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Optimal Reward-Based Scheduling of Periodic Real-Time Tasks Hakan Aydın, Rami Melhem, and Daniel Moss´ e Pedro Mej´a-Alvarez y ı Computer Science Department o o CINVESTAV-IPN. Secci´ n de Computaci´ n University of Pittsburgh Av. I.P.N. 2508, Zacatenco. Pittsburgh, PA 15260 e M´ xico, DF. 07300 faydin,mosse,melhemg@cs.pitt.edu pmejia@cs.pitt.edu Abstract improved approach to provide hard real-time guarantees to k out of n consecutive instances of a task. Reward-based scheduling refers to the problem in which The techniques mentioned above tacitly assume that a task’s there is a reward associated with the execution of a task. output is of no value if it is not executed completely. How- In our framework, each real-time task comprises a manda- ever, in many application areas such as multimedia appli- tory and an optional part, with which a nondecreasing reward cations [17], image and speech processing [4, 6, 19], time- function is associated. Imprecise Computation and Increased- dependent planning [3], robot control/navigation systems [21], Reward-with-Increased-Service models fall within the scope of medical decision making [9], information gathering [7], real- this framework. In this paper, we address the reward-based time heuristic search [12] and database query processing [20] scheduling problem for periodic tasks. For linear and con- a partial or approximate but timely result is usually acceptable. cave reward functions we show: (a) the existence of an optimal The Imprecise Computation [5, 15] and IRIS (Increased Re- schedule where the optional service time of a task is constant ward with Increased Service) [10, 13] models were proposed at every instance and (b) how to efﬁciently compute this service to enhance the resource utilization and provide graceful degra- time. We also prove that RMS-h (RMS with harmonic periods), dation in real-time systems. In these models, every real-time EDF and LLF policies are optimal when used with the opti- task is composed of a mandatory part and an optional part. The mal service times we computed, and that the problem becomes former should be completed by the task’s deadline to provide NP-Hard, when the reward functions are convex. Further, our output of minimal quality. The optional part is to be executed solution eliminates run-time overhead, and makes possible the after the mandatory part while still before the deadline, if there use of existing scheduling disciplines. are enough resources in the system that are not committed to running mandatory parts for any task. The longer the optional part executes, the better the quality of the result (the higher the 1 Introduction reward). The algorithms proposed for imprecise computation appli- In a real-time system each task must complete and produce cations concentrate on a model that has an upper bound on correct output by the speciﬁed deadline. However, if the sys- the execution time that could be assigned to the optional part tem is overloaded it is not possible to meet each deadline. In [5, 15, 18]. The aim is usually to minimize the (weighted) sum the past, several techniques have been introduced by the re- of errors. Several efﬁcient algorithms are proposed to solve search community regarding the appropriate strategy to use in optimally aperiodic scheduling problem of imprecise compu- overloaded systems of periodic real-time tasks. tation tasks [15, 18]. A common assumption in these studies One class of approaches focuses on providing somewhat is that the quality of the results produced is a linear function less stringent guarantees for temporal constraints. In [11], of the precision error; consequently, the possibility of having some instances of a task are allowed to be skipped entirely. more general error functions is usually not addressed. The skip factor determines how often instances of a given task An alternative model allows tasks to get increasing reward may be left unexecuted. A best effort strategy is introduced with increasing service (IRIS model) [10, 13] without an upper in [8], aiming at meeting k deadlines out of n instances of a bound on the execution times of the tasks (though the deadline given task. This framework is also known as (n,k)-ﬁrm dead- of the task is an implicit upper bound) and without the sepa- lines scheme. Bernat and Burns present in [2] a hybrid and ration between mandatory and optional parts [10]. A task ex- This work has been supported by the Defense Advanced Research Projects ecutes for as long as the scheduler allows before its deadline. Agency through the FORTS project (Contract DABT63-96-C-0044). Typically, a nondecreasing concave reward function is associ- y Work done while at the University of Pittsburgh ated with each task’s execution time. In [10] the problem of maximizing the total reward in a system of aperiodic indepen- lem to multiple resources and quality dimensions. Further, de- dent tasks is addressed. The optimal solution with static task pendent and independent quality dimensions are separately ad- sets is presented, as well as two extensions that include manda- dressed for the ﬁrst time in this work. However, a fundamen- tory parts and policies for dynamic task arrivals. tal assumption of that model is that the reward functions and Note that imprecise computation and IRIS models are resource allocations are in terms of utilization of resources. closely related, since the performance metrics can be deﬁned as Our work falls rather along the lines of Imprecise Computa- duals (maximizing the total reward is a dual of minimizing the tion model, where the reward accrued has to be computed sep- total error). Similarly, a concave reward function corresponds arately over all task instances and the problem is to ﬁnd the to a convex error function, and vice versa. optimal service times for each instance and the optimal sched- We use the term “Reward-based scheduling” to encompass ule with these assignments. scheduling frameworks such as Imprecise Computation and Aspects of Periodic Reward-Based Scheduling IRIS models, where each task can be decomposed into manda- tory and optional subtasks. A nondecreasing reward function Problem is associated with the execution of each optional part. The difﬁculty of ﬁnding an optimal schedule for a periodic An interesting question concerns types of reward functions reward-based task set has its origin on two objectives that must which represent realistic application areas. A linear reward be simultaneously achieved, namely: (i) Meeting deadlines function [15] models the case where the beneﬁt to the overall of mandatory parts at every periodic task invocation, and (ii) system increases uniformly during the optional execution. Sim- Scheduling optional parts to maximize the total (or average) ilarly, a concave reward function [10, 13] addresses the case reward. where the greatest increase/reﬁnement in the output quality is These two objectives are both important, yet often incom- obtained during the ﬁrst portions of optional executions. The patible. In other words, hard deadlines of mandatory parts may ﬁrst derivative of a nondecreasing concave function is nonin- require sacriﬁcing optional parts with greatest value to the sys- creasing. Linear and general concave functions are considered tem. The analytical treatment of the problem is complicated by the most realistic and typical in the literature since they ade- the fact that, in an optimal schedule, optional service times of quately capture the behavior of many application areas such as a given task may vary from instance to instance which makes those mentioned above [4, 6, 19, 3, 21, 12, 7, 17, 20]. In this the framework of classical periodic scheduling theory inappli- paper, we show that the case of convex reward functions is an cable. Furthermore, this fact introduces a large number of vari- NP-Hard problem and thus focus on linear and concave reward ables in any analytical approach. Finally, by allowing nonlin- functions. Reward functions with 0/1 constraints, where no re- ear reward functions to better characterize the optional tasks’ ward is accrued unless the entire optional part is executed, or contribution to the overall system, the optimization problem step reward functions have also received some interest in the becomes computationally harder. literature. Unfortunately, this problem has been shown to be In [5], Chung, Liu and Lin proposed the strategy of assign- NP-Complete in [18]. ing statically higher priorities to mandatory parts. This de- Periodic reward-based scheduling remains relatively unex- cision, as proven in that paper, effectively achieves the ﬁrst plored, since the important work of Chung, Liu and Lin [5]. objective mentioned above by securing mandatory parts from In that paper, the authors classiﬁed the possible application ar- the potential interference of optional parts. Optional parts are eas as “error non-cumulative” and “error cumulative”. In the scheduled whenever no mandatory part is ready in the sys- former, errors (or optional parts left unexecuted) have no effect tem. In [5], the simulation results regarding the performance on the future instances of the same task. Well-known examples of several policies which assign static or dynamic priorities of this category are tasks that periodically receive, process and among optional parts are reported. We call the class of algo- transmit audio, video or compressed images [4, 6, 19] as well rithms that statically assign higher priorities to mandatory parts as information retrieval tasks [7, 20]. In “error cumulative” ap- Mandatory-First Algorithms. plications, such as radar tracking, an optional instance must be In our solution, we do not decouple the objectives of meet- executed completely at every (predetermined) k invocations. ing the deadlines of mandatory parts and maximizing the total The authors further proved that the case of error-cumulative (or average) reward. We formulate the periodic reward-based jobs is an NP-Complete problem. In this paper, we restrict scheduling problem as an optimization problem and derive an ourselves to error non-cumulative applications. important and surprising property of the solution for the most Recently, a QoS-based resource allocation model (QRAM) common (i.e., linear and concave) reward functions. Namely, has been proposed for periodic applications [17]. In that study, we prove that there is always an optimal schedule where op- the problem is to optimally allocate several resources to the tional service times of a given task do not vary from instance various applications such that they simultaneously meet their to instance. This important result immediately implies that minimum requirements along multiple QoS dimensions and the optimality (in terms of achievable utilization) of any policy the total system utility is maximized. In one aspect, this can which can fully use the processor in case of hard-real time peri- be viewed as a generalization of optimal CPU allocation prob- odic tasks also holds in the context of reward-based scheduling (in terms of total reward) when used with these optimal ser- t reaches the threshold value oi , the reward accrued ceases to vice times. Examples of such policies are RMS-h (RMS with increase. harmonic periods) [14], EDF [14] and LLF [16] scheduling A schedule of periodic tasks is feasible if mandatory parts disciplines. meet their deadlines at every invocation. Given a feasible Following these existence proofs, we address the problem schedule of the task set T, the average reward of task Ti is: P=Pi X REWi = Pi of efﬁciently computing optimal service times and provide polynomial-time algorithms for linear and/or general concave P j =1 Ritij (2) reward functions. Note that using these optimal and con- stant optimal service times has also important practical advan- where P is the hyperperiod, that is, the least common multiple of P1; P2; : : :; Pn and tij is the service time assigned to the j th tages: (a) The runtime overhead due to the existence of manda- tory/optional dichotomy and reward functions is removed, and (b) existing RMS-h, EDF and LLF schedulers may be used instance of optional part of task Ti . That is, the average reward without any modiﬁcation with these optimal assignments. of Ti is computed over the number of its invocations during the The remainder of this paper is organized as follows: In hyperperiod P, in an analogous way to the deﬁnition of average Section 2, the system model and basic deﬁnitions are given. error in [5]. The main result about the optimality of any periodic policy The average weighted reward of a feasible schedule is which can fully utilize the processor(s) is obtained in Section then given by: X n 3. In Section 4, we ﬁrst analyze the worst-case performance of REWW = wi REWi (3) Mandatory-First approaches. We also provide the results of ex- i=1 where wi is a constant in the interval (0,1] indicating the rela- periments on a synthetic task set to compare the performance tive importance of optional part Oi. Although this is the most of policies proposed in [5] against our optimal algorithm. In general formulation, it is easy to see that the weight wi can al- Section 5, we show that the concavity assumption is also nec- ways be incorporated into the reward function fi , by replac- essary for computational efﬁciency by proving that allowing ing it by wi fi . Thus, we will assume that all weight (impor- convex reward functions results in an NP-Hard problem. We present details about the speciﬁc optimization problem that we use in Section 6. We conclude by summarizing our contribu- Pn tance) information are already expressed in the reward function formulation and that REWW is simply equal to i=1 REWi . tion and discussing future work. Finally, a schedule is optimal if it is feasible and maximizes the average weighted reward. 2 System Model A Motivating Example: We consider a set T of n periodic real-time tasks Before describing our solution to the problem, we present T1; T2 ; : : :; Tn on a uniprocessor system. The period of Ti is a simple example which shows the performance limitations denoted by Pi , which is also equal to the deadline of the cur- of any Mandatory-First algorithm. Consider two tasks where rent invocation. We refer to the j th invocation of task Ti as Tij . P1 = 4; m1 = 1; o1 = 1; P2 = 8; m2 = 3; o2 = 5. Assume All tasks are assumed to be independent and ready at t = 0. that the reward functions associated with optional parts are lin- Each task Ti consists of a mandatory part Mi and an op- ear and f1 t1 = k1 t1 ; f2t2 = k2t2 , where k1 k2. In tional part Oi . The length of the mandatory part is denoted by this case, the “best” algorithm among “Mandatory-First” ap- mi ; each task must receive at least mi units of service time be- proaches should produce the schedule shown in Figure 1. fore its deadline in order to provide output of acceptable qual- ity. The optional part Oi becomes ready for execution only 000000 111111 000000 111111 000000 111111 000000 111111 M1 M1 O1 111111 000000 when the mandatory part Mi completes. 000000 111111 000000 111111 111111 000000 0 1 4 5 6 8 Associated with each optional part of a task is a reward function Ri tij which indicates the reward accrued by task 00000000000 11111111111 Tij when it receives tij units of service beyond its mandatory M2 11111111111 00000000000 11111111111 00000000000 O2 00000000000 11111111111 11111111111 00000000000 portion. Ritij is of the form: 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 if 0 tij oi 0 1 4 6 8 t Ri tij = fii oij f i if tij oi (1) Figure 1. A schedule produced by Mandatory-First Algorithm where fi is a nondecreasing, concave and differentiable func- In Figure 1, the Rate Monotonic Priority Assignment is used tion over nonnegative real numbers and oi is the length of en- whenever more than one mandatory task are simultaneously tire optional part Oi . We underline that fi tij is nondecreas- ready, as in [5]. Yet, following other (dynamic or static) prior- ing: the reward accrued by task Tij can not decrease by al- ity schemes would not change the fact that the processor will lowing it to run longer. Notice that the reward function Rit be busy executing solely mandatory parts until t = 5 under is not necessarily differentiable at t = oi . Note also that in any Mandatory-First approach. During the remaining idle in- this formulation, by the time the task’s optional execution time terval [5,8], the best algorithm would have chosen to schedule O1 completely (which brings most beneﬁt to the system) for 1 The above constraint allows us also to readily substitute fi time unit and O2 for 2 time units. However, an optimal algo- for Ri in the objective function. Finally, we need to express rithm would produce the schedule depicted in Figure 2. the “full” feasibility constraint, requiring that mandatory parts complete in a timely manner at every invocation. Note that it is sufﬁcient to have one feasible schedule for task Ti with mi 111111 000000 000000 111111 111111 000000 000000 111111 000000 111111 000000 111111 111111 000000 000000 111111 and the involved optimal ftij g values. M1 O1 111111 000000 000000 111111 M1 O1 000000 111111 000000 111111 111111 000000 111111 000000 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 0 1 111111 000000 2 4 5 6 8 111111 000000 To re-capture all the constraints, the periodic reward-based scheduling problem, which we denote by REW-PER, is to ﬁnd ftij g values so as to: 111111 000000 000000 111111 000000 111111 111111 000000 M2 M2 111111 000000 111111 000000 O2 P P Pi P=Pi f t 111111 000000 0 2 4 6 000000 111111 7 8 n maximize P i ij (4) i=1 j =1 Figure 2. The optimal schedule subject to As it can be seen, the optimal strategy in this case consisted n P P P m + P P=Pi t P n of delaying the execution of M2 in order to be able to exe- Pi i ij (5) cute ’valuable’ O1 and we would still meet the deadlines of i=1 i=1 j =1 all mandatory parts. By doing so, we would succeed in exe- P 0 tij oi i = 1; : : :; n j = 1; : : :; Pi (6) cuting two instances of O1, in contrast to any Mandatory-First scheme which can execute only one instance of O1. Remem- A feasible schedule exists with fmi g and ftij g values (7) bering that k1 k2, one can conclude that the reward accrued by the ’best’ Mandatory-First scheme may only be around half We express this last constraint in English and not through of that accrued by the optimal one, for this example. Also, ob- formulas since the algorithm producing this schedule including serve that in the optimal schedule, the optional execution times optimal tij assignments need not be speciﬁed at this point. of a given task did not vary from instance to instance. In the Pn P m stating , ouris mainpossible to schedule mandatory Before P it not result, we underline that if next section, we prove that this pattern is not a mere coinci- i=1 Pi i dence. We further perform an analytical worst-case analysis of Pn parts in a timely manner and the optimization problem has no Mandatory-First algorithms in Section 4. solution. Note that this condition is equivalent to i=1 mii P 1, which indicates that the task set would be unschedulable, 3 Optimality of Full-Utilization Policies for Pe- Pn even if it consisted of only mandatory parts. Hence, hereafter, riodic Reward-Based Scheduling we assume that i=1 mii 1 and therefore there exists at least P one feasible schedule. The objective of the Periodic Reward-Based Scheduling problem is clearly ﬁnding optimal ftij g values to maximize Theorem 1 Given an instance of Problem REW-PER, there the average reward. By substituting the average reward expres- exists an optimal solution where the optional parts of a task sion given by (2) in (3), we obtain our objective function: Ti receive the same service time at every instance, i.e. tij = P tik 1 j k Pi . Furthermore, any periodic hard-real X P=Pi n X Pi maximize Ritij time scheduling policy which can fully utilize the processor i=1 P j =1 (EDF, LLF, RMS-h) can be used to obtain a feasible schedule The ﬁrst constraint we must enforce is that the total proces- with these assignments. sor demand of mandatory and optional parts during the hyper- Proof: Our strategy to prove the theorem will be as fol- period P may not exceed the available computing capacity: lows. We will drop the feasibility condition (7) and obtain a X P=Pi n X new optimization problem whose feasible region strictly con- mi + tij P tains that of REW-PER. Speciﬁcally, we consider a new op- i=1 j =1 timization problem, denoted by MAX-REW, where the ob- Note that this constraint is necessary, but not sufﬁcient for jective function is again given by (4), but only the constraint feasibility of the task set with fmi g and ftij g values. Next, we sets (5) and (6) have to be satisﬁed. Note that the new prob- observe that optimal tij values may not be less than zero, since lem MAX-REW does not a priori correspond to any schedul- negative service times do not have any physical interpretation. ing problem, since the feasibility issue is not addressed. We In addition, the service time of an optional instance of Ti does then show that there exists an optimal solution of MAX-REW not need to exceed the upperbound oi of reward function Rit, where tij = tik 1 j P k Pi . Then, we will return to since the reward accrued by Ti ceases to increase after tij = oi . REW-PER and demonstrate the existence of a feasible sched- Hence, we obtain our second constraint set: ule (i.e. satisﬁability of (7)) under these assignments. The re- P 0 tij oi i = 1; : : :; n j = 1; : : :; P ward associated with MAX-REW’s optimal solution is always i greater than or equal to that of REW-PER’s optimal solution, for MAX-REW does not consider one of the REW-PER’s con- Corollary 1 Optimal ti values for the Problem REW-PER can straints. This will imply that this speciﬁc optimal solution of be found by solving the optimization problem given by (8), (9) the new problem MAX-REW is also an optimal solution of and (10). REW-PER. We discuss the solution of this concave optimization prob- Now, we show that there exists an optimal solution of MAX-REW where tij = tik 1 j k Pi . P lem in Section 6. Claim 1 Let ftij g be an optimal solution to MAX-REW, 1 i n 1 j Pi = qi. Then ft0ij g where t0i1 = t0i2 = : : : = P 4 Evaluation and comparison with other ap- t0iqi = t0i = ti1 +ti2 qi:::+tiqi 1 i n 1 j qi, is also + proaches an optimal solution to MAX-REW. We showed through the example in Section 2 that the re- ward accrued by any Mandatory-First scheme [5] may only be We ﬁrst show that ft0 g values satisfy the constraints (5) ij Pq approximately half of that of the optimal algorithm. We now and (6), if ftij g already satisfy them. Since ji tij = Pqi t0 = q t0 the constraint (5) is not violated by the =1 prove that, under the worst-case scenario, the ratio of the re- j =1 ij ii ward accrued by a Mandatory-First approach to the reward of transformation. Also, by assumption, tij oi 8j , which the optimal algorithm approaches zero. implies maxftij g oi . Since t0 , which is arithmetic i j Theorem 2 There is an instance of the periodic reward-based mean of ti1; ti2; : : :; tiqi is necessarily less than or equal scheduling problem where, for any integer r 2, the ratio to maxftij g, the constraint set (6) is not violated either Reward of the best mandatory, rst scheme 2 =r j Reward of the optimal scheme by the transformation. Proof: Consider two tasks T1 and T2 such that P2=P1 = r; Furthermore, the total Preward does not decrease by this q f1t1 = k1 t1; f2 t2 = k2 t2 and k1=k2 = rr , 1. Further- transformation, since ji fi tij qi fi t0 . The =1 i 1 more, let m2 = 2 r o2 and P1 = m1 + o1 + m2 = m1 + rm21 proof of this statement is presented in the Appendix. Using Claim 1, we can commit to ﬁnding an optimal so- r , lution of MAX-REW by setting ti1 = ti2 = : : : = tiqi = m . P which implies that o1 = r r,1 2 ti i = 1; : : :; n. In this case, P=Pi fi tij = Pi fi ti and j =1 P This setting suggests that during any period of T1 , a sched- PP=Pi P j =1 tij = Pi ti . Hence, this version of MAX-REW can be uler has the choice of executing (parts of) O1 and/or M2 , in re-written as: addition to M1 . Note that under any Mandatory-First policy, the proces- P f t n sor will be continuously busy executing mandatory parts until maximize i i (8) t = P2 , P1 + m1 . Furthermore, the best algorithm among i=1 P Pt P,P Pm n n Mandatory-First policies should use the remaining idle times by scheduling O1 entirely (since k1 k2) and t2 = m2 = o2 2 Pi i i r i=1 Pi subject to (9) i=1 units of O2. The resulting schedule is shown in Figure 3. 0 ti oi i = 1; : : :; n (10) 1 0 1 0 1 0 0 1 1 0 11 00 1 0 Finally, we prove that the optimal solution t1; t2; : : :; tn of 0 1 0 1 00 11 0 1 ........ m m m 1 0 1 0 01 1 1 0 m m 00 11 o 1 0 0 1 1 1 1 01 1 1 0 1 00 111 MAX-REW above, automatically satisﬁes the feasibility con- 0 P 2.P1 3.P (r-2).P 1 (r-1).P 1 r P0 1 1 1 straint (7) of our original problem REW-PER. Having equal 0 1 optional service times for a given task greatly simpliﬁes the 0000 1111 0 1 o2 0 1 veriﬁcation of this constraint. P t1 ; t2; : : :; tn (by assump- 0000 1111 ...... Since m2 m2 m2 0000 1111 0 1 m2 2 0 1 n tion) satisfy (9), we can write i=1 P mi +ti P , or equiv- r-1 r-1 r-1 r-1 0000 1111 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 0 1 Pn mi +ti 1. Pi 0 P 2 alently, i=1 Pi This implies that any policy which can achieve 100% pro- Figure 3. A schedule produced by Mandatory-First Algorithm cessor utilization in classical periodic scheduling theory (EDF, LLF, RMS-h) can be used to obtain a feasible schedule for tasks, which have now identical execution times mi + ti at The average reward that the best mandatory-ﬁrst algorithm every instance. Hence, the “full feasibility” constraint (7) of (MFA) can accrue is therefore: RMFA = f1 o1 + f2 t2 REW-PER is satisﬁed. Furthermore, this schedule clearly max- imizes the average reward since fti g values maximize MAX- REW whose feasible region encompasses that of REW-PER. r However, an optimal algorithm (shown in Figure 4) would 2 choose delaying the execution of M2 for o1 units of time, at 1 0 1 0 ........ 1 0 0 00 1 11 0 1 0 1 11 00 11 00 00 11 1 0 11 00 1 11 0 00 0 1 usually much higher than the other ﬁve policies, BIR is used as 11 00 11 m 00 m 00 11 0 1 11 m 00 1 11 0 1 00 0 1 11 1 00 11 1 00 11 1 00 11 1 00 o m o o o m o 11 001 11 001 11 001 0 1 0 1 11 001 1 00 0 11 0 1 1 0 1 1 0 a yardstick for measuring the performance of other algorithms. 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 0 P 2.P1 3.P (r-2).P 1 (r-1).P 1 rP 1 1 1 We have used a synthetic task set comprising 11 tasks whose 0 1 total (mandatory + optional) utilization is 2.3. Individual task 0 1 m2 m2 m2 1 0 ...... 1 0 m2 m2 utilizations vary from 0.03 to 0.6. Considering exponential, r r r 1 0 0 1 r r logarithmic and linear reward functions as separate cases, we 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 0 P 2 compared the reward of six Mandatory-First schemes with our optimal algorithm (OPT). The tasks’ characteristics (including Figure 4. An optimal schedule reward functions) are given in the Table below. In our exper- iments, we ﬁrst set mandatory utilization to 0 (which corre- every period of T1 . By doing so, it would have the opportunity sponds to the case of all-optional workload), then increased it of accruing the reward of O1 at every instance. to 0.25, 0.4, 0.6, 0.8 and 0.91 subsequently. The total reward of the optimal schedule is: Task Pi mi + oi fi1 t fi2 t fi3 t T1 20 10 , 15 1 , e,3tt 7 ln20 t + 1 5t o ROPT = r f1r 1 = f1 o1 T2 T3 30 40 18 5 20 1 , e 4 1 , e,t 10 ln50 t + 1 2 ln10 t + 1 7t 2t T4 60 2 10 1 , e,0:5t 5 ln5 t + 1 4t The ratio of rewards for the two policies turns out to be (for T5 60 2 10 1 , e,0:2t 5 ln25 t + 1 4t any r 2) T6 80 12 5 1 , e,t 3 ln30 t + 1 2t T7 90 18 , 17 1 , e,tt 8 ln8 t + 1 6t T8 120 15 8 1 , e 4 ln6 t + 1 3t T9 240 28 8 1 , e,t 4 ln9 t + 1 3t RMFA = 1 + f2 t2 = 1 + 1 m2 rr , 1 = 2 T10 T11 270 2160 60 300 12 1 , e,0t:5t 5 1 , e, 6 ln12 t + 1 3 ln15 t + 1 5t 2t ROPT r f1 o1 r rr , 1 r m2 r which can be made as close as possible to 0 by appropriately In our experiments, a common pattern appears: the opti- choosing r (i.e., choosing a large value for r). mal algorithm improves more dramatically with the increase in mandatory utilization. The other schemes miss the opportu- 2 nities of executing “valuable” optional parts while constantly Theorem 2 gives the worst-case performance ratio of favoring mandatory parts. The reward loss becomes striking Mandatory-First schemes. We also performed experiments as the mandatory workload increases. Figures 5 and 6 show with a synthetic task set to investigate the relative perfor- the reward ratio for the case of exponential and logarithmic mance of Mandatory-First schemes proposed in [5] with differ- reward functions, respectively. The curves for these strictly ent types of reward functions and different mandatory/optional concave reward functions are fairly similar: BIR performs workload ratios. best among Mandatory-First schemes, and its performance de- The Mandatory-First schemes differ by the policy accord- grades as the mandatory utilization increases; for instance the ing to which optional parts are scheduled when there is no ratio falls to 0.73 when mandatory utilization is 0.6. Other mandatory part ready to execute. Rate-Monotonic (RMSO) and algorithms which are more amenable to practical implemen- Least-Utilization (LU) schemes assign statically higher priori- tations (in terms of runtime overhead) than BIR perform even ties to optional parts with smaller periods and least utilizations worse. However, it is worth noting that the performance of respectively. Among dynamic priority schemes are Earliest- LAT is close to that of BIR. This is to be expected, since task Deadline-First (EDFO) and Least-Laxity-First (LLFO) which sets with strictly concave reward functions usually beneﬁt from consider the deadline and laxity of optional parts when assign- “balanced” optional service times. ing priorities. Least Attained Time (LAT) aims at balancing Figure 7 shows the reward ratio for linear reward functions. execution times of optional parts that are ready, by dispatching Although the reward ratio of Mandatory-First schemes again the one that executed least so far. Finally, Best Incremental Re- decreases with the mandatory utilization, the decrease is less turn (BIR) is an on-line policy which chooses the optional task dramatic than in the case of concave functions. However, contributing most to the total reward, at a given slot. In other note that the ratio is typically less than 0.5 for the ﬁve prac- words, at every slot BIR selects the optional part Oij such that tical schemes. It is interesting to observe that the (impractical) the difference fi tij + , fi tij is the largest (here tij is BIR’s reward now remains comparable to that of optimal, even the optional service time Oij has already received and is the in the higher mandatory utilizations: the difference is less than minimum time slot that the scheduler assigns to any optional 15%. The main reason for this behavior change lies on the fact task). However, it is still a sub-optimal policy since it does that, for a given task, the reward of optional execution slots not consider the laxity information. The authors indicate in [5] in different instances does not make a difference in the lin- that BIR is too computationally complex to be actually imple- ear case. In contrast, not executing the “valuable” ﬁrst slot(s) mented. However, since the total reward accrued by BIR is of a given instance creates a tremendous effect for nonlinear Reward Ratio with Reward Ratio with Respect to Optimal Respect to Optimal 1.00 OPT 1.00 OPT 0.90 0.90 BIR 0.80 0.80 0.70 0.70 0.60 0.60 0.50 0.50 0.40 0.40 LLFO 0.30 RMSO 0.30 EDFO BIR 0.20 LAT 0.20 LAT RMSO 0.10 0.10 LLFO LU EDFO LU 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mandatory 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mandatory Utilization Utilization (a) Figure 7. The Reward Ratio of Mandatory-First schemes: linear reward functions Figure 5. The Reward Ratio of Mandatory-First schemes: exponential reward functions SUBSET-SUM: Given a set S = fs1 ; s2 ; : : :; sn g of posi- Reward Ratio with tive integers and the integer M, is there a set SA S such that X 1.00 Respect to Optimal OPT si = M ? 0.90 si2SA 0.80 Pn s . Now consider a set of n periodic We construct the corresponding REW-PER instance as fol- lows. Let W = i=1 i 0.70 tasks with the same period M and mandatory parts mi = 0 8i. 0.60 The reward function associated with Ti is given by: 0.50 0.40 Riti = fi ti if 0 o os = si fi oi if t i = i t i 0.30 where fi ti = t2 + W , si ti is a strictly convex and in- BIR 0.20 i LAT LLFO 0.10 RMSO EDFO creasing function on nonnegative real numbers. Notice that fi ti can be re-written as ti ti , si + W ti . LU 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mandatory (b) Utilization Also we underline that having the same periods for all tasks implies that REW-PER can be formulated as: Figure 6. The Reward Ratio of Mandatory-First schemes: P t t , s + W P t n n logarithmic reward functions maximize i i i i (11) i=1 i=1 concave functions. The improvement of the optimal algorithm Pt M n subject to i (12) would be larger for a larger range of ki values (where ki is i=1 the coefﬁcient of the linear reward function). We recall that 0 ti si (13) the worst-case performance of BIR may be arbitrarily bad with respect to the optimal one for linear functions, as Theorem 2 Let us denote by MaxRew the total reward of the optimal suggests. schedule. Observe that for 0 ti si , the quantity ti ti , si 0. Otherwise, at either of the boundary values 0 or si , 5 Periodic Reward-Based Scheduling Problem ti ti , si = 0. Hence, MaxRew WM . with Convex Reward Functions is NP-Hard Now, consider the question: ”Is MaxRew equal to WM ?”. Clearly, this question can be answered quickly if there is a As we mentioned before, maximizing the total (or average) polynomial-time algorithm for REW-PER where reward func- reward with 0/1 constraints case had already been proven to tions are allowed to be convex. Furthermore, the answer can be be NP-Complete in [15]. In this section, we show that convex P t = M and each t is equal to either 0 n positive only when i i reward functions also result in an NP-Hard problem. i=1 We now show how to transform the SUBSET-SUM prob- or si . Therefore, MaxRew equal to WM , if and only if there P lem, which is known to be NP-Complete, to REW-PER with is a set SA S such that si 2SA si = M , which implies that convex reward functions. REW-PER with convex reward functions is NP-Hard. 6 Solution of Periodic Reward-Based Schedul- When F contains nonlinear functions then the procedure ing Problem with Concave Reward Functions becomes more involved. In the next two subsections, we in- troduce two auxiliary optimization problems, namely Problem Corollary 1 reveals that the optimization problem whose so- OPT (which considers only the equality constraint) and Prob- lution provides optimal service times is of the form: lem OPT-L (which considers only the equality and lower bound P n maximize fi ti constraints), which will be used to solve OPT-LU. i=1 P n 6.1 Problem OPT: Equality Constraints subject to bi ti d Only i=1 ti oi i = 1; 2; :::n The problem OPT is characterized by: P n 0 ti i = 1; 2; :::n maximize fi ti i=1 d (the ’slack’ available for optional parts) and P n where b1; b2; : : :; bn are positive rational numbers. In this section, subject to bi ti = d i=1 where F = ff1 ; : : :; fng is the set of nondecreasing concave we present polynomial-time solutions for this problem, where each fi is a nondecreasing, concave and differentiable1 func- functions, possibly including some linear function(s). As it tion. can be seen, OPT does not consider the lower and upper bound First note that, if the available slack is large P enough to ac- commodate every optional part entirely (i.e., if n bi oi constraints of Problem OPT-LU. The algorithm which returns i=1 d), then the choice ti = oi 8 i clearly maximizes the objective the solution of Problem OPT, is denoted by “Algorithm OPT”. When F is composed solely of non-linear reward functions, function due to the nondecreasing nature of reward functions. Otherwise, the slack d should be used in its entirety since the application of Lagrange multipliers technique to the Prob- lem OPT, yields: the total reward never decreases by doing so (again due to the nondecreasing nature of the reward functions). In this 1 f 0 t , = 0 i = 1; 2; : : :; n case, we obtain a concave optimization problem with lower bi i i (18) where is the common Lagrange multiplier and fi0 ti is the and upper bounds, denoted by OPT-LU. An instance of OPT- derivative of the reward function fi . The quantity bi fi0 ti 1 LU is speciﬁed by the set of nondecreasing concave functions F = ff1 ; : : :; fng, the set of upper bounds O = fo1 ; : : :; ong and the available slack d. The aim is to: in (18) actually represents the marginal return contributed by P f t n Ti to the total reward, which we will denote as wi ti . Ob- maximize i i (14) serve that since fi ti is non-decreasing and concave by as- i=1 sumption, both wi ti and fi0 ti is non-increasing and posi- P n bi ti = d tive valued. Equation (18) implies that the marginal returns subject to (15) i=1 witi = b1i fi0 ti should be equal for all reward functions in ti oi i = 1; 2; :::n (16) the optimal solution ft1 ; : : :; tng. Considering that the equal- 0 ti i = 1; 2; :::n (17) P b t = d should also hold, one can obtain closed for- n ity i i Pn i=1 where 0 d i=1 bi oi. mulas in most of the cases which occur in practice. The closed Special Case of Linear functions: If F comprises solely formulas presented below are obtained by this method. linear functions, the solution can be considerably simpliﬁed. Note that for a function fi ti = ki ti , if we increase ti by For logarithmic reward functions of the form then total reward increases by ki . However by doing so, fi ti = lnki ti + cni , Pc c P n d+ kj , k11b1 bj we make use of bi units of slack (d is reduced by bi due j to (15)). Hence, the “marginal return” of task Ti per slack t1 = j=1 1 P j=1 n unit is wi = ki . It is therefore possible to order the functions b i b1 bj j=1 tj = b1 t1 + k1c1b1 , kjcjbj 8j 1 j n. according to their marginal return, and distribute the slack in decreasing order of marginal returns, while taking account the upper bounds. We note that this solution is analogous to the For exponential reward functions of the form one presented in [17]. The dominant factor in the time com- fi ti = ci 1 , e,ki ti , plexity comes from the initial sorting procedure, hence in the P n cj d, k1j ln c1 b1 kj bj k1 special case of all-linear functions, OPT-LU can be solved in time On logn. t1 = j=1 P k n k1 j 1 In the auxiliary optimization problems which will be introduced shortly, j=1 tj = kj 1 k1 t1 + ln cj b1 kj 8j 1 j n. the differentiability assumption holds as well. c1 bj k1 For “kth root” reward functions of the form 6.2 Problem OPT-L: Equality Constraints fi ti = ci t1=k k 1, i with Lower Bounds t1 = P bj d1 1 n c b1 cj k,1 Now, we consider the optimization problem with the j=1 equality and lower bound constraints, denoted by OPT-L. tj = t1 bj1c1 k,1 8j 1 j n. b cj 1 An instance of Problem OPT-L is characterized by the set F=ff1 ; f2; ::; fng of nondecreasing concave reward functions, and the available slack d: When it is not possible to ﬁnd a closed formula, follow- P n ing exactly the approach presented in [10, 13], we solve in Pn maximize fi ti (19) the equation i=1 bi hi = d, where hi k is the inverse i=1 function of bi fi0 ti = wi ti (we assume the existence of the 1 P n subject to bi ti = d (20) derivative’s inverse function whenever fi is nonlinear, com- i=1 plying with [10, 13]). Once is determined, ti = hi; i = 0 ti i = 1; 2; :::n (21) 1; : : :; n is the optimal solution. We now examine the case where F is a mix of linear and To solve OPT-L, we ﬁrst evaluate the solution set SOPT nonlinear functions. Consider two linear functions fi t = ki t to the corresponding problem OPT and check whether all in- and fj t = kj t. The marginal return of fi ti is witi = equality constraints are automatically satisﬁed. If this is the ki = wi and that of fj is wj tj = kj = wj . I wj wi then case, the solution set SOPT ,L of Problem OPT-L is clearly bi bj the solution SOPT . Otherwise, we will construct SOPT ,L it- the service time ti should be deﬁnitely zero, since marginal re- eratively as described below. turn of fi is strictly less than fj everywhere. After this elimina- 0 Let = fxj b1 fx 0 bi fi0 0 8ig. Remember that 1 tion process, if there are p 1 linear functions with the same 1 0 x (largest) marginal return wmax then we will consider them as a bx fx tx is the marginal return associated with fx tx and single linear function in the procedure below and evenly divide was denoted by wx tx . Informally, contains the functions2 the returned service time tmax among tj values corresponding fx 2 F with the smallest marginal returns at lower bound 0, to these p functions. wx0. Hence, without loss of generality, we assume that fn t = Lemma 1 If SOPT violates some lower bound constraints of kn t is the only linear function in F. Its marginal return is Problem OPT-L, then, in the optimal solution tm = 0 8m 2 . wntn = kn = wmax . We ﬁrst compute the optimal distri- b n bution of slack d among tasks with nonlinear reward functions f1; : : :; fn,1. By the Lagrange multipliers technique, witi , The proof of Lemma 1 is based on Kuhn-Tucker optimality = 0 and thus w1t = w2t = : : : = wn,1t ,1 = at 1 2 n conditions for nonlinear optimization problems and is not pre- the optimal solution t ; t ; : : :; t ,1. 1 2 n sented here for lack of space (the complete proof can be found Now we distinguish two cases: in [1]). The time complexity COPT n of Algorithm OPT is On (If the mentioned closed formulas apply, then the com- max. In this case, t ; t; : : :; t ,1 and tn = 0 is 1 2 n plexity is clearly linear. Otherwise the unique unknown can the optimal solution to OPT. To see this, ﬁrst remember be solved in linear time under concavity assumptions, as indi- that all the reward functions are concave and nondecreas- cated in [10, 13]). Lemma 1 immediately implies the existence ing, hence wit , wit wn = wmax i = i i of an algorithm which sets tm = 0 8m 2 , and then re- 1; : : :; n , 1 for all = 0. This implies that transferring invokes Algorithm OPT for the remaining tasks and slack (in some service time from another task Ti to Tn would mean case that some inequality constraints are violated by SOPT ). favoring the task which has the smaller marginal reward Since the number of invocations is bounded by n, the com- rate and would not be optimal. plexity of the algorithm which solves OPT-L is On2 . Furthermore, it is possible to converge to the solution in wmax . In this case, reserving the slack d solely time On log n by using a binary-search like technique on La- to tasks with nonlinear reward functions means violating grange multipliers. Again, full details and correctness proof of the best marginal rate principle and hence is not optimal. this faster approach can be found in [1]. Therefore, we should increase service time ti until witi drops to the level of wmax for i = 1; 2; : : :; n , 1, but not 6.3 Problem OPT-LU: Equality Constraints beyond. Solving hi wmax = ti for i P 1; 2; : : :; n , 1 =, with Upper and Lower Bounds d, n=11 ti and assigning any remaining slack bn i to tn (the An instance of Problem OPT-LU is characterized by the service time of unique task with linear reward func- set F= ff1 ; f2; : : :; fng of nondecreasing, differentiable, and tion) clearly satisﬁes the best marginal rate principle and 2 We use the expression “functions” instead of “indices of functions” unless achieves optimality. confusion arises. concave reward functions, the set O= fo1; o2 ; ::; ong of upper On logn. Furthermore, the cardinality of F decreases by bounds on optional parts, and available slack d: at least 1 after each iteration. Hence, the number of iterations P n is bounded by n. It follows that the total time complexity of maximize fi ti (22) Algorithm OPT-LU is On2 logn. However, in case of all- i=1 P n linear functions, OPT-LU can be solved in time On log n as subject to bi ti = d (23) shown before. i=1 ti oi i = 1; 2; :::n (24) 7 Conclusion 0 ti i = 1; 2; :::n (25) In this paper, we have addressed the periodic reward-based We ﬁrst observe the close relationship between the prob- scheduling problem in the context of uniprocessor systems. We lems OPT-LU and OPT-L. Indeed, OPT-LU has only an ad- proved that when the reward functions are convex, the prob- ditional set of upper bound constraints. It is not difﬁcult to lem is NP-Hard. Thus, our focus was on linear and concave see that if SOPT ,L satisﬁes the constraints given by Equation reward functions, which adequately represent realistic appli- (24), then the solution SOPT ,LU of problem OPT-LU is the cations such as image and speech processing, time-dependent same as SOPT ,L . However, if an upper bound constraint is planning and multimedia presentations. We have shown that violated then we will construct the solution iteratively in a way there exists always a schedule where the optional execution analogous to that described in the solution of Problem OPT-L. times of a given task do not change from instance to instance. 0 Let , = fxj b1 fx ox bi fi0 oi 8ig. Informally, , con- x 1 This result, in turn, implied the optimality of any periodic real- tains the functions fx 2 F with the largest marginal returns at time policy which can achieve 100% utilization of the proces- the upper bounds, wx ox . sor. The existence of such policies is well-known in real-time systems community: RMS-h, EDF and LLF. We have also pre- Lemma 2 If SOPT ,L violates some lower bound constraints sented polynomial-time algorithms for computing the optimal of Problem OPT-LU, then, in the optimal solution tm = service times. om 8m 2 ,. We underline that besides clear and observable reward im- provement over previously proposed sub-optimal policies, our The proof of Lemma 2 is again based on Kuhn-Tucker op- approach has the advantage of not requiring any runtime over- timality conditions and can be found in [1]. head for maximizing the reward of the system and for con- The algorithm ALG-OPT-LU (see Figure 8) which ﬁnds so- stantly monitoring the timeliness of mandatory parts. Once lution to the problem OPT-LU is based on successively solving instances of OPT-L. First, we ﬁnd the solution SOPT ,L of the optimal optional service times are determined statically by our algorithm, an existing (e.g., EDF) scheduler does not need to corresponding problem OPT-L. We know that this solution is be modiﬁed or to be aware of mandatory/optional semantic dis- optimal for the simpler problem which does not take into ac- tinction at all. This appears as another major beneﬁt of hav- count upper bounds. If the upper bound constraints are auto- ing pre-computed and optimal equal service times for a given matically satisﬁed, then we are done. However, if this is not the case, we set tq = oq 8q 2 ,. Finally, we update the sets F, O task’s invocations in reward-based scheduling. and the slack d to reﬂect the fact that the values of tm 8m 2 , In addition, Theorem 1 implies that as long as we are con- cerned with linear and concave reward functions, the resource have been ﬁxed. allocation can be also made in terms of utilization of tasks Algorithm OPT-LU(F,O,d) without sacriﬁcing optimality. In our opinion, this fact points 1 Set SOPT ,LU = ; to an interesting convergence of instance-based [5, 15] and 2 ; if F = then exit utilization-based [17] models for the most common reward 3 Evaluate SOPT ,L /* consider only lower bounds */ functions. 4 if all upper bound constraints are satisﬁed then About the tractability issues regarding the nature of reward SOPT ,LU = SOPT ,LU SOPT ,L ; exit functions, the case of step functions was already proven to be 5 compute , NP-Complete ([15]). By solving efﬁciently the case of con- 8 2 set tq = oq q , in SOPT ,LU ,P 6 cave and linear reward functions and proving that the case of 7 set d = d x 2 , bx ox convex reward functions is NP-Hard, we believe that efﬁcient 8 , set F=F , solvability boundaries in (periodic or aperiodic) reward-based 9 ,f j 2 g set O=O ox x , scheduling problem have been reached by our work in this as- 10 Goto Step 2 pect (assuming P 6= NP). We believe that considering dynamic aperiodic task arrivals, Figure 8. Algorithm to solve Problem OPT-LU fault tolerance issues and investigating good approximation al- gorithms for intractable cases such as step functions and er- Complexity: Notice that the worst case time complexity of ror cumulative jobs can be major avenues for reward-based each iteration is equal to that of Algorithm OPT-L, which is scheduling. Acknowledgements: The authors would like to thank the [17] R. Rajkumar, C. Lee, J. P. Lehozcky and D. P. Siewiorek. A Re- anonymous reviewers whose suggestions helped to improve source Allocation Model for QoS Management. In Proceedings the paper. of 18th IEEE Real-Time Systems Symposium, December 1997. [18] W.-K. Shih, J. W.-S. Liu, and J.-Y. Chung. Algorithms for scheduling imprecise computations to minimize total error. References SIAM Journal on Computing, 20(3), July 1991. [1] H. Aydın, R. Melhem and D. Moss´ . A Polynomial-time Al- e [19] C. J. Turner and L. L. Peterson. Image Transfer: An end-to-end gorithm to solve Reward-Based Scheduling Problem. Technical design. In SIGCOMM Symposium on Communications Architec- Report 99-10, Department of Computer Science, University of tures and Protocols, August 1992. Pittsburgh, April 1999. [20] S. V. Vrbsky and J. W. S. Liu. Producing monotonically im- [2] G. Bernat and Alan Burns. Combining (n,m) Hard deadlines and proving approximate answers to relational algebra queries. In Dual Priority scheduling. In Proceedings of 18th IEEE Real- Proceedings of IEEE Workshop on Imprecise and Approximate Time Systems Symposium, December 1997. Computation, December 1992. [3] M. Boddy and T. Dean. Solving time-dependent planning prob- [21] S. Zilberstein and S.J. Russell. Anytime Sensing, Planning and lems. Proceedings of the Eleventh International Joint Confer- Action: A practical model for Robot Control. In IJCAI 13, 1993. ence on Artiﬁcial Intelligence, IJCAI-89, Aug 1989. APPENDIX [4] E. Chang and A. Zakhor. Scalable Video Coding using 3-D Sub- We will show that: band Velocity Coding and Multi-Rate Quantization. In IEEE Int. qi X fi tij qi fi t0i Conf. on Acoustics, Speech and Signal processing, July 1993. [5] J.-Y. Chung, J. W.-S. Liu and K.-J. Lin. Scheduling periodic jobs (26) that allow imprecise results. IEEE Transactions on Computers, j =1 19(9): 1156-1173, September 1990. t +t +:::+tiqi and the function f is concave. where t0 = i1 i2 qi i i If fi is a linear function of the form fi t = ki t, then [6] W. Feng and J. W.-S. Liu. An extended imprecise computation model for time-constrained speech processing and generation. In Pqi 0 Proceedings of the IEEE Workshop on Real-Time Applications, j =1 fi tij = ki ti1 + ti2 + : : : + tiqi = ki qi ti and the May 1993. inequality (26) is immediately established. [7] J. Grass and S. Zilberstein. Value-Driven Information Gather- If fi is general concave, we recall that: fi x + 1 , fi y fi x + 1 , y ing. AAAI Workshop on Building Resource-Bounded Reasoning Systems, Rhode Island, 1997. (27) [8] M. Hamdaoui and P. Ramanathan. A dynamic priority assign- 8x; y and for every such that 0 1. In this case, we prove the validity of (26) by induction. If qi = 1, (26) holds ment technique for streams with (m,k)- ﬁrm deadlines. IEEE trivially. So assume that it holds for qi = 1; 2; : : :; m , 1. Transactions on Computers, 44(12): 1443-1451, Dec 1995. [9] E.J. Horvitz. Reasoning under varying and uncertain resource Induction assumption implies that: constraints Proceedings of the Seventh National Conference on Artiﬁcial Intelligence, AAAI-88, pp. 111-116, August 1988. m,1 X t + :::+ t [10] J. K. Dey, J. Kurose and D. Towsley. On-Line Scheduling Poli- fi tij m , 1 fi i1 m , 1im,1 (28) cies for a class of IRIS (Increasing Reward with Increasing j =1 Service) Real-Time Tasks. IEEE Transactions on Computers 45(7):802-813, July 1996. , Choosing = mm 1 ; x = ti +ti +:::+ti m, ; y = tim in 1 2 m,1 1 [11] G. Koren and D. Shasha. Skip-Over: Algorithms and Complex- (27), we can write: ity for Overloaded Systems that Allow Skips. In Proceedings of 16th IEEE Real-Time Systems Symposium, December 1995. m , 1 f ti1 + : : : + tim,1 + 1 f t [12] R. E. Korf. Real-time heuristic search. Artiﬁcial Intelligence, m i m,1 m i im 42(2): pp.189 -212, 1990. : fi ti1 + :m: + tim (29) [13] C. M. Krishna and K. G. Shin. Real-time Systems. Mc Graw- Hill, New York 1997. Combining (28) and (29), we get: [14] C.L. Liu and J.W.Layland. Scheduling Algorithms for Multi- programming in Hard Real-time Environment. Journal of ACM 1 X f t f ti1 + : : : + tim 8m m 20(1), 1973. m j =1 i ij i m [15] J. W.-S. Liu, K.-J. Lin, W.-K. Shih, A. C.-S. Yu, C. Chung, J. Yao and W. Zhao. Algorithms for scheduling imprecise compu- tations. IEEE Computer, 24(5): 58-68, May 1991. establishing the validity of (26). [16] A.K. Mok, Fundamental Design Problems of Distributed sys- tems for the Hard Real-Time Environment. Ph.D. Dissertation, MIT, 1983.

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