Optimal Checkpoint Placement on Real-Time Tasks with Harmonic Periods

Transcription

1 Kwak SW, Yang JM. Optimal checkpoint placement on real-time tasks with harmonic periods. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 27(1): Jan DOI /s Optimal Checkpoint Placement on Real-Time Tasks with Harmonic Periods Seong Woo Kwak 1 and Jung-Min Yang 2 1 Department of Electronic Engineering, Keimyung University, Daegu , Korea 2 Department of Electrical Engineering, Catholic University of Daegu, Gyeongbuk , Korea ksw@kmu.ac.kr; jmyang@cu.ac.kr Received September 13, 2010; revised August 29, Abstract This paper presents an optimal checkpoint strategy for fault-tolerance in real-time systems where transient faults occur in Poisson distribution. In our environment, multiple real-time tasks with different deadlines and harmonic periods are scheduled in the system by rate-monotonic algorithm, and checkpoints are inserted at a constant interval in each task. When a fault is detected, the system carries out rollback to the latest checkpoint and re-executes tasks. The maximum number of re-executable checkpoints and an equation to check schedulability are derived, and the optimal number of checkpoints is selected to maximize the probability of completing all the tasks within their deadlines. Keywords checkpoint scheme, real-time system, fault-tolerance, harmonic period, rate-monotonic (RM) scheduling 1 Introduction In real-time systems working in hostile environments or safety-critical applications, adversarial effects caused by permanent or transient faults become an unavoidable feature. In particular, due to the critical nature of tasks supported by many real-time systems, it is essential that tasks complete before their deadlines even in the presence of faults. This makes fault-tolerance an inherent requirement of real-time systems [1-2]. Among various fault-tolerance techniques, the checkpoint scheme [3-4] is a simple but reliable one commonly used for transient faults, which are usually caused by environmental changes and disappear after an active period [5]. The checkpoint scheme consists of saving intermediate states of the task in a reliable storage and, upon a detection of a fault, restoring the previously stored state. Hence, checkpointing enables to reduce the time to recover from a fault, while minimizing loss of the processing time [4]. Determining the interval between adjacent checkpoints is the critical factor that affects the performance of this scheme. A trade-off must be considered when selecting the interval, because inserting more checkpoints reduces the re-processing time after failures, but it increases the program execution time by adding the overall checkpointing overhead. The objective of this paper is to present an off-line algorithm for placement of checkpoints. Compared with the former studies [4,6-10], the main contribution of our study lies in that our scheme can be applied to multiple real-time tasks with different deadlines. To the authors knowledge, only very few researches, including a preliminary study by the authors [11], have been proposed about checkpointing for multiple tasks, possibly due to the complexity of the problem. [11] differs from this work in that it uses an approximate method to find the checkpoint intervals, so the solutios not optimal. Moreover, transient faults considered in [11] have a finite length of duration, while our study assumes zero length of duration occurrences of transient faults. [12] also dealt with checkpointing for multiple tasks, but with the rule that all the intervals between checkpoints are the same regardless of tasks, for which the optimal solution cannot be obtained. We assume that the considered multiple tasks have harmonic periods [12] with each other, and are scheduled by RM (rate-monotonic) algorithm [13] in a real-time system where transient faults occur in Poisson distribution. Every fault is supposed to be detected at the checkpoint that comes first after the fault occurrence. Checkpoints are inserted at a constant interval in the execution of a task; the width of anterval is different for each task in general. We first derive the slack time available to each task, and then determine the maximum number of re-executable checkpoints that does not violate the deadline constraint. Based on this information, a formula is presented to examine the schedulability of a given set of re-executed checkpoint by faults. Short Paper 2012 Springer Science + Business Media, LLC & Science Press, China

2 106 J. Comput. Sci. & Technol., Jan. 2012, Vol.27, No.1 The probability that all the tasks complete within their deadlines is also formulated. The optimal set of checkpoint intervals is selected such that it has the maximum probability as well as guarantees the schedulability. A numerical case study is conducted to demonstrate the procedure of the proposed scheme. With the condition of harmonic periods, the probability of task completios formulated in a closed form as will be shown later. If we relax the condition, i.e., when multiple tasks have arbitrary periods, the probability cannot be expressed in a closed form but only in aterative formula. A condensed paper on the latter topic was presented in [14]. 2 Preliminaries In RM scheduling algorithm, the priority of a task is inverse proportion to the task s deadline. Hence the task with the shortest deadline has the highest priority and vice versa for the task with the longest deadline. Supposing all the multiple real-time tasks are scheduled by RM algorithm, we define the following specifications. A1. There are m multiple tasks τ 1,..., τ m for which D i and E i (E i D i ) are the period and the execution time of τ i, respectively. A2. D 1 < D 2 < < D m and, for any D i, there exists a nonnegative integer k(i) such that D i = 2 k(i) T, where T = D 1. A3. The deadline of task τ i is the same as its period D i, and every fault can be detected at checkpoints. Assumption A2, often called harmonic period condition [12], gives a task a unique and constant slack time in each period, since a task s period is a multiple of the period of another task that has a higher priority. Under Assumptions A2 and A3, in addition, the result of RM scheduling in anstance of D m, the maximum period among D i, will be the same as in any other instance of D m. Hence, our analysis can be confined to a single period of D m. As hardware manufacturing technology continues to improve, the number of permanent faults is gradually decreasing, and instant malfunctions of systems (or computers) due to transient faults become the main reason for system failures [1,15]. The present study also focuses on transient faults only. We assume that all the transient faults are detectable and occur according to a Poisson process with rate λ. α k (λ, t), the probability that k faults occur within time range [0, t], is then that at least one fault occurs in [0, ] is (λ ) k α k (λ, ) = e λ = 1 e λ. (2) k! k=1 k=1 In each checkpoint, the program checks whether a transient fault occurs, while saving the intermediate states in a storage. If a fault is detected, the program is rolled back to the last saved state, i.e., to the last visited checkpoint, and resumes the task from that point. Saving the intermediate states and conducting fault detection procedures cause some overhead to each checkpoint. Denote by the number of checkpoints inserted in a period of task τ i and denote by t cp the checkpoint overhead. Then i, the checkpoint interval of τ i, can be written as i = E i + t cp. (3) e i, the execution time of τ i prolonged by the checkpoint overhead, is derived as e i = i = ( Ei + t cp ) = E i + t cp. (4) (4) means that since the processor should pass through checkpoints to complete τ i, the execution time increases by t cp compared with the case of no checkpoints. Note that re-execution against transient faults is not considered yet; the execution time will be prolonged further once the checkpoint scheme is activated. If there is no fault, S i, the maximum slack time task τ i can have in period D i, is constant and derived as S i = D i D i D 1 e 1 D i D i e i. (5) The constant slack time for each task is due to the harmonic period condition of Assumption A2. Fig.1 is a counterexample, where two tasks τ 1 and τ 2 are scheduled. Since D 1 = 2 (unit) and D 2 = 3, Assumption A2 is violated in Fig.1. According to RM algorithm, τ 1 is executed in the first (E 1 = 1) at t = 0, followed by the execution of τ 2 (E 2 = 0.5). At t = 2, τ 1 is executed once more since D 1 = 2. The slack time of τ 2 in the first period (D 1 = 3) is 0.5 as displayed in Fig.1. A similar assignment of τ 1 and τ 2 in the second period leaves 1.5 as the slack time, different from the value in α k (λ, t) = (λt)k e λt. (1) k! Similarly, the probability that no faults occur in time range [0, ] is α 0 (λ, ) = e λ, and the probability Fig.1. Task scheduling of τ 1 and τ 2 when D 1 = 2 (unit), D 2 = 3, E 1 = 1, and E 2 = 0.5.

3 Seong Woo Kwak et al.: Optimal Checkpoint Placement on Real-Time Tasks 107 the first period. Therefore, D 2 must be a multiple of D 1 for having a constant slack time. 3 Checkpointing for Multiple Tasks 3.1 Rollback Scheme Fault-tolerance by checkpointing is done by reexecuting the faulty part of a task during the slack time available to the task. In multi-tasking environments, the slack time of a task is determined not only by the execution time of the task but those of other tasks. Moreover, the slack time is different with respect to each kind of task in general. In this paper, therefore, we use the policy of assigning different numbers of checkpoints to each task, while fixing the interval between two adjacent checkpoints in the same task. We first derive the probability that all m tasks τ 1,..., τ m complete successfully within their deadlines D 1,..., D m even with transient faults. Then we select the number of checkpoints n 1,..., n m and their intervals, termed 1,..., m, that maximize this probability. Note that tasks are scheduled by RM algorithm in which low-priority tasks are always preempted by highpriority tasks. Thus, for obtaining S i, it is sufficient to subtract from D i the time span expended only by those tasks which have the same or higher priority than task τ i, i.e., tasks τ 1,..., τ i as in (5). When the program enters into a new period D i, tasks τ 1,..., τ i 1 are executed in the first according to RM algorithm, after which task τ i is executed. Note that τ i may be preempted by τ 1,..., τ i 1 again during the execution thereafter, since τ 1,..., τ i 1 have higher priorities. Hence the nominal feature of task execution period D i is described like Fig.2(a). But, since the considered transient faults are a Poisson process, their property of memorylessness allows us to interpret such that all the time spans of preemption by τ 1,..., τ i 1 can be shifted to their initial execution range, as shown in Fig.2(b). This equivalence clarifies the derivation of the probability of task completion. h i in Fig.2(b) is the extended execution time of τ i, implying that h checkpoints are re-executed by faults that occur in this period; h i must be never greater than the maximum slack time S i for guaranteeing schedulability. 3.2 Probability of Task Completion Denote by v i := D m /D i the number of period D i contained in the longest period D m. By Assumption A2, v i is always a power of 2. For each task τ i, define the 1 v i re-execution vector L i = [l i1,..., l ivi ] such that l ik designates the number of re-executed checkpoints in the k-th period. Under the checkpoint strategy, l ik is the same as the number of checkpoint intervals i corrupted by faults. If no transient fault occurs to task τ i, L i becomes the null vector, i.e., L i = [0,..., 0]. Let us calculate the probability that task τ i can complete in the k-th period when the re-execution vector is L i = [l i1,..., l ivi ]. Based on (2), denote by p i := e λ i the probability that no fault occurs in range i, and denote by q i := 1 e λ i the probability that at least one fault occurs in i. Since l ik checkpoint intervals are corrupted by faults during the k-th period, the program should be rolled back to l ik checkpoints, i.e., it will pass through + l ik checkpoints in total until accomplishing τ i, among which no fault is detected at checkpoints. In particular, in any case of completing task τ i, the last checkpoint, i.e., the ( + l ik )-th one, will end with no fault. Otherwise, the program should be rolled back from and re-execute the last checkpoint, which means that the total number of checkpoints the program passes through would be more than ( + l ik ). Taking into account these characteristics, ψ ik (l ik ), the probability that task τ i completes in the k-th period with l ik checkpoint intervals attacked by faults, is derived as ψ ik (l ik ) = ni+l ik 1 C lik p i q i l ik, where x C y is the y-combination from x. ψ i (L i ), the probability that task τ i can complete throughout D m with the re-execution vector L i, is obtained by multiplying ψ i1 (l i1 ),..., ψ ivi (l ivi ): Fig.2. Task execution period D i. (a) Nominal feature under RM algorithm. (b) Equivalent feature under consideration of preemption by tasks with higher priorities. ψ i (L i ) = ψ i1 (l i1 ) ψ ivi (l ivi ) =( ni+l i1 1C li1 p i q i l i1 ) ( ni+l ivi 1C livi p i q i l ivi ). (6)

4 108 J. Comput. Sci. & Technol., Jan. 2012, Vol.27, No.1 4 Optimal Checkpoint Intervals 4.1 Schedulability Indicator We now turn to developing a schedulability criterion for a given set of re-execution vectors. Assume that the re-execution vectors up to task τ i are found to be L 1,..., L i, none of which is the null vector. Recall that in the case of no fault, the maximum slack time of task τ i is S i as defined in (5). But, since there are checkpoint intervals where at least one fault occurs (L i 0), the available time for τ i should be shortened by rollback to checkpoints in each task. Hence s k i, the actual slack time of τ i in the k-th period, k = 1,..., v i, is given by s k i = S i l 1j 1 l ij i, (7) j Ω k i (1) j Ω k i (i) where Ω k i (r) {1,..., v r}, r = 1,..., i, represents the set of all the indices j such that the j-th period of task τ r is contained in the k-th period of task τ i. Fig.3 illustrates the notion of Ω k i (r). A slight reflection leads to { Ω k i (r) = j N (k 1) D i < j D } i. (8) D r D r Fig.3. Notion of Ω k i (r). If s k i 0, then task τ i is schedulable in the k-th period. For τ i to be schedulable throughout period D m, all the slack times s 1 i,..., svi i must be nonnegative. A variable γ i {0, 1} defined in the following equation represents the schedulability of τ i. γ i := u(s 1 i ) u(s vi i ), (9) where u(x) is the unit step function, i.e., u(x) = 1 for x 0 and u(x) = 0 for x < 0. According to (7) (9), γ i = 1 implies that task τ i can be scheduled in the maximum period D m under condition that the set of the re-execution vectors is L 1,..., L i. For describing the condition that multiple tasks are concurrently schedulable, define the schedulability indicator Γ (L 1,..., L i ) {0, 1} with respect to a given set of the re-execution vectors L 1,..., L i. Γ (L 1,..., L i ) := i γ r. (10) r=1 From (9), Γ (L 1,..., L i ) = 1 if and only if γ 1 = = γ i = 1, i.e., tasks τ 1,..., τ i are schedulable at the same time. Since all the tasks are scheduled by RM algorithm, the schedulability indicator can be constructed in a step-wise way, starting from τ 1 ; explicitly, for any i (2 i m), γ i = 1 only if γ i 1 = 1, i.e., the schedulability of a task can be guaranteed only if that of the task with the one-grade higher priority is guaranteed. 4.2 Overall Probability Assume, as an extreme case, that no faults occur in a period D i of task τ i. Then, the program can add checkpoints as long as the sum of checkpoint intervals is not over S i, the maximum slack time of τ i (see (5)). li, the maximum number of checkpoints the program can re-execute, is determined by li = Si i, (11) where i is the fixed checkpoint interval of τ i and x denotes the greatest integer not larger than x. l i serves as the upper boundary of elements in L i, that is, the program can be rolled back up to L i = [ l i,..., l i ] checkpoints in all v i periods of τ i. Somewhat abusing the standard notation, we define the following summation operator for connoting the number of cases of the execution vector L i = [l i1,..., l ivi ]. li L i=0 := li li l i1=0 l i2=0 li l ivi =0. (12) Assume further that the re-execution vectors for all the tasks are L 1,..., L m, with v k as the dimension of vector L k, k = 1,..., m. Recall from (6) ψ i (L i ) is the probability that task τ i can complete throughout period D m with the re-execution vector L i = [l i1,..., l ivi ], i.e., against transient faults which corrupt l ik checkpoint intervals for the k-th period, k = 1,..., v i. Hence, the probability that all the tasks τ 1,..., τ m can complete throughout period D m with the re-execution vectors L 1,..., L m is the multiplication of all ψ i (L i ), i = 1,..., m. Let ψ(l 1,..., L m ) denote this probability: m ψ(l 1,..., L m ) := ψ i (L i ). (13) i=1 By adding all ψ(l 1,..., L m ) for any possible values of L 1,..., L m, we can obtain the overall probability that all the tasks can complete within period D m. But, there are two points to consider before deriving the overall probability:

5 Seong Woo Kwak et al.: Optimal Checkpoint Placement on Real-Time Tasks 109 1) Remind from (11) that the maximum number of re-executable checkpoints for τ i cannot be greater than li irrespective of the number of fault occurrence. Thus, when obtaining the overall probability, it is sufficient to bound the re-execution vector by L i = [ l i,..., l i ] (that is the reason for introducing the shortened notation (12)). 2) Note that schedulability is not concerned in deriving ψ(l 1,..., L m ); it only quantifies the likelihood of whether the number of rollbacks to checkpoints can be tolerated within the deadline of any task. Rollbacks to checkpoints reduce the slack time, for which the scheduling of tasks with lower priority may become impossible. For eliminating any combination of L 1,..., L m that violates the schedulability condition, we should multiply ψ(l 1,..., L m ) by the schedulability indicator Γ (L 1,..., L m ) defined in (10). Embedding conditions 1) and 2), we now derive P, the probability that all the tasks τ 1,..., τ m can complete by the checkpoint scheme throughout the maximum period D m for any possible combination of fault tolerance. From (10), (12) and (13), P can be described by P = = l1 L 1=0 l1 L 1=0 lm L m=0 lm L m=0 (Γ (L 1,..., L m ) ψ(l 1,..., L m )) (Γ(L 1,..., L m ) m ψ i (L i )). i=1 4.3 Optimal Number of Checkpoints (14) The optimal set of checkpoint numbers is the solution that maximizes the probability P of the former equation. To find the solution, we should know the domain of checkpoint numbers, or how many checkpoints each task can possess given the specifications A1 A3. Suppose first the ideal case that no checkpoints are placed and all the tasks are successfully executed with no transient fault. Denote by I m the idle time left in a period D m of τ m after the execution of all the tasks. By Assumption A2, I m is derived as I m = D m D m D 1 E 1 D m D m E m. (15) Since τ m has the lowest priority, any task, including τ m itself, can use I m for posting the checkpoints. The extreme case is that I m is devoted entirely to the checkpoints of one kind of task. Hence,, the maximum number of checkpoints of task τ i, is I m = (D m /D i )t cp (Dm (D m /D 1 )E 1 (D m /D m )E m ) =. (D m /D i )t cp (16) Recall that executing a checkpoint requires the overhead t cp and there are D m /D i periods of task τ i in D m. By varying the value of each (i = 1,..., m) between 1 and, we find the optimal set of checkpoint numbers that leads to the maximum P. We can lessen the computational load for calculating P by identifying the feasible domain of checkpoint numbers. A requirement of checkpoints is that the total overhead of the posted checkpoints should not be greater than the idle time I m. Since τ i has D m /D i periods in D m and checkpoints in each period D i, the overhead of τ i s checkpoints is t cp (D m /D i ). Summing up this value for all τ i,..., τ m, we obtain the following inequality. (D m /D 1 )n (D m /D m )n m I m /t cp. (17) We finally acquire [n 1,..., n m], the optimal set of checkpoint numbers, by solving the following constrained optimization problem. [n 1,..., n m] = arg max P, (D m /D 1 )n (D m /D m )n m I m /t cp, 1 n 1 n 1,..., 1 n m n m. (18) Note that further computation reduction can be achieved by choice of proper numerical methods for solving (18). i, the optimal checkpoint interval for task τ i, is thenduced from (3) as i = E i n i + t cp, i = 1,..., m. (19) Compared with the single-task case, the probability of task completion for multi-task scheduling involves complicated calculations associated with multiplicity of tasks. As shown (7), the slack time of a task is influenced by the tasks with higher priority. Thus the schedulability indicator of a task in (10) depends on the scheduling result of the tasks with higher priority. This implies that the probability of completing a task cannot be expressed with only the parameters of the task; instead it is a function of the parameters of all the tasks with higher or equal priority (see also (15) and (16)). 5 Numerical Examples 5.1 Two-Task Scheduling As a case study, the proposed scheme is applied to

6 110 J. Comput. Sci. & Technol., Jan. 2012, Vol.27, No.1 two multi-task scheduling problems. In the first problem, m = 2, i.e., two tasks τ 1 and τ 2 are being scheduled, where D 1 = 1, D 2 = 2, E 1 = 0.4, and E 2 = 0.7. Transient faults are assumed to occur in a Poisson process with rate λ = 0.1, and the checkpoint overhead is set to be t cp = From (15), I m = I 2 is two adjacent checkpoints are set to be 1 = and 2 = 0.253, respectively. We can see that after completion of τ 1 (t = 0.46), the slack time S 1 = 0.54 remains as calculated above. As l 1 = 0.54/0.153 = 3 from (11), we can re-execute up to three checkpoints for fault-tolerance when the scheduling of τ 2 is not considered. A similar analysis will be made to task τ 2. I 2 = 2 (2/1)0.4 (2/2)0.7 = 0.5. From (16) and the value of I 2, n 1 and n 2 are derived as 0.5 n 1 = = 12, (2/1) n 2 = = 25. (2/2)0.02 Furthermore, (17) induces the following inequality between n 1 and n 2. 2n 1 + n Now that all the constraints on n 1 and n 2 are identified, we calculate P in the feasible domain. Fig.4 shows numerical values of P for parameters n 1 and n 2, where the ranges of n 1 and n 2 are confined to 1 n 1 12 and 1 n 2 25 for clarity. The optimal set of checkpoint numbers is found to be [n 1, n 2] = [3, 3], providing the maximum value P = From (19), the optimal checkpoint intervals are determined as 1 = 0.4/ = and 2 = 0.7/ = Accordingly, e 1 = 0.46, e 2 = 0.76 from (4) and S 1 = 0.54, S 2 = 0.32 from (5). Fig.5. Optimal checkpoint placement for two-task scheduling. 5.2 Three-Task Scheduling Fig.6 is the simulation result for the second problem, where three tasks τ 1 τ 3 are scheduled with D 1 = 1, D 2 = 2, D 3 = 4, E 1 = 0.6, E 2 = 0.5, and E 3 = 0.3. The Poisson rate λ and the checkpoint overhead t cp are set to be the same as the first problem. As the figure shows, the values of n 1 n 3 are confined to 1 n 1 4, 1 n 2 7, and 1 n 3 13 because in other ranges the inequality constraint (18) is not valid. The maximum value of P is when [n 1, n 2, n 3] = [1, 2, 3] (see Fig.6(a)). n 1 = 1 implies that a single checkpoint is placed at the end of τ 1. Fig.7 is the result of checkpoint placement in time range [0, 4]. The optimal checkpoint intervals are 1 = 0.72, 2 = 0.17, and 3 = Since n 1 = 1 and n 2 = 2, a single checkpoint interval of τ 1 is placed first, followed by two checkpoint intervals of τ 2. Note that since > D 1 = 1, the second checkpoint interval of τ 2 is preempted by the second interval of τ 1 invoked at time 1. This preemptios observed also at time 1 of Fig Four-Task Scheduling Table 1 is a part of the simulation result for the third problem, where four tasks τ 1 τ 4 are scheduled with the same λ and t cp as the preceding problems, and with D 1 = 1, D 2 = 2, D 3 = 4, D 4 = 8, E 1 = 0.5, E 2 = 0.4, E 3 = 0.5, and E 4 = 0.7. From (15) and (16), checkpoint numbers n 1 n 4 must satisfy the following inequality Fig.4. Probability of task completion versus number of checkpoints, where m = 2, D 1 = 1, D 2 = 2, E 1 = 0.4, and E 2 = 0.7. Fig.5 illustrates the result of checkpoint placement in time range [0, 2]. Three checkpoints are placed in each period D 1 = 1 and D 2 = 2, where the intervals between 8n 1 + 4n 2 + 2n 3 + n 4 35, which confines their ranges to 1 n 1 3, 1 n 2 6, 1 n 3 11, 1 n 4 21.

7 Seong Woo Kwak et al.: Optimal Checkpoint Placement on Real-Time Tasks 111 Fig.6. Probability of task completion versus number of checkpoints, where m = 3, D 1 = 1, D 2 = 2, D 3 = 4, E 1 = 0.6, E 2 = 0.5, and E 3 = 0.3. (a) n 1 = 1. (b) n 1 = 2. (c) n 1 = 3. (d) n 1 = 4. The maximum value of P is when [n 1, n 2, n 3, n 4] = [1, 2, 2, 1]. The checkpoint placement and calculation of relevant parameters will be analyzed in a similar manner to the former cases. Fig.7. Optimal checkpoint placement for three-task scheduling. Table 1. Probability of Task Completion Versus Number of Checkpoints, where m = 4, D 1 = 1, D 2 = 2, D 3 = 4, D 4 = 8, E 1 = 0.5, E 2 = 0.4, E 3 = 0.5, and E 4 = 0.7 (n 1, n 2, n 3, n 4 ) vs P (1, 1, 1, 1) (1, 1, 1, 2) (1, 1, 1, 3) (1, 1, 2, 1) (1, 1, 2, 2) (1, 1, 2, 3) (1, 1, 3, 1) (1, 1, 3, 2) (1, 1, 3, 3) (1, 2, 1, 1) (1, 2, 1, 2) (1, 2, 1, 3) (1, 2, 2, 1) (1, 2, 2, 2) (1, 2, 2, 3) (1, 2, 3, 1) (1, 2, 3, 2) (1, 2, 3, 3) (1, 3, 1, 1) (1, 3, 1, 2) (1, 3, 1, 3) (1, 3, 2, 1) (1, 3, 2, 2) (1, 3, 2, 3) (1, 3, 3, 1) (1, 3, 3, 2) (1, 3, 3, 3) (2, 1, 1, 1) (2, 1, 1, 2) (2, 1, 1, 3) (2, 1, 2, 1) (2, 1, 2, 2) (2, 1, 2, 3) (2, 1, 3, 1) (2, 1, 3, 2) (2, 1, 3, 3) (2, 2, 1, 1) (2, 2, 1, 2) (2, 2, 1, 3) (2, 2, 2, 1) (2, 2, 2, 2) (2, 2, 2, 3) (2, 2, 3, 1) (2, 2, 3, 2) (2, 2, 3, 3) Conclusions This paper is a novel research result on checkpoint placement in multiple real-time tasks, a topic of faulttolerance that arouses inherent complexity and yet has been studied little so far. We have modeled a multi-task environment in which transient faults occur in a Poisson process and RM algorithm governs the scheduling according to the priority of each task. The probability of task completion and a criterion on schedulability have been derived in an analytic framework, and the optimal set of checkpoint intervals has been obtained such that it maximizes the probability of task completion while guaranteeing schedulability. The significance of the proposed checkpoint scheme is that it provides antegrated solution to multiple real-time tasks by solving a single numerical optimization problem. Simulation results validate the applicability of the proposed scheme.

8 112 J. Comput. Sci. & Technol., Jan. 2012, Vol.27, No.1 References [1] Shin K G, Kim H. Derivation and application of hard deadlines for real-time control systems. IEEE Transactions on Systems, Man, and Cybernetics, 1992, 22(6): [2] Ghosh S, Melhem R G, Mosse D. Fault-tolerance through scheduling of aperiodic tasks in hard real-time multiprocessor systems. IEEE Transactions on Parallel and Distributed Systems, 1997, 8(3): [3] Young J W. A first order approximation to the optimal checkpoint intervals. Communications of the ACM, 1974, 17(9): [4] Ziv A, Bruck J. An on-line algorithm for checkpoint placement. IEEE Transactions on Computers, 1997, 46(9): [5] Siewiorek D P, Swarz R S. Reliable Computer Systems: Design and Evaluation, 3rd Edition. Massachusetts: A K Peters, [6] Shin K G, Lin T H, Lee Y H. Optimal checkpointing of realtime tasks. IEEE Transactions on Computers, 1987, 36(11): [7] Ziv A, Bruck J. Performance optimization of checkpointing schemes with task duplication. IEEE Transactions on Computers, 1997, 46(12): [8] Ziv A, Bruck J. Analysis of checkpointing schemes with task duplication. IEEE Transactions on Computers, 1998, 47(2): [9] Kwak S W, Choi B J, Kim B K. Optimal checkpointing strategy for real-time control systems under faults with exponential duration. IEEE Transactions on Reliability, 2001, 50(3): [10] Quaglia F. A cost model for selecting checkpoint positions in time warp parallel simulation. IEEE Transactions on Parallel and Distributed Systems, 2001, 12(4): [11] Kwak S W, Choi B J, Kim B K. Checkpointing strategy for multiple real-time tasks. In Proc. the 7th International Conference on Real-Time Computing Systems and Applications (RTCSA 2000), Dec. 2000, pp [12] Kim J K, Kim B K. Probabilistic schedulability analysis of harmonic multi-task systems with dual modular temporal redundancy. Real-Time Systems, 2004, 26(2): [13] Aydin H, Melhem R, Mosseé D, Mejia-Alvarez P. Optimal reward-based scheduling for periodic real-time tasks. IEEE Transactions on Computers, 2001, 50(2): [14] Kwak S W, Yang J M. Schedulability and optimal checkpoint placement for real-time multi-tasks. In Proc. IEEE Int. Conf. Industrial Engineering and Engineering Management (IEEM 2010), Dec. 2010, pp [15] Kim H, Shin K G. Design and analysis of an optimal instruction-retry policy for TMR controller computers. IEEE Transactions on Computers, 1996, 45(11): Seong Woo Kwak received his B.S., M.S., and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Korea, in 1993, 1995, and 2000, respectively. From September 2000 to February 2003, he was a research professor at the Satellite Technology Research Center (SaTReC) in KAIST. Since March 2003, he has been with the Department of Electronic Engineering, Keimyung University, Korea, where he is currently an associate professor. His research interests are in control of asynchronous sequential machines, space-borne digital electronics, and fault-tolerance in real-time systems. Jung-Min Yang received his B.S., M.S., and Ph.D. degrees in electrical engineering from KAIST in 1993, 1995, and 1999, respectively. From March 1999 to February 2001, he was a senior member of Engineering Staff at Electronics and Telecommunications Research Institute (ETRI), Korea. Since March 2001, he has been with the Department of Electrical Engineering, Catholic University of Daegu, Korea, where he is currently an associate professor. His research interests are in control of asynchronous sequential machines, fault-tolerance in real-time systems, and scheduling of casting process.