Schedulability and Optimization Analysis for Non-Preemptive Static Priority Scheduling Based on Task Utilization and Blocking Factors
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1 Schedulability and Optimization Analysis for Non-Preemptive Static Priority Scheduling Based on Task Utilization and Blocking Factors Georg von der Brüggen, Jian-Jia Chen, Wen-Hung Huang Department of Computer Science, Chair 12 TU Dortmund University, Germany 09 July 2015 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 1 / 17
2 Preemptive vs. Non-Preemptive Scheduling τ 1 Preemptive τ 2 τ 1 Non Preemptive τ 2 τ 2 blocks τ 1 Deadline Miss v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 2 / 17
3 Preemptive vs. Non-Preemptive Scheduling τ 1 Preemptive τ 2 τ 1 Non Preemptive τ 2 τ 2 blocks τ 1 Deadline Miss v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 2 / 17
4 Preemptive vs. Non-Preemptive Scheduling τ 1 Preemptive τ 2 Deadline Miss τ 1 Non Preemptive τ 2 τ 2 blocks τ 1 Deadline Miss v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 2 / 17
5 Notation τ 1 WCET τ i (C i, D i, ), U i = C i Relative Deadline Period Utilization v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 3 / 17
6 Notation τ i (C i, D i, ), U i = C i τ 1 WCET τ 2 τ 3 τ 4 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 3 / 17
7 Notation τ i (C i, D i, ), U i = C i τ 1 blocking WCET τ 2 τ 3 τ 4 B k : Blocking Time v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 3 / 17
8 Notation τ i (C i, D i, ), U i = C i τ 1 blocking WCET τ 2 τ 3 Blocking Factor γ k : B k C k B k : Blocking Time τ 4 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 3 / 17
9 Notation τ i (C i, D i, ), U i = C i τ 1 blocking WCET τ 2 τ 3 Blocking Factor γ k : B k C k B k : Blocking Time τ 4 Task Set Blocking Factor γ : max {γ k} τ k τ v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 3 / 17
10 Interesting Problem: Speedup Factors Preemptive Non-Preemptive Task Set Lower Upper Lower Upper Constraints Bound Bound Bound Bound Implicit 1/ln(2) Deadline Constrained Deadline Arbitrary Deadline Fixed Priority Scheduling Speedup Factors [R. Davis, L. George, P. Courbin. Quantifying the sub-optimality of uniprocessor fixed priority non-pre-emptive scheduling ] Ω = ln ( 1 Ω) (transcendental equation) Ω Ω v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 4 / 17
11 Interesting Problem: Utilization Bounds Utilization Bound Andersson and Tovar Total utilization bounds of RM-NP with respect to γ = max τk {max τi lp(τ k ) { Ci C k }} provided Andersson and Tovar (2009) γ v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 4 / 17
12 Response Time of τ k (Example: τ 4 ) τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t 20 The schedule according to Deadline Monotonic k 1 ( t t with 0 < t D k C k and B k + ) + 1 C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 5 / 17
13 Response Time of τ k (Example: τ 4 ) τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic k 1 ( t t with 0 < t D k C k and B k + ) + 1 C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 5 / 17
14 Response Time of τ k (Example: τ 4 ) τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) t 20 The schedule according to Deadline Monotonic k 1 ( t t with 0 < t D k C k and B k + k 1 t t with 0 < t D k C k and B k + ) + 1 C i C i t t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 5 / 17
15 Self-Pushing Phenomenon τ 1 = (3, 8) τ 2 = (3, 9) τ 3 = (3, 12) τ 4 = (2, 99) Deadline Miss Self-pushing phenomenon for τ 3 [Buttazzo] Theorem [Yao, Buttazzo, and Bertogna, 2010] The worst-case response time of a non-preemptive task occurs in the first job if the task is activated at its critical instant and the following two conditions are both satisfied: 1 the task set is feasible under preemptive scheduling; 2 the relative deadlines are less than or equal to periods. v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 6 / 17
16 Preemptive and Non-Preemptive Case Simultaneously τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic k 1 t NP: t with 0 < t D k C k and B k + k 1 t P: t with 0 < t D k and C k + C i t C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 7 / 17
17 Preemptive and Non-Preemptive Case Simultaneously τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic k 1 t NP: t with 0 < t D k C k and B k + k 1 t P: t with 0 < t D k and C k + C i t C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 7 / 17
18 Preemptive and Non-Preemptive Case Simultaneously τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic k 1 t NP: t with 0 < t D k C k and B k + k 1 t P: t with 0 < t D k and C k + C i t C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 7 / 17
19 Preemptive and Non-Preemptive Case Simultaneously τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic k 1 t NP: t with 0 < t D k C k and B k + k 1 t P: t with 0 < t D k and C k + k 1 t NP+P: t with 0 < t D k and B k + C k + C i t C i t C i t v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 7 / 17
20 Aims of the Paper NP+P: t with 0 < t D k and B k + C k + k 1 t Ti C i t Easy sufficient schedulability test Good Runtime: O(k) Utilization of higher priority tasks Blocking time Hyperbolic form Good Runtime: O(n) Speedup Factor Non-preemptive deadline monotonic Non-preemptive rate monotonic Utilization bounds for non-preemptive rate monotonic Depending on blocking factor Blocking factor 2 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 8 / 17
21 Sufficient Linear Time Schedulability Test for τ k τ 1 = (2, 8, 8) τ 2 = (2, 10, 10) τ 3 = (3, 12, 12) τ 4 = (3, 23, 35) τ b = (2, 99, 99) The schedule according to Deadline Monotonic 20 k 1 tj t j {t 1, t 2,..., t k 1, t k } and B k + C k + C i t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 9 / 17
22 Sufficient Linear Time Schedulability Test for τ k τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t The reordered task set k 1 tj t j {t 1, t 2,..., t k 1, t k } and B k + C k + C i t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 9 / 17
23 Splitting the Summation τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t j {t 1,..., t k } and releases before last B k + C k + k 1 tj last release C i t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 10 / 17
24 Splitting the Summation τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t j {t 1,..., t k } and releases before last B k + C k + k 1 tj last release C i t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 10 / 17
25 Splitting the Summation τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t j {t 1,..., t k } and t j {t 1,..., t k } and releases before last B k + C k + k 1 B k + C k + k 1 tj ti C i C i + last release j 1 C i t j t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 10 / 17
26 Splitting the Summation τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t j {t 1,..., t k } and t j {t 1,..., t k } and releases before last B k + C k + k 1 tj C i B k + C k + k 1 t i C i + last release j 1 C i t j t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 10 / 17
27 Splitting the Summation τ 3 = (3, 12, 12) τ 1 = (2, 8, 8) τ2 = (2, 10, 10) t 1 t 2 t 3 τ 4 = (3, 23, 35) τ b = (2, 99, 99) t t j {t 1,..., t k } and t j {t 1,..., t k } and t j {t 1,..., t k } and releases before last B k + C k + k 1 tj C i B k + C k + k 1 t i C i + B k + C k + k 1 t i U i + j 1 C i j 1 last release t i U i t j t j t j v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 10 / 17
28 Test for τ k in Hyperbolic Form (Theorem 1) Non-preemptive sporadic task system Constrained deadlines Fixed priority scheduling Schedulability of higher priority tasks has already been ensured Theorem 1 A task τ k is schedulable if: ( ) Bk + C k + 1 D k Runtime: O(n) τ j hp(τ k ) (U j + 1) 2 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 11 / 17
29 Speedup Factor for DM-NP and RM-NP Preemptive Non-Preemptive Task Set Lower Upper Lower Bound Upper Bound Constraints Bound Bound Previous This Paper Implicit 1/ln(2) Deadline Constrained Deadline Arbitrary Deadline v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 12 / 17
30 Utilization Bound for RM-NP Theorem 3 Suppose that { the } tasks are indexed such that +1. If γ = max Ci τi lp(τ k ) C k = B k C k, then task τ k is schedulable by RM-NP if ( ( 2 U sum 1 Theorem 4 1+γ ) 1 ) ( k 1 1+γ + (k 1)( 2 ) 1 ) k 1+γ 1 if γ 1 1+γ if γ>1 { Suppose that γ = max τk {max Ci τi lp(τ k ) C k }}. A task set can be feasibly scheduled by RM-NP if { ( ) γ γ+1 U sum + ln 2 1+γ if γ γ if γ > 1 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 13 / 17
31 Utilization Bound for RM-NP Utilization Bound Theorem 4 Andersson and Tovar Total utilization bounds of RM-NP with respect to γ = max τk {max τi lp(τ k ) { Ci C k }} Comparison: Andersson and Tovar (2009) and Theorem 4 γ v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 13 / 17
32 Tighter Schedulability Test Testing schedulability in two equations: NP: t 0 < t D k C k and B k + k 1 P: t 0 < t D k and C k + k 1 tj tj C i t C i t Theorem 6 A task τ k is schedulable by a fixed priority non-preemptive scheduling algorithm A if all higher priority tasks are schedulable and the following two conditions hold: ( ) Bk + 1 (U j + 1) 2 D k C k τ j hp(τ k ) ( ) Ck + 1 D k τ j hp(τ k ) (U j + 1) 2 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 14 / 17
33 Tighter Utilization Bound for RM-NP Theorem 9 { Suppose that γ = max τk {max Ci τi lp(τ k ) C k }}. A task set can be feasibly scheduled by RM-NP if { ln(2) if γ 1 ln(2) ln(2) U sum 1 1+γ if γ > 1 ln(2) ln(2) - UPDATED - v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 15 / 17
34 Comparison of Utilization Bounds for RM-NP Utilization Bound Theorem 9 Theorem 4 Andersson and Tovar Total utilization bounds of RM-NP with respect to γ = max τk {max τi lp(τ k ) { Ci C k }} provided by Andersson and Tovar, Theorem 4 and Theorem 9 γ - UPDATED - v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 16 / 17
35 Results First schedulability test in hyperbolic form for non-preemptive fixed priority scheduling Tighter schedulability test based on two individual equations in hyperbolic form Tight speedup factors of 1 Ω for Deadline Monotonic (DM) and Rate Monotonic (RM) non-preemptive scheduling in comparison to non-preemptive Earliest Deadline First (EDF) Utilization bounds for Rate Monotonic scheduling { depending }} on the blocking factor γ = max τk {max Ci τi lp(τ k ) C k if γ < 2 v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 17 / 17
36 Results First schedulability test in hyperbolic form for non-preemptive fixed priority scheduling Tighter schedulability test based on two individual equations in hyperbolic form Tight speedup factors of 1 Ω for Deadline Monotonic (DM) and Rate Monotonic (RM) non-preemptive scheduling in comparison to non-preemptive Earliest Deadline First (EDF) Utilization bounds for Rate Monotonic scheduling { depending }} on the blocking factor γ = max τk {max Ci τi lp(τ k ) C k if γ < 2 Thank You! v. d. Brüggen, Chen, Huang (LS 12, TU Dortmund) 17 / 17
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