Global Semi-Fixed-Priority Scheduling on Multiprocessors

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1 Global Semi-Fixed-Priority Scheduling on Multiprocessors Hiroyui Chishiro and Nobuyui Yamasai School of Science and Technology Keio University, Yoohama, Japan Abstract Current real-time systems such as robots have multiprocessors and the number of processors tends to be increased. In order to achieve these real-time systems, global real-time scheduling has been required. Many real-time scheduling algorithms are usually based on Liu and Layland s model. Compared to Liu and Layland s model, the imprecise computation model is one of the techniques to overcome the gap between theory and practice. Semi-fixed-priority scheduling is part-level fixedpriority scheduling in the extended imprecise computation model, which has a second mandatory part to terminate an optional part. Unfortunately, current semi-fixed-priority scheduling is only adapted to uniprocessors. This paper presents a global semifixed-priority scheduling algorithm, called Global Rate Monotonic with Wind-up Part (G-RMWP). G-RMWP calculates the optional deadline, the termination time of each optional part, by Response Time Analysis for Global Rate Monotonic (G-RM). The schedulability analysis shows that one tas set is schedulable by G-RMWP if the tas set is schedulable by G-RM. Simulation results show that G-RMWP has higher schedulability than G- RM. I. INTRODUCTION The research of real-time scheduling has started by Liu and Layland since 973. They presented the traditional tas model (Liu and Layland s model) and proposed the representative real-time scheduling algorithm, called Rate Monotonic (RM) [] which is the optimal fixed-priority scheduling algorithm for implicit deadline tas sets on uniprocessors. Nowadays many real-time systems such as autonomous mobile robots have multiprocessors [2], [3], [4]. In addition, the number of processors tends to be increased. In order to achieve these real-time systems, multiprocessor real-time scheduling has been required. There are mainly two multiprocessor realtime scheduling policies: partitioned scheduling and global scheduling. Partitioned scheduling assigns all tass to specific processors. In contrast, global scheduling permits tass to migrate another processor. Unlie RM on uniprocessors, Global Rate Monotonic (G-RM) is not the optimal fixedpriority scheduling algorithm for implicit deadline tas sets on multiprocessors, because traditional priority assignment policies on uniprocessors do not wor well for global scheduling, which is called Dhall s effect [5]. In order to overcome Dhall s effect, Andersson et al. proposed Rate Monotonic with Utilization Separation (RM-US) [6]. RM-US sets the following tass to the highest priority: if the CPU utilization of each tas is higher than M/(3M 2), where M is the number of processors. The disadvantage of RM-US against G-RM is that there is one tas set, which is schedulable by G-RM and is not schedulable by RM-US. In addition, the remaining time, which subtracts Actual Case Execution Time (ACET) from Worst Case Execution Time (WCET), is too much, because the current WCET analysis is too pessimistic [7], [8], [9]. In order to mae use of the remaining time (WCET - ACET), the imprecise computation model [] was presented. The imprecise computation model is one of the techniques used to cope with such uncertainty. The crucial point is that the computation is split into two parts: mandatory part and optional part. A mandatory part affects the correctness of the result and an optional part only affects the quality of the result. By restricting the execution of the optional part to only after the completion of the mandatory part, real-time applications based on the imprecise computation model can provide the correct output with lower quality, by terminating the optional part. However, the imprecise tass in autonomous mobile robots require the processing to output results after terminating or completing their optional parts. For example, object detection tass by image processing need to output proper instructions to actuators for avoiding objects. When the imprecise tass terminate or complete their optional parts, the imprecise computation model cannot guarantee to complete them by their deadlines. In order to overcome the weaness of the imprecise computation model, we use the extended imprecise computation model [], [2], which has a second mandatory part, called wind-up part. In the extended imprecise computation model, we proposed semi-fixed-priority scheduling to achieve both low-jitter and high schedulability [3]. Semi-fixed-priority scheduling fixes the priority of each part in the extended imprecise tas. When changing the part of each extended imprecise tas, its priority may be changed. In addition, we presented a semifixed-priority scheduling algorithm for implicit deadline tas sets on uniprocessors, called Rate Monotonic with Windup Part (RMWP). Unfortunately, current semi-fixed-priority scheduling is only adapted to uniprocessors. This paper presents a global semi-fixed-priority scheduling algorithm for implicit deadline tas sets on multiprocessors, called Global Rate Monotonic with Wind-up Part (G-RMWP). G-RMWP calculates the optional deadline [3], the termination time of each optional part, by Response Time Analysis (RTA) [4] for G-RM [5], [6]. The schedulability analysis

2 Discarded Mandatory part Optional part Completed Wind-up part Terminated Fig.. Extended imprecise computation model remaining execution time Ri(t) general scheduling mi+wi semi-fixed-priority scheduling mi wi τ τ2 time OD OD2 time Mandatory part Optional part Wind-up part Fig. 2. Optional deadline shows that G-RMWP is at least as effective as G-RM. The contribution of this paper is to present a global semifixed-priority scheduling algorithm for implicit deadline tas sets on multiprocessors. In addition, this paper explains that the technique for fixed-priority scheduling such as RTA can be adapted to semi-fixed-priority scheduling. The remainder of this paper is organized as follows: Section II describes the system model. Section III explains semi-fixedpriority scheduling. Section IV presents G-RMWP algorithm. The effectiveness of G-RMWP is evaluated in Section V. Section VI compares our wor with related wor. Finally we offer concluding remars in Section VII. II. SYSTEM MODEL Figure shows the extended imprecise computation model [], [2]. The extended imprecise computation model adds the wind-up part to the imprecise computation model []. The imprecise computation model assumes that the processing to terminate or complete the optional part is not required. However, image processing tass in autonomous mobile robots require the processing to output results. They must guarantee the schedulability of them so that the extended imprecise computation model has the wind-up part. The extended imprecise computation model is similar to the self-suspension model [7]. The extended imprecise computation model defers the execution of the wind-up part to execute the optional part. Each extended imprecise tas must not miss its deadline by the deferred execution. That is to say, the schedulability of extended imprecise computation model is higher than or equal to that of Liu and Layland s model. In contrast, the self-suspension model manages the worst case suspension time. Due to suspension, each self-suspension tas may miss its deadline. This paper assumes that the system has M identical processors and a tas set Γ consisted of n periodic tass with implicit deadlines. Tas τ i is represented as the following tuple (T i,d i,od i,m i,o i,w i ):wheret i is the period, D i is the relative deadline, OD i is the relative optional deadline, m i is the WCET of the mandatory part, o i is the Required Execution Time (RET) of the optional part and w i is the WCET of the wind-up part. The RET of each optional part tends to be underestimated or overestimated from time to time because general semi-fixed-priority Fig. 3. Scheduler τi τi mi mi+wi ODi Di time Mandatory part Wind-up part General scheduling and semi-fixed-priority scheduling RTQ NRTQ Higher Priority Lower Priority SQ Mandatory part Optional part Wind-up part Sleep Fig. 4. Tas queue Empty autonomous mobile robots run in uncertain environments. The relative deadline D i of each tas τ i is equal to its T i.the j th instance of τ i is called job τ i, j. The utilization of each periodic tas is defined as U i =(m i + w i )/T i. The reason why U i does not include o i is because the optional part of τ i is a non-realtime part so that completing it is not relevant to scheduling the tas set successfully. Hence, the system utilization within n tass can be defined as U = i U i /M. All tass are ordered by increasing their periods and τ has the shortest period. An optional deadline [3] is a time when an optional part is terminated and a wind-up part is released. Each wind-up part is ready to be executed after each optional deadline and can be completed if each mandatory part is completed by each optional deadline. Figure 2 shows the optional deadline of each tas. Solid up arrow, solid down arrow and dotted down arrow represent release time, deadline and optional deadline respectively. Tas τ completes its mandatory part by OD and executes its optional part until OD. After OD, then τ executes its wind-up part. In contrast, tas τ 2 does not complete its mandatory part by OD 2.Whenτ 2 completes its mandatory part, then τ 2 executes its wind-up part and does not execute its optional part. III. SEMI-FIXED-PRIORITY SCHEDULING Semi-fixed-priority scheduling [3] fixes the priority of each part in the extended imprecise tas and changes the priority of each extended imprecise tas only in the two cases: (i) when the extended imprecise tas completes its mandatory part and executes its optional part; (ii) when the extended imprecise

3 tas terminates or completes its optional part and executes its wind-up part. Figure 3 shows the difference between general scheduling with Liu and Layland s model [] and semi-fixed-priority scheduling with our model. In general scheduling, when tas τ i is released at, then remaining execution time R i (t) is set to m i +w i and monotonically decreasing until R i (t) becomes at m i + w i. In semi-fixed-priority scheduling, when τ i is released at, then R i (t) is set to m i and monotonically decreasing until R i (t) becomes at m i.whenr i (t) is at m i,thenτ i sleeps until OD i.whenτ i is released at OD i,thenr i (t) is set to w i and monotonically decreasing until R i (t) becomes atod i + w i.ifτ i does not complete its mandatory part by OD i,thenr i (t) is set to w i at the time when τ i completes its mandatory part. In both schedulings, τ i completes its wind-up part by D i. RMWP [3] is one of semi-fixed-priority scheduling algorithms with the extended imprecise computation model. As shown in Figure 4, RMWP manages three tas queues: Real- Time Queue (RTQ), Non-Real-Time Queue (NRTQ) and Sleep Queue (SQ). RTQ holds tass which are ready to execute their mandatory or wind-up parts in RM order. One tas is not allowed to execute its mandatory and wind-up parts simultaneously. NRTQ holds tass which are ready to execute their optional parts in RM order. Every tas in RTQ has higher priority than that in NRTQ. SQ holds tass which complete their optional parts by their optional deadlines or wind-up parts by their deadlines. In RMWP, each tas has the optional deadline. An optional deadline is a time when an optional part is terminated and a wind-up part is released. Each wind-up part is ready to be executed after each optional deadline and can be completed if each mandatory part is completed by the optional deadline. If each tas executes its mandatory part after its optional deadline, the tas may miss its deadline. Each optional deadline is set to the time as late as possible to expand the executable range of each optional part. The wind-up part of each tas must not miss its deadline if the system is idle or executes lower priority tass between the time when the mandatory part is completed and the wind-up part is released. We show how to calculate the relative optional deadline of each tas. Theorem (from [3]). The worst case interference time I i(i < ) which is the upper bound time when τ is interfered by τ i in RMWP is I i = T (m i + w i ). T i The worst case interference time I i is calculated when the relative optional deadline of each tas is. In this case, RMWP generates the same schedule as RM so that I i in RMWP is equal to that in RM. Next we show the relative optional deadline OD of tas τ. Theorem 2 (from [3]). The relative optional deadline OD of tas τ in RMWP is OD = max(,d w I i ). i< The relative optional deadline OD of tas τ in RMWP by theorem 2 is too pessimistic on multiprocessors. Because the worst case interference time on multiprocessors is dramatically less than I i on uniprocessors. Therefore, we extend theorem 2 for global semi-fixed-priority scheduling on multiprocessors. IV. G-RMWP ALGORITHM G-RMWP is based on and extends RMWP for global scheduling on multiprocessors. Lie RMWP based on RM, G- RMWP is based on G-RM. The overall of RMWP algorithm is shown in [3]. A. Optional Deadline We show how to calculate the relative optional deadline of each tas in G-RMWP. First we calculate the worst case interference time of each tas in G-RMWP using RTA [4] for G-RM. Next we calculate the relative optional deadline of each tas for G-RMWP. Bertogna and Cirinei showed RTA with slac for G-RM [5]. This RTA can be expressed in the following fixed-point iteration on the upper bound R ub. R ub (m + w )+ M CI, ) () i< where CI ) is the worst case interference time due to tas τ i within the worst case response time of τ given by: CI ) = min(wci i where Wi CI interval of length L given by: ),Rub (m + w )+), ) is the worst case worload of tas τ i in an Wi CI (L) = Ni CI (L)(m i + w i )+min(m i + w i, L + R ub i (m i + w i ) N CI i (L)T i ), where Ni CI (L) is the maximum number of jobs of tas τ i that contributes all of their execution time in the interval given by: L + R Ni CI ub i (m i + w i ) (L) =. T i Guan et al. improved the precision of RTA for G-RM against equation [6] using Baruah s window analysis framewor [8]. They showed that if tas τ i does not have a carry-in job, then the worst case interference time NC where I NC i ) is NC ) = min(wi ),Rub (m + w )+), Wi NC (L) = Ni NC (L)(m i + w i ) + min(m i + w i,l N NC N NC i (L) = L T i. i (L)T i )

4 The difference between I CI i DIFF ) = ICI i ) and INC i ) is ) INC i ). Using this result, the refined RTA for G-RM can be expressed in the following fixed-point iteration on the refined upper bound R ub : where ˆ I = R ub ( M (m + w )+ ˆ I, (2) i< NC )+ i<max(,m ) ) DIFF ). Now we show the worst case interference time of tas τ i in G-RMWP by equation 2. Theorem 3 (Worst Case Interference Time in G-RMWP). The worst case interference time of tas τ by higher priority tass τ i (i < ) in G-RMWP is Iˆ by equation 2. Proof: The worst case interference time of tas τ by higher priority tass τ i occurs if the relative optional deadline of each tas is equal to. In this case, G-RMWP generates the same schedule as G-RM. Moreover, there is no case that the sum of interference time of tas τ interfered by τ i is more than Iˆ by equation 2. In addition, this theorem explains that the technique for fixed-priority scheduling can be adapted to semi-fixed-priority scheduling. Next we calculate the relative optional deadline of each tas in G-RMWP. Theorem 4 (Optional Deadline in G-RMWP). The relative optional deadline OD of tas τ in G-RMWP is { max(,d w ) ( M) OD = max(,d w Iˆ ) ( > M). Proof: If M, it is clear that the wind-up part of tas τ does not miss its deadline if τ completes its mandatory part at its relative optional deadline OD.If> M, the worst case interference time of τ by higher priority tass τ i (i < ) is at most Iˆ by theorem 3. Hence, this theorem holds. The approach of calculating the relative optional deadline of each tas by theorem 4 maes use of the worst case interference time of G-RM by equation 2, which is pessimistic. If the worst case interference time of G-RM is more precise, the relative optional deadline of each tas in G-RMWP can be set to the later value to expand the executable range of each optional part. Figure 5 shows an example of schedule generated by G-RMWP and G-RM on two processors. The following tas set Γ = {τ =(5,5,4,2,,),τ 2 =(5,5,3,,,2),τ 3 = (5,5,,2,,)} is scheduled by G-RMWP and G-RM in Figure 5(a) and 5(b) respectively. Each relative optional deadline is calculated by theorem 4. This example shows that there is at least one tas set, which is schedulable by G-RMWP and is not schedulable by G-RM. Moreover, in G-RMWP, job τ, executes its optional part in [2,3). τ τ2 τ3 5 (a) Schedule successfully by G-RMWP τ τ2 τ3 5 (b) Schedule unsuccessfully by G-RM Mandatory part Optional part Wind-up part Fig. 5. Example of schedule generated by G-RMWP and G-RM on two processors B. Schedulability Analysis We analyze the schedulability of G-RMWP. First we prove that G-RMWP is at least as effective as G-RM, which is similar approach to RMWP [3]. Theorem 5 (G-RMWP is at least as effective as G-RM). One tas set is schedulable by G-RMWP if the tas set is schedulable by G-RM. Proof: This proof is shown by contraposition. We show that if one tas set is not schedulable by G-RMWP, the tas set is not schedulable by G-RM. By theorem 4, it is clear that τ i completes its wind-up part by its deadline if τ i completes its mandatory part by its optional deadline. Tas τ i misses its deadline only if τ i executes its mandatory part after its optional deadline. In this case, τ i executes its mandatory and wind-up parts continuously without executing its optional part. In G-RM, tas τ i also misses its deadline because of executing its mandatory and wind-up parts continuously. Hence, this theorem holds. By theorem 5, we next show the least upper bound of G- RMWP. Theorem 6 (Least Upper Bound of G-RMWP). The least upper bound of G-RMWP is U lub = M 2 ( U max)+u max, where U max = max{u i i =,2,3,...,n}. Proof: G-RMWP is at least as effective as G-RM by theorem 5 and generates the same schedule as G-RM if the relative optional deadline of each tas is equal to by theorem 3. Hence, the least upper bound of G-RMWP is equal to that of G-RM [9]. By theorem 5 and 6, the schedulability of G-RMWP is higher than or equal to that of G-RM. In addition, by Figure 5, G-RMWP outperforms G-RM from the aspects of both schedulability and imprecise computation. V. SIMULATION STUDIES This simulation uses, tas sets in each system utilization and compares G-RMWP with both G-RM and

5 (a) M = (b) M = 8 Fig. 6. Success ratio when U max = (c) M = (a) M = (b) M = 8 Fig. 7. Success ratio when U max = (c) M = (a) M = (b) M = 8 Fig. 8. Success ratio when U max = (c) M = 6 RM-US[M/(3M-2)]. In simulation environments, the number of processors M is selected within [4,8,6] and U max is selected within [.,.5,.]. Each U i is selected within [.2,.3,.4,...,U max ] and splits U i into two utilizations which are assigned to m i and w i respectively. In autonomous mobile robots, there are various periodic tass. Therefore, the period T i of each tas τ i is selected within [,2,3,...,3]. The system utilization U is selected from [.3,.35,.4,...,.]. The simulation length of the th tas set is the hyperperiod H. The performance metric is defined as the following equation. # of successfully scheduled tas sets = # of scheduled tas sets Figure 6, 7 and 8 show the success ratios when U max =., U max =.5 and U max =. respectively. In all results, the success ratio of G-RMWP is higher than or equal to that of G-RM by theorem 5. When U max =., RM-US has the highest success ratio in all evaluated algorithms. In contrast, when U max =.5, G-RMWP has the highest success ratio in all evaluated algorithms. G-RM outperforms RM-US when U max =.5 because the occurrence frequency of Dhall s effect [5] is less than that of avoiding the deadline miss of each tas thans to the utilization separation. When U max =., G- RMWP, G-RM and RM-US generate the approximately same success ratios. In addition, the success ratio of G-RM is equal to that of RM-US because G-RM generates the same schedule as RM-US when U max =. < M/(3M 2). Considering all simulation results, when U max is lower and lower, the success ratios of all algorithms are approximately the same. When U max =., G-RMWP has slightly lower success ratio than RM-US. However, G-RMWP is at least as effective as G-RM by theorem 5, unlie RM-US. In addition, G-RMWP supports the extended imprecise computation model. That is to say, G-RMWP can execute its optional part when CPU is idle. VI. RELATED WORK We compare our wor with related wor for imprecise computations. There are dynamic-priority scheduling algorithms such as Mandatory-First with Earliest Deadline [2] and Optimization with Least-Utilization [2] in the imprecise computation

6 model []. Unfortunately, these algorithms are only adapted to uniprocessors. Khema et al. discuss the problem of scheduling multiprocessors for imprecise computations, as a networ flow problem [22]. Yun et al. propose a heuristic scheduling algorithm of imprecise multiprocessor systems with / constraint [23]. Stavrinides and Karatza evaluate the performance of dynamicpriority scheduling in distributed real-time systems [24], [25]. However, these approaches for multiprocessor or distributed systems do not analyze the schedulability. Our wor has the theoretical contribution of imprecise computation including both optional deadline and schedulability analysis. Kobayashi and Yamasai propose two dynamic-priority scheduling algorithms for the extended imprecise computation model on uniprocessors: Mandatory-First with Wind-up Part (M-FWP) [], [2] and Slac Stealer for Optional Parts (SS- OP) [26]. M-FWP and SS-OP are too complex to be adapted to multiprocessors, because M-FWP and SS-OP calculate the assignable time of each optional part dynamically. In contrast, RMWP [3] does not calculate the assignable time of each optional part dynamically, thans to the optional deadline, so that RMWP is easy to be adapted to multiprocessors. VII. CONCLUDING REMARKS This paper presented G-RMWP, which is a global semifixed-priority scheduling algorithm for implicit deadline tas sets on multiprocessors. We show how to calculate the optional deadline of each tas in G-RMWP by maing use of RTA for G-RM to achieve global semi-fixed-priority scheduling. The schedulability analysis shows that G-RMWP is at least as effective as G-RM. Simulation results show that G-RMWP has higher schedulability than G-RM. In future wor, we will analyze the schedulability test and more precise worst case interference time for G-RMWP. In addition, the practicality of G-RMWP will be evaluated on RT-Est [27], which is a real-time operating system for semifixed-priority scheduling. ACKNOWLEDGEMENT This research was supported in part by CREST, JST. This research was also supported by Grant in Aid for the Global Center of Excellence Program for Center for Education and Research of Symbiotic, Safe and Secure System Design from the Ministry of Education, Culture, Sport, and Technology in Japan. REFERENCES [] C. L. Liu and J. W. Layland, Scheduling Algorithms for Multiprogramming in a Hard Real-Time Environment, Journal of the ACM, vol. 2, no., pp. 46 6, 973. [2] F. Kanehiro, H. Hiruawa, and S. Kajita, OpenHRP: Open Architecture Humanoid Robotics Platform, The International Journal of Robotics Research, vol. 23, no. 2, pp , 24. [3] H.S.Ahn,Y.M.Bea,I.-K.Sa,W.S.Kang,J.H.Na,andJ.Y. 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