The preemptive uniprocessor scheduling of mixed-criticality implicit-deadline sporadic task systems

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The preemptive uniprocessor scheduling of mixed-criticality implicit-deadline sporadic task systems Sanjoy Baruah 1 Vincenzo Bonifaci 2 3 Haohan Li 1 Alberto Marchetti-Spaccamela 4 Suzanne Van Der Ster 5 Leen Stougie 5 1 The University of North Carolina, Chapel Hill, USA. 2 IASI CNR, Italy. 3 MASCOTTE project INRIA, France 4 Sapienza University of Rome, Italy. 5 Vrije Universiteit, The Netherlands. (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 1 / MAS 40

Mixed-Criticality scheduling motivations In Safety-critical embedded systems, there is an increasing trend towards implementing multiple functionalities upon a single shared computing platform Examples: Integrated Modular Avionics (IMA) and AUTomotive Open System ARchitecture (AUTOSAR) This can force tasks of different importance (i.e. criticality) to share a processor and interfere with each other (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 2 / MAS 40

Mixed-Criticality scheduling motivations Example In unmanned aerial vehicles functionalities are classified into two levels of criticality: Level 1: mission-critical functionalities Certified by clients or vendors Less rigorous: consider low WCET Interested in Level 2 functionalities Level 2: flight-critical functionalities Certified by civilian Certification Authorities (CA) CAs are very conservative: consider high Worst Case Execution Time (WCET) CAs are not interested in Level 1 functionalities (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 3 / MAS 40

Each CA has its own rules to determine the WCET of a job The WCET of the same job of a flight critical functionality has two values: One lower value: WCET if we consider mission critical functionalities One higher value: WCET if we restrict to flight critical functionalities (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 4 / MAS 40

In this paper We analyze an algorithm EDF-VD for scheduling mixed-criticality task systems proposed in [Baruah et al, European Symposium on Algorithms 2011] we show that its speed-up factor is 4/3 (instead of φ) we show that it is optimal w.r.t. speed-up factor we show how to implement it in logarithmic computational time we derive utilization bounds and simulate its behavior (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 5 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 6 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 7 / MAS 40

Sporadic Task Systems A Sporadic Task System consists of a set of tasks τ 1, τ 2,..., τ n A task τ i = (c i, d i, p i ) generates a potentially infinite sequence of jobs, where: c i d i p i is the execution requirement is the relative deadline: time between a job s arrival and its deadline is the period: minimum time between two successive arrivals of jobs from this task In an implicit-deadline tasks system it holds that d i = p i for all tasks (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 8 / MAS 40

Mixed-Criticality Sporadic Task Systems A Mixed-Criticality Sporadic Task System with 2 levels consists of a set of tasks τ 1, τ 2,..., τ n Each task τ i = (χ i, c i (LO), c i (HI ), T i ) generates a potentially infinite sequence of jobs, where: χ i {LO, HI } is the criticality level c i (LO) is the execution requirement at level LO c i (HI ) is the execution requirement at level HI T i is the relative deadline and period We assume c i (LO) c i (HI ) (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 9 / MAS 40

Mixed-Criticality Sporadic Task Systems The amount of execution time a that job of task τ i needs is not known, but discovered by executing the job until it signals completion. A collection of realized values for the execution time is called a behavior By executing the tasks, we learn in which level the system is. This may change over time When the system is in level HI we need to process only the tasks that are of criticality level HI, the tasks of criticality level LO are omitted (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 10 / MAS 40

Utilization For x, y {LO, HI }, U y x = χ i =x c i (y) T i For example, UHI LO level-lo behavior is the utilization of HI criticality tasks, assuming a (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 11 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 12 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 LO-criticality behavior (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 13 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 HI -criticality behavior (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 14 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 HI -criticality behavior (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 15 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 HI -criticality behavior (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 16 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 17 / MAS 40

Overview of the Algorithm EDF-VD Preprocessing 1 Compute x as: x ULO HI 1 U LO LO 2 If ( xu LO LO + UHI HI 1 ) then declare success and compute Else declare failure and exit ˆT i : modified period ˆT i xt i for each τ i s.t. χ i = HI (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 18 / MAS 40

Overview of the Algorithm EDF-VD Run-time 1 If the system is at level LO schedule according to EDF where HI criticality tasks τ i have period ˆT i 2 If some job executes beyond its LO-criticality WCET: discard all LO-criticality jobs schedule HI -criticality tasks according to EDF with actual periods (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 19 / MAS 40

Example τ i c i (LO) c i (HI ) T i χ i U LO U HI τ 1 2 2 4 1 1/2 τ 2 1 5 6 2 1/6 5/6 Total 2/3 5/6 x = ULO HI 1 U LO LO = 1/6 1 1/2 = 1 3 xulo LO + UHI = 1 3 1 2 + 5 6 = 1 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 20 / MAS 40

Example τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 21 / MAS 40

Example τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 22 / MAS 40

Example τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 τ 1 τ 2 0 2 4 6 8 10 12 14 16 18 20 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 23 / MAS 40

Preprocessing Theorem The following condition is sufficient for ensuring that EDF-VD successfully schedules all LO-criticality behaviors: x ULO HI 1 U LO LO EDF-VD chooses the smallest value of x such that the above condition is satisfied Theorem The following condition is sufficient for ensuring that EDF-VD successfully schedules all HI -criticality behaviors: xu LO LO + UHI HI 1 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 24 / MAS 40

Dispatching 1 Γ = LO (criticality level indicator) 2 while (Γ LO) 1 when some job of task τ i arrives at time t if χ i LO the deadline is t + T i if χ i HI the deadline is t + ˆT i 2 at each instant the job with the earliest deadline is scheduled 3 if the current scheduled job executes for more than its LO-criticality WCET Γ HI 3 Once (Γ HI ) 1 add T i ˆT i to the deadline of active jobs of task τ i 2 when some job of task τ i arrives at time t, its deadline is t + T i 3 LO-criticality jobs do not receive any execution (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 25 / MAS 40

Dispatching a O(log(n)) implementation We have to implement three operation corresponding to three events: 1 Arrival of a job 2 Completion of a job 3 Switching of Γ to HI We use two priority queues: Q LO Q HI J c : current executed job We use a timer that indicate whether J c executed more than its LO-criticality WCET (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 26 / MAS 40

Dispatching a O(log(n)) implementation Arrival of a job of task τ i at time t c If χ i = LO, the job is inserted only in Q LO If χ i = HI, the job is inserted in both Q LO (with modified deadline) and Q HI (with actual deadline) If the new job is the one with the earliest deadline, set the timer to t c + C i (LO) Completion of job J c at time t c Delete J c from both Q LO and Q HI Reset the timer to t c + the LO-criticality WCET of the minimum job in Q LO The timer goes off We schedule according to Q HI (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 27 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 28 / MAS 40

Comparison with Worst-Case Reservation The worst case reservation approach (WCR) maps a mixed-criticality task system into a traditional task system τ i = (χ i, c i (LO), c i (HI ), T i ) τ i = (c i (χ i ), p i ) and schedules according to regular EDF Necessary condition U LO LO + UHI HI 1 Theorem Any task system that is correctly scheduled using WCR is also scheduled by EDV-VD Proof. As U LO LO + UHI HI 1 and x 1, then xu LO LO + UHI HI 1 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 29 / MAS 40

System where the LO-criticality WCET of HI -criticality task is 0 (U LO HI = 0) In this systems x = 0 and then the sufficient condition is U LO LO 1 U HI HI 1 i.e. LO-criticality and HI -criticality behaviors are separately schedulable ˆT i = 0 for each τ i s.t. χ i = HI i.e. HI -criticality tasks always have earliest deadline (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 30 / MAS 40

Speed-up bounds The Speed-up factor of an algorithm A is the smallest real number f such that any task system that is clairvoyantly schedulable on a unit speed processor is schedulable by A on a f -speed processor Theorem The speed-up factor of EDF-VD is 4 3 Theorem No non-clairvoyant algorithm for scheduling dual-criticality systems can have a speed-up factor smaller than 4 3 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 31 / MAS 40

Utilization bounds The two given sufficient condition are: that are satisfied if and only if x ULO HI 1 U LO LO x 1 UHI HI U LO LO that is U LO HI 1 U LO LO 1 UHI HI U LO LO ( U UHI HI 1 UHI LO LO ) LO 1 ULO LO (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 32 / MAS 40

Utilization bounds U LO U LO LO HI 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.98 0.97 0.9 0.93 0.90 0.85 0.76 0.60 0.10 0.2 0.97 0.95 0.914 0.86 0.80 0.70 0.53 0.20 0.3 0.96 0.92 0.87 0.80 0.70 0.55 0.30 0.4 0.95 0.90 0.82 0.70 0.60 0.40 0.5 0.94 0.87 0.78 0.60 0.50 0.6 0.93 0.85 0.74 0.50 0.7 0.92 0.82 0.70 0.8 0.91 0.80 0.9 0.90 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 33 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 34 / MAS 40

Input instances Randomly generated instances with the following parameters: U bound = max{ulo LO + ULO HI, UHI HI }: larger utilization of the task system [U L, U U ]: utilization of the a task [Z L, Z U ]: ratio between HI -criticality utilization and LO-criticality utilization of a task P: probability that a task is an HI -criticality task We generate the tasks one by one until the required utilization parameters are met We generated 1000 instances and measured the fraction of instances which are schedulable by algorithms EDF-VD and Regular-EDF (WCR approach) (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 35 / MAS 40

Results U L = 0.02, U U = 0.2, Z L = 1, Z U = 2, P = 0.5 U L = 0.02, U U = 0.2, Z L = 1, Z U = 8, P = 0.5 U L = 0.02, U U = 0.2, Z L = 1, Z U = 4, P = 0.5 U L = 0.02, U U = 0.2, Z L = 1, Z U = 8, P = 0.3 In any case, the system is schedulable by Regular-EDF if U bound 0.5 and by EDF-VD if U bound 0.75 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 36 / MAS 40

Results The improvement is more significant if the ratio between HI -criticality utilization and LO-criticality utilization increases Z U = 2 Z U = 8 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 37 / MAS 40

It is even more significant if the ratio between the HI -criticality utilization and LO-criticality utilization of a the system approaches 1 1.6 1.06 (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 38 / MAS 40

Outline 1 Model 2 Algorithm EDF-VD 3 Properties of EDF-VD 4 Evaluation via simulation 5 Conclusion (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 39 / MAS 40

Conclusion We analyzed algorithm EDF-VD: we have shown that its speed-up factor is 4/3 we have shown that no non-clairvoyant algorithm can can have a speed-up factor smaller than 4/3 we have shown how to implement it in logarithmic computational time we have derived utilization bounds and simulate its behavior Thank you for your attention (The University ECRTS of 2012 North Carolina, Chapel Hill, USA., IASI CNR, Italy., 40 / MAS 40

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