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 1 
 
 Bounds on Scheduling Algorithms for 
 Heterogeneous Computing Systems 
 
 by 
 
 Jane W. S. Iiu 
 C. L. Liu 
 
 June 197^ 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 THE LIBRARY OF THE 
 
 i N 1 £ 
 
UIUCDCS-R-7U-632 
 
 Bounds on Scheduling Algorithms for 
 Heterogeneous Computing Systems 
 
 by 
 
 Jane W. S. Liu 
 C. L. Liu 
 
 June 197^ 
 
 Department of Computer Science 
 
 University of Illinois at Urb ana- Champaign 
 
 Urbana. Illinois 
 
 This work was supported in part by the National Science 
 Foundation under Grants No. US NSF GJ-36265 and GJ-^1538. 
 
1. INTRODUCTION 
 
 Recent progress in hardware technology and computer architecture 
 has led to the design and construction of computer systems that contains 
 a large number of processors. Because of their capability of executing 
 several tasks simultaneously, the problem of job scheduling in a multi- 
 processor system is of both theoretical and practical interest. Several 
 authors have designed scheduling algorithms to produce schedules which 
 minimize the total execution time of a given set of tasks and thus achieve 
 optimal utilization of the processors [1-3]. Unfortunately, such 
 algorithms are known only for some special uses. Furthermore, in many 
 instances, the algorithm that produces optimal schedules is so complex that 
 the reduction in the total execution time of a set of tasks is offset by 
 the computation time required to determine an optimal schedule. For this 
 reason, simple algorithms that produce only sub-optimal schedules are often 
 used in practice. Such a choice becomes even more appealing when the 
 performance of the simple algorithms producing sub- optimal schedules can 
 be compared quantitatively with that of algorithms producing optimal 
 schedules. For this reason, lower bounds on the performance of simple 
 algorithms have been studied extensively [^, 5]. 
 
 In previous works on job scheduling, a multiprocessor computing 
 system is modelled as one containing identical processors. We introduce here 
 a more general model in which different processors have different computation 
 speeds. This model is a realistic one when we consider the possibilities of 
 replacing one or more of the processors in an existing system by faster 
 processors and of interconnecting different computers in an installation. 
 It will be described precisely in Section 2. 
 
In Section 3 } lower bounds on the performance of a class of simple 
 nonpreemptive algorithms are derived. These results will also provide 
 us with information concerning the relative merit of different computing 
 systems and the trade-off between the speeds and the number of processors 
 in a multiprocessor system. 
 
 Algorithms which produce preemptive schedules with minimum total 
 execution time have been found in some special cases [3] for systems 
 containing identical processors. Algorithms which produce optimal 
 schedules of independent tasks when the multiprocessor systems contain 
 different processors are described in Section k. The performance of 
 preemptive scheduling algorithms is compared to that of nonpreemptive 
 scheduling algorithms studied in Section 3. 
 
 An algorithm which produces schedules with minimum mean flow time 
 is described in Section 5. Performances of different computing systems, 
 using mean flow time as a criterion, are compared. 
 
2. GENERAL MODEL 
 
 2.1 A Model of Heterogeneous Computing System 
 
 By a heterogeneous computing system, we mean a multiprocessor 
 system in which processors have different computing speeds. We measure 
 the speed of a processor against that of a "standard" processor whose 
 speed is considered as 1. The speed of a processor is said to be To if 
 it is b times as fast as a standard processor. Without loss of generality, 
 we shall assume that b > 1 throughout our discussion. Let us denote a 
 multiprocessor system which contains n_ processors of speed b.., n„ 
 processors of speed b p , . .., n, processors of speed b -.by (P = (n , n , . .., 
 n, ; b , b , . .., b ). Furthermore, the N processors will be referred to 
 individually as processors P. for i = 1, 2, . .., N where N = n. + n + . . . + 
 n. In particular, the n processors of speed b. are referred to as P , 
 P , ..., P ; the n p processors of speed b are referred to as 
 
 P n +1' P n +2> •*•' P n +n ' etc * For exam P le > the system (P = (l, 3; 2, l) 
 contains four processors P , P , P , and P, whose speeds are 2, 1, 1, and 1, 
 respectively. 
 
 2.2 Definitions and Notations 
 
 Let if - {T , T , ..., T } denote a set of tasks to be processed by 
 
 a system (P . The execution time of a task is defined as the time required 
 
 to complete the task on a standard processor. We shall denote the execution 
 
 time of the task T. by ^(T. ) where |_i is a function from ^ to the reals. 
 
 In other words, (i is a function that specifies the execution times of 
 
 the tasks in Zf . Furthermore, we suppose that there is a precedence 
 
 relation < defined over ZT . That T. < T. (reads T. precedes T. or T. 
 
 1 J I J J 
 
 follows T. ) shall mean that T. cannot begin to be executed before the 
 
 J 
 
execution of T. is completed. A task is said to be executable at a certain 
 
 time if the execution of all tasks preceding it has been completed. 
 
 Formally, a set of tasks is specified by an ordered triple ( 7, \i, <). 
 
 We also describe a set of tasks {j> , \±, <) by a directed graph whose 
 
 vertices correspond to the tasks and are labeled by the names, T., and 
 
 J 
 
 their execution times, u(T.). There is a directed edge from T. to T. if 
 T. < T.. For example, a set of tasks {&, n, <) is shown in Fi^ore 2-1 
 where V = {T^ T g , T , T^, T } and ^ < T^, T^ < T , etc. 
 
 By scheduling a set of tasks on a multiprocessor system, we mean 
 to specify for each task the time interval within which it is to be 
 executed and the processor on which execution will take place. A schedule 
 can be described by a timing diagram (also known as the Gantt chart) such 
 as that shown in Figure 2-2. In this timing diagram, each horizontal 
 line is a time axis and its subdivisions give the sequence of tasks 
 executed on a processor together with the idle periods of the processor. 
 The idle periods are the time intervals within which the processor is not 
 executing any tasks. We use cp , cp , ... to denote idle periods of the 
 processors as shown in Figure 2-2. With a slight abuse in notation, we 
 use u(cpu ), |J.(cpp), ... to denote the lengths of .the idle periods. 
 
 The completion time of a schedule is the total time it takes to 
 
 t 
 execute all the tasks according to the schedule. For example, the 
 
 completion time of the set of tasks (Z/ f u, <) shown in Figure 2-1 according 
 
 to the schedule shown in Figure 2-2 is 6.5. When the completion time is 
 
 used as a criterion for comparing different scheduling algorithms, an 
 
 optimal schedule for a given set of tasks (.7", (a, <) is one with the minimal 
 
 completion time. 
 
 Throughout this report, we assume that the execution of a set of tasks 
 begin at t = 0. 
 
T,/3 
 
 T s /1 
 
 T 2 /2 
 
 Figure 2-1 
 
 Pi 
 
 T 4 i T 2 
 
 <k 
 
 1.5 ' 1 ' 1 
 
 *, 
 
 2 ' i ' 1.5 
 
 Figure 2-2 
 
For a given schedule, the flow time t. of a particular task T. 
 
 is defined to be the time at which its execution is completed. For 
 
 example, for the schedule shown in Figure 2-2, the flow time of tasks T, 
 
 and T\ are equal to 1.5 and 2.5 respectively. The mean flow time t of a 
 
 schedule for a set of tasks (f 7 , u, <) is equal to the sum of the flow 
 
 times of all tasks in ^T . That is, 
 
 m 
 t = Z t. 
 
 The mean flow time of the schedule shown in Figure 2-2 is 15. It is clear 
 that the mean flow time of a schedule represents the average waiting time 
 (turnaround time) of the tasks in t/" . This parameter will also be used 
 as a criterion for comparing the performance of different scheduling 
 algorithms. 
 
 2.3 A More General Model of Heterogeneous Computing Systems 
 
 Let us note that the model of heterogeneous computing systems 
 described in Section 2.1 can be generalized by assuming that it takes 
 different amounts of time to execute a task on different processors but 
 the difference in execution times is not uniform among different tasks. 
 For example, a lengthy computation task T can be executed on processor 
 P , which contains a fast arithmetic unit, in a smaller amount of time than 
 on processor P p . But another task T p consisting of primarily i/O operations 
 may be executed in a smaller amount of time on processor P p instead. For 
 a system (P containing N processors, the execution times of the tasks can 
 be specified by N functions u_, u , ..., u where u.(T. ) is the execution 
 time of task T. on processor P.. No general result is known for this 
 model of heterogeneous computing systems. In this report, we shall not 
 be concerned with this model. 
 
3. PRIORITY-DRIVEN SCHEDULING ALGORITHMS 
 
 Throughout this section, we shall use completion time as the 
 criterion for comparison of the performance of different scheduling 
 algorithms. We assume that once a task is assigned to a processor, it 
 will be executed until completion„ That is, preemption is not allowed. 
 We are concerned, in particular, with the class of non-preemptive 
 scheduling algorithms that never leave processors idle intentionally. 
 (We note that it is sometimes necessary to leave a processor idle 
 intentionally in order to minimize the completion time of a set of tasks. ) 
 In other words, when a processor becomes free, a task that is executable 
 at that moment is scheduled. These algorithms can be described by 
 assignments of priorities to the tasks. At the moment a processor becomes 
 free, the task that have the highest priority among all executable tasks 
 is scheduled. If several processors are free at the same time, the task 
 is executed on the processor P. with the smallest index i among the free 
 processors. Such algorithms are known as priority-driven scheduling 
 algorithms. Schedules produced by such algorithms are called priority- 
 driven schedules. As an example, for the set of tasks {Z7", \i, <) shown 
 in Figure 2-1, the priority-driven schedule according to the priority list 
 (T , T„, T , T. , T ) on the system (P = (l, 1; 2, l) is that shown in 
 Figure 2-2. 
 
 t 
 In order of decreasing priorities. 
 
3.1 Performance of Priority- Driven Scheduling Algorithm 
 
 In this section, we establish a lower bound on the performance of 
 priority- driven schedules in which priorities are assigned to tasks in a 
 completely arbitrary manner. For simplicity, let us consider first a 
 multiprocessor system (P = (l, n; b, l), that is, a system containing a 
 processor P of speed b and n processors P p , P„, . .., P , of speed 1. 
 We have, 
 Theorem 3.1 
 
 Let (J" , u, <) be a set of tasks to be executed on the system 
 (P = (l, n; b, 1). Let w and to' denote the completion times when {y , u, <) 
 is executed according to a priority-driven schedule specified by the 
 priority list L and an arbitrary schedule, respectively. Then, 
 
 — . < b + 
 
 b+n 
 
 Moreover, this bound is best possible. 
 
 Proof. In the priority-driven schedule specified by the priority 
 list L, let ZT-.., denote the set of tasks run on processor P- and 3C denote 
 the set of tasks run on processors P p , P , ..., P , . Also, let 0, and $ 
 denote the corresponding sets of idle periods of processor P and processors 
 Pp, P, ..., P _, respectively. Let 
 
 U, = Z u(T ) 
 
 V = Z u(T ) 
 i • ^2 
 
 I ± = Z u(cp i ) 
 
 I 2 = Z u(cp i ) 
 
We have 
 
 u -5a<T +u a + I i + V 
 
 1 r 1 2 b-1 /tt _ \ 12 b-1 _ _ / . \ 
 
 [-1— + TT ( u p +i p) + "IT- + IT ^ (3-1) 
 
 n+1 L b b v 2 2 y b b 1 J 
 
 Clearly, 
 
 1 V U 2 
 
 w >q^ ^ u(T ) =— - (3-2) 
 
 n D T.G^ 1 n ° 
 
 l 
 
 Also 
 
 u 2+ i 2 
 
 = OJ 
 
 n 
 
 (3-3) 
 
 To bound the magnitudes of I and I_, let us consider an idle period 
 
 cp of a certain processor, P n . Let t n and t~ denote the times at which 
 m k 1 2 
 
 cp begins and ends. We have the following observations: (i) if the 
 m 
 
 execution of a task T. on another processor begins at t where t < t < t p , 
 
 then there must exist a task T. such that T. < T. and the execution of 
 
 i i -J 
 
 T. ends at t. Otherwise, T. would have been executed prior to t on 
 i .) 
 
 processor P . (ii) If the execution of a task T. begins at t where 
 
 t < t, then there must exist a task T. such that T. < T. and the execution 
 
 of T. ends at t . Combining these two observations, we conclude that if 
 l 2 ' 
 
 9 , cp , ..., cp are the idle periods of a certain processor, there 
 exists a set of tasks "6 - (T.,, T. , ..., T. .) such that 
 
 (i) t <!•:... 
 
 -u 
 
 (ii) at any moment within any one of the idle periods cp , cp , 
 
 ..., 9 one of the tasks in iS is being executed on another processor, 
 mr 
 
 That is, 
 
10 
 
 2 n(T. ) > Z n(cp, ) 
 
 Vt? : >,^ fflV v...,U 
 
 It follows that 
 
 I 1 < b u>» (3-U) 
 
 Moreover, 
 
 Hence, 
 
 n E n(T ) > I + I 
 
 T^e k 1 2 
 
 I l +I 2 
 
 -^ <b co' (3-5) 
 
 Substituting this inequality and (3-2)-(3- J +) into eq. (3-1), we obtain, 
 
 , , , 1 r h+n , b-1 . /, ., N , n 
 
 w < — - [— — co' + — — nco + nu' + (b-1) co'l 
 
 - n+1 L b b v ' J 
 
 which simplifies to 
 
 -. < b + 
 
 oj' - b+n 
 
 That this bound is the best possible can be demonstrated by the 
 following example. Consider a set of tasks {j, \i, <) where 
 fT l' T 2 , ..., T 2n , T 2n+1 ), 
 
 H(T ± ) = b i = 1, 2, . .., n-1 
 
 M- 
 
 (T ) = b+e 
 
 n 
 
 |j.(T. ) = n i = n+1, n+2, ..., 2n 
 
 ^ (T 2n + l } = (n+b)b 
 
11 
 
 with e being a positive number and < is empty. The priority list 
 
 L= ^1' T n+1' T n+2' *••' T 2n' T 2> V '"' V T 2n+1^ 
 
 2 
 yields a schedule whose completion time is n + nb + b for very small e as 
 
 shown in Figure 3 -la. On the other hand, the priority list 
 
 L= ^ T 2n+1' T n+1' T n+2> '~> T 2n' T l' T 2> "" T n^ 
 
 yields a schedule whose completion time is n + b for very small e as shown 
 in Figure 3-lb. □ 
 
 Let us observe that the upper bound of the ratio go/ go' in Theorem 3-1 
 is approximately equal to b + 1 for large n. That is, even when there is 
 only one fast processor in a multiprocessor system with many standard 
 processors, the worst case performance of an arbitrary scheduling algorithm 
 when compared to that of an optimal scheduling algorithm still depends 
 primarily on the speed of the fast processor. For a fixed n, the upper 
 bound of oj/gj' approaches b as a limit when b increases. This fact indicates 
 that a priority-driven schedule may become very inferior for large b. As a 
 matter of fact, when b is very large, better performance can often be 
 assured by using only the fast processor to execute all the tasks in -/ . 
 In that case, the completion time approaches the minimum completion time 
 as a limit for large b. We shall return to this point in Sec. 3-3. 
 
 The result in Theorem 3.1 can be extended immediately. 
 Theorem 3.2 
 
 Consider a multiprocessor system (P = (n , n , ..., n. ; b b , ..., 
 b ). Let go and go' be the completion times of a set of tasks (J" ', \x, <) 
 when it is run on (P according to a priority-driven schedule and an 
 
T T • • • • T 
 
 p t i V i i i i n i 
 
 ■ 2 
 
 p, f 
 
 P n+1 f 
 
 p 3 
 
 P 
 
 r n+i 
 
 (a) 
 
 T 2n 
 
 +i 
 
 p l I 
 
 p I W , ]i. 
 
 T n + ; 
 
 12 
 
 A 
 
 n+l i '2n+i 
 
 1*2 n i Tn 
 
 (b) 
 
 Figure 3-1 
 
 T n+ 2 , <fe 
 
 T 2 n i ^n 
 
13 
 
 an arbitrary schedule, respectively. Suppose that b, > b„ > ... > b . 
 We have 
 
 b. 
 
 b 
 
 co' - b k 
 Is. 
 
 1 
 
 Z n. b. 
 
 .-,11 
 
 1 = 1 
 
 Moreover, the bound is the best possible. 
 
 Proof. In the priority- driven schedule specified by a priority 
 
 list L, let U. denote the sum of the execution times of all tasks executed 
 l 
 
 on the processors of speed b.. Let I. denote the sum of the lengths of 
 the idle periods on the processors of speed b.. We have 
 
 oo = 
 
 k 
 
 Z n. 
 
 U l U 2 
 
 b^ +I l +I 2 + '" \ 
 
 i=l 
 
 Z n. 
 
 1 = 1 
 
 • k k b -b. U. 
 
 ± z u. + z -±-l Ur + i-) 
 
 b n . .. i . b_ v b. i' 
 
 1 i=l i=l 1 i 
 
 > k k-1 b.-b ") 
 
 - Z I. + Z ~~ - I. 
 'l i=l x i=l b l \J 
 
 (3-6) 
 
 Clearly 
 
 k 
 
 
 Z 
 i=l 
 
 U. 
 
 l 
 
 k 
 
 
 Z 
 i=l 
 
 a. b. 
 l i 
 
 < UJ 1 
 
 (3-7) 
 
 and 
 
Ik 
 
 u. 
 
 : — +1. = n. w 
 
 D. 1 1 
 
 1 
 
 (3-8) 
 
 Using the same argument presented in the proof of Theorem 3-1, we conclude 
 
 that the total length of the idle periods of any processor must be equal to 
 
 b co' 
 
 or less than — . Hence 
 
 \ 
 
 b. I. 
 k l 
 
 < b. oo' 
 
 n. - 1 
 
 l 
 
 (3-9) 
 
 Moreover, 
 
 Consequently, 
 
 k k b oo' 
 
 Z I. < ( Z n.-l) ~^— 
 i=l i=l k 
 
 k 
 
 b n Z I. 
 
 k . , l 
 1=1 
 
 k 
 
 Z n. -1 
 
 i=l x 
 
 < b co» 
 
 (3-io) 
 
 Substituting (3-7)-(3-10) into eq. (3-6), we obtain 
 
 co < 
 
 1 
 
 - k 
 Z n. 
 i=l x 
 
 Z n. b. k b -b. 
 
 i=l 1 1 oo « + Z '■ ■. X n. oo 
 b n i=l i 
 
 b b k 
 
 + r^rr ( Z n.-l) oo- 
 
 b_ b \ _ i 
 
 1 k i=l 
 
 k-1 b.-b b n 
 
 + Z -V^ >T n - "' 
 i=l b l b k x 
 
15 
 
 Multiplying both sides of this expression by b b Z n., we obtain 
 
 1=1 
 
 1=1 1=2 
 
 i=l i=l 
 
 k-1 
 
 + b Z (b -b ) n ] u' 
 
 i=1 i k i 
 
 which simplifies to 
 
 k k k 
 
 (b, Z n. b. ) w < (b Z n. b. + b Z n. b. - b b ) w» 
 k . 1 l i' - 1 ., i i k . , i i Ik 
 
 That is , 
 
 2 < 1 + 1 . 1 
 W — b, k 
 
 k 
 
 Z n. b. 
 1=1 X X 
 
 To show that the bound is best possible,, we consider a set of tasks 
 ^/ , u, <) where the precedence relation < is empty and 
 
 ^ = fT rpql r = lf 2 > ••" k ' q = - 1 ' 2 > '"> N > and 
 
 p = 1, 2, ..., n n -l for r 1 and p = 1, 2, ..., n^ for r / 1 
 
 k-1 
 U(Tjs = 1, 2, ..., Z n +1} 
 
 s i=l x 
 
 The execution times of the tasks T are 
 
 rpq 
 
16 
 
 rpq 
 
 b r \ \ 
 
 H(T ) = < 
 
 b Id b 
 r k 2 
 
 b r \ \-l 
 
 r k k 
 
 1 < q < n ! 
 
 n + 1 < q < n., + n_ 
 1 - H - 1 2 
 
 k-2 k-1 
 
 E n.+l < q < En. 
 
 i=l 1=1 
 
 k-1 k 
 
 E n.+l < q < En. 
 
 1=1 i=l 
 
 for r = 1, 2, . .., k and p = 1, 2, ..., n -1 if r = 1 and p 
 if r / 1. The execution times of the tasks T are 
 
 = 1, 2, 
 
 n 
 
 n(T fl ) 
 
 k-1 
 s = 1, 2, . .. E n. 
 i=l ± 
 
 ^V = b l\ . z n n i b i 
 i=l 
 
 k-1 
 
 where e is a small positive number and S = E n.+l. Let us consider a 
 
 i=l 1 
 priority-driven schedule according to a priority list which assigns 
 
 higher priorities to the tasks T than the tasks T . Priorities are 
 
 rpq s 
 
 assigned to tasks T according to the lexiographical order of their 
 
 rpq * 
 
 subscripts. (That is, T ^_ has the highest priority and T-,-,p has the 
 
 next highest priority, etc. ) The priority assignment of the tasks T is 
 
 s 
 
 (T,, Tp, ..., T q ). We obtain the schedule in Figure 3-2a where 
 
 CO 
 
 = b 1 b k (i^-l) + b 2 b^ n 2 +.. 
 
 b. b, n. + t>_ E n. b 
 k k k 1 i=1 i i 
 
 k 
 
 = (b n +b n ) E n. b. - b b 
 v k 1 . n 11 Ik 
 
 i=l 
 
17 
 
 However, this set of tasks can be scheduled as shown in Figure 3-2b where 
 
 k 
 
 w' = h, Z n. b. □ 
 
 i=l 
 
 Let (P = (n , n , . .., tcl; b , b , . .., b ) be a multiprocessor 
 
 k 
 system. The quantity Z n. b. can be considered as a measure of the total 
 
 i=l 
 
 k 
 computational capacity of the system. Indeed, the sum E n b. is the 
 
 i=l ± x 
 
 maximal throughput of the system. According to the result in Theorem 3-2, 
 
 the worst case performance of a priority- driven schedule depends mainly 
 
 on the ratio of speeds of the slowest processor and the fastest processors 
 
 in the system. For systems of large maximal throughput, the bound in 
 
 b l 
 Theorem 3-2 approaches — + 1. This implies that in systems where the speed 
 
 k 
 of the fastest processors is significantly larger than that of the slowest 
 
 processors, the worst case performance of a priority-driven schedule may 
 
 become very poor in comparison with that of an optimal schedule. 
 
 3.2 Performance of Some Heuristic Scheduling Algorithms 
 
 Theorem 3.1 and 3.2 establish a lower bound on the performance of 
 priority-driven scheduling algorithms. Between the two extremes of using 
 an optimally chosen priority list and of using a completely arbitrary 
 priority list, there is the possibility of using priority lists obtained 
 by simple heuristic procedures. Such a possibility offers a good 
 compromise between the performance of the resultant scheduling algorithm 
 and the computational cost of determining the priority list. In this 
 
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 kll 
 
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 ttf 
 
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 + 
 
 T T $ 
 
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 T 21n T 2l( n;L +l) T 21( n;L +2 ) T 2l( n;L +n 2 ) T 21( n;L +n „+!] 
 
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 b k b k ■ e /b 1 
 
 l(n,-l)N T n, 1 n, 
 
 £ / b l 
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 T„„ T. 
 
 L 222 x 22n x 22(n + l) 1 22(n+2) T 22(n +n Q ) T 22 (n +n ^l) 
 
 HH — I I — I \ I I '-f^- 
 
 T 2n 2 T 2n n ^xUn^l)^ V 2 } S^l^^ S»<V°2 +1 > 
 
 2 h i 21 I 2 ' I HH I Mr- 
 
 kl2 T km T kl(n 1 +l) T kl(n 1 +2) T kl(n 1+ n 2 ) ^(Oj+i^+l) 
 
 M 
 
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20 
 
 section, we shall consider the special case in which the precedence relation 
 < is empty. We denote such a set of tasks by ( J , \i). 
 
 Let us consider the problem of scheduling a set of tasks (2T, \i) 
 on the multiprocessor system (p = (l, n; b, l). Suppose that we choose the r 
 longest tasks in the set 3 and schedule them according to a priority list 
 L . Let oo denote the corresponding completion time when they are executed 
 on (r . Let L denote a priority list obtained by appending to L an 
 assignment of priorities to the remaining tasks. Let oo denote the 
 corresponding completion time. When oo > go, we have, 
 Theorem 3.3 
 
 "r ' 1 
 
 7^< 1 + u- 
 
 M "' (b+n)M 
 where 
 
 m I • / Tr+1 r r+l [b] rbJ-lA 
 
 M = max mm — st-t > — rrr i_ H - - j_ t — \> 
 \|n+pb] ' n+[b] b b J' 
 
 r+1 
 n+b 
 
 and co' is the completion time according to an arbitrary schedule. This 
 
 bound is the best possible when b is an integer and n+b divides r. 
 
 Proof. Let T n , T~, .... T denote the r longest tasks and T n 
 1 ' 2 7 ' r r+1 
 
 denote the (r+l)st longest task in ^J . Since oo > go, tasks T,, T p , ..., T 
 
 are completed before t = co . Moreover, since each processor can have at 
 
 most one idle period immediately prior to t = w , the idle period of the 
 
 r 
 
 fast processor is equal to or less than u(T , ). The idle period of any 
 
 slow processor is equal to or less than — u(T , ) if the fast processor is 
 
 not idle at t = co and is equal to or less than li(T ) if the fast 
 processor is idle at t = w . Now, 
 
21 
 
 1 U l 
 "r " S3 f~ + U 2 + h + V 
 
 m [ ^ ( W + ¥ (U 2 +I 2 ) + k J 2 +I i] (3- 11 ) 
 
 where I , I p , U and U are as defined in the proof of Theorem 3.1. We note 
 that if L = 
 
 J 2 < ^ T r + 1> 
 
 n - b 
 
 and if I-, ^ 0, 
 
 h ± "<W 
 
 and 
 
 X 2 
 
 — T < u(T J 
 n-1 - KV r+1 
 
 Consequently, 
 
 ^I 2 + I 1 <max {•%, S^ + l)n(T ) 
 
 b 
 
 ,n±^( T ) 
 - b KV r+1 
 
 Thus eq. (3-11) becomes 
 
 . 1 r n+b , b-1 n+b-1 /m Nn 
 
 w < — - [— — w' + —— nw + — - — |j.(T , )] 
 
 r - n+1 L b b r b ^ v r+l /J 
 
 Notice that in any schedule, either r ri J of the r+1 tasks T_, T„, .... T 
 
 |n+[b]| 1 2.' ' r 
 
 are executed on a slow processor or r f -. 1 [b] - ([b]-l) of them are executed 
 on the fast processor. Consequently, we have 
 
22 
 
 P#t] n(T ,)<(•)' 
 n+[b] | KV r+l y - 
 
 (3-12) 
 
 or 
 
 Un+ bl b b / M ' U r+l j ^ 
 
 CO' 
 
 (3-13) 
 
 We also note that 
 
 (r+1) n(T p+1 ) < (n+b) o>' 
 
 or 
 
 r+1 
 
 n+b K ' VJ "r+l' — 
 
 ^(T J < 
 
 (3-1*0 
 
 Combining (3-12) - (3-1^+), we write 
 
 M ^(L i) < w ' 
 
 r+1 
 
 where 
 
 M = max min 
 
 (li 
 
 r+1 
 
 n+[b] 
 
 r+1 
 
 n+[b] 
 
 [bj |"b]-l\ r+1 
 b b /' n+b 
 
 Equation (3 -11) becomes 
 
 , 1 r n+b , b-1 n+b-1 , n 
 
 co < — - [— — co ' + — — nto + co • 1 
 
 r - n+1 L b b r bM J 
 
 which simplifies to 
 
 ^ 1 1 
 co- - M ~ (n+b )M 
 
 To show that the bound is best possible, let us note that for b and 
 
 n+b 
 
 being integers, the bound becomes 
 
 r rb+n+b (n+b ) 
 ^7 _ rb+n+b 
 
 (3-15) 
 
23 
 
 Let tT = (T^ T 2 , ..., T r+n ( n+i ) +1 ) 
 
 and n(T ± ) = (n+b)b i = 1, 2, ..., r, r + 1 
 
 ji(T. ) = 1 i = r + 2, ...., r + n(n+b) 
 
 n(T. ) = 1 + e i = r+ n(n+b) + 1 
 
 where e is a small positive number. The tasks T-,, T , . .., T. can be 
 scheduled by the priority list (T, , Tp, . .., T ). ■ If we use the priority 
 list 
 
 (T^ T 2 , . .., T^ T r+1 , T^ 2 , . .., T r+n+1 , T r+n+2 , ..., 
 
 T ") 
 
 r+n(n+b)+ I 
 
 we obtain the schedule in Figure 3-3a with completion time 
 
 w = rb + n + b 
 If we use the priority list 
 
 (T^ Tp, ..., T r , T r+2 , T r+3 , ..., T r+n(n+b) ^ T r+ n(n+b)+l> T r+1 } 
 the completion time is 
 
 oj = rb + n + b(n+b) 
 
 as shown in Figure 3-3b. □ 
 
 r 
 When b and — — are integers and when the maximal throughput of the 
 
 system, n+b, is large, the upper bound of — - given by eq. (3-15) becomes 
 
 U) 
 
 approximately 
 
 1 + - b 
 
 1 +Zk TT 
 
 ii 1 1. 
 
2k 
 
 Vn-b+l T r TV* 
 
 ? * I ^k * H^ I btlt ,ft H+tHH^ 
 
 p . T 3 Tr-n-b+3 J*"!*? T mmrt)*l 
 
 ?3 h bU * H-+-^ h T 4 T HH T H 
 
 ? *hI „,,Ir n — hv^-^v — kh^h 
 
 b(»+b) b(ntb) i i 
 
 -y -^"— 
 
 rb n+b 
 
 y A V ' 
 
 w 
 
 T, T hvb W^i TV Trfz T„ B(ritb)tl ^ 
 
 ?, Ht-^ I " I * I I I * 1 : I .I. I IWh-H^ 4 1 
 
 h+b *+b mb *+b -r -r i+e 
 
 P D 
 
 b Tr+| 
 
 V z i ^ pv I """" ^ i- : r4-A l , I 1 V- 
 
 b(n+b) b(*+b) ' ' b(n+b) 
 
 % I ^ f-A 1 Tr "" E> ^| | 1 V- 
 
 bln+b) bCn+b) I I I 
 
 ? nM l \ hV-l \ h-H H 1 1 ^ 1 
 
 b(r\+b) bl"+b} I I I 
 V v /\ y A v J 
 
 rb n b(^4b) 
 
 Figure 3-3 
 
25 
 
 Hence, the ratio of w to the minimum completion time of the tasks in ^J is 
 
 upper bounded by 
 
 1 + 
 
 n+b 
 
 when the minimum completion time of the r longest tasks is less than w . 
 
 We now generalize Theorem 3.3 to the problem of scheduling a set 
 
 of tasks (J, u.) on a multiprocessor system Q= (n , ru, . .., tl ; 
 
 b , b , . .., b ). Again, let us choose the r longest tasks in 3. Let 
 
 to denote the completion time of these tasks when they are executed on (? 
 
 according to a priority list L . Let L denote the priority list obtained 
 
 by appending to L an arbitrary assignment of priorities to the remaining 
 
 tasks in 3 • Let w denote the corresponding completion time, and let w' 
 
 denote the completion time according to an arbitrary schedule. When 
 
 w > oj and b, =1, we have, 
 r k ' ' 
 
 Theorem 3.^- 
 
 U) 
 
 7 < i 
 
 1 
 + Q 
 
 Q Z n. b. 
 1-1 X X 
 
 where 
 
 Q = max 
 
 min 
 l<j<k| 
 
 "■ 
 
 
 ~ * 
 
 
 r+1 
 
 
 k 
 
 
 
 Z 
 1=1 
 
 n. fb 
 l 1 i 
 
 1 
 
 ill 
 
 M- 1 
 
 \ 
 
 r+1 
 
 ' k 
 
 / E 
 ' 1=1 
 
 E n. b. 
 l l 
 
26 
 
 k 
 Moreover, the "bound is "best possible when "b. are integers and Z n. b. 
 
 i=l X X 
 divides r. 
 
 Proof. The proof is similar to that of Theorem 3.3. Let T,, T~, .... T 
 
 1 2 ' r 
 
 denote the r longest tasks and T - denote the (r+l)st longest task in *7. 
 It is clear that 
 
 rU. U. 
 
 w = 
 r k 
 
 1 / u l 2 \ 
 
 (b- + b7 + -- + \ +I i + I 2. + ••• +I k) 
 
 Z n. 
 1=1 1 
 
 where U n , LL, .... II and I_, I_, .... I are as defined in the proof of 
 1' 2 3 k 1 '2' ' k 
 
 Theorem 3.2. This expression of co can be rewritten as 
 
 r 
 
 1 
 
 co = 
 
 r k 
 
 Z n. 
 
 1=1 
 
 • 1 k k b -b. .U. x k b. "] 
 
 z U + Z -^ ^ + I ± ) + Z ^ I (3.16) 
 1 i=l 1=1 i V i / i=l 1 J 
 
 Moreover „ , 
 
 , Z n. b. 
 
 1 k 1=1 X 1 
 
 - ^ U <— co- (3 .1 7 ) 
 
 1 1=1 1 
 
 and 
 
 U. 
 
 (r-i + I.) = n. co (3-18) 
 
 b. i i r w / 
 
 i 
 
 To bound the third term in the right hand side of eq. (3-l6), we note that 
 if there is no idle period in a processor of speed b. then 
 
 I - M-(T ^ ) 
 and 
 
 V 1 " b i 
 
27 
 
 Hence 
 
 k b. 
 
 Z ri I. < max 
 i=l 1 "" Ki<k 
 
 k ^ ^(T r+1 ) | V^-1) ,(T r+1 ) 
 
 b 
 1=1 1 
 
 & 
 
 & 
 
 L i^ 
 
 = max 
 KKk 
 
 Z n b. 
 i=l i X 
 
 - i 
 
 «Vl> 
 
 Z n. b. - 1 
 1=1 
 
 l l 
 
 b. 
 
 "'W 
 
 Thus eq. (3-l6) becomes 
 
 co < — = 
 r — k 
 
 2 n. 
 i=l X 
 
 Z n. b. . , , 
 
 . , 11 k b -b. 
 
 i=l . 1 l 
 
 ■ co' + Z — : n. co 
 
 1 i=l 1 
 
 Z n. b.- 1 
 i=l X 1 
 
 
 (3-19) 
 
 We note that in any schedule, there must be 
 
 r+1 
 
 Z njb.l 
 l i 
 
 i=l 
 
 b^-Cfb^-l) 
 
 of the r+1 tasks T.,, T , . .,, T , being executed on a processor of speed b 
 for some I, < £ < k. Consequently, we have 
 
 w ' > ^ T r+ 1 } 
 
 r+1 
 
 Z n.[b." 
 i=l x X 
 
 r^i I^L- 1 
 
 (3-20) 
 
 for some g, = 1, 2, ..., k. Since 
 
 k 
 
 (r+1) |i(T , , ) < ( Z n.b. ) co' 
 r+1 — . , 11 
 i=l 
 
28 
 
 and eq. (3-20), we write 
 
 QuOr,-,) < 
 
 r+1' - 
 
 where 
 
 Q, = max 
 
 mm 
 
 Equation (3-19) becomes 
 
 •• 
 
 r+1 
 
 k 
 
 Z 
 
 i=l 
 
 n.[b.] 
 
 1 X 
 
 '1A !hl±) _-'- 
 
 b. b 4 |' k 
 
 Z n.b. 
 1=1 X X 
 
 co < — 
 r k 
 
 Z n. 
 i=l x 
 
 k 
 
 , , , Z n. b. - i 
 
 , k b n - b. . t l l 
 
 s n . b . HI + E _2^_i n . u + izi «L 
 
 i=l X X b l 1=1 b l X r \ Q 
 
 which simplifies to 
 
 Z n. b. oj < 
 . -, l i r — 
 i=l 
 
 ■ k 
 
 z 
 
 , n. b. + 
 i=l i l 
 
 k 
 
 Z n. b. 
 
 i=l X X 
 
 Q 
 
 . ±1 
 
 QJ 
 
 Q ' ' 
 
 That is 
 
 "Wi 1 
 
 Q Z n. b. 
 .-,11 
 i=l 
 
 For integer values of b. and Z n. b. divides r, the bound becomes 
 
 l . , l l 
 
 1=1 
 
 k 
 
 /., , n Z n.b. - b 
 co rb n + (b^+l) .,ii 1 
 r < 11 i=l 
 
 co' - k 
 
 rb_ + Z n.b. 
 1 i=1 xi 
 
 (3-21) 
 
 and is best possible. That the bound in eq. (3-21) is best possible can be 
 demonstrated by an example similar to the one shown in Figure 3-3. 
 
29 
 
 Another simple heruistic for assigning priorities to the tasks in 
 (fT, (j.) is to assign higher priorities to longer tasks. Let w denote the 
 completion time when a set of tasks {??, \x) is executed on the multiprocessor 
 system (P = (l, n; b, l) using a priority assignment according to the lengths 
 of the tasks. Let W be the completion time of an arbitrary schedule. We 
 have 
 Theorem 3.5 
 
 - t < ^b) 
 
 oj' - b+2 ' - 
 
 and 
 
 u , < 2?, D > 2 
 
 oo' - 2 
 
 Proof. Consider the priority-driven schedule corresponding to the priority 
 assignment according to the lengths of the tasks. Let U, and LL be the sums 
 of the execution times of the tasks run on the fast processor and the standard 
 processors, respectively. Let I, and I p denote the sums of the lengths of the 
 idle periods of the fast processor and the standard processors, respectively. 
 The completion time is 
 
 u = 57i [T + u 2 + I i + y (3-22) 
 
 We consider two cases : 
 
 Case 1. I = 0, that is, the fast processor is never idle. In 
 
 this case, 
 
 1 U i 
 oj = —. r-± t L + I 
 n+l ' b 2 2 J 
 
 [-TT^ + ^Co+^+ry (3-23) 
 
 n+l L b b v 2 2 J b 2- 
 
 t 
 
 Note that the case I, - I Q = needs not be of concern to us since 
 
 when I 
 
 , = I p = 0, we have co = co ' . 
 
30 
 where 
 
 i (U 1+ U 2 ) <^o>' (3-21+) 
 
 and 
 
 U 2 + I 2 = n w (3-25) 
 
 Let us note that if the fast processor executes only one job according to 
 
 this schedule, then co < co'. Therefore, we can assume that the fast processor 
 
 executes two or more jobs. Let T be the last job executed in the fast 
 
 processor and cp be the longest idle period among the idle periods in the slow 
 s 
 
 processors as illustrated in Figure 3-*+. Since 
 
 ^ (T r } >n(y.) 
 
 and 
 
 u(T r ) < w - u(cp s ) 
 
 We have 
 
 Moreover, 
 
 Hence 
 
 w > n(T ) + u(cp ) > (b+1) u(cp ) 
 
 *Hi(<P s ) > I 2 
 
 n oj > (b+1) I 2 
 
 Substituting this expression and eqs. (3-2*0 and (3-25) into eq. (3-23), we 
 
 obtain 
 
 . 1 r n+b , b-1 nco _ 
 
 - n+1 L b b b(b+l) J 
 
 which simplifies to 
 
 (o (b + l)(n + b) (3 _ 26) 
 
 co' - b(n+b+l) v ° ' 
 
31 
 
 ^ — i — h 
 
 T 
 r 
 
 n(T p )/b) 
 
 ^ 1- 
 
 ^\ v 
 
 n+1 
 
 (a) 
 Case 1: T is executed on the fast processor 
 
 T 
 
 r 
 
 H v 
 
 <t> 
 
 3 
 
 (b! 
 
 Case 2: T is executed on a slow processor 
 
 Figure 3-^ 
 
32 
 
 Case 2. 1^0, that is, at least one of the slow processors is never idle. 
 
 Suppose that a processor that is never idle is P.. We consider the two 
 
 J 
 oo 
 possibilities, b > 2 and b < 2. For b < 2, I > — implies that at most one 
 
 task is executed on processor P.. Consequently, the first task executed in 
 
 the fast processor is not the longest task which is a contradiction to the 
 
 assumption that higher priorities are assigned to longer tasks. Therefore, 
 
 for b < 2, I < |. Thus, eq. (3-22) becomes 
 
 u - m: £ <W + ¥ ( U 2 +I 2> + t + y 
 
 „ 1 r n+b , b-1 n-1 w 
 
 < — - [— — to' + — — n w + ^— (o + -] 
 - n+1 L b b b 2 J 
 
 That is 
 
 a) 2(n+b) 
 
 co T - b+2 
 
 Similarly, we claim that I, < — — w for b > 2. (if I_ > — f— oo, each 
 
 1 — b — lb 
 
 slow processor executes at most one job. Let T. be the task whose 
 
 execution time u(T. ) is equal oj. Let T be the longest task in 5^. Since 
 
 T is executed in the fast processor and 
 r 
 
 ■<V _ i 
 
 < =■ 00 
 
 b b 
 
 we have 
 
 u(T r ) < u(T ± ) 
 which is impossible. ) Therefore, 
 
 . 1 r n+b , b-1 n-1 b-1 _ 
 
 oo < — — - r-r— oo' + — — n oo + — — oo + — — oo] 
 
 — n+1 L b b b b J 
 
 or 
 
 - < n+1 ° 
 oo' - 2 
 
33 
 
 Combining case 1 and case 2, we have for b < 2 
 
 - < 2 (n+b) 
 
 to' - b+2 
 
 since for n > 1 and b < 2, 
 
 (b+l)(n+b) < 2 (n+b ) 
 b(n+b+l) - b+2 
 
 Similarly, we have for b > 2 
 
 w n+b 
 go' - 2 
 
 Since for n > 1 and b > 2 
 
 b(n+b+l) - 2 u 
 
 3.3 Scheduling on Different Systems 
 
 We now extend our discussion to the case of executing a set of 
 tasks on different multiprocessor systems. Let {^T , \i, <) be a given set 
 of tasks and (P = (n., n , . . . , n ; b , b , ..., b ) and (P ' = (n', n', ..., 
 n'; b', b', . .., b') be two multiprocessor systems. Let co be the completion 
 
 iC -L 2 i£ 
 
 time when ( Zf , u, <) is executed on (P according to a priority-driven 
 schedule. Let to' be the completion time when {^ , u, <) is executed on 
 (P* according to an arbitrary schedule. Extending the result in Theorem 3.2, 
 we obtain 
 Theorem 3.6 
 
 u)' - h 
 k 
 
 £-> 11 . >~> . 
 .,11 
 1=1 
 
 b i 
 
 k 
 
 Z n. b. 
 i=l X X 
 
 k 
 
 Z n. b. 
 i=l X 1 
 
3^ 
 
 Moreover, the "bound is the best possible. 
 
 Proof. The proof of this theorem is similar to that of Theorem 3.2 and 
 will only be outlined here. As was discussed in the proof of Theorem 3.2, 
 the completion time of the priority- driven schedule is given .by eq. (3-6). 
 Again, we have 
 
 U. 
 
 r^- + I. = n. to 
 b. l l 
 
 l 
 
 and 
 
 k 
 
 Z U. 
 i=l : 
 
 k« 
 
 Z n' b! 
 i=l ± 1 
 
 < co' 
 
 (3-27) 
 
 Similar to eqs. (3-9) and (3-10), the sum of the lengths of the idle 
 periods of any processor must be such that 
 
 b. I. 
 k l 
 
 < b' w< 
 
 n. - i 
 
 i 
 
 (3-28) 
 
 and 
 
 k 
 b Z I. 
 k i=l X 
 
 < b* w* 
 
 - 1 
 
 Z n. - 1 
 i=l X 
 
 (3-29) 
 
 Thus, eq. (3-6) becomes 
 
 to < 
 
 k 
 
 Z n. 
 i=l x 
 
 "k r 
 
 Z n! b.' , , . 
 
 . -, i i k b -b. 
 
 i-l , „ 1 i 
 
 = W + Z — : — - n. OJ 
 
 b n . , b, l 
 
 1 i=l 1 
 
 k b.-b, b' 
 i k 1 
 
 Z n.-l w f + Z — r — t— ■ n. W 
 
 b b • -, i I ■ i b, b. l 
 
 1 k \i=l / i =1 1 k 
 
 b n b' 
 k 1 
 
35 
 
 That is 
 
 k k' k 
 
 b n Z n. b. oo < (h Z nj b.' + b' Z n. b. - b n b ' ) u> f 
 
 k.,11 - v k . n 1 1 I.-.11 k 1' 
 
 i=l i=l i=l 
 
 or 
 
 Z n.' b! 
 bj . . i i b 
 
 W <^ + i= 1 
 
 oo» - b k k 
 
 Z n. b. Z n. b 
 1=1 x 1 i=l x x 
 
 The example in Figure 3-5 demonstrates that this bound is best possible. □ 
 
 k* k 
 
 Let us note that if Z n! b.' = Z n. b., the bound in Theorem 3-6 
 
 i=l x 1 i=l x * 
 
 becomes 
 
 b' b' 
 
 ". < TT + 1 " T — ( 3 " 3 °) 
 
 oo' - b k 
 
 k Z n. b. 
 
 i=l x X 
 
 That is, for two multiprocessor systems with the same maximal throughput, 
 the bound is mainly a function of b' and b . Thus, if we hold the value 
 
 J_ K. 
 
 of b T and b fixed, we can trade a smaller number of fast processors for 
 a large number of slow processors or conversely without changing the bound 
 
 on the worst case performance of priority- driven scheduling algorithms. 
 
 k 
 
 For example, suppose b' is approximately equal to b and Z n. b. is 
 
 1 k i=1 i i 
 
 very large, then the ratio -, is approximately upper bounded by 2. In this 
 
 case, P is a system containing a small number of fast processors and (P ' 
 
 is a system containing a large number of slow processors. The result in (3-30) 
 
 says that any arbitrary priority- driven schedule for (P is never worse than 
 
 the best possible schedule for ^?' by a factor of 2. This indeed is a 
 
36 
 
 
 
 
 
 
 
 
 
 
 
 
 rH 
 
 
 OJ 
 
 - • • • «; 
 
 H 
 
 H 
 + 
 
 OJ 
 + 
 
 sf 
 
 - • • •+ ^ 
 rT 1 
 
 • • • «: 
 
 
 ..»•«; 
 
 
 © 
 
 
 O 
 
 © 
 
 H 
 
 a 
 
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 rH 
 
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 •°.r, « 
 
 © 
 
 
 
 
 
 r7 
 
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 OJ 
 
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 EH 
 
 
 + 
 H 
 S3 
 — EH — 
 
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 EH 
 
 rH 
 
 A! W II 
 
 •H 
 
 
 
 H 
 
 H _ 
 
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 111 , 112 
 
 b. Z n.b. 
 k i=1 
 
 e/b. 
 
 T Hn n T ll(n + 1)T 11( +2} T 11( } H(n 1+ n 2+ l) T^ T 2 $ 
 
 Vk b l\ b l b k ' b 2 b k ' b 2 b k b 2 b k ' b 3\ 
 
 T 1 Cr,f. 
 
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 — 1 — . — uj — i — h — i — i 
 
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 211 i 212 
 
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 221 212 
 
 e/b 2 
 
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 1 1 — M l I \-±-\ h-H 
 
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 T n' T 
 
 2 2 2n'2 
 
 T 2n'n T 2ni(n, + l) T2n 2 ( V 2) ^VV ^2 ^l +n 2 +l) ^ V^ \ + *Z 
 
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 k'n,' , k»n»t 
 
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 i=l 1 
 
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 N' = L n! 
 1-1 i 
 
 w' = b, L n.b. 
 k 1=1 A i 
 
 Figure 3 -5b 
 
38 
 
 quantitative confirmation of the fact that a processor of speed b is more 
 desirable than b slow processors of speed 1. 
 
 There is another interesting interpretation of the result in 
 Theorem 3.6. Suppose that we are to execute a set of tasks on the system 
 f) = (n , ru, . .., tl ; b , b , . .., b ). Instead of using all the 
 processors in the system, one might choose only to use the n fastest 
 processors. Let w„ denote the completion time when a set of tasks 
 ( 5^, \i, <) is executed on the n fastest processors according to a 
 priority- driven schedule and let W be the completion time when the set 
 (5^, u, <) is executed on (P according to an arbitrary schedule. According 
 to Theorem 3.6, we have 
 
 —r < r— Z n. b. + 1 
 
 oj' — n n b., . ., ix n n 
 1 1 i=l 1 
 
 To compare the upper bound of — with the upper bound of — given in 
 Theorem 3.2, we plot in Figure 3-6 the values of b for which the upper 
 
 bound of — , becomes larger, than that of —7 for fixed values of n n and 
 
 k 
 
 Z n. b.. Without loss of generality, assume that b n = 1. When b n is 
 .,11 k 1 
 
 i=l 
 
 larger than that shown in Figure 3-6 one is assured of an improvement on 
 the worst case performance of a priority-driven scheduling algorithm by 
 using only the fastest processors in the system. 
 
 Another interesting problem is to compare the execution of a set 
 of tasks on two different multiprocessor systems using a priority- driven 
 schedule specified by the same priority list. This is a realistic 
 situation when some of the processors in an existing system were replaced 
 by processors of different speed yet the priorities assigned to the tasks 
 to be executed were not changed. To illustrate the point, we investigate a 
 
39 
 
 10 12 14 
 
 k 
 
 MAXIMAL SYSTEM THROUGHPUT 2 n L b; 
 
 i = l 
 
 Figure 3-6 
 
ko 
 
 special case in which a set of tasks (u, u) is to be executed on two 
 multiprocessor systems (P = (n+1; 1) and (P* = (.1, n; b, l) according to 
 the same priority list L . Let w and co f denote the respective completion 
 times. 
 Theorem 3.7 
 
 The ratio -, is bounded by 
 
 CO 
 
 00 
 
 2(n+b) 
 n+2 
 
 b < 2 
 
 b > 2 
 
 Proof. Suppose that there are n+1 or fewer tasks in *f . Clearly, 
 
 " = n(T x ) 
 and 
 
 W >^ u(T i ) 
 
 where T- is a task with the longest execution time. It follows that 
 
 w 
 
 -. < b 
 
 U) 1 — 
 
 Suppose that there are more than n+1 tasks in J . Let U denote the 
 sum of the execution times of the tasks in 7 and let I denote the sum of 
 the lengths of idle periods of all processors when {tF, u) is executed on 
 the system (P = (n+1, l). Clearly, 
 
 CO 
 
 = FT1 ^ U+I ) (3-3D 
 
 Let T be the last task on a processor which is never idle in the time 
 interval (0, co). The length of the idle period of any processor must be 
 equal to or less than |i(T ). Hence 
 
I < n ti(T r ) 
 
 But 
 
 U 
 
 H 
 
 (T ) < 
 
 "r / - n + 2 
 Combining these two inequalities with eq. (30), we have 
 
 , 1 /tt nU \ 
 
 U) < (U + ;r.) 
 
 - n + 1 v n+2' 
 
 nU 
 
 n + 2 
 
 Since 
 
 go 
 the ratio — , is bound by 
 
 <*• > u 
 
 - n + b 
 
 Since 
 
 u 2 ( n +b ) 
 u» - n + 2 
 
 2 (n+b) 
 
 1 < -, < 2 
 -co' — 
 
 for any priority list L and the bound is best possible. 
 
 iH 
 
 for b > 2, the theorem follows immediately. □ 
 
 Corollary 1. For the case of b = 2 and n = 2, we have 
 
k2 
 
 k. PERFORMANCE OF PREEMPTIVE SCHEDULING ALGORITHMS 
 
 In this section, we study algorithms which produce preemptive 
 schedules with minimum completion time. By a preemptive schedule, we mean 
 one in which the execution of a task may he interrupted before its 
 completion. We assume that the cost associated with task preemption is 
 negligible throughout our discussion. This assumption is justified in 
 systems where task preemption does not require the reloading and removal 
 of tasks in and out of the main memory. 
 
 k.l Performance Improvement Obtainable by Preemptive Scheduling 
 
 The bound on improvement in completion time of a set of tasks 
 executed on a system (P = (n , n , . .., ri ; b , b , . .., b ) according to 
 a preemptive schedule over that using an arbitrary priority-driven schedule 
 can be obtained directly from Theorem 3.2. Indeed, this theorem can be 
 restated as 
 Theorem ^t-.l 
 
 Let w be the completion time when a set of tasks {^7 , u, <) is 
 executed on the system (P = (n.,, n , . .., ri ; b , b ? , ..., b ) according 
 to a priority-driven schedule and w be the completion time when the set 
 is executed on (P according to an optimal preemptive schedule. Then 
 
 w t> t b n b_, 
 
 - co — "b k 
 
 P k Z n. b. 
 i=l X 1 
 
 where b > b > ...>b . Moreover, the bound is best possible, 
 
J+3 
 
 As noted in Section 3> for systems of large maximal throughput 
 
 k b 
 
 ( Z n. b. ), this bound approaches — + 1. Hence, when the speed of the 
 i=l k 
 
 fastest processor in the system is very large compared to that of the 
 slowest processor, significant amount of reduction in completion time of 
 the set {7 ', u, <) can be achieved by allowing task preemption. 
 
 Let us note again, that in a priority-driven schedule, processors 
 are never idle when there are tasks ready to be executed. We expect the 
 improvement in completion time attained by preemption to be less for 
 non-preemptive schedules in which processors are allowed to idle when there 
 are executable tasks. 
 
 k.2 Performance of Preemptive Schedule for Independent Tasks 
 
 Let us for the moment consider the special case in which the 
 precedence relation < over the set of tasks is empty. That is, the tasks 
 in Z7* = {T,, T , ..., T } are independent. Suppose that 
 u(T ) > u(T ) > u(T ) ... > u(T ). Let w denote the completion 
 time of the set (&" , \i) when it is executed on the system 0-* = (1, n; b, l) 
 according to an optimal preemptive schedule. We have 
 
 Lemma k . 1 
 
 oj > max j max 
 P- 
 
 f 
 
 r J 
 
 n<V 
 
 ) max E l „ ±_ 
 
 ^l<j<n+lLi=l b+j-1 )> /^ n+b 
 
 m n(T, ) 
 
 Proof. It is clear that 
 
 m u(T ) 
 
 w x, ; 2 ±- 
 
 P- i=1 n + b 
 
kh 
 
 since no schedule can have a completion time shorter than the one which keeps 
 all of the processors busy. To show that 
 
 w > max 
 
 P l<j<n+l 
 
 ■ i ,(v 
 
 _i=l b+j-l„ 
 
 let us look at the individual terms inside the brackets. They are 
 
 u(T 1 ) u(T 1 )+u(T 2 ) u(T 1 )+u(T 2 )+u(T 3 ) ud-^+u^ )+. . .+u(T n+1 ) 
 — b > b+1 ' b+2 > •"> b+n 
 
 is equal to the time required to execute task T-, on the fast 
 
 — - _ ~^^~ . ~~ *.~^~.~ ~ -— ^ 1 
 
 processor, w is equal to or larger than this quantity since there is 
 
 no way to complete the execution of T., in shorter time. Similarly, 
 
 t— r [u(T-.) + |_i(T p )] is equal to the shortest possible time to complete 
 
 the execution of tasks T-, and Tp and , . j [|i(T 1 ) + u(Tp) + ... + \i(T. )] 
 
 is equal to the shortest possible time to complete the execution of tasks 
 
 T, , T , . .., T. . Clearly, w is lower bounded by the maximum of these 
 
 quantities since no schedule can have a completion time shorter than the 
 
 time required to execute the i longest task on i processors. □ 
 
 Indeed, the lower bound on the optimal completion time in Lemma h.l. 
 
 can be achieved. In other words, we have 
 Theorem k.2. 
 
 co = max / max 
 
 ^ l<j<n+l 
 
 Li=l b+j-U 
 
 i=l n+b J 
 
 Proof. We shall prove the theorem by finding a schedule for the tasks 
 (fT, u) on the system (P = (l, n; b, l) whose completion time is equal to 
 a) given by eq. (U-l). We consider the three cases: 
 
 We note that in any valid schedule, a task cannot be assigned to two 
 different processors simultaneously at any time. 
 
^5 
 
 Case 1. 
 
 m n(T ) j n(T ) 
 
 Z J" max Z . . = (U-2) 
 
 n+b - . ^._ -, . -, b+n-1 
 i=l i<o<n+l i=l d 
 
 In this case, we want to find a schedule whose completion time is equal to 
 
 m |i(T ) 
 i=l 
 
 To do so, let us suppose that u(T. ) < w for all i = 1, 2, . .., m. Let 
 
 p be an integer which is such that 
 
 P x -1 P-l 
 
 E = ^ T i)< W P<^. Z n ^ T i) 
 i=l i=l 
 
 We assign the tasks T-., T p , . .., T -, to the processor P, . Let 
 
 T, = r Z u(T.). We assign the task T to the processor P, in the time 
 1 b . , l p, 1 
 
 1=1 ^1 
 
 interval (t,, w p ) and to the processor P p in the time interval (0, t-J ) 
 where 
 
 Ti-H(^ ) - b( VTl ) 
 
 The tasks T -, T p , . .., T are assigned to the processor P 
 
 P-i |"-L P-i" 1 "^ P-i 'Pp — -L £- 
 
 p is such that 
 
 where 
 
 P -l p. 
 
 t' + Z u(T J .j)<w T3 <T' f Z u(T .,) 
 
 _. -, P n+l — P 1 • . P,+l 
 
 ^ H(T .)< a) <T» *■ Z 
 i=l V 1 P X i=l 
 
 Let 
 
 P 2 -l 
 
 T - T« + Z u(T ) 
 
 i=l p l 
 
 and 
 
ke 
 
 T 2 = ^ T p 1+ p 2 >- <VV 
 
 The task |_l(T ) is assigned to the processor P in the time interval 
 p l p 2 d 
 
 (^2> w p) and "to the processor p in the time interval (0, t'). Similarly, 
 we let the integer p, be such that 
 
 where 
 
 T„ = t' + E u(T ^ . ) 
 and 
 
 We assign the tasks T ,, T , . . . , T to thp 
 
 P-L+. • -P^_! +1 Pl +P 2 +P i-1 +2 ' p^.. . tp^_! t0 tiie 
 
 processor P and the task T to the processor P. in the time 
 i> p l l> & 
 
 interval (i g > w p ) and to the processor P„ 1 in the time interval (0, t«). 
 
 We proceed in this manner until the m tasks are all assigned. The 
 
 resultant schedule is as shown in Figure k-1 where T is completed at the 
 
 time w^ on the processor P _ . 
 P ntl 
 
 Suppose that the execution times of some of the tasks are larger 
 than w In particular, suppose that u(T-,) > u(Tp) > ... > \x(T ) > co 
 
 We now show that 
 
 m 
 w p < Z p.(T i ) < hw p 
 i=n+l 
 
hi 
 
>+8 
 
 Since 
 
 and 
 
 we obtain 
 
 That 
 
 m 
 
 Z Ji(T.) 
 
 i=i - = CO 
 
 b+n P 
 
 n 
 
 Z H(T.) > nw 
 i=l 
 
 m 
 
 Z u(T ) <bw 
 i=n+l 
 
 m 
 co < z [i{l ) 
 
 r i=n+l x 
 
 follows from eq. (^-2), that is, 
 
 n m 
 
 Z n(T.) Z n(T ) 
 
 i=l i=l 
 
 b+n-1 - " b+n W P 
 
 m 
 
 If we let T' , be a new task such that u(T' .. ) = Z u(T. ), we can 
 n+1 ^ n+l y . ., ^ i 
 
 i=n+l 
 
 schedule the n+1 tasks T n , T„, .... T , T' n as shown in Figure h-2. where 
 
 1 2' 7 n n+1 
 
 M-(T ± ) - w p 
 Aj_ = ^z± i = 1* 2 j •••> n 
 
 and 
 
 ^ v n+l y P 
 \+l ~ b-1 
 
 Since 
 
 bA. + u _ - A. = n(T. ) i = 1, 2, ..., n 
 
 l P l^i 
 
h9 
 
 <f 
 
 <r 
 
 ■r 
 
 3* 
 
 <r 
 
 
 <f 
 
 <^ 
 
 <f 
 
 <f 
 
 i 
 
50 
 and 
 
 b Vl + W P - Vl = ^ T n + 1> 
 
 and 
 
 n+1 
 
 E A. = w„ 
 . -, i P 
 
 the schedule in Figure k-2 is a valid one. 
 
 Suppose that there is only j tasks with execution time longer 
 
 than Up, and j < n. That is, ^(T-) > u(T„) > ... > u(T.) > u but 
 
 u(T. t) < u In this case, we let T! , be a new task such that 
 ^ v 3+l J - P 0+1 
 
 m 
 u(T' ) = Z u(T.). We can use the schedule shown in Figure h-3> where 
 ' J i=j+l X 
 
 H(T ± ) - ^ p 
 \ = £T[ i = 1* 2 , . . ., j 
 
 and 
 
 3 t 
 
 3+1 P i=1 i 
 
 Since 
 
 b^ + w p - /\ ± = n(T ± ) i = 1, 2, ..., j 
 
 f 
 This can be done because . 
 
 Z u(T ) - ju 
 i=l 
 A j+1 = W P " b^l 
 
 J 
 
 (b+j-1) Up - Z n(T ± ) 
 
 _ i=l 
 
 b-1 
 
 3 
 
 2 [i(T ) 
 
 i=l 
 
 follows from u > — . — 
 
 P - b+j-1 
 
 > 
 
51 
 
 <?> 
 
 3 ft 
 
52 
 
 and 
 
 V + w p"V + (n " J) w p 
 
 = (la-1) A. +1 + (n-d+1) w p 
 
 2 n(T ± ) - jw p 
 = (b-1) [«p - — ^1 3 + ( n "J +1 ) w t 
 
 = (b+n) w p - Z n(T ± ) 
 i=l 
 
 m 
 
 Z H(T.) 
 i=j+l X 
 
 the schedule in Figure k~3 is a valid one, 
 Case 2. 
 
 Suppose that 
 
 hCf-l) 
 
 "P ~ ~" b 
 
 w. 
 
 From eq. (k-l), we note that 
 
 (l^) + n(T 2 ) u(T 1 ) 
 
 b+1 
 
 < 
 
 It follows that 
 
 ■ (To) 
 
 H(TJ 
 
 2' - b 
 
 (M) 
 
 Also, 
 
 m u(T. ) 
 Z — ±- 
 i=2 n 
 
 m u(T. ) ,, 
 x n+b 
 
 i=2 
 
 n+b n 
 
 n+b 
 n 
 
 m u(T ) 
 
 Z 
 
 \l(\) 
 
 Li=l 
 
 n+b n+b 
 
53 
 
 n+b 
 — n 
 
 ' k i(T 1 ) k i(T 1 )' 
 
 - b 
 
 n+b _ 
 
 h(T-l) 
 
 (WO 
 
 Therefore, the schedule shown in Figure k-h can be used. In this schedule, 
 
 T-, is assigned to the fast processor P and the. remaining tasks are 
 
 assigned to P p , V , ..., P , . The relation in (h-3) and (h-k) guarantee 
 
 the validity of the schedule. 
 
 Case 3. 
 
 3 H(T.) 
 
 for some j, 1 < j < n+1. It is clear that 
 
 ^(T.)<03 p 
 
 1=1, 2, . .., j 
 
 0*-5) 
 
 However, since 
 
 which can be rewritten as 
 
 j-l fi(T.) 
 
 j-l n(T.) 
 
 ifi b+ J- 2 
 
 b+j-l 
 
 b+j-2 
 
 
 < «p 
 
 (i(T.) 
 
 oj - 
 
 P b+j-1 
 
 we have 
 
 w p < u(Tj) 
 
 (U-6) 
 
^ 
 
 ef 
 
 ef 
 
 b/" 
 
On the other hand it follows from 
 
 55 
 
 0+1 n(T.) 
 
 i=l 
 
 b+o ~ P 
 
 (That is, 
 
 b+j-1 
 b+j 
 
 3 
 
 z 
 
 Li=l 
 
 ^ T i) 
 n+b 
 
 b+,1-1 
 
 < 
 
 — < 
 
 J 
 
 P 
 
 that 
 
 Moreover, from 
 
 H<Vl> ^ «p 
 
 m |i(T. ) 
 
 E — < w 
 
 . _ n+b - P 
 i=l 
 
 (W7) 
 
 we have 
 
 b+j-1 
 n+b 
 
 J [i 
 
 (Tj 
 
 i=l 
 
 b+j-1 
 
 n-j+1 
 n+b 
 
 m 
 Z 
 
 i=J+l 
 
 H(T. ) 
 
 — r=r < w t, 
 
 n-j+1 - P 
 
 or 
 
 v • n . , m ^(T. ) 
 
 EStl u + n=fl+l z ULil < a, 
 
 n+b P n+b . . . n-j+1 - P 
 
 1=0+1 
 
 which simplifies to 
 
 m 
 i-J+1 
 
 n-j+1 
 
 f,i 
 
 (h-Q) 
 
 Therefore, the schedule shown in Figure U-5 can be used. In this schedule, 
 the tasks T,, T , ..., T. are assigned to the first j processors. The 
 
 1. d. J 
 
 execution time of the portion of T. assigned to P. , A., is given by 
 
 J P 
 
 n(T. ) - t 
 
 *1 
 
 b-1 
 
56 
 
 <r 
 
 <f 
 
 <^ 
 
 <^ 
 
57 
 
 for i = 1, 2, ..., j. The remaining tasks are assigned to processors 
 
 P. ■ _, P. , .... P , . The relations in ( h r-^)-(k-8) and 
 3+1 J+2 n+1 
 
 Z 
 
 i=l 
 
 E V U I 
 
 guarantee the validity of the schedule. 
 
 The result in Theorem h.2 can he extended immediately. Let oj 
 denote the completion time of a set of tasks (^, u) executed on the system 
 (P = (n , n , ..., n ; b , b f ..., b ) according to an optimal preemptive 
 schedule. Similar to Theorem k.2, we have 
 
 co - max 
 P 
 
 max 
 
 Z u(T ) 
 i=l x 
 
 l<j<n 1 +n 2 +...n k £ 
 
 i+1 
 
 m 
 
 2 u(T ) 
 
 k 
 
 Z n. b 
 
 i=l 
 
 i i 
 
 where i is an integer such that 
 
 I £+1 
 
 Z n. < j < Z n. 
 
 i=l 
 
 i=l 
 
58 
 
 5. SCHEDULING ALGORITHMS TO MINIMIZE 
 MEAN FLOW TIME 
 
 In this section, we study the performance of scheduling algorithms 
 using mean flow time as the criterion for comparison. We shall consider 
 only the special case in which the precedence relation < over the set of 
 tasks is empty. Algorithms producing schedules with minimal mean flow 
 times will be described. A bound on the minimum mean flow times for 
 different multiprocessor systems will be derived. These bounds will also 
 provide us with information concerning the relative merit of different 
 multiprocessor systems. 
 
 5.1 Optimal Scheduling Algorithms 
 
 We present now an algorithm for constructing schedules with 
 
 minimum mean flow time when a set of independent tasks ( ?7 , u) is executed 
 
 on the system (P = (l, n; b, l). We note that the mean flow time of a 
 
 schedule, t, defined in Sec. 2,2 can be written as 
 
 m 
 
 t = Z w u(T ) (5-1) 
 
 i=l 
 
 m. 
 
 where w. = — if task T. is executed on the fast processor and is followed 
 x b i 
 
 by m -1 tasks in that processor and w. = m. if task T. is executed on a 
 
 i x x x 
 
 standard processor and is followed by m.-l tasks in that processor. 
 
 Hence, a schedule that minimizes the mean flow time is one that minimizes 
 
 the weighted sum, in eq. (5-1) overall valid sets of weights {w,, w~, ..., w } 
 
 Suppose that the execution times of T, , T„, ..., T are such that 
 l_i(T n ) > u(T ) ... > u(T ). Let r be an integer such that 
 
 [rbj + rn < m < [ (r-i-l)bj + (r+l) n (5-2) 
 
59 
 
 where [xj denotes the integer part of x. We have, 
 Theorem 5«1 
 
 When a set of tasks {f7, u) is executed in the system (P = (l, n; b, l), 
 a set of weights {w,, w p , . .., w } that minimizes the mean flow time is 
 
 1 
 b 
 
 w. = 
 
 x-sn 
 
 v 
 
 i = 1, 2, ..., [bj 
 
 i = [bj + 1, [bj + 2, ..., [bj + n 
 
 i = [sbj + sn + 1, [sbj + sn + 2, . .., |_(s+l)bj + sn 
 
 and s = 1, 2, . .., r-1 (5-3) 
 
 i = [sbj + (s-1) n + 1, [sbj + (s-l) n + 2, ..., 
 [sbj + sn, and s = 2, 3, •••> ^ 
 
 and 
 
 w. 
 
 i-rn 
 
 i = |_rbj + rn + 1, [rbj + rn + 2, ..., 
 
 in 
 
 if m - [rbj - rn < [(r+l)bj - [rbj, and 
 
 i-rn 
 
 i = [rbj + rn + 1, [rbj + rn + 2, ..., [ (r+l)bj + rn 
 
 w. = 
 l 
 
 r+1 i = [(r+l)bj + rn + 1, [ (r+l)bj + rn + 2, ..., m 
 
 (5-4) 
 
 otherwise. 
 Proof. 
 
 We show that there is a schedule for which the weights w. are 
 
 l 
 
 that given by eqs. (5-3) and (5-*0. To construct this schedule, we 
 partition the tasks in *7 into r+1 blocks, B,, B p , ..., B where 
 
6o 
 B l= {T 1> T 2' •••' T M' T [bJ+l' T LbJ+2' ••" T Lbj+n 3 
 
 B 2 = {T [bJ+n+l' T [bj+n+2> •" T [2bJ+n' T [2bJ+n+l' "" T [2bJ+2n } 
 
 B r = {T [(r-l)bJ+(r-l)n+l' T [ (r-l)b]+(r-l)n+2' ' " T [rbJ + (r-l)n' 
 
 T LrbJ+(r-l)n+l^ '" T LrbJ + (r-l)n+n-l' T [rb]+rn 3 
 
 J r+1 = {T [rbj+rn+l' T |.rbJ+rn+2' •"' T [ (r+l)bj+rn' T [ (r+l)bj+rn+l' 
 
 
 In the schedule, the task ^(r+OObJ+rn' T [ (r+l)bj+rn-l' ~" T [rbj+rn+l in 
 
 B are assigned to the fast processor in that order and the remaining tasks 
 r+1 
 
 in B are assigned to any m-l (r+l)bj-rn of the standard processors one in each 
 r+1 
 
 processor. Next, tasks ^ rb j + (r _ l)n , T [ rb j + ( r _i) n .r •" T [ (r-l)bj+(r-l)n+l 
 
 in B are assigned to the fast processor in that order while the remaining 
 r 
 
 n tasks in B are assigned to the n standard processors one in each processor, 
 r 
 
 This procedure is repeated until the tasks Ti fe i, T j -(-, ! _i/ '"> T 2' T l in B l 
 
 are assigned to the fast processor and the remaining n tasks in B_ L are 
 assigned to the n standard processors. The resultant schedule is shown in 
 Figure 5-la. We note that when b=l, this schedule is just a shortest 
 execution time first schedule. 
 
 To show that the set of weight w. in eqs. (5-3) and (5-k) indeed 
 minimizes the mean flow time -x, we note that for any q tasks assigned to the 
 fast processor, the associated weights are 
 
 f We assume that m-[rbj-rn > [b(r+l)J-[brJ . Whenwe have m-LrbJ-rn < [ (r+l)bj-|_rbj ; 
 
 the tasks in B n are assigned to the fast processor. The resultant schedule 
 r+1 
 
 is as shown in Figure 5-lb. 
 
61 
 
 -a 
 
 EH 
 
 
 .a 
 
 EH 
 
 © 
 
 
 1111 1 
 
62 
 
 V b ' •••' b' b 
 
 Similarly, for any q tasks assigned to a standard processor, the associated 
 weights are 
 
 Hence, the smallest m weights are 
 
 12 LbJ [bj+l [b]+2 \2bj_ 
 
 h' h' ' ' ' ' b ■* . > '"> ' >) ' >) > *••> -u > *-) ^t ' • • > ^> 
 
 . . . , r, r, . . . , r, 
 
 n n 
 
 l rb J +1 !■*>]-* I (r*D* 1 (r+1) (r+1) 
 
 b ' b ' "•' b 
 
 n 
 which are exactly that given by eqs. (5-3) and (5-*+). The weighted sum 
 in eq. (5-1) is minimized follows from the assignment of smaller weights 
 to longer tasks. 
 
 The result in Theorem 5.1 can be generalized to the case when the 
 set of tasks is executed on the system (P = (n, , n„, ..., n ; b , b_, ..., b ), 
 Suppose that b > b > . ,.>b , let 
 
 b. 
 d, = r± i = 1, 2, .,., k-1 (5-6) 
 
 - 1 - Vl 
 
 We have 
 
 Theorem 5»2 
 
 When a set of tasks (fT, \i) is executed on the system 
 
 P = (n , n , . .., n^j b^ b g , . .., b^ the weights w^ w g , ..., w ffi 
 
 that minimize the mean flow times are given by 
 
n l n l [ 
 
 63 
 
 b ' b ' "*' b ' b ' b ' •"■' b ' ***' b ' b ' '"' b ' 
 
 D l D l D l 1 1 1 D l u l D l 
 
 n 2 X - 1 
 
 £ A ^ UJ [dj+l [dj+l [dJ+2 [dJ+2 d 1+ 2 
 
 y bj' ••" bT/ ~b^~' b x ' "••' b 1 ' b 1 ' b x ' "•' ~b^~' 
 
 *1 
 
 A 
 
 , A / ^ 
 
 [2dJ [2d 1 J [2d 1 J ■ 2 
 
 ' ~~ b ' ~~ b ' * * * ' ~~ b ' b~~ ' ~ ' ' ' ' ' b~~ ' 
 1 1 1 2 2 2 
 
 n l r^ 
 
 [d 2 d x j [d 2 d x j Ld 2 d x j [d 2 j [d 2 j [d 2 j 
 
 b i ' \ ' "" \ ' V V "" V 
 
 "3 
 
 b ' b ' ' "' b ' 
 
 3 3 3 
 
 (5-6) 
 
 The proof of this theorem being similar to that of Theorem 5.1 is 
 not repeated here. 
 
 5.2 Comparison of Systems 
 
 We observed in Sec. 3.^- that on the basis of the worst case 
 performance of priority-driven schedules, a processor of speed b is more 
 desirable than b processors of speed 1. We now show that the same 
 conclusion can be reached when the performance of two multiprocessor 
 systems are compared in the basis of minimum mean flow time. Let us 
 consider two multiprocessor systems (P = (l, n; b, l) and tf> x = (n+b; 1) 
 where b is an integer. Let t and t' denote the minimum mean flow times 
 
6k 
 
 when a set of tasks {ZT , u) is executed on the systems (P and <^>', 
 respectively. Again, we suppose that u(T-.) > u(T ) > ... > u(T ). We have 
 Theorem 5.3 
 
 b-1 
 
 ,. .,.- m 
 
 t<t- - — — b 
 
 J "<V 
 
 the bound is best possible when — — is an integer and u(T-, ) = u(T p ) 
 
 u(T ). In this case 
 r m 
 
 b-1 
 T - T 2 
 
 where co is the minimum completion time of the tasks ( ^ , u) on the 
 
 system (P . 
 
 Proof. 
 
 Consider the optimal schedule for the set of tasks (^ u) on 
 the system (P = (l, n; b, l) shown in Figure 5-2a. The mean flow time t 
 can be written as 
 
 r+1 
 
 t = Z f. (5-7) 
 
 i=l 
 
 where 
 
 f. -i f[(i-l)b + l] .(T (:Ul)(b+n)+1 ) + [(1-1)^2] ^ (i _ l)(b+n)+2 ) 
 
 + ... + i b H(l 11)+(i . l)n )) + i[^I lb+(1 .i) n+1 ) + ^ T i D+ (i.l)n +2 ) 
 
 and 
 
65 
 
 (i-l)b tasks on P n 
 
 "(i-3)(b+n)+2 ' 
 
 l T ib+(i-l)n V T (i-l)(b+n)+J 
 
 H h-V-P — t 
 
 i! 
 
 T T 
 
 T ib+(i-l)n+l 
 
 Ht—4 — h 
 
 b+1 
 
 I « 
 
 ib+(i-l)n+2 
 
 I I * I 
 
 b+2 
 
 T. 
 
 (b+n) 
 
 ■H-^ 
 
 b+n 
 
 H— : 1 
 
 n+1 
 
 y — 
 
 Block B. 
 
 (i-1) tasks on each of 
 the standard processors 
 
 (a) 
 
 (i-1) tasks on each processor 
 K 
 
 '(i-l)(b+n)+l 
 
 ■h-* 1" 
 
 ■(i-l)(b+n)+2 
 
 f-^ 1" 
 
 1 
 
 ■(l-l)(b+n)+3 
 
 *-^ V 
 
 i(b+n] 
 
 ■*M — H 
 
 b+n 
 
 b+n-1 
 
 b+n 
 
 Block B. 
 
 (b) 
 
 Figure 5-2 
 
66 
 
 r 
 
 \ [[rb+1] ^(T r(b+n)+1 ) + (rb+2) ^(T r(b+n)+2 ) + ... + (m-rn) ^TJ) 
 
 f = 
 r 
 
 if m-r(n+b) < b 
 
 I \ {(rb + l) ^(T r(b+n)+1 + (rb + 2) n(T r(b+n)+2 ) + 
 
 + (r + D b u(T (r+l)b+rn )} 
 
 + (r+l) ^(T (r+l)b+rn+1 ) + 
 
 . + (r+l) n(T ) 
 
 if m-r(n+b) > b 
 
 (5-9) 
 
 Similarly, an optimal schedule for the set of tasks {Zf , u) on the system (p • 
 is shown in Figure 5-2b. Let 
 
 r+l 
 t* = E f.* 
 i=l x 
 
 where 
 
 f i - 1 [^ T (i-l)(n + b) + l) + ^ T (i-l)(n + b) +2 ) + ••• + ^ T i(nA)> ] 
 
 1 — X^ £- ^ • • • y X 
 
 and 
 
 f r+l = r ^ (T r(n+b)+l ) + ^ (T r(n+b)+2 
 
 ) + • 
 
 • + ^ T m^ 
 
 The 
 
 expressions for f. in eqs. (5-8) and (5-9) can be written as 
 
 f, - f. - £ d-i) ^ (i . 1)( , +n)+j ) i = 1, 2, ..., r (5-10) 
 
67 
 
 Similarly, 
 
 a 
 
 f ^ = f \i ~ 2 (1-J-) u(T - v .) (5-11) 
 
 r+1 r+1 . -. b' ^ r(b+n)+n 
 3=1 
 
 where 
 
 a = min[b, m-r(n+b)] 
 
 Substituting eqs. (5-10) and (5-11) into eq. (5-7)> we obtain 
 
 r b a . 
 
 t = t' - Z Z (l-£) n(T/. lW _ x .) - Z (1-J) u(T ,. v .) 
 
 .-,.-. b 7 (i-l)(b+n)+3 y . . b y ^ v r(b+n)+j ' 
 1=1 3=1 3=1 ' u 
 
 Since u(T ) < u(T. ) for i = 1, 2, ..., m-1, we have 
 
 r b CK 
 
 T < T* - Z Z (l-£) u(T ) - Z (l-£) u(T ) 
 — . , . -, b ^ m . -, b ^ m 
 i=l j-l 3=1 
 
 < t' - r ^ n(T m ) 
 But 
 
 I m I 
 r = — r 
 L n+b J 
 
 Therefore 
 
 T <T ' " T L-?hJ ^(T ) 
 — 2 L n+b J ^ m 
 
 ^g is an integer and ^(Tj = nCT^) = ... = nC^), 
 
 m u(T ) 
 m 
 
 ; = w^ 
 
 n+b 
 
 and 
 
 when 
 
68 
 
 ACKNOWLEDGEMENT 
 
 The authors wish to thank Mr. Henry Shapiro for his suggestions 
 and comments. 
 
 REFERENCES 
 
 [1] E. G. Coffman, Jr. and R. L. Graham, "Optimal Scheduling for 
 Two-Processor Systems, " Acta Information , Vol. 1, No. 3> 1972, 
 pp. 200-213. 
 
 [2] T. C. Hu, "Parallel Sequencing and Assembly Line Problems, " 
 Operations Research , Vol. 9, No. 6, I96I, pp. 8^1-8i+8. 
 
 [3] R. R. Muntz and E. G. Coffman, Jr., "Preemptive Scheduling of 
 Real-Time Tasks on Multiprocessor Systems, " Journal of the ACM , 
 Vol. 17, No. 2, 1970, pp. 32^-338. Also, "Optimal Preemptive 
 Scheduling on Two-Processor Systems, " IEEE Transaction on Computers , 
 Vol. e-18, No. 11, 1969, pp. 10li+-1020. 
 
 [h] R. L. Graham, "Bounds on Multiprocessing Timing Anomalies, " 
 
 SIAM J. on Applied Math , Vol. 17, No. 2, 1969, pp. U16-U29. Also, 
 "Bounds for Certain Multiprocessing Anomalies, " Bell System Tech . 
 Journal , 1966, pp. I563-I58I. 
 
 [5] M. R. Garey and R. L. Graham, "Bounds for Multiprocessor Scheduling 
 with Resource Constraints, " (to appear). 
 
iLIOGRAPHIC DATA 
 
 1. Report No. 
 
 UIUCDCS-R-7^-632 
 
 3. Recipient's Accession No. 
 
 "itle and Subtitle 
 
 Bounds on Scheduling Algorithms for Heterogeneous 
 Computing Systems 
 
 5- Report Date 
 
 June I97J+ 
 
 iuthor(s) 
 
 Jane W. S. Iiu and C. L. Iiu 
 
 8. Performing Organization Rept. 
 No. 
 
 'erforming Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract/Grant No. 
 
 GJ-36265 
 GJ-^1538 
 
 Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 Supplementary Notes 
 
 Abstracts 
 
 The problem of job scheduling in a computing system containing 
 processors of different operation speeds is studied. In particular, bounds 
 on the worst case performance of same scheduling algorithms that can be 
 implemented easily are obtained. Such bounds also provide us with 
 information concerning the effect of the speeds of the processors and the 
 maxmal throughput of the system on the performance of these scheduling 
 algorithms. The trade-off between the speed and the number of processors 
 in the system is also discussed. Optimal scheduling algorithms which 
 produce preemptive schedules with minimal completion times and schedules 
 with minimal mean flow times for independent tasks are described. 
 
 Key Words and Document Analysis. 17a. Descriptors 
 
 multiprocessor scheduling, non-preemptive scheduling, 
 preemptive scheduling, priority-driven scheduling 
 
 Identifiers/Open-Ended Terms 
 
 • ( OSATI Field/Group 
 
 Availability Statement 
 
 IM NTIS-38 110-701 
 
 19. Security Class (This 
 Report) 
 
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 20. Security Class (This 
 
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 21. No. of Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7I 
 
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