New Schedulability Test Conditions for Non-preemptive Scheduling on Multiprocessor Platforms

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1 New Schedulability Test Conditions for Non-reemtive Scheduling on Multirocessor Platforms Technical Reort May 2008 Nan Guan 1, Wang Yi 2, Zonghua Gu 3 and Ge Yu 1 1 Northeastern University, Shenyang, China 2 Usala University, Usala, Sweden 3 Hong Kong University of Science and Technology, Hong Kong, China Abstract We study the schedulability analysis roblem for nonreemtive scheduling algorithms on multirocessors. To our best knowledge, the only known work on this roblem is the test condition roosed by Baruah [1] (referred to as [BAR- EDF n ]) for non-reemtive EDF scheduling, which will reject a task set with arbitrarily low utilization if it contains a task whose execution is equal or greater than the minimal relative deadline among all tasks. In this aer, we firstly derive a linear- test condition which avoids the roblem mentioned above, by building uon the work in [2] for reemtive multirocessor scheduling. This test condition works on not only non-reemtive EDF, but also any other work-conserving non-reemtive scheduling algorithms. Then we imrove the analysis and resent test conditions of seudo-olynomial comlexity for Non-reemtive Earliest Deadline First scheduling (EDF n ) and Non-reemtive Fixed Priority scheduling (FP n ) resectively. Exeriments with randomly generated task sets show that our roosed test conditions, esecially the imroved test conditions, have significant erformance imrovements comared with [BAR-EDF n ]. 1 Introduction Comared with reemtive scheduling, non-reemtive scheduling and schedulability analysis have received considerable less attention in the research community. However, nonreemtive scheduling is widely used in industry ractice, and it may be referable to reemtive scheduling for a number of reasons [3]: Non-reemtive scheduling algorithms are easier to imlement and have lower run overhead than reemtive scheduling algorithms; the overhead of reemtive scheduling algorithms is more difficult to characterize and redict than that of non-reemtive scheduling algorithms due to inter-task interference caused by caching and ielining. These benefits of non-reemtive scheduling are even more imortant on multirocessor latforms, since the task migration overhead is higher and more difficult to redict. However, this roblem is much less severe for non-reemtive scheduling, where each task instance runs to comletion on one rocessor, and task migration only haens at task instance boundaries. Non-reemtive scheduling is considered inferior for critical systems (artially) because of its oor resonsiveness. On a single rocessor latform, the most urgent task may not get the rocessor for quite a long due to non-reemtive blocking, which could be deadly harmful for hard real systems. However, the natural arallelism of multirocessors can mitigate the enalty of non-reemtive blocking. Even if several rocessors are long occuied by tasks with large execution, urgent tasks can execute on other rocessors. We have conducted simulation exeriments to comare the erformance of two well-known reemtive scheduling algorithms EDF and DM, and their non-reemtive versions EDF n and DM n on multirocessors (details shown in Section 9). To our surrise, for task sets in which the range of task execution is not very wide, the erformance of EDF n is very close to EDF, and DM n actually erforms better than DM. Note that we even have not accounted the context switch overhead in the simulations. So we believe that, for a considerable art of reallife alications on multirocessor latforms, non-reemtive scheduling could be a better choice with resect to real erformance. This is yet another motivation for us to study non-reemtive scheduling on multirocessors. In this work, we study the schedulability analysis roblem for soradic task sets on identical multirocessors with nonreemtive scheduling. We focus on work-conserving scheduling algorithms, i.e., it is not allowed to idle a rocessor if there is some task instance awaiting for execution. Note that in the context of multirocessor scheduling, a work-conserving algorithm is necessarily a global [4] algorithm. The only known work on this roblem is the test condition roosed by Baruah [1] (referred to as [BAR-EDF n ]) for non-reemtive EDF

2 scheduling (referred to as EDF n ), which has quite oor erformance and suffers from the disadvantage of that task sets with arbitrarily low utilization will be rejected if C max D min, where C max is the maximal execution among all tasks and D min is the minimal relative deadline. In this aer we develo new sufficient schedulability tests by building uon the work in [5, 2] for reemtive scheduling algorithms. At first we derive a test condition of linear- comlexity, which works on any work-conserving non-reemtive scheduling algorithm, and can avoid the [BAR-EDF n ] s disadvantage mentioned above. Then we derive imroved test conditions for EDF n and FP n (Non-reemtive Fixed Priority scheduling) resectively, which are both of seudo-olynomial -comlexity, but have significant erformance imrovement comared with [BAR-EDF n ]. The aer is structured as follows: Section 2 resents the related work. Section 3 introduces the system model and our analysis framework. We resent our first general test condition in Section 4, and then imrove it for EDF n and FP n in Section 5 and Section 6 resectively. The robust roerty of the roosed tests is roved in Section 7. Section 8 resents erformance evaluation results and Section 9 use simulation to rovide aroximate erformance comarison of reemtive and non-reemtive scheduling algorithms. Finally, conclusions are resented in Section Related Work Preemtive Scheduling. All scheduling algorithms mentioned in this aragrah are within the context of global reemtive, for examle, we refer to the global reemtive EDF as EDF for short. Goossens et al. [6] introduced a schedulability test of olynomial -comlexity for eriodic task sets scheduled by EDF based on the resource-augmentation techniques [7]. Similar techniques are also used in [8], to derive schedulability tests for tasks with limited utilization scheduled by RM. Baker [5] resented schedulability tests of both EDF and DM by assuming that a given task τ k s task instance misses deadline, and then determining necessary conditions on the arameters of all the tasks to cause such a deadline miss. Based on Baker s idea, Bertogna et al. [9] observed that the work done in arallel with a task instance do not need to be accounted into its interference, and rovided a new test condition of olynomial -comlexity, which can occasionally outerform Baker s test condition. Baruah [2] extended Baker s aroach to reduce the over-estimation of the so-called carryin, and rovided a test condition of seudo-olynomial comlexity, which has much higher accetance ratio than revious test conditions for task systems satisfying the following conditions: the number of tasks n is significantly larger than the number of rocessors m (i.e., n m), or the arameters of different tasks have widely varying orders of magnitude. Recently, Baruah et al. [10, 11, 12] have develoed a new aroach based on Baker s idea, which rovides test conditions (of seudo-olynomial -comlexity or olynomial comlexity deending on different accuracy degrees) for both EDF and DM, as well as some interesting resource augmentation bounds. Andersson et al. [13] first used the aroximate resonse analysis for multirocessor scheduling, which was later imroved by Bertogna et al. [14] by utilizing their observation in [9] and exloring task slack to reduce the essimistic degree in the comutation of the aroximate resonse. Similar to the exact resonse analysis for singlerocessor scheduling, the -comlexity of their aroaches is seudo-olynomial. Non-Preemtive Scheduling. For single-rocessor scheduling, Jeffay et al. [3] considered non-reemtive algorithms for scheduling eriodic or soradic task systems with relative deadline equal to eriod under the work-conserving assumtion and resented a exact schedulability test of seudoolynomial -comlexity for eriodic or soradic task set with the EDF n scheduling algorithm on a single rocessor. George et al. [15] study general task models in which relative deadlines and eriods are not necessarily related, and established exact schedulability tests for both EDF n and FP n on a single rocessor of seudo-olynomial -comlexity. Recently, Baruah et al. [16] studied the schedulability analysis for non-reemtive recurring tasks, which is the general form of non-reemtive soradic tasks, and showed that the nonreemtive schedulability analysis can be reduced to a olynomial number of reemtive schedulability analysis roblems. For multirocessor scheduling, Baruah [1] roosed a sufficient but not necessary olynomial- schedulability test condition [BAR-EDF n ] for global EDF n for eriodic task sets, which can be easily generalized to soradic task sets. [BAR-EDF n ] used the technique similar to [6] and took into account the extra interference caused by non-reemtion. [BAR-EDF n ] showed that a task set τ is EDF n schedulable on m rocessors if V sum (τ) m (m 1)V max (τ) (1) where V sum (τ) = τ i τ V i, V max (τ) = max τ i τ V i V i = { D i C max D i > C max D i C max and C max is the maximum execution among all tasks. It is obvious that a task set with arbitrarily low utilization can not ass the test if C max D min, where D min denotes the minimal D i among all tasks. Intuitively, it means that for any task instance J k, if there is some task with execution large enough to cover its relative deadline D k, J k will definitely miss its deadline, which is true for single rocessor scheduling, but not necessarily true for multirocessor scheduling, since even if there are some rocessors long occuied by a task instance with large, other task instances can execute on other rocessors to meet their deadlines.

3 cu3 cu2 cu1 at least one rocessor is idled t o r k deadline miss occurs l k S k Problem Window execution of J k execution of other task instances carry - in t o... body Problem Window r k carry -out l k cu3 cu2 cu1 (a) the roblem window in non-reemtive scheduling at least one rocessor is idled t o r k deadline miss occurs D k Problem Window (b) the roblem window in reemtive scheduling execution of J k execution of other task instances Figure 1. the roblem window in reemtive and non-reemtive scheduling. 3 System Model and Analysis Framework We adot the discrete model in this aer, i.e., any value t involved in scheduling and any task arameter is assumed to be a non-negative integer value. We assume that a multirocessor latform consisting of m identical rocessors. A soradic task set τ consists of n soradic tasks. A soradic task is denoted by τ i = (, D i, ), where is the worst-case execution, D i is the relative deadline and is the minimum inter-release searation, which is also referred to as the eriod of the task, and we assume D i. We define S i = D i. The utilization of task τ i is defined as U i = Ci, and we use U(τ) to denote the sum of U i of all τ i τ. Such a soradic task τ i generates a otentially infinite sequence of task instances (also known as jobs) with successive releases searated by at least units. We use J i to denote the th instances of τ i. We also use J i to denote τ i s instance in general if we do not want to secify which instance it is. Each task instance has a release (arrival ) r i and a absolute deadline d i = r i + D i. We use l i = d i to denote the latest start of task instance J i, i.e., if J i starts execution after l i, it must miss its deadline. For fixed riority scheduling, we use P (τ i ) to denote τ i s riority. We assume that every task has a unique riority in our task system, and use P (τ i ) > P (τ j ) to denote that τ i s riority is higher than τ j s. We aim to analysis the schedulability of a soradic task set on multirocessors with non-reemtive scheduling. As shown in [3], exact feasibility-analysis of eriodic task systems uon non-reemtive systems is highly intractable, even uon singlerocessors. So our goal is to obtain sufficient, rather than exact, conditions for schedulability test. In the following we will introduce the general framework of our analysis, which is closely related to the work in [2] for re- Figure 2. body, carry-in and carry-out of a task in the roblem window [t o, l k ]. emtive scheduling. Suose a task set τ is non-schedulable, and let J k be the first task instance that misses deadline. Let t o denote the latest -instant earlier than r k at which at least one rocessor is idle and let = r k t o. Since all rocessors are idle when the system starts, so there always exists such a t o. Since reemtion is not allowed, once a task instance starts execution, it must run to comletion without interrution. So if J k starts to execute before its latest start l k, it must be able to finish execution before deadline. Therefore, in order for J k to miss its deadline, all m rocessors must be continuously busy in the interval [t o, l k ]. What haens after l k has no effect on the schedulability of J k. We name the interval [t o, l k ] as roblem window, as shown in Figure 1-(a). The definition of roblem window here is different from the one in [2] for reemtive scheduling, where all m rocessors must be continuously busy in the interval [t o, r k ], but does not have to be continuously busy in the interval [r k, ] as long as the sum of the busy segments (shadow area in the figure) is large enough to cause τ k to miss its deadline, as shown in Figure 1-(b). The necessary condition for the deadline miss to occur is that the worst-case work done in the roblem window by all other task instances in the task set τ excet J k, is larger than ( + S k ) m (the shadow area in Figure 1-(a)). Since there is no critical instant in multirocessor scheduling, it is not ossible to find the worst-case situation without exhaustively simulating the system. So we will comute the worst-case work done by each task in the roblem window, denoted by I(τ i ), and use the sum of each I(τ i ) as an uer bound of the overall worst-case work done in the roblem window. The work done by of a task τ i in the roblem window can be categorized into three tyes: body: the contribution of all task instances (called body instance) with both release and deadline in the roblem window; each task instance contributes to the workload in that interval with a comlete execution C k ; carry-in: the contribution of at most one task instance (called carry-in instance) with release earlier than t o and deadline in the roblem window; this task instance contributes with the fraction of its execution actually executed in the roblem window. carry-out: the contribution of at most one task instance (called carry-out instance) with release in the roblem window and deadline later than l k ; this task instance

4 contributes with the fraction of its execution actually executed in the roblem window. We always consider the work of a carry-in instance is executed as late as ossible and a carry-out instance is executed as early as ossible, as shown in Figure 2. This is a essimistic but safe aroximation to account the work done by a task. Since there is at least one rocessor idled at t o, so at most m 1 tasks may cause the carry-in, and the remaining (n m + 1) tasks has no carry-in. We use I 1 (τ i ) to denote I(τ i ) if τ i has no carry-in instance, and use I 2 (τ i ) to denote I(τ i ) if τ i has a carry-in instance, and define: (τ i ) = I 2 (τ i ) I 1 (τ i ) (2) We sort all (τ i ) in a non-increasing list, and use m 1 to denote the sum of the first (m 1) elements in this list, then we can get a sufficient condition for τ to be schedulable: Lemma 1. A task set τ is schedulable with work-conserving non-reemtive scheduling algorithms on m rocessors, if for any task τ k and any 0 the following condition is satisfied: τ i τ I 1 (τ i ) + m 1 < ( + S k ) m (3) 4 The First Schedulability Test To use Lemma 1 for schedulability test, we should comute the LHS of Inequality 3 as well as solve the unknown variable in the inequality (The LHS of the inequality also imlicitly contains ). Simle uer bounds of I 1 (τ i ) and I 2 (τ i ) can be obtained by essimistically accounting both the carry-in and carry-out of a task τ i as and accounting the work done by its body instances as +S k. At the same, we observed that as increases, the roortion of the carry-in and carry-out in the overall work done by a task in the roblem window tends to decrease, which imlies that the adverse effect of the overestimation of the carry-in and carry-out is more severe with small + S k values, of which the extreme case is = 0. We get our first test condition based on the observations above: Theorem 1. [TEST-1] A task set τ is schedulable with workconserving non-reemtive scheduling algorithms on m rocessors if: n i U(τ) < m + m 1 (4) S min where n i is the sum of all tasks, S min is the minimal S i among all tasks; we sort all in a non-increasing list, and use m 1 to denote the sum of the first (m 1) elements in this list, Proof. We rove the theorem by contradiction. Assume a task set τ satisfies Inequality 4 but it is non-schedulable, and with a task τ k missing its deadline. Since τ k is non-schedulable, by Lemma 1 we have: I 1 (τ i ) + m 1 ( + S k ) m (5) τ i τ The number of τ i s body instances in the roblem window is at most +S k, and the carry-in and the carry-out are both at most, so by the definition of I 1 (τ i ) and I 2 (τ 2 ), we have: I 1 (τ i ) + S k + + S k + and I 1 (τ i ) ( + S k )U i + (6) I 2 (τ i ) + S k S k + 2 I 2 (τ i ) ( + S k )U i + 2 (7) so (τ i ) =, so we have m 1 Therefore, we have ( + S k )U(τ) + n i = m 1. + m 1 ( + S k ) m (8) and since 0 and S k S min, we get n i U(τ) m + m 1 S min which contradicts our assumtion that τ satisfies Inequality 4. Note that [TEST-1] does not suffer from the disadvantage in [BAR-EDF n ] that any task set with C max D min will be rejected. [TEST-1] works on any work-conserving non-reemtive scheduling algorithm, since it does not rely on any scheduling algorithm-secific roerty excet requiring that no rocessor can be idle if there is some task instance awaiting for execution. U i, n i and S min all can be comuted in linear, and we can use linear- selection [17] to comute m 1, so [TEST-1] is with linear- comlexity, which is the same as [BAR-EDF n ]. [TEST-1] is still deely essimistic, since it used very coarse bounds on I 1 (τ i ) and I 2 (τ i ). In Section 5 and 6, we will resent less-essimistic test conditions for EDF n and FP n by deriving more recise bounds on I 1 (τ i ) and I 2 (τ i ), where we assume each task τ i exactly executes for and its instances are exactly released with searations of, and later in Section 7 we will show the test conditions are still correct if this assumtion is broken. 5 The Imroved Test for EDF n In last section, we essimistically assume that every carryout instance contributes to the overall work in the roblem window. Actually, a carry-out instance can execute in the roblem

5 window only if it can interfere with J k, otherwise, it must execute after l k. Now we discuss the ossible interference on a task instance J k in EDF n. We assume the riority ties are broken arbitrarily in the EDF n scheduler. Lemma 2. For EDF n, if D i > D k, the necessary condition for J i to cause interference to J k is r i < r k, i.e., J i must be released earlier than J k ; if D i D k, the necessary condition for J i to cause interference to J k is d i, i.e., J i s absolute deadline must be no later than that of J k. Proof. There are two tyes of interference in non-reemtive scheduling: 1. Interference caused by the riority order. A task instance J i can cause this kind of interference to J k only if d i. Note that since we assume the riority ties are arbitrarily broken, J i may interfere with J k if d i =. 2. Interference caused by non-reemtion blocking. A task instance J i can cause this kind of interference to J k only if J i has already been running when J k is released, which imlies r i < r k. If D i > D k, suose r i r k, then J i can not cause the second tye of interference; since D i > D k, J i s deadline must be later than, so it also can not cause the first tye of interference. If D i D k, suose d i >, then J i can not cause the first tye of interference; since D i D k, J i s release must be later than r k, so it also can not cause the second tye of interference. 5.1 Comuting I 1 (τ i ) for EDF n +D k a 1 D i l k k S k (a) α 1 > + D k +D k a 1 l k k Sk (b) α 1 + D k ( +S k ) mod D i Figure 4. the worst case of I 1 (τ i ) if D i D k. We use J i to denote the first body instance of τ i. At the left end of the roblem window, J 1 i s release is t o + x after being moved rightwards for x. Since x <, J 1 i s release is still earlier than t o, and by the definition of I 1 (τ i ), J 1 i has no contribution to I 1 (τ i ) after being moved, so there is no increase to I 1 (τ i ) at the left end of the roblem window. At the right end of the roblem window, trivially there is no increase to I 1 (τ i ). So moving all τ i s releases rightwards for a x (x < ) will not increase I 1 (τ i ). Now we comute I 1 (τ i ) (i k) in this worst case: k D k Figure 3. the worst case of I 1 (τ i ) if i = k. At first we comute I 1 (τ i ) with i = k, i.e., the worst-case work done by τ k s body instances. As shown in Figure 3, the number of τ k s body instances is, so we have I 1 1 (τ i ) = C k (9) Next we will comute I 1 (τ i ) with i k. The following Lemma shows the worst case of I 1 (τ i ) of a task τ i with i k. Lemma 3. The worst case of I 1 (τ i ) (i k) occurs when one of τ i s instances is released at the -instant t o. Proof. To rove the worst case of I 1 (τ i ) occurs when one of τ i s instances is released at the -instant t o, we should rove that based on this case, moving all τ i s releases rightwards for a distance x (x < ) will not increase I 1 (τ i ) 1. 1 Moving τ i s releases for any distance can be transformed to an equal form of moving them rightwards (or leftwards) for x (x < ). 1. D i D k. By Lemma 2, we know a task instance of τ i with D i D k can interfere with J k only if its deadline is no later than. We use α 1 to denote the distance between t o and the deadline of the τ i s last instance released before l k : α 1 = + S k + D i (10) (a) α 1 > + D k. As shown in Figure 4(a), the deadline of τ i s last instance released in the roblem window is later than, so it has no contribution to I 1 (τ i ). The number of τ i s instances contributing to I 1 (τ i ) is +S k. So we have: I 2 1 (τ i ) = + S k (11) (b) α 1 + D k. As shown in Figure 4(b), the deadline of τ i s last instance released in the roblem window is no later than, so it contributes to I 1 (τ i ), and the contribution is bounded by both

6 and ( + S k )mod. So we have: I 3 1 (τ i ) = + S k +min(, ( +S k )mod ) a 2 D i (12) by: 0 D i > D k = 0 I1 1 (τ i ) i = k I 1 (τ i ) = I1 2 (τ i ) (i k D i D k α 1 > + D k ) (D i > D k α 2 > 0) I1 3 (τ i ) otherwise (14) where I1 1 (τ i ), I1 2 (τ i ), I1 3 (τ i ), α 1 and α 2 are defined in Equation 9, 11, 12, 10 and 13 resectively. l k k Sk 5.2 Comuting I 2 (τ i ) for EDF n (a) α 2 > 0 ( +S k ) mod ( +D k ) mod a 2 D i k D k l k k (b) α 2 < Sk Figure 5. the worst case of I 1 (τ i ) if D i > D k. 2. D i > D k By Lemma 2, we know an instance of τ i with D i > D k can interfere with J k only if its release is earlier than r k. We use α 2 to denote the distance between t o and the release of τ i s last instance released in the roblem window: α 2 = + S k (13) (a) = 0. If = 0, then t o = r k. Since D i > D k, any task instance released no earlier than t o has deadline later than, so can not interfere with J k. So in this case I 1 (τ i ) = 0. (b) α 2 > 0. As shown in Figure 5(a), the release of τ i s last instance released in the roblem window is no earlier than r k, so it can not interfere with J k. The number of τ i s instances contributing to I 1 (τ i ) is +S k. So in this case I 1 (τ i ) is comuted by Equation 11. (c) α 2 <. As shown in Figure 5(b), the release of τ i s last instance released in the roblem window is earlier than r k, so it contributes to I 1 (τ i ), and its contribution is bounded by both and ( + S k ) mod. So in this case I 1 (τ i ) is comuted by Equation 12. By the discussions above, we can comute I 1 (τ i ) for EDF n Figure 6. the worst case of I 2 (τ i ) if i = k. At first we comute I 2 (τ i ) with i = k, i.e., the worst-case woke done by τ k s carry-in and body instances. As shown in Figure 6, if we take the interval between the deadlines of two adjoining instances of τ i as a unit, there are +D k such units in the interval [t o, ]. The carry-in is bounded by both C k and ( + D k )mod. At the same, the work done by J k itself should be subtracted. So for τ i with i = k, we have: I2 1 (τ i ) = + D k C k +min(c k, ( +D k )mod ) C k (15) In the following we will comute I 2 (τ i ) with i k in Lemma 4, 5 and 6. ( +D k ) mod S i k l k D k r i C k d i Figure 7. the worst case of I 2 (τ i ) if D i D k S i > C k. Lemma 4. The worst case of I 2 (τ i ) (i k) occurs when one of τ i s instance has its deadline at, if D i D k S i > C k (16) and in this case we can comute I 2 (τ i ) by: I 2 2 (τ i ) = + D k +min(, ( +D k )mod ) (17)

7 Proof. Let J i be the instance with its deadline at, and by Lemma 2, we know J i may interfere with J k. As shown in Figure 7, since S i > C k, J i s contribution to I 2(τ i ) is. Now we examine whether I 2 (τ i ) will be increased if we move all τ i s releases leftwards for x (x < ). After moving leftwards for x (x < ), at the right end of the roblem window, the contribution of J i is still ; the deadline of J +1 i is at + x, and since x <, it is still later than, so has no contribution to I 2 (τ i ), so I 2 (τ i ) will not be increased at the right end of the roblem window. At the left end, trivially there is no increased to I 2 (τ i ). So I 2 (τ i ) will not be increased after moving τ i s releases leftwards for x (x < ), so d i = is the worst case for I 2 (τ i ) for tasks with D i D k S i > C k. As shown in Figure 7, we take the interval between the deadlines of two adjoining instances of τ i as a unit, then there are +D k such units in the interval [t o, ]. The carryin is bounded by both and ( + D k )mod. So we can comute I 2 (τ i ) by Equation 17. earlier than r k, so J +1 i has no contribution to I 2 (τ i ). Therefore, I 2 (τ i ) will not be increased at the right end of the roblem window. At the left end, trivially the contribution will not be increased. So I 2 (τ i ) will not be increased after moving τ i s releases leftwards for x (x < ), so r i = r k 1 is the worst case for I 2 (τ i ) for tasks with D i > D k S k. If > 0, as shown in Figure 8, the number of τ i s body instances is 1, the carry-out is, the carry-in is bounded by both and the distance between t o and the deadline of the carry-in instance, which equals to max(0, ( 1)mod ( D i )). If = 0, I 2 (τ i ) = 1. So we can comute I 2 (τ i ) by Equation 19. max(0, ( +S k - ) mod - ( -D i )) -D i k l k S k r i S i d i C k max(0, ( -1) mod - ( - D i )) (a) D i D k S i C k max(0, ( +S k - ) mod - ( -D i )) -D i t o -1 r i r k l S k k C k d i -D i r i k l k S k d i d C k k Figure 8. the worst case of I 1 (τ i ) if D i > D k S k. Lemma 5. The worst case of I 2 (τ i ) (i k) occurs when one of τ i s instance is released at r k 1, if D i > D k S k (18) and in this case we can comute I 2 (τ i ) by: where { I2 3 Ci 1 = 0 (τ i ) = ( 1 + 1) + υ > 0 (19) υ = min(, max(0, ( 1)mod ( D i ))) (20) Proof. Let J i be τ i s task instance released at r k 1, as shown in Figure 8. By Lemma 2 we know J i can interfere with J k. Since S k, the contribution of J i is. Now we examine whether I 2 (τ i ) will be increased if we move all τ i s releases leftwards for x (x < ). After moving leftwards for x (x < ), at the right end of the roblem window, the contribution of J i is still ; the release of J +1 i is r k 1+ x, since x <, it is still not (b) D i > D k S k < Figure 9. the worst case of I 2 (τ i ) if D i D k S i C k or D i > D k S k <. Lemma 6. The worst case of I 2 (τ i ) (i k) occurs when one of τ i s instances is released at l k, if (D i D k S i C k ) (D i > D k S k < ) (21) and in this case we comute I 2 (τ i ) by: { I2 4 Ak + S k + S k (τ i ) = +S k + + ω + S k > (22) where ω = min(, max(0, ( + S k )mod ( D i ))) (23) Proof. At first we will show that if r i = l k, J i contributes to I 2 (τ i ): D i D k S i C k. As shown in Figure 9(a), since S i C k, we have d i, so J i can interfere with J k, and since S i C k, the contribution of J i is.

8 D i > D k S k <. As shown in Figure 9(b), since S k, we have r i r k, so J i can interfere with J k, and since S k, the contribution of J i is. Now we examine whether I 2 (τ i ) will be increased if we move all τ i s releases leftwards for x (x < ): If we move τ i s releases leftwards for x 1 (x 1 < ), at the left end of the roblem window, trivially the contribution can not be increased; at the right end of the roblem window, the contribution of J i is still, the release of the J +1 i is l k +( ) x 1, and since x 1 <, it is still later than l k after moving, so J +1 i has contribution to I 2 (τ i ). Therefore there is no increase of I 2 (τ i ) at the right end of the roblem window. So moving τ i s release leftwards for x 1 (x 1 < ) will not increase I 2 (τ i ). Moving τ i s releases leftwards for x 2 ( x 2 < ) has the same effect as moving τ i s release rightwards for x 2. Since x 2 ( x 2 < ), we have 0 < x 2. So at the right end of the roblem window, the contribution of J i is decreased by x 2 after being moved rightwards for x 2 ; at the left end of the roblem window, I 2 (τ i ) is at most increased by x 2. So I 2 (τ i ) will not be increased after moving τ i s releases rightwards for x 2 ( x 2 < ). In summary, I 2 (τ i ) will not be increased if we move τ i s releases leftwards for x (x < ). So r i = l k is the worst case for I 2 (τ i ). If A s + S k, it is easy to see I 2 (τ i ) = + S k. If A s + S k >, the number of body instance is +S k ; the carry-out is ; the carry-in is bounded by both and the distance between t o and the deadline of the carry-in instance, which equals to max(0, ( + S k )mod ( D i )). So we can comute I 2 (τ i ) by Equation 22. By the discussions above, we can comute I 2 (τ i ) for EDF n by: I2 1 (τ i ) I2 I 2 (τ i ) = 2 (τ i ) I2 3 (τ i ) I2 4 (τ i ) i = k i k D i D k S i > C k D i > D k S k otherwise (24) where I 1 2 (τ i ), I 2 2 (τ i ), I 3 2 (τ i ) and I 4 2 (τ i ) are defined in Equation 15, 17, 19, 22 resectively. 5.3 A New Test Condition for EDF n By now we have obtained a sufficient schedulability test condition for EDF n by Lemma 1 and the comutation of I 1 (τ i ) and I 2 (τ i ) above: Theorem 2. [TEST-EDF n ] A task set is EDF n schedulable on m identical rocessors if for any task τ k and for any we have: I 1 (τ i ) + m 1 < ( + S k ) m (25) τ i τ where I 1 (τ i ) and I 2 (τ i ) are defined in Equations14 and 24. For any given τ k and, the LHS of Condition 25 can be evaluated in linear with n, since the for comuting I 1 (τ i ), I 2 (τ i ) and (τ i ) for each τ i is O(n), and we can use linear- selection [17] to comute m 1. The next theorem tells us the range of that should be tested: Theorem 3. For any task set with U(τ) < m, if condition 25 is to be violated for any, then it is violated for some that satisfies the condition below: n i + m 1 S k (26) m U(τ) Proof. As shown in Section 4 (Inequality 6, 7), we know: I 1 (τ i ) ( + S k ) U i + I 2 (τ i ) ( + S k ) U i + 2 If Condition 25 is violated, it must be true that n i + m 1 + ( + S k )U(τ) ( + S k )m n i + m 1 m U(τ) S k Since is a non-negative integer, Condition 25 can be checked in seudo-olynomial to the task arameters, for all task systems τ for which U(τ) is bounded by a constant strictly less than the number of rocessors m. As mentioned in Section 4, the effect of the over-estimation of the carry-in and carry-out is more severe with smaller + S k values. So one should check the tasks in increasing order of their S k s, and in increasing order of s for each task, to advance the testing efficiency. We have tested task sets consisting of one hundred of tasks with each uniformly distributed in [10, 2000]. The tests of 1,000,000 such task sets are finished in several minutes. We believe this rates suggest that [TEST- EDF n ] is not only alicable to off-line schedulability test, but also a good candidate for on-line admission control with moderate-scale task systems. 6 The Imroved Test for FP n The following lemma shows the ossible interference on caused by lower-riority tasks in FP n. Lemma 7. A task instance J i with P (τ i ) < P (τ k ) can interfere with J k of τ k only if J i is released before r k.

9 6.1 Comuting I 1 (τ i ) for FP n If i = k, the comutation of I 1 (τ i ) is the same as in Equation 9: I 1 1 (τ i ) = C k Now we comute I 1 (τ i ) with i k. The conclusion of Lemma 3 also works for FP n, i.e., the worst case of I 1 (τ i ) occurs when one of τ i s instance is released at t o. Now we comute I 1 (τ i ) in this worst case. a 2 D i 2. P (τ i ) > P (τ k ). If P (τ i ) > P (τ k ), any τ i s instance released in the roblem window can contribute to I 1 (τ i ). So we can comute I 1 (τ i ) by Equation 12. I 3 1 (τ i ) = + S k + min(, ( + S k )mod ) So we have : 0 P (τ i ) < P (τ k ) = 0 I1 I 1 (τ i ) = 1 (τ i ) i = k I1 2 (τ i ) P (τ i ) < P (τ k ) α 2 > 0 I1 3 (τ i ) otherwise 6.2 Comuting I 2 (τ i ) for FP n (27) l k k Sk When i = k, the comutation of I 2 (τ i ) is the same as Equation 15. a 2 (a) α 2 ( +S k ) mod D i I2 1 (τ i ) = + D k C k +min(c k, ( +D k )mod ) C k In the following we will comute I 2 (τ i ) with i k in Lemma 8 and 9. Their roofs are similar with the roofs in Section 5.2. (( -1) 0 mod - ( - D i )) 0 l k k Sk (b) α 2 < -D i r i n d i n Figure 10. P (τ i ) < P (τ k ). 1. P (τ i ) < P (τ k ) We still use α 2 to denote the distance between t o and the release of τ i s last instance released in the roblem window: α 2 = + S k (a) = 0. If = 0, then t o = r k. Since P (τ i ) < P (τ k ), any instance released at or later than t o can not interfere with J k. So we know in this case I 1 (τ i ) = 0. (b) α 2 > 0. In this case, the release of the last instance released in the roblem window is not earlier than r k, so by Lemma 7 we know it can not interfere with J k, so we can comute I 1 (τ i ) by Equation 11. I1 2 (τ i ) = + S k (c) α 2 <. In this case, the release of the last instance that released in the roblem window is earlier than r k, so by Lemma 7 we know it can interfere with J k, so we can comute I 1 (τ i ) by Equation 12. I 3 1 (τ i ) = + S k +min(, ( +S k )mod ) t o -1 r k l S k k C k Figure 11. P (τ i ) < P (τ k ) S k. Lemma 8. For task with P (τ i ) < P (τ k ) S k, the worst case of I 2 (τ i ) occurs when one of τ i s instances is released at the -instant r k 1. We can comute I 2 (τ i ) by Equation 19: { I2 3 Ci 1 = 0 (τ i ) = ( 1 + 1) + υ > 0 where υ = min(, max(0, ( 1)mod ( D i ))) Lemma 9. For tasks with P (τ i ) > P (τ k ) or P (τ i ) < P (τ k ) > S k, the worst case of I 2 (τ i ) occurs when one of the τ i s instance is released at the -instant l k. We can comute I 2 (τ i ) by Equation 22: { Ak + S k + S k I 4 2 (τ i ) = where +S k + + ω + S k > ω = min(, max(0, ( + S k )mod ( D i )))

10 (( +S k - ) 0 mod - ( -D i )) 0 S k C k -D i n r i k l k S k n d i d C k k Figure 12. P (τ i ) > P (τ k ) or P (τ i ) < P (τ k ) > S k execution as x 1 execution as x 2 k x 1 l k x 2 x 1 -x 2 So we have : I2 1 (τ i ) I 2 (τ i ) = I2 3 (τ i ) I2 4 (τ i ) i = k P (τ i ) < P (τ k ) S k otherwise (28) Figure 13. decreasing execution Theorem 5. The schedualbiility tests [TEST-EDF n ] and [TEST-FP n ] are both execution and inter-release searation robust. 6.3 A New Test Condition for FP n By now we have obtained a sufficient schedulability test condition for FP n by Lemma 1 and the comutation of I 1 (τ i ) and I 2 (τ i ) for FP n above: Theorem 4. [TEST-FP n ] A task set is FP n schedulable on m rocessors if for any task τ k and for any we have: I 1 (τ i ) + m 1 < ( + S k ) m (29) τ i τ where I 1 (τ i ) and I 2 (τ i ) are comuted by Equation 27 and 28 resectively. Its comutational comlexity is the same as [TEST-EDF n ]. 7 Robustness A scheduling algorithm is said to be execution robust (inter-release searation robust) if decreasing the execution s (increasing the inter-release searation) of task instances in a schedulable task set does not lead to deadline violations. The robustness roerty is imortant. If a scheduling algorithm is robust, then the system designer only needs to consider the boundary values ( and ) to determine if the system is schedulable, rather than consider the every ossible execution in the interval [BCET, WCET] or consider the infinitely many ossible inter-release searations in [, ). Non-reemtive scheduling of eriodic tasks is neither execution robust nor inter-release searation robust if the scheduling algorithm is work-conserving. Therefore, any exact schedulability test for work-conserving scheduling algorithms, is necessarily non-robust. However, a sufficient but not necessary schedulability test can be robust in the sense that if a soradic task system is guaranteed by this test to meet all deadlines with their WCETs and minimal inter-release searations, then it is guaranteed to continue to meet all deadlines even if some of the execution requirements are decreased or some of the release searations are increased. In the following we will show that the tests roosed in this aer are robust. Proof. At first, when we test the schedulability of a task τ k, we only need to consider S k = D k C k in the test conditions, rather than consider every ossible value of S k. This observation is quite straightforward: if a task instance can start execution no later than l k, it must be schedulable if it s actual execution is smaller than C k. In the following, we will show when we test the schedulability of a task τ k, it is adequate to only consider the WCET and minimal inter-release searation for the interfering task τ i. It is easy to see that I1 1 (τ i ), I1 2 (τ i ), I1 3 (τ i ), I2 1 (τ i ), I2 2 (τ i ), I2 3 (τ i ) are all monotonically non-decreasing with resect to and monotonically non-increasing with resect to, and I2 4 (τ i ) is monotonically non-increasing with resect to. The only case we will elaborate on is that I2 4 (τ i ) is monotonically non-decreasing with resect to. As shown in Figure 13, when the execution of τ i is decreased from x 1 to x 2, τ i s releases should be moved rightwards for x 1 x 2 to kee the release of τ i s carry-out instance being l k x 2, so at the left end of the roblem window, I2 4 (τ i ) is at most increased by x 1 x 2. At the same, the contribution of each body instance and the carry-out instance is decreased by x 1 x 2, so I2 4 (τ i ) will not be increased. So I2 4 (τ i ) is monotonically nondecreasing with resect to. 8 Performance Evaluation The only known schedulability test condition for EDF n is [BAR-EDF n ], and there is no known test condition for FP n to our best knowledge. So we will comare [BAR-EDF n ] with our roosed test conditions [TEST-1], [TEST-EDF n ] and [TEST-FP n ]. [TEST-EDF n ] and [TEST-FP n ] are suerior to [TEST-1], which means a task set acceted by [TEST- 1] can also be acceted by [TEST-EDF n ] and [TEST-FP n ]. In general, our new test conditions is incomarable to [BAR- EDF n ], i.e., we can construct a task set acceted by our test conditions but rejected by [BAR-EDF n ], as well as a task set acceted by [BAR-EDF n ] but rejected by ours. In the following we will use randomly generated task sets to comare the average erformances, in terms of accetance ratio, of these

11 test conditions. Additionally, we will also comare the erformance of the test conditions for EDF n with the test condition (referred as to [BASE-EDF]) in [2] for reemtive EDF, which is the base of the analysis in this aer. Accetance Ratio (%) Accetance Ratio (%) Accetance Ratio (%) BASE-EDF 80 BAR-EDF 8 0 n T EST T EST -EDF n T EST -FP n T otal Utilization (a) 0.1 U i 0.4, BASE-EDF BAR-EDF n T EST T EST -EDF n T EST -FP n T otal Utilization (b) 0.1 U i 0.6, BASE-EDF BAR-EDF n T EST T EST -EDF n T EST -FP n T otal Utilization (c) 0.1 U i 0.4, Figure 14. erformance comarison of the test conditions We follow the method in [18] to generate task sets: A task set of m + 1 tasks was generated, and tested. Then we increase the task number by 1 to generate a new task set, and all the schedulability tests were run on the new task set. This rocess was reeated until the total rocessor utilization exceeded m. The whole rocedure was then reeated, starting with a new task set of m + 1 tasks, until 1,000,000 task sets have been generated and tested. This method of generating random task sets roduces a fairly uniform distribution of total utilizations, excet at the extreme end of low utilization. The task arameter setting in Figure 14(a) is as follows: The rocessor number is 6; for each task, is uniformly distributed in [10, 20], the ratio between D i and is uniformly distributed in [0.8, 1], and U i is uniformly distributed in [0.1, 0.4]. In this exeriment, [TEST-1] erforms slightly better than [BAR- EDF n ], and they are clearly outerformed by the imroved test conditions. [BASE-EDF] outerforms all test conditions for non-reemtive scheduling, since more interference caused by non-reemtion blocking should be taken into account in the tests for non-reemtive scheduling. In Figure 14(b), we change the range of U i to [0.1, 0.6] and kee other settings same as in Figure 14(a). It is shown that as the average task utilization increases, the erformance of [BAR-EDF n ] degrades, while the effect on our roosed test conditions is much smaller. The reason is that [BAR-EDF n ] is more sensitive to the C max value. A interesting henomena shown in this exeriment is that [TEST-EDF n ] erforms very close to (even a little better than) [BASE-EDF]. The reason is that the length of the roblem window in the analysis of nonreemtive scheduling is shorter than in the analysis of reemtive scheduling, which comensates the extra interference caused by non-reemtion blocking in some degree. As discussed in Section 6.3, most of the failures in the testing occurs with small + S k values, so this comensation is significant when S k is relatively small. In Figure 14(c), we change the range of to [10, 100] based on the setting in Figure 14(a), which means different tasks have a wider varying scale range. In this case, the erformance of all test conditions for non-reemtive scheduling degrades raidly, while the effect on [BASE-EDF] is trivial. This result accords with the intuition that tasks with long execution are harmful to the schedulability of short-urgent tasks due to the nonreemtive blocking. By the above exeriments, we can see that the test conditions roosed in this aer, esecially the imroved test conditions, have a significant erformance imrovement comared with [BAR-EDF n ]. In general, the erformance of our roosed test conditions is inferior to the test condition [BASE- EDF], which is for reemtive scheduling, while in some secial cases, the erformance of [TEST-EDF n ] is close to (or better than) [BASE-EDF]. 9 Simulation Exeriments In the following, we will comare the erformance of different scheduling algorithms by simulations. For hard real systems, the erformances of the test conditions are usually considered to make more sense than the absolute erformance of the scheduling algorithms. However, studying the erformance characteristics of the scheduling algorithms will disclose their otentials and may insirit the develoment of

12 Accetance Ratio (%) Accetance Ratio (%) Accetance Ratio (%) EDF DM NP-EDF n 10 NP-DM 1 0 n Total Utilization (a) EDF DM NP-EDF n DM NP-DM n Total Utilization (b) EDF DM NP-EDF n NP-DM n Total Utilization (c) Figure 15. erformance comarison of reemtive and non-reemtive scheduling by simulations new scheduling and analysis techniques. Figure 15 shows the comarison of the erformance of two well-known reemtive scheduling algorithms, EDF and DM, and their non-reemtive versions, EDF n and DM n. In the simulation of each task set, we set all tasks release offsets as 0, task inter-release searation as and task execution as the, and check the task set till its hyer-eriod. Even though these assumtions do not guarantee to generate the worst-case scenario in terms of schedulability, they are adoted since it is not comutationally feasible to try all ossible task release offsets, task inter-release s or task execution s. Therefore, the simulation result can at best be viewed as an aroximation of the real erformance of a scheduling algorithm without any guarantee of correctness. The arameter setting of the exeriment in Figure 15(a) is as follows: is uniformly distributed in [10, 30]; the ratio between D i and is uniformly distributed in [0.8, 1]; U i is uniformly distributed in [0.1, 0.4]. From Figure 15 (a), we can see that the erformance of EDF n is very close to EDF, while DM n is actually better than DM. We also tried other riority assignment olicies for fixed-riority scheduling (such as random assignment, slack monotonic, comutation monotonic, inverse comutation monotonic etc.), and with all of them FP n outerforms FP more or less. Note that we even have not taken the context switch overhead into account in these simulations. In Figure 15(b) and 15(c), we examine their erformance when different tasks have wider varying scale ranges. It is not feasible to have uniformly distributed in a very wide range, which may lead to very large hyer-eriods. Instead, we set each task eriod as 2 e, where e is uniformly distributed in [4, 7] and [4, 9] in Figure 15(b) and 15(c) resectively, corresondingly, the task eriods are in the range of [16, 128] and [16, 512] resectively. Again, the results obtained by this aroach are only aroximations of the real erformances of the scheduling algorithms. By these simulation exeriments, we can see that for task sets with narrow task scale ranges, non-reemtive scheduling is otentially a better choice. As the range of task scales being enlarged the erformance of non-reemtive scheduling degrades. This observation encourages us to study new scheduling olicies such as grouing tasks with similar scales and then scheduling each grou on a subset of the rocessors, which could be our future work. 10 Conclusions As observed by Baruah [2], global scheduling is fundamentally different from, and seems much more difficult than artitioned scheduling. In this aer, we take another stab at this tough roblem by resenting new schedulability test conditions for work-conserving (necessarily global) non-reemtive scheduling on multirocessor latforms, by building uon the techniques of Baker [5] and Baruah [2]. We firstly derive a linear- test condition which works on any work-conserving non-reemtive scheduling algorithms. Then we imrove the analysis and resent test conditions of seudo-olynomial comlexity for EDF n and FP n, which significantly outerform the existing result. The essimism of the analysis in this aer mainly comes from the assumtion that worst-case interferences by all tasks haen simultaneously, which is actually not necessary. As future work, we lan to emloy ILP or SAT methods to identify imossible scenarios, in order to obtain less-essimistic schedulability tests.

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