Workflow multivariate optimization problem and calculate a near-optimal App HA enhancement recommendation using the Lagrange multi- plier method [10],which is computationally more effective Dependency chain that Janakiraman et al.'s approach. Assume the number of workflows whose availability re- Group Resource quirements have not yet been met is n.For workflow Wi Reso■ce Resource DB Instance we define the enhancement parameter PWi as the amount WAS WAS by which that workflow's current availability needs to be ↓ enhanced to meet the availability requirement P: 03 03 OS Hardware Haoae Hardware P PW:=P(Wi) (7) Fig.8.Availability calculation example By definition,PWi>1.We also define the enhancement parameter for each resource as PC1,PC2,...,PCm.This C.Weak-Point Analysis Algorithm yields the following constraints: Given the workflow-resource relationship matrix,we can calculate the current availability capability for each business workflow.As seen in Section II.B,we denote the availability PW1≤PCB,1.PC2.PCRm of the m IT resources as P(C1).P(C2),P(C3),...P(Cm). These availabilities can be calculated based on availability PW2≤PCR.PCa2.…PCRm characteristics of the individual resource components,which 4 (8) are derived from historical measurements or manufacturer's PW≤PC1.PC2PCRm evaluation data.As captured in the workflow-resource re- lationship matrix,a workflow can depend on a given IT PWn≤PCR1.PC2.PCRm resource in several ways.Assume that the relevant IT resources appear several times in an IT resource workflow.We can then calculate the current availability for each workflow by using In other words.the overall availability enhancement for Equation 6: the IT resources within the workflow should be no less than the availability enhancement requirement for the workflow. P(w)=ΠI(P(C)R) To simplify the calculations,we take the logarithm of the (6) inequalities in Equation 8. j=1 where P(Wi)is the current availability capability for work- flow Wi,and Rij is the number of times resource Ci is referenced by workflow Wi.We then compare the calculated n(PWi)≤R1,1·ln(PC)+…+R1,m·n(PCm) availability with the workflow availability requirement Pi:if n(PW2)≤R2.1·ln(PC1)+.+R2,m·n(PCm） P(Wi)>P,the requirement is met;otherwise,the avail- ability requirement is unsatisfied,and some resources in the n(PW)≤Ri.1·ln(PC)+.+R.m·ln(PCm) (9) resource list of workflow Wi need to have their availability enhanced using some HA pattern.This is an optimization In (PWn)<Rn.1.In (PC1)+...+Rn,m.In(PCm) problem: Find which resources should be enhanced for the availability to meet the availability requirements, For notational convenience,we replace In(PC1), while keeping HA enhancement cost as low as In (PC2)...In(PCm)by X1,X2....Xm:there exists possible. 0≤X,≤h(pa)because1≤PC,≤p:Fora A simple method of addressing an optimization problem failover HA pattern where only one primary server and one is by enumerating all possible solutions and comparing their standby server exist in the cluster,we can adjust the upper cost;however,this is computationally expensive for all but bound to In().For cluster HA pattems,weca the simplest problems.Janakiraman et al.[9]propose an adjust the lower bound from 0 to In()if we P(C)】 approach to search for the optimal solution through multi-tier want the initial cluster size to be n;instead of 1,and we system design,based on exhaustive iteration.In our weak- substitute B1.B2....,Bn for In(PW1),In(PW2),...,In(PWn). point analysis methodology,we represent the problem as a Therefore the following constraints should be satisfied:Fig. 8. Availability calculation example C. Weak-Point Analysis Algorithm Given the workflow-resource relationship matrix, we can calculate the current availability capability for each business workflow. As seen in Section II.B, we denote the availability of the m IT resources as P(C1), P(C2), P(C3), ..., P(Cm). These availabilities can be calculated based on availability characteristics of the individual resource components, which are derived from historical measurements or manufacturer’s evaluation data. As captured in the workflow-resource re￾lationship matrix, a workflow can depend on a given IT resource in several ways. Assume that the relevant IT resources appear several times in an IT resource workflow. We can then calculate the current availability for each workflow by using Equation 6: P(Wi) = Ym j=1 (P(Cj ) Ri,j ) (6) where P(Wi) is the current availability capability for work- flow Wi , and Ri,j is the number of times resource Cj is referenced by workflow Wi . We then compare the calculated availability with the workflow availability requirement Pi : if P(Wi) ≥ Pi , the requirement is met; otherwise, the avail￾ability requirement is unsatisfied, and some resources in the resource list of workflow Wi need to have their availability enhanced using some HA pattern. This is an optimization problem: Find which resources should be enhanced for the availability to meet the availability requirements, while keeping HA enhancement cost as low as possible. A simple method of addressing an optimization problem is by enumerating all possible solutions and comparing their cost; however, this is computationally expensive for all but the simplest problems. Janakiraman et al. [9] propose an approach to search for the optimal solution through multi-tier system design, based on exhaustive iteration. In our weak￾point analysis methodology, we represent the problem as a multivariate optimization problem and calculate a near-optimal HA enhancement recommendation using the Lagrange multi￾plier method [10], which is computationally more effective that Janakiraman et al.’s approach. Assume the number of workflows whose availability re￾quirements have not yet been met is n. For workflow Wi we define the enhancement parameter PWi as the amount by which that workflow’s current availability needs to be enhanced to meet the availability requirement Pi : PWi = Pi P(Wi) (7) By definition, PWi ≥ 1. We also define the enhancement parameter for each resource as P C1, P C2, ..., P Cm. This yields the following constraints:    PW1 ≤ P CR1,1 1 · P CR1,2 2 · ... · P CR1,m m PW2 ≤ P CR2,1 1 · P CR2,2 2 · ... · P CR2,m m ... PWi ≤ P CRi,1 1 · P CRi,2 2 · ... · P CRi,m m ... PWn ≤ P CRn,1 1 · P CRn,2 2 · ... · P CRn,m m (8) In other words, the overall availability enhancement for the IT resources within the workflow should be no less than the availability enhancement requirement for the workflow. To simplify the calculations, we take the logarithm of the inequalities in Equation 8.    ln (PW1) ≤ R1,1 · ln (P C1) + ... + R1,m · ln (P Cm) ln (PW2) ≤ R2,1 · ln (P C1) + ... + R2,m · ln (P Cm) ... ln (PWi) ≤ Ri,1 · ln (P C1) + ... + Ri,m · ln (P Cm) ... ln (PWn) ≤ Rn,1 · ln (P C1) + ... + Rn,m · ln (P Cm) (9) For notational convenience, we replace ln (P C1), ln (P C2),..., ln (P Cm) by X1, X2,...,Xm: there exists 0 ≤ Xi ≤ ln ( 1 P (Ci) ) because 1 ≤ P Ci ≤ 1 P (Ci) . For a failover HA pattern where only one primary server and one standby server exist in the cluster, we can adjust the upper bound to ln( 1−(1−P (Ci))2 P (Ci) ). For cluster HA patterns, we can adjust the lower bound from 0 to ln( 1−(1−P (Ci))ni P (Ci) ) if we want the initial cluster size to be ni instead of 1, and we substitute B1, B2,...,Bn for ln(PW1), ln(PW2),..., ln(PWn). Therefore the following constraints should be satisfied: