**7.3 ATSRABSR algorithm**

62 Grid Computing – Technology and Applications, Widespread Coverage and New Horizons

It is important to note that there is nothing that prevents a node to be part of both computefarm and data-farm. For example if the solution matrix has x3=1 and y3=1 and then n3 is used both in data-farms and compute-farms. Once the costs calculated with all the Architectural Templates are calculated, the minimum of them is chosen. If costmin= cost4-tier ,

For small number of nodes, ATSRAorg algorithm performs well. But as the number of nodes increases the time taken by the algorithm to run becomes considerable. ATSRASSR is proposed to improve performance for large number of nodes. It is based on finding a lower bound for the cost minimization problem formulated in Equation (7). The basic idea is to replace a "difficult" minimization problem by a simpler minimization problem whose

1. Enlarge the set of feasible solutions so that one optimizes. If the set of feasible solutions

2. Replace the minimum objective function of Equation (7) by a function that has the same

For the ATSRASSR, we have chosen the first approach. We formulate a new optimization problem called the relaxation of the original problem, using the same objective function but

that includes P as a subset. Because ��

Tenlarge the set of feasible solutions, constraint relaxation technique is used. In ATSRASSR, the constraint relaxation technique is used at the Architectural Template selection stage only. Once an Architectural Template has been chosen, exact LP formulation is used for resource allocation. It is thus named as ATSRA Single Stage Relaxation (SSR) or ATSRASSR,

such that ����

.

as well. This relaxed cost is denoted by cost4-tier-relaxed..

contains P, any solution

then resources are allocated according the values of the variables in solution matrix.

For the "relaxed" problem to have this property, there are two possibilities.

as the constraint relaxation is applied only to first stage of the algorithm.

*9 choose Architectural Template associated with cost*min

*10 Allocate resources for Architectural Template associated with costmin* 

**Summary of ATSRA SSR algorithm** 

*2 calculate cost*2-Tier-a *and cost*2-Tier-b *3 costmin = min(cost2-Tier-a, cost2-Tier-b)* 

*8 cost* min*= min(cost*4-Tier*,cost*min*)* 

*4 calcuate cost4-Tier-relaxed 5 If cost4-Tier-relaxed < costmin <* 

 *calculate cost*4-Tier

*6 goto step 9*

*7 else* 

xi, yi ���������� � ��to n

wij �������������� � ��to n, j=1 to k

**7.2 ATSRASSR algorithm** 

OR

optimal value is at least as small as cost4-tier.

or a smaller value everywhere.

which belongs to P, also belongs to ��

*1 initialize* 

a larger feasible region ��

is represented by P, then it means to find ��

In ATSRABSR algorithm we apply relaxation of the constraints at both Architectural Template Selection and resource allocation stages.

A summary of ATSRABSR is as follows:


The important thing to note is that in ATSRABSR, the constraint relaxation technique is used at both the Architectural Template selection stage and the relaxed solution matrix is used for actual resource allocation*.* For relaxation, constraints 3 and 4 are replaced by

 3 � � ≤ �� ≤ 1 4 � � ≤ �� ≤ 1 

Note that the constraint relaxed in ATSRASSR produces an invalid solution matrix. By dropping fifth constraint (i.e. ��� ≤ �� ), the variable wij can be assigned a non-zero value even if the corresponding data-farm node yi is not assigned. Thus the resultant solution matrix cannot be used in resource allocation. But in ATSRASSR, we are using this relaxation only for the selection of Architectural Template and if 4-tier is chosen then the exact LP formulation is used for actual resource allocation. For ATSRABSR, we have chosen such constraints for relaxation that do not produce an invalid solution matrix. Thus the same resultant solution matrix is used for resources allocation as well.

Note that as we move from ATSRAorg to ATSRASSR and then to ATSRABSR, following are some of the considerations.


The decrease in algorithm running time is the benefit of using this relaxation
