This paper considers the problem of optimal distribution of computational resources of multiprocessor systems based on the principles of parallelization and conveyorization. The necessary conditions and criteria of efficiency and optimality of systems of identically distributed competing processes are obtained that take into account the overheads required in the context of their run time

Kapitonova, Yu.V.

Kovalenko, N.S.

Pavlov, P.A.

Repository of Polessky State University (Репозиторий Полесского государственного университета)

Cybernetics and Systems Analysis, Vol. 41, No. 6, 2005CYBERNETICSOPTIMALITY OF SYSTEMS OF IDENTICALLYDISTRIBUTED COMPETING PROCESSESYu. V. Kapitonova,aN. S. Kovalenko,band P. A. PavlovbUDC 681.3.06:519This paper considers the problem of optimal distribution of computational resources of multiprocessorsystems based on the principles of parallelization and conveyorization. The necessary conditions andcriteria of efficiency and optimality of systems of identically distributed competing processes areobtained that take into account the overheads required in the context of their run time.Keywords: system of competing processes, distributed processing, optimality and efficiency criteria.Optimal distribution of computational resources of multiprocessor systems (MSs) based on the principles ofparallelization and conveyorization is one of the most important problems in creating efficient system and applicationsoftware [1, 2]. An important place in solving it is occupied by problems of optimal organization of competing processesusing common program resources. The solution of these problems determines not only the efficiency of using MSs but alsotheir capabilities to solve complicated problems from various areas of knowledge.It is relevant to note that, in solving problems of optimal organization of competing processes, mathematical modelsbased on lumped processing are most developed [2]. At the present time, MSs based on distributed processing havecontinued to play an increasingly important part [3–5]. In this connection, the problems of construction and investigation ofmathematical models of optimal organization of competing processes during distributed processing are especially topical.As in [5], a mathematical model of distributed processing of competing processes includes the following parameters:p p,  2 , is the numbers of processors of a multiprocessor system, n n,  2 , is the number of competing processes, s s,  2 ,is the number of blocks Q j of a structured program resource (PR), j s1, , and [ ], , , ,t i n j sij  1 1 , is the matrix of runtimes of execution of blocks with the help competing processes. It is assumed that all n processes use one copy of the PRstructured in the form of blocks.Let us introduce the parameter   0 that characterizes the time of additional system expenditures for the organizationof parallel use of PR blocks by a set of distributed competing processes.In this article, the necessary conditions and efficiency and optimality criteria of systems of identically distributedcompeting processes are considered, taking into account the overheads required depending on their run time.1. RUN TIME OF DISTRIBUTED COMPETING PROCESSESDefinition 1. We call a system of competing processes identically distributed if the run times of execution of all PRblocks Q j sj , ,1 , with the help of each ith process, i n1, , are equal to one another, i.e., we havet t t ts11 12 1 1    . . . ,t t t ts21 22 2 2    . . . ,. . . . . . . . . . . .t t t tn n ns n1 2    . . . ,where ( , , . . . , )  t t t n1 2 are run times of execution of each block Q j sj , ,1 , with the help of all n processes.7931060-0396/05/4106-0793©2005 Springer Science+Business Media, Inc.aV. M. Glushkov Cybernetics Institute, Kiev, Ukraine, kap@d105.icyb.kiev.ua.bByelorussian State EconomicsUniversity, Minsk, Republic of Belarus, kvm@bseu.minsk.by; pin2535@tut.by. Translated from Kibernetika i SistemnyiAnaliz, No. 6, pp. 3-10, November-December 2005. Original article submitted March 16, 2005.Definition 2. We call an identically distributed system of competing processes stationary if we have      t t t tn1 2 . . . .As is shown in [5], for an identically distributed system of processes competing in the asynchronous and secondsynchronous modes, the minimal total run time can be computed by the formulasT TT s t s p s p T ptkn n n nopasopmax maxwhen or but     21( ) ;T p t s kp k T ptk T rn n n nn      ( ) , , and ;( ) (1 11max maxwhen1 1 1) , , , and .t s kp r k r p T ptn n nmax maxwhen      	Here, T tnini1is the total run time of execution of each block Q j with the help of n processes andt tni nimax max 1.For the first synchronous mode that provides the continuous execution of PR blocks within each process, the minimaltotal run time of a given amount of computations in the MS being considered is specified by the relationshipsTT s t t tnnini iopmax{ } when1211 0      ( ) , s pkT p n p k s kp kkT   ;( , , ) ( ) , , ;opomin{ } when11 21 1 p opmin{ } min{ }w1 11 2 1 21( , , ) ( , , ) ( ) , , ,p n p T p n r k      hen s kp r k r p    	, , .1 1Here, we haveT p n p T p t t tnnini iop max{ }1211 0( , , ) ( ) ,       ,(1) 1 1 2111       ( ) , , ( , , )p t t T p n p p tni nimin{ } maxop ,T p n r T r t t tnnini iop max{ }1211 0( , , ) ( ) ,       ,(2)1 11      ( ) , ( )r t t p r tn nmin { } ,2111 1     T p n p T p i p T p i r ri nop op opmax( , , ) ( ( , , ) ( , , ) t i ) .In this case, T p i pop1( , , ) and T p i rop1( , , ) are found by formulas (1) and (2) after substitution of i for n.Taking into account overheads   0 when s p , the minimal total run time of systems of identically distributedcompeting processes in the asynchronous and second synchronous modes is determined by the formulaT p n s T p n s T p n s T s tn( , , , ) ( , , , ) ( , , , ) ( )      opasop21max, (3)where T t t t t tninii ni i i        11, , andmaxmax .In the case of an identically distributed stationary system of competing processes, their minimal total run time for allthe three basic modes is found, taking into account an additional system expenditures , by the formulas794Tn s t p n skn p t p n s s kp     ( ) , , ;( ) , , ,11min{ }min{ } , ;(( ) ) , , , , , .kk n r t p n s s kp r k r p        	 11 1 1 1min{ }(4)Here, t T n T ntn n  / , . In formulas (3) and (4), the parameter  is explicitly taken into account.2. EFFICIENCY AND OPTIMALITY OF SYSTEMS OF COMPETING PROCESSESWHEN THE NUMBER OF PROCESSORS IS SUFFICIENTDefinition 3. An identically distributed system of competing processes with a fixed p sand  2 is called efficient ifthe value of ( ) ( , , , )n sT T p n sn   0 , where sTnis the time of sequential execution of s blocks with the help of all nprocesses.Definition 4. An identically distributed efficient system is called optimal if the value of ( )n is maximum.We will show that it suffices to search for the optimal identically distributed system among identically distributedefficient systems.THEOREM 1. For any identically distributed efficient system of competing processes, there exists a more efficientidentically distributed stationary system when s p and   0 .Proof. According to Definition 3, the condition of efficiency of an identically distributed system of competingprocesses is of the form( ) ( )( ) ( )n s T t n sn n      1 1 0max(5)or of the form( ) ( )( ) ( )n s T t n sn      1 1 0 , where t T nn / , (6)when such an identically distributed system is stationary.Let us show that the inequality   ( ) ( )n n is true. To this end, we consider the difference  ( ) ( ) ( )( )n n s t tn   1max. The case when  ( ) ( )n n 0 takes place only if t tnmax . The latter inequality cannot betrue since the fact of nonstationarity of an identically distributed system would implyinint nt T   1but, by the condition,we haveinint T  1. We obtain the contradiction T Tn n , which proves the theorem.THEOREM 2. In order that an identically distributed efficient system of competing processes be optimal when s p ,it is necessary and sufficient that it be stationary.Theorem 2 follows from Theorem 1 and Definition 4.THEOREM 3. An identically distributed system with n  3 competing processes in an MS with p  3 processors isefficient when n s  3 , s p , and    min1 i nit if and only if the condition sn n s  2 1( ) is true.The proof of the theorem is based on the analysis of the inequalities presented below. According to condition (5), theefficiency condition is equivalent to the inequalityT t n ssn n max11.By virtue of the condition    min1 i nit of the theorem, the following inequality holds true:T tnn n max1.Moreover, the condition sn n s  2 1( ) of the theorem is equivalent to the inequalitynn ss  111.795Let us formulate the necessary and sufficient conditions (criteria) of existence of an efficient system of identicallydistributed competing processes with a sufficient number of processors s p , depending on overheads   0.THEOREM 4. In order that an identically distributed efficient system of competing processes exist with givenp Tn 3 0, , and  when s p , it is necessary and sufficient that the following conditions be true: ( )( [ ]) ( [ ])11 2s ss f s sif is integer;max{ } if is not integer.	Here, ( )( ) ( )( ), [ ]xs T xx x sxn  1 11is the greatest integer that does not exceed x.Proof. The necessity follows from Definition 3 and Theorem 1. In fact, by virtue of formula (6), the efficiencycondition is of the form ( ) ( )( ) ( )n sT T s T t n sn n        1 1 0that is equivalent to the following inequality:   ( ) ( )( ).s T nn n sn1 11(7)Introducing the function ( ) ( ) ( ) / ( )x s T x x x sn    1 1 1 , x  0 , we can easily show that it reaches its maximumwhen x s 1 .Choosing n x s  1 if s is integer or n x s  {1 [ ], 2  [ ]s }if s is not integer, we obtain an efficient systemof identically distributed processes, which proves the necessity.The sufficiency follows from inequality (7) since ( )x reaches its maximum when n x s  1 if s is integer orwhen n x s  {1 [ ], 2  [ ]s } if s is not integer.Comment. When p s  2 , an identically distributed system of competing processes is efficient if the relationTnn nn11( )is true.In Fig. 1, the plot of the function y x x ( ), 0 , is presented for fixed s Tn  50 7 5, , and  . The existence of anidentically distributed efficient system of competing processes is determined by the domain . All the integer points of thesegment  are the values of n for which the system is efficient; here, we have x s11 8  [ ] , x s*, 1 andx s22 9  [ ] .The solution of the problem of optimality of an identically distributed system that consists of n competing processesfor a sufficient number of processors follows from the theorem formulated below.THEOREM 5. In order that an identically distributed efficient system of competing processes be optimal with s pand given p Tn 2 0, , and  , it is necessary and sufficient that it be stationary and the number of processes n in the systembe equal to the value of( ) ( )s T s Tn n1 11 or for which the function ( )x assumes its maximum. Here, [ ]xis the greatest integer that does not exceed x.Proof. By virtue of Theorem 2, one should search for the optimal identically distributed system in the class ofstationary systems. Then it is obvious that we have ( ) ( ) ( / ) ( )n s T n n sn     1 1 1 1 .Necessity. Let us introduce a function of a real argument x in the form( ) ( ) ( / ) ( ) ,x s T x x s xn      1 1 1 1 1.796According to Definition 4, an identically distributed system is optimal at the point x at which ( )x reaches itsmaximum. Let us show that this is possible at the point xs Tn( )1. In fact, we have      ( )( )and ( )( )xs TxxT sxn n1 2 102 3since s x 2 0and .Hence, the function ( )x reaches its maximum at the point at which  ( )x 0, i.e., we have xs Tn*( )1. Theinteger points at which the maximum of the function ( )x is reached are n x [ ]*or n x [ ]*1. Thus, any of the numbers( ) ( )s T s Tn n1 11 or for which the function ( )x assumes its maximum can be chosen in the capacity of n.The sufficiency follows from the properties of convexity of the function ( )x on the segment [ , ]2 n when s p .3. CRITERIA OF EFFICIENCY OF SYSTEMS OF IDENTICALLY DISTRIBUTEDCOMPETING PROCESSES WHEN THE NUMBER OF PROCESSORS IS LIMITEDLet us formulate the necessary and sufficient conditions (criteria) of efficiency and optimality of identicallydistributed systems of competing processes when the number of processors is limited.It has been shown in Sec. 1 that, taking into account the parameter   0 that characterizes the time of additionalsystem expenditures for the organization of parallel use of blocks by a set of distributed competing processes, forcomputation of the minimal total time T p n s( , , , ) in the asynchronous and second synchronous modes, the followingformulas take place for the class of identically distributed competing processes:T p n sk T p t s kp k T ptn n n n( , , , )( ) , , ;(     1 1max maxwhenk T r t s kp r k r p T ptn n n n        	1 1 1 1) ( ) , , , ,max maxwhenwhere T tnini1is the total run time of execution of each block Q j with the help of n processes andt tni nimax max 1.797Fig. 1The theorem formulated below is true.THEOREM 6. If we have p n s n s, , ,  3 3, and    min1 i nit , then an identically distributed system consisting of ncompeting processes in a multiprocessor system with p processors is efficient if and only if the following conditions hold:snkn p s kp kk n r s kp r k        2 1 12 1 1 1( ) , ;(( ) ) , ,ifif 1 	 r p .Proof. 1. In the case when s kp k and 1, the efficiency condition is equivalent to the inequality( ) ( )s k T p tkn pn n    11max.According to the condition    min1 i nit of the theorem, the following inequality is true:( ) ( )( )s k T p ts k n pn n     11max.These inequalities imply that sn kn p  2 1( ) .2. When s kp r k r p    , , and1 1 , the following inequalities hold true:( ) ( )( )s k T r tk n rn n      1 11 1max,( ) ( )( )s k T r ts k n rn n       1 11 1max,whence sn k n r   2 1 1(( ) ) .Let us formulate and prove the criteria of existence of an efficient system of identically distributed competingprocesses, depending on overheads   0 , when the number of processors is limited.THEOREM 7. In order that an identically distributed efficient system of competing processes exist for givenp Tn 2 0, , and  , it is necessary and sufficient that the following conditions be true.1. When s kp , we have k 1,~~111 11pkpkpkif is integer;max	 , 1111pkifpkis not integer,	where 11 1 1( ) ( ) ( ) / ( )x p T kx x kx pn     and [ ]x is the greatest integer that does not exceed x.2. When s kp r k r p    , , and1 1 , we have~~ 22 21( )([ ]), ([ ] )x xx x xif is integer;max{ } if is not integer,	where 21 1 11 1( )[( ) ( )( )][( ) ]xp kx r x Tx k x rn     and [ ]x is the greatest integer that does not exceed x xrp k r,( )  !11 11 11 11   ( )p k rk.798Proof. By virtue of formulas (4), when s kp and 1, the efficiency condition is of the form( ) ( )( ) ( )n p kT t kn pn      1 1 0, which is equivalent to the inequality    ( ) ( )( ).p T knn kn pn1 11The function 11 1 1 0( ) ( ) ( ) / ( ),x p T kx x kx p xn      , reaches its maximum when xpk1. Hence, whens kp k , 1, for an identically distributed efficient system, the number of processes must be n xpk 1if1 pkisinteger and n xpkpk 	 1 11, if1 pkis not integer.When s kp r k r p    , ,1 1 , the efficiency condition is of the form      [( ) ( )( )][( ) ].p kn r n Tn k n rn1 1 11 1After introducing the function 21 1 1 1 1( ) [( ) ( )( )] / [( ) ] ,x p kx r x T x k x rn        x  0 , it may be shown that itreaches its maximum whenxrp k rp k rk     11 11 11 11( )( ).Consequently, an identically distributed system is efficient if the number of its processes equals n x when x is integerand n x x x  { }[ ], [ ] 1 when x is not integer.The proof of sufficiency is similar to that used in Theorem 4.The obtained criteria of efficiency and optimality of systems of identically distributed competing processes can beused in designing system and application software for MSs and networks during the solution of problems of optimal use ofcomputational resources.REFERENCES1. Yu. V. Kapitonova and A. A. Letichevsky, Mathematical Theory of Computer System Design [in Russian], Nauka,Moscow (1988).2. Yu. V. Kapitonova, P. S. Klyaus, and N. S. Kovalenko, “Mathematical model of conveyor organization of competingprocesses,” Dokl. Akad. Nauk SSSR, 293, No. 6, 1320–1324 (1987).3. V. V. Voyevodin and Vl. V. Voyevodin, Parallel Computations [in Russian], Nauka, St. Petersburg (2002).4. G. R. Andrews, Foundations of Multithreaded, Parallel, and Distributed Programming [Russian translation], Mir,Moscow (2003).5. N. S. Kovalenko and S. A. Samal’, Computational Methods of Realization of Intelligent Models of Complex Systems[in Russian], Belarusk. Navuka, Minsk (2004).799

Optimality of systems of identically distributed competing processes

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Optimality of systems of identically distributed competing processes

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