availability requirement of workflow Wi.In this way,the F(X1,X2,,Xm,)=f(X1,X2,,Xm) priority list of resources can be determined according to the weight.Those resources which support more workflows and +入·g(X1,X2,,Xm)】 (9) more availability-critical workflows will have higher weights. By calculating the following partial derivatives according According to the priority list,the top k resources can be to the Lagrange multiplier method.we can finally get the selected to calculate the HA solution:the calculated solution optimal solution (X1,X2....Xm).(Fdenotes to calculate will be a near optimal solution for only the k candidate re- the partial derivative function for F according to the variable sources which are taken into consideration,but the calculation X.) complexity can be greatly reduced according to the selected 是FX,X2…,Xm,)=0 number k. a成F(X1,X2,,Xm,)=0 D.Computational Complexity Analysis (10) In this section,we analyze the computational complexity 泉F(X,X2,,Xm,)=0 of the conventional exhaustive iteration method and our op- According to the optimal solution for resource HA enhance- timal solution calculation method.Assume that there exist ment (X1,X2,...,Xm),we can get the enhanced availabilities n candidate resources which need to be HA enhanced,and (P(C1),P(C2),...,P(Cm)),and the exact HA solutions we set the upper bound for the cluster size of any resource can be found (e.g,whether a cluster should be constructed to k (which is necessary for the iteration method but not and what is the size of cluster).Assume there should be n for our optimal solution calculation method).Then,for the members to support the HA cluster,the availability capability iteration method,the computational complexity to arrive at the for the cluster should be as follows: optimal solution is·k·..·k,that is,.O(k").For our optimal solution calculation method,since the solution is calculated P(C)=1-(1-P(C)m (11) by solving the equations 10,the computational complexity is According to the above formula,the size n of the cluster only bound by the number of variables in the equations,which can be calculated as follows: have the computational complexity of a polynomial:that is, O(nm),where m is a constant.Apparently,our method is n(1-P(C) scalable to the size of candidate resources,and has much lower n=[in(1-P(C)) (12) computational complexity than the iteration method when the Leveraging the domain information for the component,the number of candidate resources is large. HA cluster pattern can be generated and deployed into the E.Alternative Resource Selection topology. For some HA requirement analysis cases users may not C.Weight-based Optimization Approach to Reduce Calcula-be able to confirm the exact components of the resource,for tion Complexity example,for DB2 HA solution,the user is not sure whether Because the number of candidate resources for availabil-a hotstandby solution with X86 platform will well satisfy ity enhancement over the IT infrastructure can be large,it the HA requirement or a mainframe solution is better,user increases the computational complexity of calculating the may specify several candidate resources types for the exact optimal solutions by solving equations 10.Therefore,we pro- resource.Based on the above analysis,we further propose an pose a method to effectively reduce the number of candidate algorithm with alternative resource selection,as algorithm I resources,in order to simplify the calculation. shows.Here we abstract our availability weak point analysis The principle of our weight-based optimization approach is methodology into a function WeakPointAnalysis(ResourceList, to select a subset of the IT resources,based on weight,for use UtilityFunction,Topology.WorkflowList).As shown in al- in the optimal solution calculation.We note that,for those gorithm 1,we first generate all possible resource lists and resources which are involved in more workflows with more relevant utility functions according to the various candidate critical availability requirements,enhancing the availablity of resource types specified by user,then we leverage function these resources will yield better overall HA enhancement WeakPointAnalysis to calculate various solutions according to for the workflows,in a cost-efficient manner.Therefore,we those various resource lists.Thus we can finally decide the propose a weight-based method to select relevant resources best solution among those candidate solutions. as follows:for resource Ci,we define the weight for Ci F Example calculated as: In this section,we show a detail example to depict our optimal solution calculation work.Fig.4 shows an example W(C)= 〉(R·P) (13) topology for HA enhancement,there exist two candidate 1=1 resources standalone resource Ci and C2 over the original In the above formula,R.;denotes the Integer value defined topology which need to be availability enhanced,and resource in the workflow-resource mapping matrix.P denotes the C3 has been supported by a mainframe which needs no HAF(X1, X2, ..., Xm, λ) = f(X1, X2, ..., Xm) +λ · g(X1, X2, ..., Xm) (9) By calculating the following partial derivatives according to the Lagrange multiplier method, we can finally get the optimal solution (X1, X2, ..., Xm). ( ∂ ∂X F denotes to calculate the partial derivative function for F according to the variable X.)    ∂ ∂X1 F(X1, X2, ..., Xm, λ) = 0 ∂ ∂X2 F(X1, X2, ..., Xm, λ) = 0 ... ∂ ∂λ F(X1, X2, ..., Xm, λ) = 0 (10) According to the optimal solution for resource HA enhance￾ment (X1, X2, ..., Xm), we can get the enhanced availabilities (P 0 (C1), P0 (C2), ..., P0 (Cm)), and the exact HA solutions can be found (e.g., whether a cluster should be constructed and what is the size of cluster). Assume there should be n members to support the HA cluster; the availability capability for the cluster should be as follows: P 0 (Ci) = 1 − (1 − P(Ci))n (11) According to the above formula, the size n of the cluster can be calculated as follows: n = d ln(1 − P 0 (Ci)) ln(1 − P(Ci)) e (12) Leveraging the domain information for the component, the HA cluster pattern can be generated and deployed into the topology. C. Weight-based Optimization Approach to Reduce Calcula￾tion Complexity Because the number of candidate resources for availabil￾ity enhancement over the IT infrastructure can be large, it increases the computational complexity of calculating the optimal solutions by solving equations 10. Therefore, we pro￾pose a method to effectively reduce the number of candidate resources, in order to simplify the calculation. The principle of our weight-based optimization approach is to select a subset of the IT resources, based on weight, for use in the optimal solution calculation. We note that, for those resources which are involved in more workflows with more critical availability requirements, enhancing the availablity of these resources will yield better overall HA enhancement for the workflows, in a cost-efficient manner. Therefore, we propose a weight-based method to select relevant resources as follows: for resource Cj , we define the weight for Cj calculated as: W(Cj ) = Xn i=1 (Ri,j · Pi) (13) In the above formula, Ri,j denotes the Integer value defined in the workflow-resource mapping matrix. Pi denotes the availability requirement of workflow Wi . In this way, the priority list of resources can be determined according to the weight. Those resources which support more workflows and more availability-critical workflows will have higher weights. According to the priority list, the top k resources can be selected to calculate the HA solution; the calculated solution will be a near optimal solution for only the k candidate re￾sources which are taken into consideration, but the calculation complexity can be greatly reduced according to the selected number k. D. Computational Complexity Analysis In this section, we analyze the computational complexity of the conventional exhaustive iteration method and our op￾timal solution calculation method. Assume that there exist n candidate resources which need to be HA enhanced, and we set the upper bound for the cluster size of any resource to k (which is necessary for the iteration method but not for our optimal solution calculation method). Then, for the iteration method, the computational complexity to arrive at the optimal solution is k · k · ... · k | {z } n ; that is, O(k n). For our optimal solution calculation method, since the solution is calculated by solving the equations 10, the computational complexity is only bound by the number of variables in the equations, which have the computational complexity of a polynomial: that is, O(n m), where m is a constant. Apparently, our method is scalable to the size of candidate resources, and has much lower computational complexity than the iteration method when the number of candidate resources is large. E. Alternative Resource Selection For some HA requirement analysis cases users may not be able to confirm the exact components of the resource, for example, for DB2 HA solution, the user is not sure whether a hotstandby solution with X86 platform will well satisfy the HA requirement or a mainframe solution is better, user may specify several candidate resources types for the exact resource. Based on the above analysis, we further propose an algorithm with alternative resource selection, as algorithm 1 shows. Here we abstract our availability weak point analysis methodology into a function WeakPointAnalysis(ResourceList, UtilityFunction, Topology, WorkflowList). As shown in al￾gorithm 1, we first generate all possible resource lists and relevant utility functions according to the various candidate resource types specified by user, then we leverage function WeakPointAnalysis to calculate various solutions according to those various resource lists. Thus we can finally decide the best solution among those candidate solutions. F. Example In this section, we show a detail example to depict our optimal solution calculation work. Fig.4 shows an example topology for HA enhancement, there exist two candidate resources standalone resource C1 and C2 over the original topology which need to be availability enhanced, and resource C3 has been supported by a mainframe which needs no HA