Fair Integrated Scheduling of Soft Real-time Tardiness Classes on Multiprocessors

Transcription

1 Fair Integrated Sheduling of Soft Real-time Tardiness Classes on Multiproessors UmaMaheswari C. Devi and James H. Anderson Department of Computer Siene, The University of North Carolina, Chapel Hill, NC Abstrat Prior work on Pfair sheduling has resulted in three optimal multiproessor sheduling algorithms, and one algorithm, EPDF, that is less expensive but not optimal. EPDF is still of interest in soft real-time systems, however, due to its ability to guarantee bounded tardiness. In partiular, it has been shown that a tardiness bound of t quanta is possible under EPDF if all task weights (i.e., shares or utilizations) are restrited to a value speified as a funtion of t. In an atual system, however, different tasks may be subjet to different tardiness bounds. If suh a system is sheduled under EPDF, then the tardiness of a task with a higher bound may ause the tardiness bound of a task with a lower bound to be violated; that is, temporal isolation among the various tardiness lasses may not be guaranteed. In this paper, we propose an algortihm based on EPDF for sheduling task lasses with different tardiness bounds on a multiproessor. Our algorithm provides temporal isolation among lasses, allows the available proessing apaity to be fully utilized, and does not require that previously established per-task weight restritions be made more stringent. Work supported by NSF grants CCR , ITR , CCR 0043, and CCR

2 Introdution Pfair sheduling, originally introdued by Baruah et al. [4], is the only known way of optimally sheduling reurrent real-time tasks on multiproessors. Under Pfair sheduling, eah task must exeute at an approximately uniform rate, while respeting a fixed-size alloation quantum. A task s exeution rate is defined by its weight (or utilization). Uniform rates are ensured by subdividing eah task T into quantum-length subtasks that are subjet to intermediate deadlines. To avoid deadline misses, ties among subtasks with the same deadline must be broken arefully. In fat, tie-breaking rules are ruial when devising optimal Pfair sheduling algorithms. As disussed by Srinivasan and Anderson [9], overheads assoiated with tie-breaking rules may be unneessary or unaeptable for many soft real-time systems. A soft realtime task differs from a hard real-time task in that its deadlines may sometimes be missed. If a job (i.e., task instane) or a subtask with a deadline at time d ompletes exeuting at time t, then it is said to have a tardiness of max(0, t d). As disussed in [6], the results produed by a soft real-time job are of dereasing usefulness after its deadline. Thus, an impliit bound exists on the tardiness that suh a job an tolerate. Systems with quality-of-servie requirements, suh as multimedia appliations, are examples where bounded deadline misses may be tolerable. Here, fair resoure alloation is neessary to provide servie guarantees, but oasional deadline misses often result in tolerable performane degradation. Hene, an extreme notion of fairness that preludes all deadline misses is usually not required. In dynami systems that permit tasks to join or leave, the overhead introdued by tie-breaking rules may be unaeptable. In suh a system, spare proessing apaity may beome available. To make use of this apaity, task weights must be hanged on-the-fly. It is possible to reweight eah task so that its next subtask deadline is preserved [9]. If no tie-breaking information is maintained, suh an approah entails very little overhead. However, weight hanges an ause tie-breaking information to hange, so if tie-breaking rules are used, reweighting may neessitate a Ω(N log N) ost for N tasks, due to the need to re-sort the sheduler s priority queue. This ost may be prohibitive if load hanges are frequent. The observations above motivated Srinivasan and Anderson to onsider the viability of sheduling soft real-time appliations using the simpler earliest-pseudo-deadline-first (EPDF) Pfair algorithm, whih uses no tie-breaking rules. They sueeded in showing that EPDF an guarantee a tardiness of k quanta for every subtask of a feasible task system, k in whih eah task s weight is at most k+ [9]. In reent work [], we showed that this ondition an be improved to k+ k+. With either ondition, the greater the tardiness allowed, the less stringent the weight restrition. Class Class Class k Subtask Window quanta k quanta I Tardiness is for a subtask of Class. All subtasks sheduled at t have their deadlines prior to t. t quanta Figure. Under Pfair sheduling, eah of a task s subtasks has an assoiated window in whih it should be sheduled; the end of a subtask s window is its deadline. In this figure, a shedule for different lasses of soft-real-time tasks on M proessors under EPDF is depited. Tasks in Class i are allowed to miss their deadlines by up to i quanta. For larity, only a few subask windows have been shown; for eah subtask shown, an denotes where it is sheduled. At time t, more than M subtasks of Classes and higher with deadlines prior to t have not yet been sheduled. As a result, a subtask of Class with a deadline at t annot be sheduled at t, and hene, misses its deadline by two quanta, i.e. its miss threshold is exeeded. Contributions. In the work summarized above, all tasks are assumed to have equal tolerane to tardiness. However, as disussed in [6], the usefulness of results produed by different soft real-time appliations may derease with tardiness at different rates; thus, different appliations an be expeted to have different tardiness bounds. To support multiple bounds, different tardiness lasses must be temporally isolated from one another so that deadline misses in one lass do not ause tardiness bounds to be exeeded in other lasses. Preserving temporal isolation is espeially important when multiplexing separately developed appliations onto a multiproessor. (Temporal isolation is a key virtue of fair sheduling.) The tardiness bound that an be guaranteed to a task system under EPDF depends on the largest task weight. Hene, if tasks with varying tardiness bounds and weights are present in a system and are sheduled using EPDF, then it may not be possible to guarantee every task its bound. As illustrated in Fig., breaking deadline ties in favor of tasks with more stringent tardiness bounds also may not be helpful. An obvious next solution would be to partition the tasks into lasses by their tardiness bounds and shedule eah lass independently on disjoint sets of proessors. Unfortunately, if the total utilization of a lass is not integral, then this approah will lead to wasted proessing apaity. For example, onsider a task system omprised of two tardiness lasses with utilizations M +δ and M + δ, respetively, time

3 where 0 < δ <, M + M + = M, and M is integral. Under partitioning, M + M + = M + proessors will be required to shedule the two lasses. Thus, proessing apaity equivalent to a full proessor would be wasted. In general, with q tardiness lasses, q additional proessors may be required. In this paper, we propose a new algorithm, based on EPDF, for supporting lasses with different tardiness requirements. Our algorithm provides temporal isolation among lasses, allows all available proessing apaity to be fully utilized, and does not require that previously established per-task weight restritions be made more stringent. Our algorithm is desribed in Se. 3 after first giving needed definitions in Se.. An experimental evaluation of it is presented in Se. 4. Pfair Sheduling In this setion, we summarize the onepts of Pfair sheduling and some prior results from [4,, 3,, 8]. To begin with, we limit attention to periodi tasks, eah of whih begins exeution at time 0. A periodi task T with an integer period T.p and an integer exeution ost T.e has a weight wt(t ) = T.e/T.p, where 0 < wt(t ) <. A task is light if its weight is less than /, and heavy otherwise. Pfair algorithms alloate proessor time in disrete quanta; the time interval [t, t + ), where t N (the set of nonnegative integers) is alled slot t. (Hene, time t refers to the beginning of slot t.) A task may be alloated time on different proessors, but not in the same slot (i.e., interproessor migration is allowed but parallelism is not). The sequene of alloation deisions over time defines a shedule S. Formally, S : τ N {0, }, where τ is a task set. S(T, t) = iff T is sheduled in slot t. On M proessors, T τ S(T, t) M holds for all t. Lags and subtasks. The notion of a Pfair shedule is defined by omparing suh a shedule to an ideal fluid shedule, whih alloates wt(t ) proessor time to task T in eah slot. Deviation from the fluid shedule is formally aptured by the onept of lag. Formally, the lag of task T at time t is lag(t, t) = wt(t ) t t u=0 S(T, u). A shedule is defined to be Pfair iff ( T, t :: < lag(t, t) < ). () Informally, the alloation error assoiated with eah task must always be less than one quantum. (For oniseness, we leave the shedule impliit and use lag(t, t) instead of lag(t, t, S).) The lag bounds above have the effet of breaking eah task T into an infinite sequene of quantum-length subtasks, T, T,.... Eah subtask has a pseudo-release r(t i ) and a pseudo-deadline d(t i ), where i r(t i ) = d(t i ) = wt(t ) i wt(t ). () (For brevity, we often omit the prefix pseudo-. ) To satisfy (), T i must be sheduled in the interval w(t i ) = [r(t i ), d(t i )), termed its window. For example, in Fig. (a), r(t ) = 0 and d(t ) =. Therefore, T must be sheduled at either time 0 or time. Soft real-time sheduling. The notion of tardiness disussed in Se. for soft real-time jobs an be extended in a straightforward manner to subtasks of soft real-time tasks. The tardiness of a subtask T i is defined as tardiness(t i ) = max(0, t d(t i )), where t is the time that T i ompletes exeution. The tardiness of a task system is then defined as the maximum tardiness among all of its subtasks in any shedule [9]. The earliest-pseudo-deadline-first (EPDF) algorithm [9] is the algorithm that we onsider for sheduling soft tasks, for the reasons disussed in the introdution. EPDF prioritizes subtasks by their deadlines, and resolves any ties arbitrarily. Although EPDF is not optimal on more than two proessors [], as disussed earlier, it an ensure a tardiness of at most k quanta for eah subtask, provided ertain per-task weight restritions hold. Task models. In this paper, we onsider the intra-sporadi (IS) and the generalized-intra-sporadi (GIS) task models [3, 8], whih provide a general notion of reurrent exeution that subsumes that found in the well-studied periodi and sporadi task models. The sporadi model generalizes the periodi model by allowing jobs to be released late ; the IS model generalizes the sporadi model by allowing subtasks to be released late, as illustrated in Fig. (b). That is, an IS task is obtained by allowing a task s windows to be shifted right from where they would appear if the task were periodi. Let θ(t i ) denote the offset of subtask T i, i.e., the amount by whih w(t i ) has been shifted right. Then, by (), we have the following. i i r(t i) = θ(t i)+ d(t i) = θ(t i)+ (3) wt(t ) wt(t ) The offsets are onstrained so that the separation between any pair of subtask releases is at least the separation between those releases if the task were periodi. Formally, k > i θ(t k ) θ(t i ). (4) Eah subtask T i has an additional parameter e(t i ) that speifies the first time slot in whih it is eligible to be sheduled. It is assumed that e(t i ) r(t i ) and e(t i ) e(t i+ ) for all i. Additionally, no subtask an beome eligible before its predeessor ompletes exeution, i.e., h < i e(t i) < r(t i) S(T h, u) = u < e(t i). ()

4 T 3 T 4 T T 6 T 7 T 8 T 3 T 4 T T 6 T 7 T 8 T 3 T T 6 T 7 T 8 T T T T T T (a) (b) () Figure. (a) Windows of the first job of a periodi task T with weight 8/. This job onsists of subtasks T,..., T 8, eah of whih must be sheduled within its window, or else a lag-bound violation will result. (This pattern repeats for every job.) (b) The Pfair windows of an IS task. Subtask T beomes eligible one time unit late. () The Pfair windows of a GIS task. Subtask T 4 is absent and T 6 is one time unit late. The interval [r(t i ), d(t i )) is alled the PF-window of T i and the interval [e(t i ), d(t i )) is alled the IS-window of T i. A shedule for an IS system is valid iff eah subtask is sheduled in its IS-window. (Note that the notion of a job is not mentioned here. For systems in whih subtasks are grouped into jobs that are released in sequene, the definition of e would prelude a subtask from beoming eligible before the beginning of its job.) The IS model is suitable for many appliations in whih proessing steps may be jittered. For example, in an appliation that proesses pakets arriving over a network, pakets may arrive late or in bursts. The IS model treats these possibilities as first-lass onepts: a late paket arrival orresponds to an IS delay, and if a paket arrives early (as part of a bursty sequene), then its eligibility time will be less than its Pfair release time. Note that its Pfair release time determines its deadline. Thus, in effet, an early paket arrival is handled by postponing its deadline to where it would have been had the paket arrived on time. Generalized intra-sporadi task systems. A generalized intra-sporadi (GIS) task system is like an IS task system, exept that a task may omit some of its subtasks. Speifially, a task T, after releasing subtask T i, may release subtask T k, where k > i+, instead of T i+, with the following restrition: r(t k ) r(t i ) is at least. k wt(t ) i wt(t ) In other words, r(t k ) is not smaller than what it would have been if T i+, T i+,...,t k were present and released as early as possible. For the speial ase where T k is the k first subtask released by T, r(t k ) must be at least wt(t ). Fig. () shows an example. If T i is the most reently released subtask of T, then T may release T k, where k > i, as its next subtask at time t, if r(t i )+ t. k wt(t ) i wt(t ) If a task T, after exeuting subtask T i, releases subtask T k, then T k is alled the suessor of T i and T i is alled the predeessor of T k. As shown in [3], an IS or GIS task system τ is feasible on M proessors iff wt(t ) M. (6) T τ Shares and lags in IS and GIS task systems. lag(t, t) is defined for IS and GIS tasks as before [8]. Let ideal(t, t) denote the proessor share that T reeives in an ideal fluid (proessor-sharing) shedule in [0, t). Then, t lag(t, t) = ideal(t, t) S(T, u). (7) u=0 Towards defining ideal(t, t), we define share(t, u), whih is the share assigned to task T in slot u. share(t, u) is defined in terms of a funtion f(t i, t) that indiates the share assigned to subtask T i in slot t. f(t i, t) is defined as follows. i wt(t ) ( i ( i wt(t ) + ) wt(t ) (i ), if t = r(t i) ) wt(t ), if t = d(t i) wt(t ), if t (r(t i), d(t i) ) 0, otherwise (8) Fig. 3 shows some f values for a task of weight /6. Given f, share(t, u) an be defined as share(t, u) = i f(t i, u). (9) As shown in Fig. 3, share(t, u) usually equals wt(t ), but in ertain slots, it may be less than wt(t ). Thus, ( T, t 0 : share(t, t) wt(t )). (0) We an now define ideal(t, t) as t u=0 share(t, u). Hene, from (7), lag(t, t + ) = t (share(t, u) S(T, u)) u=0 3

5 (a) Figure 3. Fluid shedule for the first five subtasks (T,..., T ) of a task T of weight /6. The share of eah subtask in eah slot of its PF-window is shown. In (a), no subtask is released late; in (b), T and T are released late. Note that share(t, 3) is either /6 or /6 depending on when subtask T is released. (b) = lag(t, t) + share(t, t) S(T, t). () Similarly, the total lag for a shedule S and task system τ at time t +, denoted LAG(τ, t + ), is as follows. (LAG(τ, 0) is defined to be 0.) LAG(τ, t+) = LAG(τ, t)+ (share(t, t) S(T, t)). () T τ 3 Integrating Tardiness Classes k k+ In this setion, we present Algorithm I-EPDF, whih shedules a soft real-time task system τ omprised of tasks with different tardiness bounds. We let tasks that an be guaranteed a tardiness of quanta omprise Class. Thus, if every task an be guaranteed a tardiness of at most q ( ), then there are at most q lasses. Any lass, exept Class q may be empty. If M denotes the total utilization of τ, then I-EPDF sheduled τ on at most M proessors. Without loss of generality, we assume that M is integral. If neessary, this property an be ensured by adding a dummy task of weight M M to τ. The algorithm onsists of three phases: (i) a lassifiation phase, (ii) a distribution phase, and (iii) a sheduling phase. In the lassifiation phase, the tardiness lass of eah task is identified, based on its weight. As already mentioned, Srinivasan and Anderson established a per-task weight restrition of for ensuring a tardiness of k quanta under EPDF [9], whih we later improved to k+ k+ []. In this paper, we assume that task weights are restrited for eah lass using the k k+ ondition, as the proof is simpler. Our goal in this paper is only to illustrate the idea of integrated sheduling. Our approah is still orret when the k+ k+ ondition is used. Classifiation phase. We inlude in Class all tasks with weights in the range (, + ], i.e., tasks that an be ensured a tardiness of quanta under EPDF. Note that this has the advantage that a task T with wt(t ) + that an tolerate a tardiness of d > an be assigned to Class and guaranteed a lower tardiness bound, without impating the tardiness of other tasks. Letting τ denote the set of all tasks in Class, this lassifiation ensures the following property. (W) ( T τ : < wt(t ) + ) Property (W) an be ensured in Θ(N) time by simply pla- ing task T in Class T.e T.p T.e. We denote the total utilization of τ by M. Thus, q τ = τ, M = q wt(t ), and M = M. (3) = T τ = Distribution phase. The goal of this phase is to distribute the M proessors among the lasses and to define how proessors are shared. The proessors are divided into q groups of sizes P,..., P q, with the i th group assigned to Class i. Beause the number of proessors assigned to Class i is integral, whereas its total utilization M i may not be, eah lass is allowed to borrow proessing apaity from at most one lower-indexed lass. To ensure orretness for eah lass, this borrowing is subjet to a number of rules given below. Later, in Se. 3., we present an algorithm for defining an assignment that satisfies these rules. Before stating these rules, we introdue some relevant notation. Notation. w i denotes the amount of proessing apaity that Class i borrows from some lower-indexed lass. Sup i denotes the lower-indexed lass that supplies to Class i; if w i = 0, then Sup i = 0. f i denotes the frational part of the utilization of τ i, i.e., f i = M i M i. (4) To enable the different lasses to share proessors at runtime, a donor task D j ( j q) of weight w j > 0 may be reated; D j is added to Class i, where i = Sup j. (The manner in whih D j is used to share proessors is explained in Se. 3..) The set of all donor tasks added to Class i is 4

6 Class (i) τ i M i w i Sup i λ i i ˆM τ {D, D 3, D 4, D 6 } τ τ τ 4 3 τ τ τ τ τ P i {D } {D 7 } {D 8 } 4 7 {D 9 } (a) D i w i D 0 D 4 D 3 0 D 4 D 7 0 D 6 0 D 7 4 D 8 D (b) 4/ /0 /0 / /0 /4 3 7 / 4 8 3/4 4 9 () Figure 4. (a) Distribution of proessors to the nine tardiness lasses of a soft real-time task system. Column headings refer to various terms mentioned in the text. (b) Weights of donor tasks. () Tree representation of the task system in (a). Labels within nodes indiate lass indies. The integer adjaent to a node denotes the number of proessors assigned to the lass that the node represents. Edge (a, b) defines the supplier/borrower relation beween Classes a and b; if a < b, then Class a supplies a proessing apaity of w(a, b) to Class b, where w(a, b) is the weight of (a, b). denoted λ i. ˆτ i extends τ i by inluding these tasks: Correspondingly, we define ˆτ i = τ i λ i. () ˆM i = M i + T λ i wt(t ). (6) Proessor sharing rules. The sharing of proessors among lasses is governed by the following rules. (R) The proessing apaity that Class i borrows is at most the frational part of its utilization, i.e., 0 w i f i. (7) (R) Class i borrows proessing apaity from at most one lass with a lower tardiness bound (or a lower-indexed lass), and lends to zero or more lasses. In other words, the following hold. ( i : i :: Sup i < i) (8) ( i : i :: {j D i λ j } = {Sup i }) (9) ( i :: {j : Sup j = i} 0) (0) Some of these rules are somewhat tehnial in nature. They are inluded to address ertain ases that arise in showing that I-EPDF is orret. (R3) Class i, where i 3 and f i /3, does not lend any proessing apaity to other lasses. If 0 < f i /, then Class i borrows a apaity of f i from Class ; if f i ranges between / and /3, then it borrows f i from Class. If f i = 0, then Class i does not borrow. ( i : i 3 f i = 0 :: w i = 0 Sup i = 0) () ( i : i 3 0 < f i / :: w i = f i Sup i = ( j : Sup j i)) () ( i : i 3 / < f i /3 :: w i = f i Sup i = ( j : Sup j i)) (3) (R4) The proessing apaity that Class 3 or higher borrows is less than what it lends, i.e., ( i : i 3 :: ( j : Sup j = i w i < w j )). (4) (R) The number of proessors assigned to the various lasses must satisfy the following. ( i : P i = M i w i + w j ) () {j:sup j =i} P = M + w + {j:((j 3) (f j /))} wj (6) P = M + {j:((j 3) (/<f j /3))} wj (7) ( i : i 3 :: P i = M i M i ) (8) q P i = M (9) i=

7 ALGORITHM I-EPDF(τ) E,..., E q: integer; /* to denote the number of tasks eligible at time t in eah lass */ P,..., P q: integer; D,..., D q : Tasks; τ,..., τ q : GIS task sets; ˆτ,..., ˆτ q : GIS task sets; λ,..., λ q : GIS task sets initially ; /* Set of all donor tasks added to a lass */ Sup,..., Sup q : integer initially 0; tight,..., tight q : boolean initially TRUE Classifiation Phase Group tasks into at most q tardiness lasses. Task sets τ i, where i q, are known at the end of this phase. Distribution Phase Determine w i and reate D i. Determine λ i, ˆτ i, and P i, for all i. Sheduling Phase 3 t := 0; 4 while TRUE do for i := q downto do 6 if ˆτ i then 7 for eah D in λ i do 8 if E P then 9 s := index of next eligible subtask of D ; 0 if r(ds ) t d(d s ) > t + then if r(ds) < t) then s := fi; r(ds ), e(d s ) := t +, t + fi fi od; 3 E i := # of eligible tasks in ˆτ i, exluding those that are early-released (it suffies to determine if P i + are eligible) fi od; 4 for i := to q do /* The lowest-indexed lass is always tight */ if tight i then 6 maxshedulable := P i else 7 maxshedulable := P i + fi; 8 Shedule at most maxshedulable tasks of ˆτ i using EPDF (for Class 3 and higher, break ties involving the heavy donor task, if any, in favor of the donor task); 9 for eah D in λ i do 0 if D is sheduled then tight := FALSE else tight := TRUE fi od; 3 t := t + od Figure. Algorithm I-EPDF detailed pseudo-ode for the sheduling phase. The supplier/borrower relationship among lasses an be represented as a weighted tree in whih nodes represent lasses. An edge of weight w between nodes i and j, where i < j, implies that w j = w and Sup j = i. As an example, Fig. 4 shows an assignment of proessors to lasses and a supplier/borrower relation among lasses that onforms to the rules above for a task system omprised of nine tardiness lasses. The following properties follow from (R) (R). (L) There are at most two tasks in ˆτ with weights exeeding /. (L) The weight of every task in ˆτ is in (, 3 ]. (L3) ( i : i 3 λ i f i > 3.) 3. Sheduling Phase of I-EPDF Assuming that proessors are assigned to lasses per the rules above, we now explain the sheduling phase. A proof that it is orret is given in Appendix C. As mentioned earlier, Se. 3. presents an algorithm for reating suh an assignment. In the sheduling phase, a separate instantiation of EPDF is used to shedule eah ˆτ i. The pseudo-ode for this phase is shown in lines 3 3 in Fig.. Note that ˆτ i inludes the donor tasks in λ i, whih ompete with tasks in τ i at every time instant. If at time t, a donor task D j in λ i is sheduled, then one of the proessors of Class i is handed down to Class j. Thus, Class j has P j + proessors for sheduling the tasks in ˆτ j at time t, zero of more of whih may be handed down to higher-indexed lasses, reursively. In the first part of the sheduling phase, given by the for loop of lines 3, the number of eligible subtasks of ˆτ at time t is identified, when i =. Beause the for loop onsiders the lasses in dereasing index order, the number of eligible tasks in lasses with higher indies than are known at this time. Therefore, if ˆτ inludes donor task D k and the number of eligible tasks in ˆτ k is at most P k, then the release time of the next subtask D k s of D k is postponed to t +, if its deadline is greater than t +. We do this beause Class k is not able to use an extra proessor that it would be given, and hene, by postponing the release time of the next subtask of its donor task under the onditions speified, Class k may be provided with an extra proessor sooner in the future than may otherwise be possible. We refer to this sheduling rule in lines 8 as the postponement rule. Another related rule in line is that if the release time of D k s before the postponement was earlier than t, then D k s is replaed by D k with r(d k ) set to t +. This rule shall be referred to as the reset rule. As disussed later, this rule does not impat the 6

8 Tasks in τ pros. 3 (Total util. of τ ) D 4 3 pros. Donor tasks D, D 3, D 4, and D 6 3 NT Non tight slots for Class 4 PT Pseudo tight slots for Class 4 T Tight slots for Class 4 Class Class 4 T NT T NT PT PT T NT T Tasks in τ 4 (Total util. of donor tasks) Figure 6. Classes and 4 of the example task system in Fig. 4. Class is assigned five proessors and supplies proessing apaity to Classes, 3, 4, and 6. Class 4 is assigned three proessors and borrows a proessing apaity of / from Class. An additional proessor is handed down to Class 4 from Class when donor task D 4 is sheduled in Class. The example partial shedule shows the first three subtasks of D 4, whih are sheduled in the slots marked by an. The release of the third subtask is postponed from time 0 to time. Thus, slots 4, 8, and 4 are non-tight for Class 4, slots 0 and are pseudo-tight (see the appendix), and the rest are tight. tardiness of other tasks. The seond part of the sheduling phase, given by the for loop in lines 4 3, determines the maximum number of tasks of ˆτ that an be sheduled (maxshedulable) at t, and shedules those with the highest priority. For all but the lowest-indexed lass, maxshedulable is either P or P +, based on whether D is sheduled. If P proessors are available for sheduling tasks in ˆτ, then t is said to be a tight slot for ˆτ ; otherwise, t is a non-tight slot for ˆτ. An example is given in Fig. 6, in whih Classes and 4 from Fig. 4 are onsidered. Seleting the highest priority tasks to shedule dominates the per-slot time omplexity of this phase, and hene, it is the same as that of EPDF, i.e., O(M log N). 3. Distribution Phase of I-EPDF (R) (R) an be expressed as linear onstraints and the problem solved using integer or mixed integer programming. (Floors and eilings in the expressions an easily be eliminated.) However, as we now show, a solution in linear time is possible. Fig. 7 presents the detailed pseudo-ode for the distribution phase. In desribing this ode, we refer to Class j as an i-borrower if Class j borrows proessing apaity from Class i. The omputation here proeeds in three steps. The for loop in lines 0 0 omprises the first step, whih is responsible for ensuring that (R3) holds, and together with the third step (see below), that (8) is satisfied. In this step, Class i, where i 3, is set to borrow its entire frational utilization of f i from Class, if f i /, or from Class, if / < f i /3. This is done by setting w i = M i M i = f i in line 0, followed by the addition of a donor task D i of appropriate weight to Class or Class. Every lass that is made a - or a -borrower at the end of Step, is onsidered finished, and does not partiipate in future distribution steps. This is marked by setting the boolean variable done i to TRUE. Suh a lass is assigned M i proessors, whih are not shared with other lasses. For example, onsider Fig. 4. The total utilization of τ 4 is 3, with a frational part f 4 = / < /. Therefore, at the end of the step just desribed, a donor task D 4 of weight w 4 = / is added to Class, three proessors are assigned to Class 4, and it is marked finished. Lines 8 omprise the next step, whih ensures (7). In this step, Class is made a -borrower by letting it borrow a proessing apaity of w = (M + T λ wt(t )) (M + T λ wt(t )) from Class. It is then marked finished. In the example in Fig. 4, the frational part of the utilization of no lass is between / and /3. Therefore, no donor tasks are added to Class in the previous step. Thus, w = 4/, a donor task D of weight 4/ is added to Class, and Class is marked finished. The while loop in lines 4 40 onstitutes the third step in the distribution phase. This step is responsible for ensuring (4), (6) and (8). In this step, every lass that is not yet finished is onsidered in inreasing index order. The goal of the i th iteration is to determine at most two higherindexed lasses with whih Class i an share its spare apaity (given by spare i = ˆM i w i ( ˆM i w i )). (To ensure the tardiness bound of the borrowing lass, it is neessary to ensure that it does not borrow from a lass with a larger bound.) Class i is also marked finished at the end of the i th iteration. Thus, at the beginning of iteration i, every lass with a lower index than i is already finished. Note that for Class 3 and higher, ˆM i = M i holds at the beginning of the iteration in whih it is onsidered. This is beause these lasses are not augmented with donor tasks prior to this point. Line 9 identifies Class l with the lowest index greater than i that is not finished that an be made an i-borrower. To ensure (R), if f l spare i holds, then Class l is made to borrow a proessing apaity of f l from Class i and is marked finished in the if blok in lines 7. Line 7 identifies and sets l to the next higher-indexed lass that is not yet finished. Irrespetive of whether the test in line sueeded, at 7

9 PROCEDURE ADDDONORTASK(i, w, sup) D i := donor task of weight w; Sup i := sup; 3 ˆτ sup, λ sup := ˆτ sup {D i }, λ sup {D i }; 4 ˆM sup := ˆM sup + w i ; return ALGORITHM I-EPDF(τ) P,..., P q: integer; w,..., w q : rational; D,..., D q : GIS tasks; τ,..., τ q : GIS task sets; ˆτ,..., ˆτ q : GIS task sets; λ,..., λ q : GIS task sets initially ; /* Set of all donor tasks added to a lass */ Sup,..., Sup q : integer initially 0; spare,..., spare q : rational initially 0; done,..., done q+ : boolean initially FALSE Classifiation Phase /* Group tasks into at most q tardiness lasses. Task sets τ i, where i q are known at the end of this phase. */ Distribution Phase for i := 3 to q do if M i M i /3 then 3 w i := M i M i ; 4 if w i > 0 then if w i / then Sup i := else Sup i := fi 6 ADDDONORTASK(i, w i, Sup i ) fi; 7 P i, done i := M i, TRUE fi od; 8 w := ˆM ˆM ; 9 if w > 0 then ADDDONORTASK(, w, ) fi; 0 P, done := ˆM, TRUE; if ˆM = ˆM then P, done := ˆM, TRUE fi; 3 i := ; 4 while i q do /* done q+ handles the boundary ondition */ while done i do i := i + od; 6 if i q then 7 avail := spare i := ˆM i w i ( ˆM i w i ); 8 l := i + ; 9 while done l do l := l + od; 0 if l q then if avail > 0 (M l M l ) avail then w l := M l M l ; 3 ADDDONORTASK(l, w l, i); 4 P l, done l := M l, TRUE; avail := avail w l ; 6 l := l + ; 7 while done l do l := l + od fi; 8 if avail > 0 /* l q */ then 9 w l := avail; 30 ADDDONORTASK(l, w l, i); 3 d, j := l, i; 3 while w d < w j do 33 ˆτ j, λ j := ˆτ j {D d }, λ j {D d }; 34 w j, ˆM j := w j w d, ˆM j w d ; 3 ˆτ Supj := ˆτ Supj {D d }; 36 λ Supj := λ Supj {D d }; 37 if w j < w d then d := j fi; 38 j := Sup j od fi; 39 P i, done i := ˆM i, TRUE; fi 40 i := l fi od Figure 7. Algorithm I-EPDF detailed pseudo-ode for the distribution phase. line 8, f l > avail holds. This is learly the ase if the test in line failed. On the other hand, if this test sueeded, then beause f l > /3 holds for every lass of index three or higher that is not finished by Step, w l > /3 holds at line. Beause spare i, as omputed in line 7, is less than one, avail < /3 holds at the end of line. Hene, for the same reason that f l > /3 holds for every lass of index exeeding three that is not finished by Step, f l > avail holds at line 8. Beause the amount of proessing apaity that Class l borrows is set to min(avail, f l ), Class i an have at most two donor tasks added to it in the i th iteration. l is the unfinished lass with the lowest index at line 40. Therefore, i is updated to l so that Class l is onsidered for the addition of donor tasks in the next iteration. Using our example (Fig. 4), spare at the beginning of the first iteration of the while loop in lines 4 40 is ˆM ˆM = 4 = 3. (Beause D4 of weight w 4 = and D of weight w = 4 were added to Class in Steps and, respetively, ˆM = M +w 4 +w = = 4, and hene, ˆM =, at the end of Step.) In this iteration, the first unfinished lass with a higher index than one, whih is Class 3, is made a -borrower (lines 8 30). Thus, l = 3 in this ase. Hene, at the end of the first iteration, w 3 is set to 3/ (line 9) and a donor task D 3 of weight w 3 is added to Class (line 30). Class is then marked finished (line 39). The unfulfilled utilization of Class 3 is now = 4 0. Therefore, Class 3 has a spare apaity spare 3 = 9 0. Beause Class 3 is the next unfinished lass, lasses with whih its spare apaity is shared are identified in the next iteration. One final adjustment is performed in lines If the weight of the donor task w l that is added to Class i is less than w i, i.e., the proessing apaity that Class i borrows in turn from its supplier j = Sup i, then Class j is made 8

10 Additional Proessing Capaity(%) Additional Proessing Capaity by Total Utilization Total Utilization (a) Average Worst Case Additional Proessing Capaity(%) Additional Proessing Capaity by No. of Classes No. of Classes (b) Average Worst Case Figure 8. Empirial determination of additional proessing apaity, expressed as a perentage of total utilization, that may be required if I-EPDF is not used to shedule soft real-time task sets omprised of multiple lasses. (a) Additional apaity vs. Total utilization. (b) Additional apaity vs. Number of lasses. Class l s supplier, too. This is done by moving D l to Class j from Class i. w i is appropriately redued so that the total apaity that Class j supplies remains the same. As a result of this adjustment, Class j will now have two donor tasks D i and D l in plae of D i (it is possible that Class j has some other donor tasks, whose weights are not altered in this iteration), and it is possible for one of them to be lighter than D j. If this is the ase, then the while loop in lines 3 38 moves the lighter of the two donor tasks, D i and D l, up the supplier hain, to ensure (R4). In our example, as desribed earlier, spare 3 = 9 0 at the end of the iteration for Class. This holds at the beginning of the iteration for Class 3, when i = 3. Reall that Class 4 is already finished, and hene, Class is the next unfinished lass. Beause f = 7 0, whih is less than spare 3, the ode in lines 7 makes Class borrow the entire frational part of its utilization from Class 3, and redues avail to (in line ). The next unfinished lass is identified 0 = 0 to be 6 in line 7. Hene, l = 6 at line 8, and the ode in lines 8 30 makes Class 6 borrow a apaity of 0 from Class 3. Thus, at the end of line 30, donor tasks D and D 6, of weights w = 7 0 and w6 = 0, respetively, are added to Class 3. Reall that Class 3 already borrows a proessing apaity of 3 from Class. Therefore, w3 = 3, and hene, w 6 < w 3. As a result, D 6 is moved to Class, the supplier of Class 3, and the weight of D 3 (w 3 ) is redued by w 6 to 0. In other words, at the end of the iteration for Class 3, Class 3 borrows only 0 from Class and is augmented with only one donor task, D. Fig. 4 shows the final distribution and sharing of proessors among lasses. It an be shown that the omplexity of the above algorithm is Θ(q). A proof that it ensures (R) (R) is presented in Appendix B. 4 Experimental Evaluation In this setion, we report results of our empirial evaluation of the additional proessing apaity that may be required, when lasses do not share proessors. The evaluation proedure was as follows.,000,000 task sets were generated at random, with total utilization M in the range..64. The tasks in eah task set were divided into q tardiness lasses based on their weights. The total number of proessors P required to shedule the task set was then omputed, assuming that eah tardiness lass has exlusive aess to the proessors that it requires. The differene E = P M, whih represents the additional proessing apaity required, was then determined. The average value of E (expressed as a perentage of M) with respet to total utilization (M) and the number of lasses (q) is plotted in Fig % onfidene intervals are also shown on these plots. The figure also depits the worst-ase observed values of E, for eah value of M and q. The graphs show that the average perentage of loss is quite high (over 30%), for small values of M and q, and dereases with inreasing M and q. The reason for this is as follows. In Fig. 8(a), the value of q for a given M is the average over all task sets with that value of M, and in Fig. 8(b), the value of M for a given q is the average over all task sets with that value of q. 9

11 As mentioned in the introdution, E is at most q. Also, beause a lass an span multiple proessors, q inreases at a lower rate than M. Therefore, E, expressed as a perentage of M, dereases with inreasing q and M. (However, for small q, q may inrease at a higher rate than M beause the minimum value of M is, while that of q is. This explains the initial rise in Fig. 8(b).) Even though E dereases as M and q inrease, the loss is still more than % for large M and q, whih suggests signifiant waste. Conlusion We have presented a new algorithm for integrating soft real-time tardiness lasses on a multiproessor. Our algorithm provides temporal isolation among lasses, allows available proessing apaity to be fully utilized, and does not require that previously established per-task weight restritions for a given tardiness threshold be lowered. Our experiments indiate that the proposed algorithm allows a substantial amount of proessing apaity to be relaimed. The algorithm presented ould be extended to allow hard tasks, in addition to soft tasks. In that ase, an optimal algorithm (with tie breaks) is used for sheduling hard tasks, while a separate instantiation of EPDF is used for eah soft lass. However, it may be required to promote a few soft tasks to the hard ategory. As disussed in Se., one motivation for using EPDF is the ability to reweight tasks effiiently in dynami systems. However, reweighting a task may alter the tardiness bound that an be guaranteed to it, and hene, may require that the task be migrated to a different tardiness lass. Redistributing proessors to the redefined lasses an be done in onstant time. It only remains to be proved that the tardiness bounds of individual tasks an still be guaranteed. We are urrently working on this problem. Aknowledgements: We are grateful to Phil Holman for his suggestions on improving the presentation of this paper and for his omments on earlier drafts. Referenes [] J. Anderson and A. Srinivasan. Mixed Pfair/ERfair sheduling of asynhronous periodi tasks. Journal of Computer and System Sienes. To appear. [] J. Anderson and A. Srinivasan. Early-release fair sheduling. In Proeedings of the th Euromiro Conferene on Real- Time Systems, pages 3 43, June 000. [3] J. Anderson and A. Srinivasan. Pfair sheduling: Beyond periodi task systems. In Proeedings of the 7th International Conferene on Real-Time Computing Systems and Appliations, pages , De [4] S. Baruah, N. Cohen, C.G. Plaxton, and D. Varvel. Proportionate progress: A notion of fairness in resoure alloation. Algorithmia, :600 6, 996. [] U. Devi and J. Anderson. Improved onditions for bounded tardiness under EPDF fair multiproessor sheduling. In Submission, November 003. [6] J.W.S. Liu. Real-Time Systems. Prentie Hall, 000. [7] A. Srinivasan. Effiient and Flexible Fair Sheduling of Realtime Tasks on Multiproessors. PhD thesis, University of North Carolina at Chapel Hill, Deember 003. [8] A. Srinivasan and J. Anderson. Optimal rate-based sheduling on multiproessors. In Proeedings of the 34th ACM Symposium on Theory of Computing, pages 89 98, May 00. [9] A. Srinivasan and J. Anderson. Effiient sheduling of soft real-time appliations on multiproessors. In Proeedings of the th Euromiro Conferene on Real-time Systems, pages 9, July 003. A Pfair Sheduling Additional Properties In this appendix we state additional properties of pfair sheduling that is needed for the orretness proof of I- EPDF presented in Appendix C. The first three lemmas onern lengths of windows of subtasks. Lemma [] The length of eah window of a task T is either wt(t ) or wt(t ) +. Lemma [] The following properties hold for any task T. (a) If (i ) is a multiple of T.e, then w(t i ) =. (b) If b(t i ) = 0, then w(t i ) = w(t i+ ). wt(t ) () If b(t i ) = 0, then w(t i ) is a minimal window of T. (d) If T is heavy and b(t i ) = 0, then w(t i ) =. Lemma 3 If the length of the window of a subtask T i of T is k, then wt(t ) /k. Proof: By Lemma, w(t i ) = wt(t ) or w(t i ) = +. Therefore, = k or + = k. wt(t ) wt(t ) wt(t ) If the former holds, then wt(t ) k, while if the latter holds, then wt(t ) k, whih implies that wt(t ) k. The next lemma summarizes some general properties of the f values of (8). Lemma 4 [7] Let f be as defined by (8). Then, the following hold. (a) In any time slot u 0, at most two onseutive subtasks of a task may have positive values for f. The f value of a subtask is also referred to as the flow that the subtask reeives in an ideal fluid shedule. 0

12 (b) If f(t i, r(t i )) < wt(t ), then b(t i ) =. () If f(t i, d(t i ) ) < wt(t ), then b(t i ) =. (d) If b(t i ) = and T i exists, then f((t i, d(t i )))+ f(t i, r(t i )) = wt(t ). B Corretness Proof for the Distribution Phase In this appendix, we state and prove lemmas onerning properties that hold at the end of the distribution phase of Algorithm I-EPDF. For oniseness, by I-EPDF, we refer to its distribution phase in this setion. All referenes to line numbers are with respet to the pseudo-ode in Fig. 7. Lemma Donor tasks are not added to Class, where 3, before the iteration of the while loop in lines 4 40, in whih i =. Proof: Inspetion of ode in lines shows that donor tasks are not added to Class in that part. The while loop in lines 4 40 onsiders lasses in inreasing index order. (The value of i for the next iteration of the loop is updated to l in line 40, and it an be verified that l > i.) In the i th iteration, new donor tasks are added only to Class i in lines 3 and 30. The while loop in lines 3 38 moves donor tasks aross lasses, but only aross those lasses that are already augmented with suh tasks. Therefore, donor tasks are not added to a lass with an index greater than i. Lemma 6 If donor task D is reated, then done = FALSE holds prior to its reation. Proof: Donor tasks are reated in lines 6, 9, 3, and 30. We onsider eah ase. The for loop in lines 7 onsiders Class 3 and higher in inreasing index order exatly one. Beause the done array is initialized to FALSE, and is not altered for Class until after D is reated (whih is done at most one) done = FALSE holds before the reation of D. In line 9, D is reated. It an easily be verified that done = FALSE at this time. In lines 3 and 30, donor tasks D l are reated, where l is determined in lines 9 and 7 suh that done l = FALSE holds. Lemma 7 If i =, where q, holds for an iteration of the while loop in lines 4 40, then done = true holds at the end of the iteration. Proof: Within the while loop, the value of i is modified in lines and 40 only. If modified in line, then done = TRUE already holds. Before i is updated in line 40 for the next iteration, done is updated to TRUE in the previous line (line 39). Note that 39 is always exeuted if q. Lemma 8 D, where, is reated at most one. Proof: By Lemma 6, donor task D is reated only if done = FALSE holds prior to its reation. Therefore, it suffies to show that done is updated to TRUE either immediately after the reation of D, or before an attempt an be made to reate it again. Donor tasks are reated in lines 6, 9, 3, and 30. If reated in line 6, 9, or 3, done is updated to TRUE in the subsequent line. If reated in line 30, then l = holds in that line. It an be verified that i is updated to l, i.e., in line 40 for the next iteration of the while loop.by Lemma 7, done = TRUE will hold at the end of the next iteration. It an also be verified that in an iteration of the while loop referred to, D i is not reated. Therefore, D will not be reated for a seond time in the future. Lemma 9 D is reated and added to a lower-indexed lass before donor tasks are added to Class, where 3. Proof: Follows from Lemmas, 6, and 7, and the fat that D is not reated in that iteration of the while loop in lines 4 40, for whih i = holds. Lemma 0 If D is reated in line 3 or 30, then 3 and f > /3. Proof: D is never reated (or does not exist), D is reated in line 9, and if 3 f /3, then D is reated in line 6. By Lemma 8, a donor task is reated exatly one. The lemma follows from these fats. Lemma Let w = f, where 3. Then, Class is not augmented with donor tasks. Proof: D, where 3, may be reated in lines 6, 3, or 30. We onsider two ases. Case : D is reated in line 6 or 3. In this ase, w = f, and done = TRUE holds immediately after. By Lemma 9, Class is not augmented with donor tasks before D is reated, and by Lemma, the augmentation ours only in the iteration of the while loop in lines 4 40, for whih i = holds. However, beause done = TRUE holds, by line, Class is skipped from onsideration in the while loop, and hene, is not augmented with donor tasks. Case : D is reated in line 30. For this ase, we first show that w < f holds at the time of reation of D. Consider the iteration of the while loop in whih D is reated in line 30. Prior to the reation of D, a different donor task D d may have been reated in line 3in the same iteration. We onsider both possibilities. If no other donor task was

13 reated, then the test in line should have failed. Beause D is reated in line 30, the test in line 8should have sueeded, whih implies that M l M l = f l > avail holds. Beause w l is set to avail in line 9before D l is reated, by = l, it implies that f > w holds. On the other hand, if D d was reated in line 3prior to the reation of D in line 30we reason as follows. The value of avail as omputed in line 7is at most one. By Lemma 0, f d /3, and beause w d is set to f d in line and avail is redued by w d in line, avail < /3 holds at line 9. Again, by Lemma 0, f /3, whereas w is set to avail, whih is less than /3. It an be verified that the only piee of ode that may alter the weights of the donor tasks is the while loop in lines 3 38, and that the weights are only dereased. Therefore, w an never equal f. Lemma 0 w f, where. Proof: w is initialized to zero, modified when D is reated, and may later be modified in the while loop in By Lemma 8, D is reated exatly one. Inspetion of the ode shows that D is reated in lines 6, 9, 3, or 30. If reated in lines 6, 9, or 3, w = f. Therefore, it remains to be shown that w f holds, if D is reated in line 30. By Lemma 0, if D is reated in line 30, then 3 f > /3. (30) We onsider exeution from the beginning of the iteration of the while loop in lines 3 38 in whih D is reated. It is easy to see that avail = spare i holds at the end of line 7. If the test in line sueeded, then another donor task D d, where d < is reated in line 3 before D in the same iteration. By Lemma 0, f d > /3. Beause w d = f d holds by the assignment in line and avail is updated to avail w d is line, avail < /3 holds at line 8. Therefore, by (30), f > avail holds at line 8. If the test in line failed, while that in line 8 sueeded, then it is easy to see the f > avail holds. w is set to avail, whih by the argument just onluded is less than f, before D is reated in line 30. One set to a non-zero value, the value of w is modified only in the while loop in lines However, the value is never inreased, and if dereased, the redution is by an amount that is stritly less than its present value. Therefore, if set to a non-zero value, 0 < w f holds. If never set, then w = 0 holds. Lemma 3 Let Class, where 3, be augmented with donor tasks. Then, the value of spare, as omputed in line 7, is equal to f + w, whih is greater than w. Proof: spare as omputed in line 7 is given by ˆM w ( ˆM w ). If 3, then by Lemma, Class is not augmented with donor tasks until the iteration of the while loop in lines Therefore, by (6), at line 7, when i =, ˆM = M. Therefore, spare is given by M w (M w ) = M + f w ( M + f w ) (from (4)) = M + f w M f + w = f w f + w = f + w. The final step is by Lemmas and. If spare is less than or equal to w, then it would imply that w f + w, or that f, whih is a ontradition to (4). Lemma 4 The sum of the weights of the donor tasks added to Class in that iteration of the while loop in lines 4 40 for whih i = is exatly equal to spare, as omputed in line 7 either at the end of line 7 or line 30. Proof: The proof follows from the fat that M is integral. Lemma w = 0. Proof: Straightforward. Lemma 6 If the sum of the weights of the donor tasks added to Class j equals f j + w j prior to the exeution of an iteration of the while loop in lines 3 38, then the same relation holds at the end of the iteration. Proof: At the beginning of the iteration, we have {k:sup k =j} wk = f j +w j. Within the while loop, one donor task D d is moved from Class j to Class Sup j. Therefore, {k:sup k =j} wk = f j +w j w d, at the end of the iteration. However, w j is redued by w d to w j w d in the same iteration. Therefore, {k:sup k =j} wk = f j + w j, still holds at the end of the iteration. Lemma 7 The values of ˆM j Supj and ˆM at the end the exeution of an iteration of the while loop in lines 3 38 are the same as their values at the beginning of that iteration. Proof: By Lemmas and, and the test in line 3, we have j. It an also be easily shown that j, and hene, j 3 holds. Therefore, by Lemmas 4, 3, and 6, we have w k = f j + w j. (3) {k:sup k =j} We first show that ˆM j is not altered. ˆM j at the beginning of an iteration of the loop is given by ˆM j = M j + {k:sup k =j wk } = M j + f j + w j (from (3)) (3) = M j + + w j (from (4)) (33) = M j +. (34)