Schedulability Analysis for Fixed Priority Real-Time Systems with - - PowerPoint PPT Presentation

Schedulability Analysis for Fixed Priority Real-Time Systems with Energy- Harvesting

Yasmina Abdedda¨ ım1 Younes Chandarli1,2 Rob Davis 3 Damien Masson 1

(1)Universit´

e Paris-Est, LIGM UMR CNRS 8049, ESIEE Paris, France

(2)Universit´

e Paris-Est, LIGM UMR CNRS 8049, Universit´ e Paris-Est Marne-La-Vall´ ee, France

(3)Real-Time Systems Research Group, University of York

RTNS’14, 10 October 2014

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Outline

1

Motivation

2

Model

3

Schedulability Analysis

4

Experiments

5

Conclusion and Future Work

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Motivation

Energy Harvesting Systems

Energy-Harvesting The process by which energy is captured from a system’s environment and converted into usable electric power.

Energy Harvester Sun Wind Vibrations

. . . . . .

Solar Panel Wind Turbine Piezoelectric Harvester Storage Unit Reusable Alkaline Battery Li-ion polymer Battery Supercapacitor

. . .

Real-Time system Environmental Energy Source Sensing tasks Data Processing Tasks Transmission tasks

. . .

Harvesting Replenishing Consuming 3 / 25

Motivation

Energy Harvesting Systems

Energy-Harvesting The process by which energy is captured from a system’s environment and converted into usable electric power.

Energy Harvester Sun Wind Vibrations

. . . . . .

Solar Panel Wind Turbine Piezoelectric Harvester Storage Unit Reusable Alkaline Battery Li-ion polymer Battery Supercapacitor

. . .

Real-Time system Environmental Energy Source Sensing tasks Data Processing Tasks Transmission tasks

. . .

Harvesting Replenishing Consuming 3 / 25

Model

Energy Model

Harvester Energy Source Energy Storage Unit Emin Emax E(t) Pr(t) Energy Source Model

Energy Sources: solar, thermal, mechanical, vibration, . . . Harvester: transform the environmental energy into electrical power.

Energy Storage Unit Model

Energy Unit: battery, super-capacitor, . . . Store the harvested energy: Pr(t) is the energy replenishment function. Constant rate of replenishment: Pr(t) = Pr The energy stored may vary between two levels Emin and Emax.

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Model

Energy Model

Harvester Energy Source Energy Storage Unit Emin Emax E(t) Pr Energy Source Model

Energy Sources: solar, thermal, mechanical, vibration, . . . Harvester: transform the environmental energy into electrical power.

Energy Storage Unit Model

Energy Unit: battery, super-capacitor, . . . Store the harvested energy: Pr(t) is the energy replenishment function. Constant rate of replenishment: Pr(t) = Pr The energy stored may vary between two levels Emin and Emax.

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Model

Task Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks Emin Emax E(t) Pr A set of sporadic tasks τi(Ci, Pi, Ei, Ti, Di)

Ci: worst-case execution time, Pi: worst-case power consumption, Ei = Pi × Ci: worst-case energy consumption, Ti: minimal inter arrival time, Di: relative deadline (Di ≤ Ti).

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

Consuming Task: P1 > Pr

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

Consuming Task: P1 > Pr

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

Consuming Task: P1 > Pr

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Model

The Model

Harvester Energy Source Energy Storage Unit

Processor

Real−time Tasks

Pr E(t) Emax Emin

Pr = 3 Emax = 3 Emin = 0 τ1 : C1 = 2 P1 = 6 E1 = 12 T1 = D1 = 4 τ2 : C2 = 1 P2 = 1 E2 = 1 T2 = D2 = 5

1 2 3 4 5 6 1 2 3

time energy

τ2

Emin Emax

τ1

Consuming Task: P1 > Pr Gaining Task: P2 ≤ Pr

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Model

The Scheduling Problem

The Model Storage Unit: Constant rate replenishment Pr and Emax, Emin the maximal and minimal level of energy. A set Γ = Γc ∪ Γg of sporadic tasks τi = (Ci, Pi, Ei, Ti, Di) in priority order with Di ≤ Ti:

Consuming Tasks: Γc= {τi ∈ Γ, Pi > Pr} Gaining Tasks: Γg = {τi ∈ Γ, 0 ≤ Pi ≤ Pr}

Feasibility A task set is feasible if all the tasks meet their deadlines: timing constraints and ∀t ≥ 0 the energy level is between Emin and Emax: energy constraints.

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Model

Related Work

An algorithm for Frame-Based Model,

• A. Allavena and D. Moss´

e, “Scheduling of Frame-based Embedded Systems with Rechargeable Batteries”, Workshop in conjunction with RTAS, 2001.

LSA Algorithm assumes variable execution time,

• C. Moser, D. Brunelli, L. Thiele and L. Benini,

“Real-time scheduling with regenerative energy”, ECRTS, 2006.

EDeg Algorithm based on EDF priority assignment.

• H. EL Ghor, M. Chetto and R. Chehade,

“A real-time scheduling framework for embedded systems with environmental energy harvesting”, Computers & Electrical Engineering journal, 2011.

PFPASAP Algorithm

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Model

Related Work

An algorithm for Frame-Based Model,

• A. Allavena and D. Moss´

e, “Scheduling of Frame-based Embedded Systems with Rechargeable Batteries”, Workshop in conjunction with RTAS, 2001.

LSA Algorithm assumes variable execution time,

• C. Moser, D. Brunelli, L. Thiele and L. Benini,

“Real-time scheduling with regenerative energy”, ECRTS, 2006.

EDeg Algorithm based on EDF priority assignment.

• H. EL Ghor, M. Chetto and R. Chehade,

“A real-time scheduling framework for embedded systems with environmental energy harvesting”, Computers & Electrical Engineering journal, 2011.

PFPASAP Algorithm

• Y. Abdedda¨

ım, Y. Chandarli and D. Masson, “The Optimality of PFPASAP Algorithm for Fixed-Priority Energy-Harvesting Real-Time Systems”, ECRTS, 2013.

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Model

The PFPASAP Algorithm

Execute tasks whenever there is enough energy available in the battery. Replenish as long as needed to execute one time unit of the highest priority active task. PFPASAP is an Energy Work-Conserving FPPS Algorithm The processor is idle only if there is insufficient energy to schedule at least

• ne time unit of the highest priority active task.

Optimality PFPASAP is optimal in the class of energy work conserving fixed priority pre-emptive scheduling algorithms in the case where all the task consume energy (Γ = Γc). Our Goal Provide a schedulability test for PFPASAP when the set of tasks is composed of both consuming and gaining tasks.

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Model

The PFPASAP Algorithm

Execute tasks whenever there is enough energy available in the battery. Replenish as long as needed to execute one time unit of the highest priority active task. PFPASAP is an Energy Work-Conserving FPPS Algorithm The processor is idle only if there is insufficient energy to schedule at least

• ne time unit of the highest priority active task.

Optimality PFPASAP is optimal in the class of energy work conserving fixed priority pre-emptive scheduling algorithms in the case where all the task consume energy (Γ = Γc). Our Goal Provide a schedulability test for PFPASAP when the set of tasks is composed of both consuming and gaining tasks.

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Schedulability Analysis

Classical Response Time Analysis

Method

1

Find the wost-case scenario: scenario where τi is subject to the maximum possible delay,

2

Compute Ri the longest response time of task τi: is the response time of τi in the worst-case scenario,

3

Exact schedulability test: If ∀τi, Ri ≤ Di the task set is schedulable. Work-Conserving FPPS with Di ≤ Ti

1

Worst-case scenario for task τi: Synchronous release of all the tasks,

2

Ri is given by the smallest t > 0 that satisfies t = F(i, t) with: F(i, t) = Ci + Maximum interference from higher priority tasks in [0, t) F(i, t) =

• h≤i

t Th

• × Ch

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Schedulability Analysis

Classical Response Time Analysis

Method

1

Find the wost-case scenario: scenario where τi is subject to the maximum possible delay,

2

Compute Ri the longest response time of task τi: is the response time of τi in the worst-case scenario,

3

Exact schedulability test: If ∀τi, Ri ≤ Di the task set is schedulable. Work-Conserving FPPS with Di ≤ Ti

1

Worst-case scenario for task τi: Synchronous release of all the tasks,

2

Ri is given by the smallest t > 0 that satisfies t = F(i, t) with: F(i, t) = Ci + Maximum interference from higher priority tasks in [0, t) F(i, t) =

• h≤i

t Th

• × Ch

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Schedulability Analysis

Response Time Analysis for PFPASAP

Response Time of a task τi Ci + Replenishment time + Interference from higher priority tasks.

Pr = 3 Emax = 10 Emin = 0 τ1 : C1 = 2 P1 = 1 E1 = 12 T1 = 8 D1 = 3 τ2 : C2 = 3 P2 = 5 E2 = 15 T2 = 10 D2 = 9

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Schedulability Analysis

Response Time Analysis for PFPASAP

Response Time of a task τi Ci + Replenishment time + Interference from higher priority tasks.

Pr = 3 Emax = 10 Emin = 0 τ1 : C1 = 2 P1 = 1 E1 = 12 T1 = 8 D1 = 3 τ2 : C2 = 3 P2 = 5 E2 = 15 T2 = 10 D2 = 9

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2 Response Time τ2 = 6 Response Time τ2 = 7

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Schedulability Analysis

Response Time Analysis for PFPASAP

Worst-Case Scenario The synchronous release of all the tasks is no longer the worst-case scenario.

Pr = 3 Emax = 10 Emin = 0 τ1 : C1 = 2 P1 = 1 E1 = 12 T1 = 8 D1 = 3 τ2 : C2 = 3 P2 = 5 E2 = 15 T2 = 10 D2 = 9

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2 Response Time τ2 = 6 Response Time τ2 = 7

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Schedulability Analysis

Response Time Analysis for PFPASAP

worst-case scenario is unknown, cannot compute exactly Ri, the worst-case response time of task τi, cannot provide an exact schedulability test. Upper bound Ri to build a sufficient schedulability test.

Pr = 3 Emax = 10 Emin = 0 τ1 : C1 = 2 P1 = 1 E1 = 12 T1 = 8 D1 = 3 τ2 : C2 = 3 P2 = 5 E2 = 15 T2 = 10 D2 = 9

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2 Response Time τ2 = 6 Response Time τ2 = 7

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Schedulability Analysis

Response Time Analysis for PFPASAP

worst-case scenario is unknown, cannot compute exactly Ri, the worst-case response time of task τi, cannot provide an exact schedulability test. Upper bound Ri to build a sufficient schedulability test.

Pr = 3 Emax = 10 Emin = 0 τ1 : C1 = 2 P1 = 1 E1 = 12 T1 = 8 D1 = 3 τ2 : C2 = 3 P2 = 5 E2 = 15 T2 = 10 D2 = 9

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2

1 2 2 4 8 6

energy

6 5 4 7 8 10

time

1 1 2 2 3 3 4 5 6 7 8 10 8 7 6 5 4 3 10

τ1 τ2 Response Time τ2 = 6 Response Time τ2 = 7

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Schedulability Analysis

Bounding Response Time

We require a monotonically non-decreasing function F(i, w) that upper bounds the length of the worst-case response time of task τi within an interval of length w, The upper bound RUB

i

• f the worst-case response time Ri corresponds to

the smallest w > 0 that satisfies F(i, w) = w. We define for every task τi a virtual scenario that:

1

Maximizes the amount of interference from higher priority tasks,

2

Maximizes the amount of replenishment time needed.

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Schedulability Analysis

Bounding Response Time

We require a monotonically non-decreasing function F(i, w) that upper bounds the length of the worst-case response time of task τi within an interval of length w, The upper bound RUB

i

• f the worst-case response time Ri corresponds to

the smallest w > 0 that satisfies F(i, w) = w. We define for every task τi a virtual scenario that:

1

Maximizes the amount of interference from higher priority tasks,

2

Maximizes the amount of replenishment time needed.

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Schedulability Analysis

Maximal Interferences

For every task τi released in a window of length w, the maximal number of higher priority jobs that are active in this window is

• h<i

w Th

• =
• h<i,τh∈Γc

w Th

• +
• h<i,τh∈Γg

w Th

• 1

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

w

Consuming Jobs Gaining Jobs

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Schedulability Analysis

Maximal Replenishment

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

w

3 Gaining Consuming Jobs Jobs 1 3 2 2 1 To upper bound the replenishment in a window of length w we consider a virtual sequence where:

1

The battery is empty at the beginning of the window,

to minimize the energy budget of interval w

2

All the consuming jobs are before all the gaining jobs.

to maximize replenishment periods

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Schedulability Analysis

Upper Bound RUB1

i

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

w

3 Gaining Consuming Jobs Jobs 1 3 2 2 1 FUB1(i, w) =    

• h≤i,τh∈Γc

w Th

• × ((Ph − Pr) × Ch/Pr)

   

• +
• h≤i,τh∈Γc

w Th

• × Ch
• maximum replenishment needed

consuming jobs +

• h≤i,τh∈Γg

w Th

• × Ch
• gaining jobs

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Schedulability Analysis

Sufficient Schedulability test UB1

FUB1(i, w) is a monotonically non-decreasing function of w and FUB1(i, w) > Ci RUB1

i

the upper bound of the longest response time of task τi is given by the smallest t > 0 that satisfies w = FUB1(i, w) with: FUB1(i, w) =        

• h≤i,τh∈Γc

w Th

• × Eh

Pr         +

• h≤i,τh∈Γg

w Th

• × Ch

Sufficient Schedulability test UB1: ∀τi, RUB1

i

≤ Di

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Schedulability Analysis

Necessary Schedulability Test LB1

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

w

1 2 3 Consuming Jobs Gaining Jobs 3 2 1 To lower bound the worst-case response time of a task τi released in a window of length w we consider a virtual sequence where:

1

The battery is empty at the beginning of the window,

2

All the gaining jobs are before all the consuming jobs.

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Schedulability Analysis

A Tighter Upper Bound RUB2

i

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

w

3 Gaining Consuming Jobs Jobs 1 3 2 2 1 Consuming jobs as soon as possible, Gaining jobs as late as possible

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Schedulability Analysis

A Tighter Upper Bound RUB2

i

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 Consuming jobs as soon as possible, Gaining jobs as late as possible

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence:

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 1

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 1 1

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 2 1 1

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 2 1 1 2

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 2 1 1 3 2

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Schedulability Analysis

Sufficient Schedulability Test UB2

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

3

w

Jobs Jobs Consuming Gaining 1 2 1 3 2 To compute FUB2(i, w), we compute the response time of the virtual sequence: 2 1 1 3 3 2

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Experiments

Performance Comparison

Competitors

UTZ: the exact test for FPPS ignoring energy constraints, SIM: an empirical necessary test based on simulating the schedule of PFPASAP over more than twice the hyper-period, UB1: sufficient schedulability test based on the upper bound RUB1, UB2: sufficient schedulability test based on the upper bound RUB2, LB1: necessary schedulability test based on the lower bound RLB1.

Input Data:

40000 task sets randomly generated using UUniFast-Discard algorithm coupled with a technique of hyper-period limitation, Processor utilization varied from 0.05 to 1, Energy utilization varied from 0.05 to 1, Percentage of gaining tasks varied from 0% to 100%.

Simulation tool: YARTISS Real-time systems simulator

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Experiments

Varying the Processor Utilization

20 40 60 80 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Schedulable Tasksets % U % UTZ LB1 SIM UB2 UB1

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Experiments

Varying the Energy Utilization

20 40 60 80 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Weighted Schedulability Ue % UTZ LB1 SIM UB2 UB1

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Experiments

Varying the Gaining Tasks Ratio

20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Weighted Schedulability gainers % UTZ LB1 SIM UB2 UB1

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Experiments

Varying the Gaining Tasks Ratio

20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Weighted Schedulability gainers % UTZ LB1 SIM UB2 UB1

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Experiments

Varying the Gaining Tasks Ratio

20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Weighted Schedulability gainers % UTZ LB1 SIM UB2 UB1

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Conclusion and Future Work

Conclusion and Future Work

1

In this paper we

Showed that under PFPASAP, the worst-case scenario for task sets with both consuming and gaining tasks is not necessarily synchronous release with all

• ther tasks,

Derived two sufficient schedulability tests based on two upper bounds on task response times, Derived a necessary schedulability test based on a lower bound on task response time, Evaluated the performance of the sufficient tests in comparison with a number of necessary tests.

2

As future work we plan to:

Investigate the problem of optimal priority assignment, Investigate analysis for more complex replenishment functions and additional costs of entering and exiting low power modes needed for energy replenishment.

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Conclusion and Future Work

Conclusion and Future Work

1

In this paper we

Showed that under PFPASAP, the worst-case scenario for task sets with both consuming and gaining tasks is not necessarily synchronous release with all

• ther tasks,

Derived two sufficient schedulability tests based on two upper bounds on task response times, Derived a necessary schedulability test based on a lower bound on task response time, Evaluated the performance of the sufficient tests in comparison with a number of necessary tests.

2

As future work we plan to:

Investigate the problem of optimal priority assignment, Investigate analysis for more complex replenishment functions and additional costs of entering and exiting low power modes needed for energy replenishment.

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Questions ?

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Battery Capacity

For the upper bounds to be valid, the battery capacity must be sufficient to store the maximum amount of energy needed in the virtual sequences.

1

For UB1: Emax must be sufficient to store the energy needed to execute

• ne time unit of the most consuming task:

EUB1

max ≥ max(max ∀i (Ei/Ci) − Pr, Pr)

2

For UB2: Emax must be sufficient to store the energy needed to execute consuming jobs in any possible energy busy period: EUB2

max ≥ max

• ∀i

max∀j(Dj) Ti

• × max (Ei − Ci × Pr, 0) , Pr