schedulability

Schedulability A real-time system is schedulable if all the tasks composing the system meet their deadline while executing Scheduling conditions are defined by comparing the resource request (workload or resource demand) and the resource provisioning Comparison: timing guarantees (hard or soft)

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Real-Time Analysis: an Example of Schedulability

i.e. Earliest Deadline First (EDF) scheduling paradigm, a task set Γ, receiving an amount of resource sbf can be guaranteed schedulable (its deadline can be guaranteed) if and only if ∀t dbfΓ(t) ≤ sbf(t) With the bounded-delay linear approximation, the feasibility condition becomes a sufficient condition ∀ t bdf(t) ≤ sbf(t) i.e. with α, ∆ approximation ∀t ∈ D : dbf(t) ≤ α(t − ∆) D is the set of deadlines the application schedulability has to be checked

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Real-Time Analysis: Feasibility Space

Schedulability conditions also relate to schedulability regions in representation space Example: the (α, ∆)-space - possible to define feasibility regions where the task set is schedulable with the (α, ∆) resource assignment i.e. EDF, the application feasibility region is defined by ∀t ∈ D : dbf(t) ≤ α(t − ∆) ∀t ∈ D : ∆ ≤ t − dbf(t) α ∆ ≤ min

t∈D

• t − dbf(t)

α

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Real-Time Analysis: Feasibility Space

1 2 3 4 5 0.2 0.4 0.6 0.8 1 Earliest Deadline First

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Outline

1

Problem Statement

2

Real-Time Analysis

3

Motivations to Probabilities

4

Probabilities

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Motivations to Probabilities

◮ Worst-case timing analysis to validate the system

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Motivations to Probabilities

◮ Worst-case timing analysis to validate the system HOWEVER, a worst-case analysis provides pessimistic results (WCET ≡ pessimism) that not all real-time systemsems can afford

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Motivations to Probabilities

◮ Worst-case timing analysis to validate the system HOWEVER, a worst-case analysis provides pessimistic results (WCET ≡ pessimism) that not all real-time systemsems can afford The pessimism can be decreased by using probabilistic approaches.

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Motivations to Probabilities

◮ Worst-case timing analysis to validate the system HOWEVER, a worst-case analysis provides pessimistic results (WCET ≡ pessimism) that not all real-time systemsems can afford The pessimism can be decreased by using probabilistic approaches. What else? ◮ Reliability analysis, used to estimate the imperfection of reality ◮ Unreliable nature of the system environment and the system elements

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Motivations to Probabilities

◮ Worst-case timing analysis to validate the system HOWEVER, a worst-case analysis provides pessimistic results (WCET ≡ pessimism) that not all real-time systemsems can afford The pessimism can be decreased by using probabilistic approaches. What else? ◮ Reliability analysis, used to estimate the imperfection of reality ◮ Unreliable nature of the system environment and the system elements Guarantee timing constraints needed for hard and soft real-time systems/applications

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Outline

1

Problem Statement

2

Real-Time Analysis

3

Motivations to Probabilities

4

Probabilities

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A Probabilistic Model

Γ = {τ1, τ2, . . . , τn} τi = (Ci, Di, Ti) Ci - random variable on the execution time with a known probability function fCi(·) (fCi(C) = P(Ci = C)) Ci = C max

i

= C 0

i

C 1

i

· · · C min

i

= C m

i

fCi(C max

i

) fCi(C 1

i )

· · · fCi(C min

i

)

• j

fCi(C j

i ) = 1

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Probabilistic Analysis

Probabilistic functions and probabilistic approximation results in probabilistic bounds to the system behaviour

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Probabilistic Analysis

Probabilistic functions and probabilistic approximation results in probabilistic bounds to the system behaviour Probabilistic dbf dbfi,j(t) = max

• 0,

t − Di Ti + 1

• C j

i

• Each dbfi,j has a bounding probability

pi,j = 1 −

k : C k

i ≤C j i fCi(C k

i ).

Probabilities: probability of bounding the resource demand

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Probabilistic Analysis

Probabilistic functions and probabilistic approximation results in probabilistic bounds to the system behaviour Probabilistic dbf dbfi,j(t) = max

• 0,

t − Di Ti + 1

• C j

i

• Each dbfi,j has a bounding probability

pi,j = 1 −

k : C k

i ≤C j i fCi(C k

i ).

Probabilities: probability of bounding the resource demand Probabilistic functions: dbfi,j, pi,j

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Probabilistic Analysis

Probabilistic functions and probabilistic approximation results in probabilistic bounds to the system behaviour Probabilistic sbf Periodic service provisioning (Q, P), the worst-case/minimum resource supply [0, t) sbf(t) = max{0, (k − 1)Q, t − (k + 1)(P − Q)} k = ⌈t − (P − Q) P ⌉ Probabilities: probability of bounding the resource provisioning

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Probabilistic Analysis

Probabilistic functions and probabilistic approximation results in probabilistic bounds to the system behaviour Probabilistic sbf Periodic service provisioning (Q, P), the worst-case/minimum resource supply [0, t) sbf(t) = max{0, (k − 1)Q, t − (k + 1)(P − Q)} k = ⌈t − (P − Q) P ⌉ Probabilities: probability of bounding the resource provisioning Probabilistic functions:sbfk, pk

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Probabilistic Feasibility Space: an Example

Test Case Given a probabilistic task τ = (0,

• 1

2 3 0.6 0.3 0.1

• , 10, 10), The

possible probabilistic demand bound curves are dbf1 = t − 10 10 + 1

• 1

→ (α1, ∆1) dbf2 = t − 10 10 + 1

• 2

→ (α2, ∆2) dbf3 = t − 10 10 + 1

• 3

→ (α3, ∆3)

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Probabilistic Feasibility Space: an Example

5 10 15 20 25 30 35 40 0.2 0.4 0.6 0.8 1

∆ α

EDF probabilistic feasibility region

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Remarks

Flexibility of the probabilistic model and analysis allows to: ◮ tackle with hard real-time ◮ easily extend to the common soft real-time case ◮ efficiently face many source of pessimism of the real-time analysis (WCET, approximations, etc.)

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Increase the Complexity

With a complete probabilistic task model: τi = (Ci, Di, Ti) ◮ Ci - random variable on the execution time with a known probability function fCi(·) (fCi(c) = P(Ci = c)) ◮ Ti - random variable on the period with a known probability function fTi(·) (fTi(T) = P(Ti = T))

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Increase the Complexity

With a complete probabilistic task model: τi = (Ci, Di, Ti) ◮ Ci - random variable on the execution time with a known probability function fCi(·) (fCi(c) = P(Ci = c)) ◮ Ti - random variable on the period with a known probability function fTi(·) (fTi(T) = P(Ti = T)) Ci =

• C max

i

= C 0

i

C 1

i

· · · C min

i

= C m

i

fCi (C max

i

) fCi (C 1

i )

· · · fCi (C min

i

)

• ;
• j

fCi(C j

i ) = 1

Ti =

• T max

i

= T 0

i

T 1

i

· · · T min

i

= T m

i

fTi (T max

i

) fTi (T 1

i )

· · · fTi (T min

i

)

• ;
• j

fTi(T j

i ) = 1

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Some Questions

1 How to derive conclusions from the probabilistic (α, ∆)-space? 2 What are other possible conclusions from the analysis of the

α, ∆)-space?

3 How to study the probabilistic (α, ∆)-space in the general

case? How to derive conclusions from the general case of probabilistic (α, ∆)-space?

4 Extend the model to other aspects of the real-time analysis? 5 A new model for probabilistic real-time?

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