An Optimal Task Allocation Model for System Cost Analysis in Heterogeneous Distributed Computing Systems: A Heuristic Approach

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1 An Optial Tas Allocation Model for Syste Cost Analysis in Heterogeneous Distributed Coputing Systes: A Heuristic Approach P. K. Yadav Central Building Research Institute, Rooree , Uttarahand (INDIA) M. P. Singh Guruul Kangri University, Haridwar , Uttarahand (INDIA) Kuldeep Shara* Krishna Institute of engg. and Technology, Ghaziabad , U.P(INDIA) ABSTRACT In Distributed coputing systes (DCSs), tas allocation strategy is an essential phase to iniize e syste cost (i.e. e su of execution and counication costs). To utilize e capabilities of distributed coputing syste (DCS) for an effective parallelis, e tass of a parallel progra ust be properly allocated to e available processors in e syste. Inherently, tas allocation proble is NP-hard in coplexity. To overcoe is proble, it is necessary to introduce heuristics for generating near optial solution to e given proble. This paper deals wi e proble of tas allocation in DCS such at e syste cost is iniized. This can be done by iniizing e inter-processor counication cost (IPCC). Therefore, in is paper we have proposed an algori at tries to allocate e tass to e processors, one by one on e basis of counication lin su (). This type of allocation policy will reduce e inter-processor counication (IPC) and us iniize e syste cost. For an allocation purposes, execution cost of e tass on each processor and counication cost between e tass has been taen in e for of atrices. Keywords Distributed coputing syste, tas allocation, execution cost, counication cost, counication lin su. 1. INTRODUCTION To eet e requireent of faster coputation, one approach is to use distributed coputing systes (DCSs).Distributed coputing syste (DCS) not only provide e facility for utilizing reote coputer resources or data not existing in local coputer systes but also iniize e syste cost by providing e facilities for parallel processing.[1, 8, 24]. A distributed coputing syste (DCS) consists of a set of ultiple processors (which are geographically distributed) interconnected by counication lins. A very coon interesting proble in DCS is e tas allocation. This proble deals wi finding an optial allocation of tass to e processors so at e syste cost (i.e. e su of execution cost and counication cost) is iniized wiout violating any of e syste constraints [3]. In DCS, an allocation policy ay be eier static or dynaic, depending upon e tie at which e allocation decisions are ade. In a static tas allocation, e inforation regarding e tass and processor attributes is assued to be nown in advance, before e execution of e tass [1]. We shall be considering static tas allocation policy in is paper. Tas allocation proble is nown to be NP- hard proble in coplexity, when we required an optial solution to is proble. The easiest way to finding an optial solution to is proble is an exhaustive enuerative approach. But it is ipractical, because ere are n ways for allocating - tass to n- processors [3]. Much research efforts on e tas allocation proble have been identified in e past wi e ain concern on e perforance easures such as iniizing e total su of execution and counication costs [1-4,6,7,11 ] or iniizing e progra turnaround tie [8, 10, 22], e axiization of e syste reliability [12-19] and safety [16]. A large nuber of techniques to tas allocation in DCSs have been reported in [1-4, 5-8, 10-19, 21-24]. They can be broadly classified into ree categories: graph eoretic technique [8, 9], integer prograing technique [6-8] and heuristic technique [1-3, 23, 24]. Graph eoretic and integer prograing techniques yields an optial solution at all e ties. But ese techniques are restricted to e sall size probles. If e proble size is very large, it is necessary to use e heuristic technique to get near optial solutions. The choice of a particular technique depends on e structure of e proble [14]. In is paper, we have developed a tas allocation odel and have proposed a heuristic algori for tas allocation at will find a near optial solution to e proble. The proposed algori try to iniize e inter processor counication cost (IPCC) by assigning ose tas first, which has e heaviest counication lin su (). Using is approach it has been seen at e syste cost will iniize ore an oer heuristic. The rest of is paper is organized as follows: section-2 forulates e tas allocation proble for iniizing e overall syste cost; section-3 discusses in detail e proposed allocation technique and algori; section-4 gives an ipleentation of e proposed algori. In e last section-5 concludes e paper. 2. PROBLEM FORMULATION In e past, different tas allocation odels and techniques for iniizing e overall syste cost have been widely investigated in e literature. In is paper, we follow (1-4, 6, 7, 10) to forulate e tas allocation proble. 30

2 2.1 Proble stateent The proble being addressed in is paper is concerned wi an optial allocation of e tass of a parallel application on to e processors in DCS. An optial allocation is one at iniizes e syste cost function subject to e syste constraints. In is paper, we have considered a distributed coputing syste ade up by two sets, P= {P 1, P 2,..,P n }of heterogeneous processors, interconnected by counication lins and T = {t 1, t 2,.,t } of progra tass, which collectively for a coon goal[1]. The execution costs of a tas running on different processors are different and it is given in e for of a atrix of order n, naed as execution cost atrix ECM (,). Siilarly, e inter tas counication cost between two tass is given in e for of a syetric atrix naed as inter tas counication cost atrix ITCCM (,) of order. Now, an allocation of tass to processors can be defined by a function X as follows: X: T P, such at X(i)= ; if i tas is allocated to processor. The purpose of defining e above function is to allocate each of e - tass to one of e n- processors such at e overall syste cost is iniized. 2.2 Notations T : e set of tass of a parallel progra to be executed. P : e set of processors in DCS. n : e nuber of processors. : e nuber of tass foring a progra. t i : P : i tas of e given progra. processor in P. x i : e decision variable such at allocated to processor, x i =0, oerwise. ec i : incurred execution cost (EC), if processor. x i =1, if i i tas is tas is executed on : incurred inter tas counication cost between tas ti andt j, if ey are executed on separate processors. ECM (,) ITCCM (,) : execution cost atrix. : inter tas counication cost. T {} ass : a linear array to hold assigned tass. T {} : a linear array to hold non assigned tass. T {} : a linear array to hold e tas aording to eir counication lin su. 2.3 Definitions Execution cost (EC) The execution cost eci of a tas t i, running on a processor P is e aount of e total cost needed for e execution of ti on at processor during e execution process. If a tas is not executable on a particular processor, e corresponding execution cost is taen to be infinite ( ) Counication cost (CC) The counication cost ( ); incurred due to e inter tas counication is e aount of total cost needed for exchanging data between ti and t j residing at separate processor during e execution process. If two tass executed on e sae processor en = Counication lin Su () It is an iportant characteristic of ITCCM (,), denoted by, which easures how counication intensive a tas is. The of a tas t i : 1 i, can be easily deterined by finding e su of counication costs of all e tass which t in ITCCM (,) are interacting wi i. Therefore, in inter tas counication cost atrix, e counication lin su of a tas ti can be coputed as: ( t i ) = j = 1 for i = 1, 2,,. (1) 2.4 Assuptions To allocate e tass of a parallel progra to processors in DCS, we have been ade e following assuptions: The processors involved in e DCS are heterogeneous and do not have any particular interconnection structure The parallel progra is assued to be e collection of - tass at are free in general, which are to be executed on a set of n- processors having different processor attributes Once e tass are allocated to e processors ey reside on ose processors until e execution of e progra is copleted. Whenever a group of tass is assigned to e processor, e inter tas counication cost (ITCC) between e is zero It is also assued at e nuber of tass to be allocated is ore an e nuber of processors (>>n) as in real life situation. 2.5 Tas allocation odel for syste cost 31

3 In is section, we have developed a tas allocation odel to get an optial syste cost. We can achieve is objective by aing tas allocation properly. Therefore, an efficient tas allocation of e progra tass to processor is iperative. However, obtaining an optial allocation of tass of a rando progra to any arbitrary nuber of processors interconnected wi non-unifor lins is a very difficult proble [1]. Hencefor, in order to allocate e tass of such progra to processors in DCS, we should now e inforation about e input such as tass attributes [e.g execution cost, inter tas counication cost etc] and processor attributes [e.g. processor topology, inter processor distance etc] etc. since obtaining such inforation is beyond e scope of is paper erefore, a deterinistic odel at e required inforation is available before e execution of e progra is assued [20]. In e present tas allocation odel, ere are two types of costs to be considered for is syste Processor execution cost (PEC) For given a tas allocation, X: T P, X(i) =, e execution cost ec i represent to execute tas t i on processor P and used to control e corresponding processor allocation. Therefore, under a tas allocation X, e processor execution cost, needed to execute all e tass assigned to processor can be coputed as: PEC ( X ) = = i 1 ec i xi Inter- processor counication cost (IPCC) Inter processor counication cost is incurred when e data is transitted fro tas to tas if ey are residing on separate processors, due to e inter tas counication. Therefore, inter processor counication cost (IPCC) is proportional to inter tas counication cost [6, 24]. Therefore, under a tas allocation X, e inter processor counication cost for processor can be coputed as: IPCC( X ) = i = 1 jfi ( ) x In is odel, bo e costs are application dependent and taes play an iportant role in tas allocation. Now, e total cost on processor is e su of e processor execution cost (PEC) and IPCC for processor, under a given tas allocation X T Cost ( X ) = PEC( X ) + IPCC( X ) (4) and e total cost of e syste is coputed by: S n Cost ( X ) = = TCost ( X ) 1 (5) Syste cost odel Wi syste resources constraints taen into aount, e tas allocation odel for syste cost ay be forulated as follows: i x jb (2) (3) in. S Cost ( X ) n s.t. = = 1 1 i x i= 1, 2, 3,.,. (6) { 0,1} x i,. (7) i In is odel, constraint-6, states at each tas should be assigned to exactly one processor. Constraint-7, guarantees at, xi is being decision variable. The above odel defines an integer prograing proble and is nown to be NP- hard proble [2-4, 7, 12-19]. An optial solution to is proble can be found by enuerating all possible allocations. But is technique requires O(n ) tie coputations. This is prohibitive even for sall size probles. Hence, in is paper, we present a heuristic algori to find quicly e solution of high quality, by ordering e tass aording to eir and ade allocation of ese tass to processors in at order. The proposed technique has been given in next section at will find near optial solution to e entioned proble at all e ties. 3. PROPOSED TASK ALLOCATION TECHNIQUE AND ALGORITHM The technique by which e tass coprising e progra are allocated to e available processors in DCS are essential, to iniize e syste cost. To achieve is objective, e order in which e tass in a progra are considered for allocation is a critical factor affecting e optiality of e resulting allocations [21]. We have selected all e tass for allocation aording to eir. The of each tas can be coputed by using ( t i ) = j = 1 for i = 1, 2,,. Now, all e tass are sorted in e onotonically decreasing order of eir in a linear array T { } and ey are considered for allocation in at order. Tie breaing is done randoly i.e. one of e tass wi equal is selected randoly. If e of a tas t i is very high an oer tas i.e. e inter tas counication oft i becoes ore intensive as copared to oer tass. In is case, syste cost derived could be lower due to involveent of ore counication lins. Therefore, first we have to allocate such tass to e processors, to iniize e IPC [1, 2]. Thus, e result will decrease in syste cost. Initially, we assue at e linear array Tass{} Φ and T T {}.Now, for an optial allocation of tass {} to processors in DCS, we have defined two rules. Rule-1 This rule is incorporated to e selection of suitable processors, whose capabilities is ost appropriate for e tas. We apply is rule as: 32

4 Pic up e tas i t fro T {} P {= 1,2, 3,..,n} for which and assign ti to processor eci is iniu. Suppose t to e r at processor is P r. Therefore, we have assigned i processor. Rule-2 It is anoer rule incorporated to IPC caused by e inter tas counication (ITC). Tas t i, which has been assigned to e r processor (say) using rule-1 in DCS, rule-2 is used to add e effect of counication cost of e executing tas t i P r, wi oer tass residing on P 1, P 2, P 3,..,P n-1 except e r processor as: Select e i colun of ITCCM (,) and add is colun to all e coluns of ECM (,) except e r colun. Now, we have odified bo e atrices ECM (,) and ITCTM (,) by deleting e i row and colun. Thus, we store e tas t i in a linear T and e linear array T {} array {} ass deleting i tas fro T {}. is odified by Hence, in e above anner, bo e rules will be repeated for each tas of T {}, until and unless T {} Φ and T {} {t 1, t 2,.,t n }= T ass{}. The detailed process of allocating e tass to e processors is given below in e for of algori. 3.1 Proposed algori Our algori consists of e following steps. Step-0: input:, n, ECM (,), ITCCM (,). Step-1: copute e counication lin su () of each tas using ( t i ) = j = 1 for i = 1, 2,,. Step-2: sort all e tass in T {} aording to decreasing order of eir. Step-3: initialize: {} T T {} Tass{} Φ Step-4: pic up e i tas (say i en t ) fro {} T and and 4.1: assign t i to e appropriate processor by using Rule-1 and Rule : T {} T { } { t } i T {} ass Φ { t i }= { t i }. Step-5: for all tass fro T {}, repeat step-4 until and unless we get Step-6: copute: T Φ and T {} T {} {} Cost ass T ( X ) = PEC( X ) + IPCC( X ) n and SCost ( X ) = = TCost ( X ) 1. Step-7: End. 3.2 Algori coplexity Using e eod suggested by H. Ellis et al [25], e run tie coplexity of e proposed algori can be analyzed as follows: step-1, executed in O() tie operations. Step-2, has a worst case tie coplexity of O( log ). In step-4, a single tas requires O(1(n)) tie operations. Therefore, for - tass step-5 requires O((n)) tie operations. Thus, e overall tie coplexity of e proposed algori is O( + log + n). Since n, erefore, e run tie coplexity of e proposed algori is O(n). 4. IMPLEMENTATION OF THE MODEL In is section, we give two nuerical exaples to illustrate e forulation and solution procedure of e proposed tas allocation odel. To show e perforance of our allocation technique for better allocation, we have tested e proposed algori on ese exaples. 4.1 Exaple-1 In is exaple, we have considered a typical progra ade up by 9- executable tass {t 1, t 2, t 3, t 4, t 5, t 6, t 7, t 8, t 9 } to be executed on e DCS having ree processors {P 1, P 2, P 3 }. We have taen e execution cost of each tas on different processors and ITCC between e tass in e for of atrices ECM (,) and ITCCM (,) respectively. Bo e atrices have been given in Table-1 and Table-2 respectively. We have applied e proposed algori on is exaple in e following anner: Step-0: Input: = 9, n = 3, ECM (,), ITCCM (,). Using ese inputs, e proposed algori traces e following output. Step-1: first of all, we have to calculate e counication lin su () of each tas using ( t i ) = j = 1 for i = 1, 2,,. Thus, we get ( i t ) = 29, 18, 18, 19, 18, 15, 17, 31, 27 corresponding to i= 1, 2, 3,, 9. 33

5 Step-2 and 3: sort all e tass in T {}, aording to decreasing order of eir ( t i ) T {} = { t8, t1, t9, t4, t2, t3, t5, t7, t6} and initially we T T {} assue, {} = { t8, t1, t9, t4, t2, t3, t5, t7, t6} Tass{} Φ Step-4: Now, pic up e first tas 8 t fro T {}. Apply Rule-1 fort 8.Since execution cost for t 8 is iniu on t and add e effect of processorp 2. Therefore, assign 8 P2 counication to processor P2 by using Rule-2. Thereafter, odify ECM (,) and ITCCM (,) by reoving t 8 fro ECM (,) and ITCCM (,).Thus, odified ECM (,) and ITCCM (,) have been given in table-3 and table-4, respectively. Thus, T { } = {} Tnon _ ass{} / t 8 { t1, t9, t4, t2, t3, t5, t7, t6 T {} ass Φ { t 8 } } Step-5: For all tass of T {}, repeat step-4, until and unless we gett Φ. {} { } T { }= T t, t, t, t, t, t, t, t, } ass Step-6: Processor wise total costs are: ( X ) 91 { t6 T = COST 1 {i.e. Total cost of processor P 1 } ( X ) 137 T = COST 2 {i.e. Total cost of processor P 2 } ( X ) 300 T = COST 3 {i.e. Total cost of processor P 3 } and total syste cost i.e. S ( X ) = TCOST ( X ) 1 + TCOST ( X ) 2 TCOST ( X ) 3 COST + Step-7: End. = = 528. Table-5 shows an optial allocation of tass to processors in DCS, for e present tas allocation odel. For is exaple, t5, t7 P1 ; t8, t9, t2, t3 P2 and t1, t4, t6 P3. Thus, e optial processor cost of P 1, P 2, and P 3 are 91, 137 and 300 respectively and e optial value of syste cost 528 wi e proposed algori. 4.2 Exaple-2 The efficacy of e proposed algori has been shown by solving e sae running exaple as in [26]. In is exaple, we have consider a DCS consists of ree processors P = {P 1, P 2, P 3 } and a typical progra ade up by 4- executable tass T = {t 1, t 2, t 3, t 4 }. Table 6 and table- 7 shows, e execution cost of each tas on processors and ITCC respectively. The results obtained wi e proposed algori and e algori presented in [26], for is exaple has been given below in table-8. In table -8, our algori shows at e proposed algori tries to iniize IPCC uch ore an e algori of H. Kuar et al [26]. Thus, e proposed algori produces lower syste cost in coparison to e algori presented in [26]. As we can observe fro table-8, e syste cost is iniized by 11.11% an at of [26] for is exaple. Hence, e proposed algori produces near optial allocation at all e ties. 5. CONCLUSION In is paper, we have looed at e proble of tas allocation in DCS. But, tas allocation proble is nown to be NP- hard proble in coplexity, when we required an optial solution to is proble. Therefore, we have proposed an efficient algori, which finds near optial syste cost for e DCS, having arbitrary structure of processors. We have used static tas allocation policy to achieve is objective. One of e best options to iniize e syste cost, is e iniization of IPC. Therefore, e proposed algori tries to allocate e tass to e processors on e basis of and found at, it is a good heuristic to iniize e syste cost. The perforance of e proposed algori is copared wi [26]. Also, e run tie coplexity of e proposed algori is O (n), which is very tie saving as copared to e coplexities of e algoris presented in [5,7,26]. Whose coplexities are O (n ), O (n ) and O ( 2 + n), respectively. For several sets of input data (, n), a coparison between e coplexities of e proposed algori and e coplexities of e algoris presented in [5, 7, 26], has been given in table-9 and figure-1and it is found at e proposed algori is suitable for a DCS having arbitrary inter connection of processors wi rando progra structure and worable in all e cases. Tass Table1. Execution cost atrix (,) t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t t t t t t t t t

6 Processors Tass Table.3 Modified execution cost atrix Table.5 An optial allocation of tass Optial allocation Tass Processors Total optial processor s cost t 5, t 7 P 1 91 t 8, t 9, t 2, t 3 P 1 P 2 P 3 t t t t t t t t t Processors P 1 P 2 P 3 Tass t t t t t t t t P t 1, t 4, t 6 P Total optial syste cost 528 Table 2. Inter tas counication cost atrix Table.4 Modified inter tas counication cost atrix Τass t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 9 t t t t t t t t Table.6 Execution cost atrix Processors Table.7 Inter tas counication cost atrix Τass t 1 t 2 t 3 t 4 t t t t Table.8 Coparison between e proposed algori t 2 Tass Nil t 3,t 4, t 1 Tass and e algori of H. Kuar et al [26] Proposed algori Processors P 1 P 2 P 3 P 1 P 2 P 3 t t t t Optial syste cost 24 H. Kuar et al. algori [26 ] Tass Processors t 2 P 1 Optial syste cost t 1, t 4 P 2 27 t 3 P 3 35

7 Table. 9 Results of run tie coplexity of e algoris S.No. Input cobination (Tass, Processors) 1 (4, 3) 2 (5, 3) 3 (6, 4) 4 (7, 4) 5 (8,5) 6 (9, 5) 7 (10, 6) 8 (11, 6) 9 (12,7) 10 (13,7) Run tie coplexity of e algoris Richard et al.[7], Peng et al. [5]O(n ) H. Kuar et al. [26] O( 2 +n) Proposed algori O(n) Figure 1.Coparision between e coplexities of e algoris 6. REFERENCES [1] G. Sagar and A.K. Sarje, Tas Allocation Model for Distributed Syste, Int. J. Systes Sci. Vol. 22, 9(1991). pp [2] A.K.Sarje and G.Sagar, Heuristic Model for Tas Allocation in Distributed Coputer Systes, IEE Proceedings-E, Vol.138, 5(1991). [3] Aji To P. and C. Siva Ra Mury, An Iproved Algori For Module Allocation in Distributed Coputing Systes, Journal of Parallel and Distributed Coputing Systes, 42 (1997),pp [4] M. Kafil and I. Ahad, Optial Tas Assignent in Heterogeneous Coputing Systes, /97$ IEEE [5] Peng, Dar- Tezen, Shin, K.G. and Abdel, Zoher,T.F., Assignent Scheduling Counication periodic Tass in Distributed Real Tie Syste, IEEE Transactions on Software Engineering, SE-13, (1997), pp [6] W.W.Chu, Leslie J.Holloway, Min-Tsung Lan, and Keal Kfe, Tas Allocation in Distributed Data Processing, IEEE Concurrency, Noveber 1980.pp [7] P-Y. Richard MA, Edward Y.S. Lee and Masahiro Tsuchiya, A Tas Allocation Model for Distributed Coputing Systes, IEEE Transactions on Coputers, Vol.C-31,1(1982), pp [8] C. C. Shen and W.H. Tasi, A Graph Matching Approach to Optial Tas Assignent in Distributed Coputing Systes Using a Miniax Criterion, IEEE Transactions on Coputers, Vol. C- 34, 3(1985). [9] Stone, H.S., Critical Load Factors in Two- Processor Distributed Syste, IEEE Transactions on Software Engrg. 4(May 1978),pp [10] Itiaz Ahad Muhaad K.Dhodhi and Arif Ghafoor, Tas Assignent in Distributed Coputing Systes, IEEE Concurrency (1995), pp [11] Cheol-Hoon Lee, Dongyun Lee and Myunghwan Ki, Optial Tas Assignent in Linear Array Networs, IEEE Transactions on Coputers, Vol.41, 7(1997). [12] Sol M.Shatz, Jia-Ping Wang, and Masanori Goto, Tas Allocation For Maxiizing Reliability of Distributed Coputer Systes, IEEE Transactions on Coputers, vol.41.9(1992). [13] S.Karti and C.Siva Ra Mury, Tas Allocation Algoris for Maxiizing Reliability of Distributed Coputing Syste, IEEE Transactions on coputers, Vol.46,6 (1997). [14] D.J. Chen et al., A Heuristic Algori for e Reliability- Oriented File Assignent in a Distributed Coputing Syste, Coputers Ma. Applic. Vol (1995), pp , [15] Peng-Yeng Yin, Shiuh-Sheng Yu, Pei-Pei Wang, Yi-Te Wang, Tas Allocation for Maxiizing Reliability of a Distributed Syste using Hybrid Particle Swar Optiization, The Journal of Systes and Software, 80(2007).pp [16] Sanana Srinivasan and Niraj K.Jha, Safety and Reliability Drivan Tas Allocation in Distributed Systes, IEEE transactions on Parallel and Distributed Systes, Vol.10. 3(1999). [17] D.P.Vidayari and A.K.Tripai, Maxiizing reliability of Distributed Coputing Syste wi Tas allocation using Siple Genetic Algori, Journal of Syste Architecture, 47(2001), pp [18] Pradeep Kuar Yadav, M.P. Singh and Kuldeep Shara, Tas Allocation Model for Reliability and Cost optiization in Distributed Coputing Syste, International Journal of odeling, siulation and scientific coputations, vol-2, 2(2011), pp

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