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 UILU-ENG 77 1720 
 
 PERFORMANCE ANALYSIS OF MULTIPROCESSOR SYSTEMS 
 CONTAINING FUNCTIONAL DEDICATED PROCESSORS 
 
 by 
 
 Jane W. S. Liu and Chung L. Liu 
 
 May 1977 
 
 ia Library 
 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
K. 
 
 % 
 
UIUCDCS-R-77-835 
 
 PERFORMANCE ANALYSIS OF MULTIPROCESSOR SYSTEMS 
 CONTAINING FUNCTIONAL DEDICATED PROCESSORS 
 
 by 
 
 Jane W. S. Liu 
 
 Chung L. Liu 
 
 May 1977 
 
 Department of Computer Science 
 University of Illinois at Urbana-Champaign 
 Urbana, Illinois 61801 
 
 This work was supported by the National Science Foundation 
 under Grant NSFDCR 72-03740 NSFMCS 73-03408. 
 
 

 I 
 
 ii 
 
I . INTRODUCTION 
 
 In previous works on job scheduling, a multiprocessor system 
 was modeled as one containing functionally identical processors. 
 A task in a given set of tasks can be assigned to any one of the 
 processors in the system. The time required to complete a task may depend 
 only on the execution time of the task in the case where all processors 
 are identical or may also depend on the speed of the processor on which 
 the task is being processed in the case where processors have different 
 speeds. For such multiprocessor systems, efficient algorithms which 
 yield schedules with minimum completion time or mean flow times have been 
 found for some special cases [1-5]. Worst case performance bounds of many 
 heuristic suboptimal scheduling algorithms have also been obtained [6-8]. 
 
 In this report, we introduce two general models of multiprocessor 
 systems containing different types of processors. A task can be assigned 
 only to a processor of certain types. We described in Section II 
 a model corresponding to systems in which processors are functionally 
 dedicated [91. That is, a task can be executed only on a particular type 
 of processors. (For example, some processors are front end processors for 
 I/O functions, some processors are to perform program compilation, some 
 processors are designed specially for merging/sorting operations, and other 
 are designed for extensive numerical calculations.) Clearly, the model of 
 multiprocessor systems with identical processors is a special case of our 
 model. This model also includes the job shop problem in which there is 
 exactly one processor of each type. 
 
 In Section III, a more general model is presented. In this 
 general model, each type of functionally dedicated processors is further 
 
I 
 
 divided into subtypes with a partial ordering relation defined over the 
 processor subtypes. Thus, multiprocessor systems in which some pro- 
 cessors are functionally identical but have dedicated memories of differ- 
 ent sizes can be modeled. 
 
 r 
 
 K 
 
 :: 
 ;> 
 'i 
 
 i 
 
 f\ 
 
 Q 
 
 :: 
 ;> 
 
 > 
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II. MULTIPROCESSOR SYSTEMS WITH FUNCTIONALLY DEDICATED PROCESSORS 
 
 Consider a model of a multiprocessor system in which there 
 are r different types of functionally dedicated processors. We assume 
 in this section that processors of the same type are identical. We shall 
 refer to these r types of processors as type 1, type 2, ... , type r pro- 
 cessors. A multiprocessor system with m type 1 processors, m type 2 
 
 processors, ... , and m type r processors can be represented by the 
 
 r 
 
 ordered r-tuple P = (m, , m„ , ... , m ). We let m = 2 m.. 
 
 12 r . , j 
 
 3=1 
 
 Let T = {T n , T., ... , T } be a set of tasks to be executed on 
 12 n 
 
 a system P. A task T. is said to be a type j task if it can only be 
 
 executed on a type j processor. Formally, we define a function A from 
 
 T to the set {1, 2, ... , r} such that A(T.) is the type of task T.. 
 
 We denote the time required to complete a task T. (on a type j processor) 
 
 by u(T.) where u is a function from T to the reals. u(T.) shall be referred 
 
 to as the execution time of T.. 
 
 l 
 
 We suppose that there is a precedence relation < defined over 
 
 the set T. That T. precedes T. (or T. follows T.) is written as T. < T. 
 i 3 1 i i J 
 
 and means that the execution of T. cannot begin before the execution of T. 
 
 J i 
 
 is completed. A task is said to be executable at a certain time if the 
 tasks preceding it have been completed. Formally, a set of tasks are 
 represented by ordered quadruple (T, A, u, <) . We shall also use the 
 notation (T., A(T.), u(T.)) to represent to a particular task T. . 
 
 We can depict a set of Tasks (T, A, y, <) graphically as il- 
 lustrated by the example in Figure 1, where T , T~, and T,. are type 1 
 tasks with execution time 3, 4 and 2, respectively. T„ and T, are type 2 
 tasks with execution time 4 and 2, respectively. 
 
 We want to determine the performance of a class of scheduling 
 
algorithms in which processors are never left idle intentionally. These 
 algorithms are known as priority driven scheduling algorithms and can be 
 
 I 
 
 
 t: 
 
 :; 
 
 ! 
 
 : 
 > 
 
 ! 
 
 > 
 >■ 
 
 la 
 
 18 
 
 (T r 1, 3) 
 
 (T 2 , 1, 4) 
 
 (T 4 , 2, 2) 
 
 (T 3 , 2, 4) 
 
 (T 5 , 1, 2) 
 
 Figure 1. An example of a Set of Tasks (T, A, u, <) . 
 
described by the priorities assigned to the tasks. At any moment when a 
 type i processor is free, the task that has the highest priority among all 
 executable type i tasks is scheduled. For example, for the set of tasks in 
 Figure 2(a), the schedule in Figure 2 (b) is obtained according to the 
 priority list (T. , T , T„, T. , T c , T , , T.., T , T n ) in which tasks appear 
 in decreasing priorities. Schedules produced by priority driven scheduling 
 algorithms are known as priority driven schedules. 
 
 Let a) and to' denote the completion times of a set of tasks 
 (T, A, u, <) when executed on a system P = (m , m , ... , m ) according to 
 a priority-driven schedule and an arbitrary schedule, respectively. We 
 have 
 Theorem 1 
 
 - ,< 1 + r 
 
 or — 
 
 min/ 1 \ 
 
 (1) 
 
 Moreover, the bound is the best possible. 
 
 Proof : Let t. denote the sum of the execution times of all type j tasks 
 
 in (T, A, u, <) . (For example, for the set of tasks in Figure 2, t, = 5 -j 
 
 and t„ = 3-^-). Let (b(n n , n„ , ... , n ) denote the sum of idle times in 
 2 2 1 2 r 
 
 all processors during which there are n idle processors of type 1, n~ 
 idle processors of type 2, ... , and n idle processors of type r in a 
 priority-driven schedule. (In the example shown in Figure 2, 4>(1,0) =1, 
 ((.(2,0) =| + |=1, (j>(l,l) =2 + 2 = 4, <K0,0) = 0, 4>(0,1) = 0, and <j>(2,l) 
 = 0.) Clearly, the total idle time, $, is given by 
 
 m„ 
 
 m. 
 
 m 
 
 = 2 2 
 n^O n 2 =0 
 
 2 4)(n 1 , n 2 , ... , n r ) 
 
 n =0 
 r 
 
 (2) 
 
 with (fi (m , m , ... , m ) = 0. The completion time of the priority- 
 
 Note that <f>(m , m . 
 
 m ) = in a priority-driven schedule, 
 
V. 
 
 (T^ 1,1) 
 
 (T 2 , 1.1) 
 
 (T 3 , 2,1) 
 
 (T 6 ,l,f) 
 
 (T ? ,l,|) 
 (T Q ,2,1) 
 
 (T 9 ,2,f) 
 
 (T 4 , 1,1) 
 
 (T 5 ,l,l) 
 
 :; 
 
 ji 
 
 :: 
 
 Type 1 
 
 T l 
 
 T 
 2 
 
 ////////A 
 
 1 T 4 
 
 T 5 
 
 1 
 
 1 
 
 ////////A 
 
 1 
 
 1 
 
 'J 
 
 :> 
 
 Type 1 
 
 T 6 
 
 T 7 
 
 V/////////////A 
 
 3 
 
 2 
 
 3 
 2 
 
 Y/////////////A 
 
 Type 2 
 
 Figure 2. Example of a priority driven 
 schedule on a multiprocessor system con- 
 taining different types of processors. 
 
driven schedule to is given by 
 
 a) = - ( 2 t.+ *) 
 m j=l * 
 
 (3) 
 
 To find an upper bound to the total idle time, $, let us 
 examine the terms 
 
 m, 
 
 m. 
 
 I = s 
 
 n =1 n =1 
 
 m 
 
 n =1 
 r 
 
 ((>(n 1 , n 2 , 
 
 • n ) 
 r 
 
 in the sum in Eq . (2) . During the time periods corresponding to the 
 terms in the sum I, there is at least one idle processor among the pro- 
 cessors of each type. Therefore, there is a chain of tasks in T that is 
 executed sequentially during these idle periods, and must be executed 
 sequentially in any schedule. Thus, we conclude that 
 
 I £ (m-l)oj' (4) 
 
 since no schedule can have a completion time oj' less than the length of 
 this chain. 
 
 Let K. denote the sum of lengths of idle periods during which 
 
 all type j processors are busy. We have 
 
 ra-m. 
 
 K. < '- 
 
 J - m 
 
 I 
 
 m.-n . 
 _J 1 
 
 l<n.<m.-l 
 
 - J- J 
 
 0<n <m for u^i 
 
 — u— u 
 
 n 1 +n„+. . .+n 
 12 r 
 
 (n.. ,n~, . . . ,n^_) 
 
 m-m . m-m . 
 
 1 t 1 
 
 m j m 
 
 < 
 
 I 
 
 m_ l l<n.<m. , 
 
 - J- 1-1 
 
 0<n <m for u^i 
 
 — u— u 
 
 (n 1 ,n 2 , . . . ,n r ) 
 
 Furthermore, 
 
 2 K. < 
 
 J-l 
 
 r m-m. 
 J-l 
 
 m , 
 
 t . - 
 
 r m-m. 
 S 
 
 j-l 
 
 m , 
 
 1 
 
 m-1 
 
 I 
 
 l<n.<m.-l 
 ~ J- J 
 
 0<n <m for u^i 
 — u— u J 
 
 (n ,n 2 , 
 
 •V 
 
f 
 
 I 
 
 A 
 
 i\ 
 
 i 
 
 I) 
 
 '1 
 
 13 
 13 
 
 I 
 
 :c 
 ;> 
 
 ! 
 
 8 
 
 
 r m-m. . m-m. 
 v -\ I rain i 
 
 — . n m. i m-l l<j<r m. 
 J=l J J 
 
 (5) 
 
 Since 
 
 
 r 
 
 
 $ < I + 2 K. 
 
 
 we have from Eq. (5), 
 
 
 r m-m. 
 
 $ < 2 1 t. + I 
 
 — . - m. j 
 J=l J 
 
 i . m-m . ^ 
 , 1 mm 
 
 m-l l<j<r m j 
 
 • 
 
 
 Combining this expression with the inequality in Eq. (4) and that 
 
 
 t . 
 
 -J- < a)' j = 1, 2, ... , r. 
 ra. — 
 3 
 
 
 We have 
 
 
 , / min i \ 
 $ < m(r-l)to' + raw' 1 1 - 1<j<r — 1 
 
 (6) 
 
 Furthermore, since 
 
 
 i r 
 
 - .2, t. < a)'. 
 
 (7) 
 
 We have the inequality Eq.(l) by substituting Eqs. (6) and (7) into 
 
 
 Eq. (3). 
 
 
 The example shown in Figure 3 demonstrates that the upper 
 
 bound 
 
 in Eq. (1) is best possible. In this example, we have 
 
 
 „ _ max 
 ra, - m. 
 1 l<j<r j 
 
 
 There are ra (m -1) + 2 type 1 tasks which are denoted as T^, T^* . 
 
 • • > 
 
 T , , , N „^ . Their execution times are 
 1, (m 1 (m 1 -l)+2) 
 
 M(T U ) - e 
 
 u(T..)=- i = 2, 3,... ,m (m -1) + 1 
 li m n J- - 1 - 
 
 There are m. + 1 type j tasks for j = 2, 3, . . . , r which we shall denote as 
 
V V 
 
 • ' T j,(.yi) for J = 2 - 3 ' 
 
 and 
 
 r. Their execution times are 
 
 u(T j£ ) =1 j = 2, 3, ... , 
 
 I = 1, 2, ... , m. 
 
 y(T j,(m.+l) )=£ J = 2, 3, ... , r 
 J 
 
 Furthermore, the precedence relation is as shown in Figure 3a. The 
 priority schedule in Figure 3b is obtained according to the priority list 
 
 T 12' T 13' '•• ' T U(m 1 (m 1 -l)+l)» T ll» T 21' T 22 "• T 2,(m 2 +1) ' T 31' T 32' 
 
 T r,(m r +1)' T l,(m 1 (m 1 -l)+2)/ 
 
 It's complete time is 
 
 m -1 
 a) = r -\ 1- re 
 
 m„ 
 
 while the completion time of the schedule shown in Figure 3c is 
 
 u) 1 = 1 + re 
 Hence, the bound in Eq. (1) is achieved for very small e 
 
 When the processors in the system are all identical, r 
 The result in Theorem 1 reduces to the well-known result [6]: 
 Corollary 1 : For a system containing m identical processors 
 
 ^< 2-i 
 
 0)' — m 
 
 On the other hand, for a job shop problem, m.. = m_ = . . . = m = 1 
 
 In this case, we have, 
 
 Corollary 2 : For a r-processor job shop 
 
 = 1 
 
 rr < r, 
 
<s> 
 
 10 
 
 I 
 
 i 
 
 
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 i: 
 
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 :> 
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 o 
 
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 ^N> 
 
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11 
 
 ii To 
 Ti,jyi[Cw r i>i 
 
 T, 
 
 1 \ i \n 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 i i { i 1 1 niiuiiiiin Ti^m-O^ 
 
 I | | \mmiiiiim I mil n nun < s in n ii 1 1 ii 1 1 1 1 1 1 1 ii \ 
 
 \ i% fas 
 
 7m — r 
 
 '////////// A J 
 
 \ {i milium n iii ii mil mi i, < j 1 1 1 h 1 1 1 1 1 1 1 1 1 1 il 1 1 ^ \ 
 
 lYPk 
 
 I I ////// / / / / /^ /////////////////////// | -j 
 
 4 
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 // ////////| l2^U | ////////////// ( , ////////////////////// ! j 
 
 
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 >Tifc3 
 
 ?doc. 
 
 I III I ' I III I 1 1 7 I I '1 I 1 1 // III 71 
 
 T^ 
 
 | /////^//////////////////// | , 
 
 ///////////////////////////////////// / ///i | |//////// 
 
 # 
 
 / 
 
 '//////////// /////////// //////////,, //A T*Z »,/////// 
 
 it 
 
 '// ///////////////////////////// //ft// 
 
 A 
 
 > rrr) r \// /////// a 
 
 TYPtf 
 
 m -1 
 
 a) ■ r H + r e 
 
 m« 
 
 Figure 3b. Worst Case Priority-Driven Schedule. 
 
12 
 
 I 
 
 ; 
 
 ii 
 
 ill 
 (3 
 !> 
 
 >f 
 
 '1 
 
 :: 
 ;> 
 
 :: 
 
 /// /////// 
 
 /////////////i 
 
 P/?OC £550^5 
 
 T, 
 Tit, Tji 
 
 ,th 
 
 1<H,-M 
 
 u\\ tfi, i^i 
 
 I //////////// 
 
 
 /" 
 
 
 Tt;tB 2 rl 
 
 |^ZJ I /// // // I 
 
 T. 
 
 Zl 
 
 \ l ill < 1 1 1 1 1 1 1 
 
 < 
 
 lit 
 
 \ 1 1 1 1 , 1 1 / 1 1 / 1\ 
 
 I 
 
 ir/i t 
 
 Imv 1 
 
 
 | ///// \////A 
 
 1/ ///////// /I 
 
 T 
 
 Ji 
 
 Tu 
 
 < 
 
 | / ///////// / 
 
 h 
 
 3013 
 
 T c 
 
 TTIr-f-l 
 
 f>#ocB&oR5 
 
 ///// ///// 
 
 T 
 
 /1 
 
 I/ // ////// 
 
 A,d 
 
 TVi 
 
 K 
 
 /////////// 
 
 Tftfli 
 
 a) = 1 + r e 
 Figure 3c. Optimal Schedule, 
 
 
13 
 
 III. GENERALIZATION 
 
 The model of a multiprocessor system in which each processor 
 has a dedicated memory of a certain fixed capacity was studied in [8]. 
 In this case, the execution of a task requires a certain amount of memory 
 space and can only take place on a processor whose memory capacity is larger 
 than or equal to its memory requirement. Our model in Section II can be ex- 
 tended such that every processor is characterized by its type as well as 
 by its memory capacity. We shall, however, present a more general model. 
 
 Consider a multiprocessor system consisting of processors of r 
 different types. Each type of processors is further divided into sub- 
 types. Thus, we shall refer to a processor as a type (j,k) processor when 
 it is a type j processor of subtype k. A partial ordering relation < is 
 defined over the processor types such that 
 
 (i) (j,k) and (n,v) are incomparable if j ^ n. 
 (ii) (j,k) < (j,v) means that if a task can be executed on a 
 tyP e (j>k) processor then it can also be executed on a type (j ,v) processor. 
 A multiprocessor system can then be represented as P = (m , m _,..., m ; 
 
 ^,) ni n ,,.. M m 0(] » . . . m _ . m _. ... , m ', 4 ) where m #1 is the number of 
 
 '21' "22"'" m 2V- m rl' m r2' 
 
 rl 
 
 jk 
 
 tyP e (j>k) processors and ^ is a partial ordering relation over the 
 processor types. We shall let 
 
 m 
 
 = 2 
 
 m. 
 
 k=l J k 
 
 m 
 
 = 2 
 j=l 
 
 m. 
 
 For example, the multiprocessor system represented by the directed graph 
 
 t 
 
 4. 
 
 Neither (j,k) < (u,v) nor (u,v) < (j ,k) . 
 
 ■L 
 
 There is an edge from (j,k) to (j ,v) means (j,v) -< (j,k). 
 
t. 
 
 : 
 
 :; 
 
 ! 
 
 \ 
 
 
 
 14 
 
 
 
 
 /- > 
 
 
 
 
 CN 
 
 
 
 
 co 
 
 
 
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 4-1 >v 
 
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 /"~\ 
 
 
 
 
 
 
 
 
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 iH 
 
 
 CN 
 
 
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 <N 
 
 
 CN4 CM 
 
 
 ^m^ 
 
 
 w v — ' 
 
 
 0) 
 
 
 QJ QJ 
 
 
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 D- 
 
 
 Pb 
 
 
 
 <r 
 
 
 
 >^ 
 
 
 >, 
 
 
 
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 4J 
 
 
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15 
 
 in Figure 4 contains three types of processors. There are three type 1 pro- 
 cessors of the same subtype. The 3 processors of type 2 having dedicated 
 memories of different sizes are linearly ordered as shown. For the 4 pro- 
 cessors of type 3, neither (3.1) < (3,2) nor (3,2) < (3.1) but (3,3) < (3,1) 
 and (3,3) •< (3,2). This system can be represented as P=(3;l,l,l; 2, 1,1, ). 
 
 Let T = {T.. , T„, ... , T } be a set of tasks to be exectued on 
 1 £. n 
 
 a system P. A task T. is said to be a type (j,k) task if it can be executed 
 on any type (j,v) processor for (j,k) -K. (j,v) and on no other type of pro- 
 cessors. Thus, a set of tasks can be specified by an ordered 4-tuple 
 (T, A, y, <) where A is a function from T. to the processor types so that 
 A(T.) specifies the type of T., and y and < are as defined in Section II. 
 
 Again, we use the notation (T., A(T.), y (T . ) ) to represent a particular task T., 
 
 Consider all type (j,k) processors for a fixed j. The smallest sub- 
 set of types {(j,k ), (j,k 2 ), ... (j,k )} is said to be the dominating set if 
 for any type (j,k), (j,k) -< (j,k ) for some 1 <_ p <_ q. In other words, any 
 tyP e (j»k) task can be executed on a processor whose type belongs to the 
 dominating set. For example, in the multiprocessor system shown in Figure 4, 
 {(1,1)}, {(2,1)} and {(3,1), (3,2)} are dominating sets. We also refer to a 
 type (j,k) as a maximal type if it is in the dominating set. 
 For a given dominating set, let 
 
 m._ = min{m.. |(i,k ) is in the dominating set} 
 jO jk ' p 
 
 In other words, m. n is the minimum of the numbers of processors among all 
 types of processors in the dominating set. Therefore, m #n is equal to 3, 1, 
 and 1 for j = 1, 2, and 3, respectively, in our example. The result in 
 Theorem 1 can be generalized to 
 Theorem 2 
 
 (0 
 
 m 
 
 , < 1 + E 
 a)' — . , m 
 
 J. _ 
 
 mm 
 
 l<j<r m.„ 
 
 j-i -jo -- JO 
 
 (8) 
 
& 
 
 13 
 
 i; 
 
 !» 
 Q 
 
 :: 
 
 ! 
 
 16 
 
 Proof. Again, let t. denote the total execution times of all type (j,k) 
 tasks for k = 1, 2, ... , £. . For a priority-driven schedule, let 
 
 <Kn n , n 1 2""» n u l ' n 21' 
 
 , ... , n „ , ... , n , ... , n f£ ) denote the 
 2 r 
 
 sum of idle times in all processors in P during which there are n idle 
 processors of type (j,k) for j = 1, 2, ... r and k = 1, 2, ... , I . . And 
 
 the total idle time 
 
 11 
 
 '12 
 Z 
 
 m 
 
 n u -0n 12 -0 
 
 r£ 
 
 n n =0 
 r 
 
 *(n n , n 12 , 
 
 r£ 
 
 r 
 
 (9) 
 
 Hence the completion time of the priority driven schedule is 
 
 1 
 
 oj = — 
 
 m 
 
 J = l 
 
 as given by Eq. (3). 
 
 Let I be the sum of all terms in Eq. (9) corresponding to 
 idle periods during which at least one of each maximal type processor is 
 idle. That is 
 
 r J 
 1 = 22 2 
 j=l k=l n.. e S 
 
 jk jk 
 
 (n u , n 12 ,..., n r£ ) 
 
 where S = {1, 2, ... , m - 1} if (j,k) is a maximal type and, S , = 
 
 J k J K J K- 
 
 {0, 1, 2, ... , m } if (j,k) is not a maximal type, 
 jk 
 
 Again, there is a chain of task in T such that during these 
 idle periods one of the other processors is executing a task in the chain, 
 
 Hence 
 
 I < (ra-lW ( 1Q ) 
 
 Let K denote the sum of lengths of idle periods during which all the 
 
 j 
 processors of one or more maximal types of the form (j,*) are busy. We 
 
 have 
 
17 
 
 m-m 
 
 JO 
 
 K.< 
 
 jO 
 
 t . 
 
 1 
 
 r J 
 
 J-l k=l n 
 
 J Jl J 2 
 
 •- n j£ 
 
 jk jk 
 
 n u +n 12 + . 
 
 +n 
 
 rH 
 
 (J)(n,^, ^2' 
 
 r 
 
 m " m iQ 
 
 — m 
 
 JO 
 
 t.- 
 
 2 2 
 
 i m-1 " Z *( n ir n l2' 
 
 3=1 k=l n. k aS. k 
 
 r 
 
 .J 
 
 Hence 
 
 r r m-m . _ 
 
 2 K. < 2 ^ t 
 
 J-i J " J-i m jo 
 
 T . m-m . _ 
 -l min jO 
 
 j m-1 l<j<r m 
 
 JO 
 
 and 
 
 <_ I + E K. 
 J-l 
 
 r m-m Q 
 < E — t. + I 
 
 J-l JO 
 
 . m-m 
 1 mm jO 
 
 m-1 l<j<r m 
 
 JO 
 
 Since 
 
 m. — 
 
 J = 1, 2, ... , r 
 
 and 
 
 - E t. < <d' 
 
 as well as Eq. (10) , we obtain 
 
 r m. 
 
 : „. + £■< E (m-m ) ^ + (m-1) 
 J-l JO 
 
 -, . m-n . _ 
 
 1 mm jO 
 
 m-1 l<j<r m 
 
 JO 
 
 r m, , r , 
 
 1 + E — J - - - E m. + 1 - kj,*-. ~ — 
 j=l m J0 m j=l J ^'"jO 
 
i 
 
 = to' 
 
 r m. 
 I + y. -J— - m i n 1 
 i_ n m i n Kj<r m. 
 
 which is the bound given by Eq. (8). When the dominating sets contains 
 only one subtype for all j - 1, 2, .... r, (that is, there is an unique 
 
 maximal subtype for all types), we have 
 
 Corollary 3: The bound given by (8) is the best possible. 
 
 That this bound is best is illustrated by the example in Figure 5, 
 When the multiprocessor system contains only one type of pro- 
 cessors, we have m. = for i=2, 3, ... , r. The bound given by Theorem 
 
 2 is simply 
 Corollary 4 : 
 
 a) m i 
 
 771 1 + 
 
 m. 
 
 m. 
 
 11 11 
 
 The bound derived in [ 8] for processors with different storage capacities 
 is a special case of our result with m =1. 
 
 • 
 
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 [l]. T. C. Hu, Parallel Scheduling and Assembly Line Problems, 
 Oper. Res. 9, No. 6 (1961), pp. 841-848. 
 
 [2], E. G. Coffman, Jr. and R. L. Graham, Optimal Scheduling for 
 
 Two Processor Systems, Acta Inf ormatica , 1, (1972), pp. 200-213. 
 
 [3]. R. R. Muntz and E. G. Coffman, Jr., Preemptive Scheduling of Real 
 Time Tasks on Multiprocessor Systems, J. ACM 17, No. 2, (1970) 
 pp. 324-338. 
 
 [4]. J. Bruno, E. G. Coffman, Jr. and R. Sethi, Scheduling Independent 
 Tasks to Reduce Mean Finishing Time, Comm of ACM , 17, No. 7, 
 (1974), pp. 382-387. 
 
 [5]. J. W. S. Liu and A. Yang, Optimal Scheduling of Independent Tasks 
 on Heterogeneous Computing Systems, Proc . 1974 ACM National 
 Conference , pp. 38-45. 
 
 [6]. R. L. Graham, Bounds on Multiprocessing Timing Anomalies, SIAM 
 Journal on Applied Mathematics , 17, No. 2, (1969) pp. 263-269, 
 also Bounds on Multiprocessing Anomalies and Related Packing 
 Algorithms, AFIPS Conf . Proc. , 40 (1972), pp. 205-217. 
 
 [7]. J. W. S. Liu and C. L. Liu, Bounds on Scheduling Algorithms for 
 Heterogeneous Computing Systems, IFIP Conf. Proc. (1974) 
 pp. 349-353. 
 
 [8]. D. G. Kafura and V. Y Shen, Scheduling Independent Processors with 
 Different Storage Capabilities, Proc. ACM National Conference 
 (1974), pp. 161-166, also K. L. Krause, V. Y. Shen, and H. D. 
 Schwetman, Analysis of Several Task Scheduling Algorithms for a 
 Model of A Multiprogramming Computer System, J. ACM, 22, No. 4, 
 (1975) pp. 522-550. 
 
 [9]. G. L. Chesson, Communication and Control in Cluster Network, Proc . 
 ACM National Conference (1974) pp. 509-514. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-77 -835 
 
 3. Recipient's Accession No. 
 
 4. Title and Subtitle 
 
 Performance Analysis of Multiprocessor Systems Containing 
 Functional Dedicated Processors 
 
 5. Report Date 
 
 May , 1 977 
 
 7. Author(s) 
 
 Jane W. S. Liu and Chung L. Liu 
 
 8- Performing Organization Rept. 
 No. 
 
 ?. Performing Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaing 
 
 Urbana, IL 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract/Grant No. 
 
 MCS-73-03408 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington DC 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 General models of multiprocessor systems in which processors are functionally 
 
 I dedicated are described. In these models, processors are divided into different 
 types. Clearly, the model of multiprocessor systems with identical processors 
 is a special case of our models. These models also include the job shop problem 
 in which there is exactly one processor of each type. Worst case performance 
 bounds of priority-driven schedules are obtained. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 worst case performance, multiprocessors systems, scheduling algorithms, 
 and functionally dedicated processors. 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSAT1 Field/Group 
 
 18. Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 22. Price 
 
 FORM NTIS-38 (10-70) 
 
 USCOMM-DC 40329-P7 1 
 

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 Implementation ot the language CLEOPATRA 
 
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