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3.2 Rate Monotonic Analysis Assumptions – – – – – 1 A1. No nonpreemptible parts in a task, and negligible preemption cost A2. Resource constraint on CPU time only A3. No precedence constraints among tasks A4. All tasks periodic A5. Relative deadline = period Spring 2002 Real-Time Systems (Shin) Rate-Monotonic Scheduling(RMS) Overview – rate monotonic priority – – the higher rate, the higher priority schedulability guaranteed if utilization rate is below a certain limit for feasible schedules fi 2 = 1/Ti : frequency (=rate) ci or Ci : execution time n c i 1 i fi 1 Spring 2002 Real-Time Systems (Shin) (continued) – If the total utilization rate has least upper bound n(21/n - 1) where n = #tasks, there exists a feasible rate monotonic schedule. That is, n c f i i – n(2 1) U (n) i 1 U(1) = 1.0, U(2) = 0.828, U(3) = 0.779, U(∞) = ln2 only sufficient condition priority inversion problem 3 1 n if a more critical task has a longer period Spring 2002 Real-Time Systems (Shin) Rate Monotonic Analysis(RMA) Basic theory with periodic tasks Introduction – Rate monotonic analysis(RMA) 4 a method for analyzing sets of real-time tasks based on rate monotonic scheduling theory analytic formulas to determine schedulability framework for reasoning about system timing behavior separation of timing and functional concerns provides an engineering basis for designing real-time systems Spring 2002 Real-Time Systems (Shin) (continued) – Basic theory applies only to independent, periodic tasks, but has been extended to address: – Why are deadline missed? 5 priority inversion task interactions aperiodic tasks preemption execution blocking Spring 2002 Real-Time Systems (Shin) A Sample Problem Periodics Servers Aperiodics Emergency 100 msec 50 msec 1 20 msec Data Server 5 msec 2 msec 150 msec 20 msec Deadline 6 msec after arrival 2 40 msec Comm Server 350 msec 10 msec 10 msec 3 Routine 40 msec 2 msec 100 msec Desired response 4 msec average 2’s deadline is 20 msec before the end of each period 6 Spring 2002 Real-Time Systems (Shin) Periodic tasks – – task’s CPU utilization Ui = Ci/Ti total CPU utilization U = Ui + U2 + … + Un Utilization Bound(UB) Test – A set of n independent periodic tasks scheduled by the rate monotonic algorithm will always meet its deadlines, for all 1 task phasings, if Cn C1 C2 ... U (n) n(2 n 1) T1 T2 Tn – 7 for harmonic task sets, the utilization bound is U(n) = 1.00 for all n. Spring 2002 Real-Time Systems (Shin) Sample Problem: Applying UB Test C T U Task 1 20 100 0.200 Task 2 40 150 0.267 Task 3 100 350 0.286 – Total utilization: 0.200 + 0.267 + 0.286 = 0.753 < U(3) = 0.779 – 8 The periodic tasks in the sample problem are schedulable according to the UB test. Spring 2002 Real-Time Systems (Shin) Timeline for Sample Problem 0 100 200 300 400 1 2 3 Scheduling Points 9 (scheduling point: a point in time when new work arrives worst-case phasing: all tasks are ready to execute at t=0) Spring 2002 Real-Time Systems (Shin) 10 Exercise: Applying the UB Test Given: Task C T 1 4 τ 1 2 6 τ 2 1 10 τ 3 U a. What is utilization for each task? b. Is the task set schedulable? c. Draw the timeline. d. What is the total if C3=2? Springutilization 2002 Real-Time Systems (Shin) Extension of UB test – UB test has three possible outcomes. – – 11 0 ≤ U ≤ U(n) success U(n) < U ≤ 1.00 inconclusive 1.00 < U overloaded UB test is conservative A more precise test can be applied. Spring 2002 Real-Time Systems (Shin) Schedulability: CT Test – Theorem – For a set of independent, periodic tasks, if each task meets its first deadline, with worst-case task phasing, the deadline will always be met. Completion Time Test Let Wi = completion time of taski. Wi may be computed by the W i (n) following iterative formula: W (n 1) C C where W (0) 0 i 12 i j i Tj j i Task i is schedulable if its completion time is before its Spring 2002 Real-Time Systems (Shin) deadline. That is, Wi ≤ Ti Example: Applying CT Test – – Taking the sample problem, we increase the compute time of T1 from 20 to 40. Is the task set still schedulable? Utilization of first two tasks: 0.667 < U(2) = 0.828 – Utilization of all three tasks: 0.953 > U(3) = 0.779 13 first two tasks are schedulable by utilization bound test utilization bound test is inconclusive need to apply completion time test Spring 2002 Real-Time Systems (Shin) – 14 (continued) 0 W3 (1CT ) test C3 todetermine 100 Use 3 meets its first C j ifC3task deadline j 3 T j 100 W3 (2) C3 C j j 3 T j 100 100 100 (40) (40) 180 100 150 180 180 W3 (3) 100 (40) (40) 260 100 150 Spring 2002 Real-Time Systems (Shin) W3 (3) 260 260 260 (continued) W3 (4) 100 ( 40 ) ( 40 ) 300 100 150 300 300 W3 (4) 100 (40) (40) 300 Done! 100 150 W3 300 T3 350 Task 3 is schedulable using CT test. 15 Spring 2002 Real-Time Systems (Shin) Example : Using Schedulability Points 0 100 200 300 400 1 2 3 Scheduling Points 16 Spring 2002 Real-Time Systems (Shin) Exercise: Applying CT Test Task τ1: C1 = 1 Task τ2: C2 = 1 Task τ3: C3 = 1 17 T1 = 4 T2 = 4 T3 = 4 a. Apply UB test. b. Draw timeline. c. Apply CT test. Spring 2002 Real-Time Systems (Shin) Summary – – – Utilization bound test is simple but conservative Completion time test is more exact but also more complicated To this point, UB and CT tests share the same limitations. 18 all tasks run on a single processor all tasks periodic and noninteracting deadlines always at the end of the period no interrupts rate monotonic priorities assigned zero context switch overhead tasks do not suspend themselves Spring 2002 Real-Time Systems (Shin) Rate Monotonic Analysis(RMA) Extensions to basic theory Extending the schedulability tests – – – Task interactions – 19 nonzero task switching times preperiod deadlines priority inversion (blocking) synchronization(mutual exclusion) is required Spring 2002 Real-Time Systems (Shin) Integrating task switching times and preperiod deadlines – Task switching time can be modeled as execution time – Preperiod deadline can be modeled as “dormant time” 20 for task switching time S, Ui = (Ci+2S)/Ti instead of Ui = Ci/Ti for preperiod deadline D, Ui = (Ci+D)/Ti applied for schedulability test on τi only C D T T Spring 2002 time Real-Time Systems (Shin) Example for task switching and preperiod deadline – 1 21 applying (C the2SUB ) Test to the sample problem 1 T1 U (1) 2 (C1 2S ) (C2 2S D2 ) U (2) T1 T2 3 (C1 2S ) (C2 2S ) (C3 2S ) U (3) T1 T2 T3 Spring 2002 Real-Time Systems (Shin) Schedulability with interrupts – Interrupt processing can be inconsistent with rate monotonic priority assignment – Effects of interrupt processing must be taken into account in schedulability model. 22 interrupt handler executes with high priority despite its period interrupt processing may delay execution of tasks with higher rate monotonic priority (shorter periods) higher priority tasks regard the interrupt as blocking factor. (source of priority inversion) Spring 2002 Real-Time Systems (Shin) Priority inversion – – – 23 Delay to a task’s execution caused by interference from lower-priority tasks is known as priority inversion. Priority inversion is modeled by blocking time. Identifying, modeling and reducing sources of priority inversion is central to schedulability analysis. Spring 2002 Real-Time Systems (Shin) Sources of priority inversion non-preemptable regions of code – interrupts – non-unique priorities for some tasks (if there are not enough priority levels) – non-rate-monotonic assignment of task priorities – FIFO queues – synchronization and mutual exclusion – 24 Spring 2002 Real-Time Systems (Shin) Accounting for priority inversion – Recall that task schedulability is affected by: – 25 preemption: potentially many times per period execution: once per period blocking: at most once per period for each source The schedulability formulas are modified to add a “blocking” or “priority inversion” term to account for inversion effects. Spring 2002 Real-Time Systems (Shin) Adding blocking time to schedulability test 0 1 100 For 1 I 0 100 200 1 For 2 2 I 0 100 200 300 400 1 2 Interrupt I I For 4 4 26 Spring 2002 Real-Time Systems (Shin) UB test with blocking – Cn C1 C2priority inversion, Before considering we simply tested the ... U ( n) total utilization for n tasks: T T T 1 – – n Now we must test each task for schedulability. k 1general schedulability test is: The C C B T i 1 27 2 i i k Tk k Tk Spring 2002 U (n) , k 1, 2, ..., n Real-Time Systems (Shin) CT test with blocking – – A set of periodic tasks, scheduled according to the rate monotonic policy, is schedulable if each task meets its first deadline. That is, if completion time of each task, Wi, is less than or Wi (k ) equal to its deadline. Wi (k 1) Bi Ci Cj j i T j blocking execution preemption 28 Spring 2002 Real-Time Systems (Shin) Exercise: Schedulability with priority inversion Tasks C T D U B Given: (where 1 τ3 is4 nonpreemptive 0.250) τ τ τ 29 1 2 3 2 1 6 10 1 0.334 0.100 a. fill in column B b. construct a schedulability model c. which tasks are schedulable? Spring 2002 Real-Time Systems (Shin) Synchronization problem – – analyze the effects of task interactions on schedulability in this discussion 30 synchronization = mutual exclusion Spring 2002 Real-Time Systems (Shin) Sample Problem : Synchronization Periodics Servers 100 msec Aperiodics Emergency 50 msec 1 20 msec Data Server 5 msec 2 msec 150 msec 2 20 msec 40 msec Comm Server 350 msec 10 msec 10 msec 3 Deadline 6 msec after arrival Routine 40 msec 2 msec 100 msec Desired response 4 msec average 2’s deadline is 20 msec before the end of each period. 31 Spring 2002 Real-Time Systems (Shin) Synchronization Model Legend Critical section Executing 1:{…P(S)…V(S)…} 3:{…P(S)…V(S)…} attempts to lock S (blocked) Blocked S locked B S unlocked τ1(H) τ2(M) S locked S unlocked τ3(L) 32 Time Spring 2002 Real-Time Systems (Shin) Priority Inversion in Synchronization Legend 1:{…P(S1)…V(S1)…} 3:{…P(S1)…V(S1)…} Executing S1 locked attempts to lock S1 (blocked) τ1(H) τ2(M) S1 locked S1 unlocked Blocked B S1 locked S1 unlocked τ3(L) 33 Time Spring 2002 Real-Time Systems (Shin) B Synchronization protocols – – – – 34 no preemption highest locker’s priority basic priority inheritance priority ceiling Each protocol prevents unbounded priority inversion Spring 2002 Real-Time Systems (Shin) Nonpreemption Protocol Legend 2:{…P(S1)…V(S1)…} 4:{…P(S1)…V(S1)…} Ready S1 locked Executing Blocked B τ1(H) Ready τ2 Ready τ3 S1 locked S1 unlocked τ4(L) Time 35 Spring 2002 Real-Time Systems (Shin) B Highest Locker’s Priority Protocol 2:{…P(S1)…V(S1)…} 4:{…P(S1)…V(S1)…} Legend S1 locked Executing Blocked τ1(H) Ready τ2 B B Ready τ3 S1 locked S1 unlocked τ4(L) Time 36 Spring 2002 Real-Time Systems (Shin) B Basic Inheritance Protocol (BIP) 2:{…P(S1)…V(S1)…} 4:{…P(S1)…V(S1)…} Legend S1 locked Executing Blocked τ1(H) Attempts to lock S1 S1 locked S1 unlocked Ready B τ2 Ready τ3 S1 locked S1 unlocked τ4(L) Time 37 Spring 2002 Real-Time Systems (Shin) B Priority Ceiling Protocol (PCP) 2:{…P(S1)…V(S1)…} 4:{…P(S1)…V(S1)…} Legend S1 locked Executing Blocked τ1(H) Attempts to lock S1 S1 locked S1 unlocked Ready B τ2 Ready τ3 S1 locked S1 unlocked τ4(L) Time 38 Spring 2002 Real-Time Systems (Shin) B Priority ceiling – priority ceiling of a semaphore S – simply the priority of the highest priority task that may lock semaphore S “system” ceiling: the maximum ceiling of all semaphores currently locked by other tasks the idea behind PCP τ (high) to create a total priority ordering of executing and suspended critical sections τ (low) S priority ceiling of semaphore S is the priority of τ (high) 39 Spring 2002 Real-Time Systems (Shin) Priority ceililng protocol(PCP) rules – – – Preemption: A task with a higher execution priority always preempts tasks with lower execution priorities. Ceiling: A task cannot enter its critical section unless its priority is higher than the system ceiling. Inheritance: A lower priority task that blocks a higher priority task Th inherits the priority of task Th. [When there is only one semaphore, PCP works just like BIP.] 40 Spring 2002 Real-Time Systems (Shin) PCP rules - revisited – 1. when T wants to enter the c. s. – 2. while running inside the c. s. – can preempt a lower priority task 4. when T completes execution normally 41 always inherits highest priority among the blocked 3. When T does not want to enter the c. s. – its priority must be higher than system ceiling when exiting, wake up the highest priority task among the blocked if there are many tasks of same priority ready for execution, schedule the task blocking other tasks Spring 2002 Real-Time Systems (Shin) PCP Example J0 = { …, P(S0), …, V(S0), … } J1 = { …, P(S1), …, P(S2), …, V(S2), …, V(S1), … } J2 = { …, P(S2), …, P(S1), …, V(S1), …, V(S2), … } Critical section guarded by S0 S0 locked S1 S2 S0 unlocked J0 S1 locked Blocked by J2 S2 locked S2 unlocked S1 unlocked (attempt to lock S1) J1 S1 unlocked S1 locked S2 unlocked S2 locked J2 t0 42 t1 t2 t3 t4 t5 t6 Spring 2002 t7 t8 t9 t10 t11 t12 t13 time Real-Time Systems (Shin) PCP Example J0 = { …, P(S0), …, V(S0), …, P(S1), …, V(S1), ... } J1 = { …, P(S2), …, V(S2), … } J2 = { …, P(S2), …, P(S1), …, V(S1), …, V(S2), … } Critical section guarded by S0 S1 S2 S0 unlocked S1 locked S1 unlocked S0 locked J0 blocked by J2 (attempt to lock S1) (attempt to lock S0) blocked by J2 S2 locked S2 unlocked J1 S2 locked S1 locked S1 unlocked S2 unlocked J2 t0 43 t1 t2 t3 t4 t5 t6 Spring 2002 t7 t8 t9 t10 t11 t12 t13 t14 time Real-Time Systems (Shin) Example Of Chained Blocking (BIP) 1:{…P(S1)…P(S2)…V(S2)...V(S1)…} 2:{…P(S1)…V(S1)…} 3:{…P(S2)…V(S2)…} Legend S1 locked S2 locked Blocked Attempts to lock S1 (blocked) τ1(H) τ2(M) B B S1 locked B Attempts to lock S2 (blocked) B S1 unlocked S2 locked S2 unlocked τ3(L) Time 44 Spring 2002 Real-Time Systems (Shin) Blocked At Most Once (PCP) 1:{…P(S1)…P(S2)…V(S2)...V(S1)…} 2:{…P(S1)…V(S1)…} 3:{…P(S2)…V(S2)…} Legend S1 locked S2 locked S1 locked S2 unlocked S1 locked Ceiling S1 unlocked Attempts to lock S1 (blocked) τ1(H) Attempts to lock S1 (blocked) S1 locked S1 unlocked τ2(M) S2 locked S2 unlocked τ3(L) Time 45 Spring 2002 Real-Time Systems (Shin) Deadlock: Using BIP Legend 1:{…P(S1)…P(S2)…V(S2)...V(S1)…} 2:{…P(S2)…P(S1)…V(S1)...V(S2)…} S1 locked S2 locked Blocked B attempts to lock S2 (blocked) locks S1 τ1(H) B S2 locked Attempts to lock S1 (blocked) τ2(M) Time 46 Spring 2002 Real-Time Systems (Shin) Deadlock Avoidance: Using PCP 1:{…P(S1)…P(S2)…V(S2)...V(S1)…} 2:{…P(S2)…P(S1)…V(S1)...V(S2)…} Legend S1 locked S2 locked Ceiling B locks S2 attempts to lock S1 (blocked) locks S1 τ1(H) locks S2 locks S1 unlocks S1 unlocks S2 τ2(M) Time 47 Spring 2002 Real-Time Systems (Shin) Summary of synchronization protocols – No preemption – Highest locker’s priority – when a lower priority task blocks the execution of a higher priority task, it inherits the priority of the task it blocks Priority ceiling 48 execute critical sections with the priority of the highest priority task that may lock the semaphore Priority inheritance – do not allow preemption during execution of critical sections priority inheritance plus priority ceiling rule for locking semaphores Spring 2002 Real-Time Systems (Shin) Bounded (continued) Blocked at Deadlock priority Summary most once aviodance Protocol of synchronization protocols inversion Nonpremptible Yes1 Yes1 Yes critical sections Highest locker’s Yes1 Yes1 Yes priority No No Yes Basic inheritance Yes Yes2 Yes Priority ceiling 1 Only if tasks do not suspend within critical sections 2 PCP is not affected if tasks suspend within critical sections. 49 Spring 2002 Real-Time Systems (Shin) C T D B Sample Problem: Using BIP T1 20 100 30 T2 40 150 T3 100 350 20 10 D: preperiod deadline 50 Spring 2002 Real-Time Systems (Shin) Schedulability model using BIP C1 T1 C1 T1 C1 T1 51 B1 T1 100 C2 D2 T2 C2 20 U (1) C3 B2 T2 T T 2 U (2) 3 100 0.50 1.0 20 20 100 Spring 2002 30 10 20 100 U (3) 40 150 0.667 0.828 150 40 100 150 350 0.753 0.779 Real-Time Systems (Shin)