B1≤R1,1·X1+…+B1,m·Xm f(X1,X2,,Xm)=fi(n1,X1)+f2(n2,X2）+ B2≤R2,1·X1+…+R2.m·Xm m +fm(nm;Xm)= ∑f(n,X) (11) B:≤R.1·X1+.+R,m·Xm i=1 For example,the utility function fi(ni,Xi)can be defined as: Bn≤Rn,1·X1++Rn,m·Xm (10) 0≤X1≤ln(pca】 1 fi(ni,Xi)=Ei(n;-ni) (12) 0≤X2≤ln(pC) where n;denotes the cluster size of resource Ci after HA 0≤Xm≤ln(pa) enhancement,and E denotes the HA enhancement cost per unit (e.g.,it can include the initial fixed cost for purchasing hardware and software,and the annual maintenance cost).The The above constraints form a continuous region for the utility function is then determined by the business service solutions in the multi-dimensional space S(X1,X2.X3..... providers who want to provide appropriate IT resources to Xm).Let the utility function f be the overall cost of the HA support their business services at an appropriate cost;it may enhancement.Let us prove that the closed lower boundaries vary according to their demands.We can now calculate n; of the solution space include the optimal solution for the according to Xi. minimum enhancement cost (i.e.,we can achieve the optimal solution for the utility function f subject to the constrained solution space of the closed lower boundaries). P'(C)=1-(1-P(C)4 Theorem 1:The closed lower boundaries of the solution P(C)=P(C)·PC region in the multi-dimensional space S(X1,X2.X3.....Xm) Xi=In(PCi) include the optimal solution Ppt .In(1-P(Ci).ex) Proof:Assume there exists an optimal solution point P(X1,X2,...,Xm)in the constraint space beyond the closed →n4=[1m(1-PC) (13) lower boundaries;we need to prove that there exists a solution In the above formula,P(C:)denotes the enhanced avail- point that is a better solution than point P,and therefore the ability for resource Ci and P(Ci)denotes the availability of optimal solution Popt is located in the closed lower boundaries one single resource.Therefore the optimized recommendation of the constraint space. can be calculated with the utility function subject to the We definex,as the mapping from point constraint with the equation g(X1,X2.....Xm)=0.By using Pi(X1:...Xi,....Xm)to point Pi(X1,...,Xi,....Xm) the Lagrange multiplier method [10],we construct the auxil- in the closed lower boundary B:along decreasing direction iary function F(X1,X2,...,Xm,A)to calculate the optimized in the X;dimension: recommendation (see Equation 14),where f(X1,X2,...,Xm) B(X1,,X,,Xm）→x4P(X1,,X,,Xnm） denotes the utility function and g(X1,X2,...,Xm)denotes the .0<X:<X;and the utility function f always has function for the constraint space: positive correlation with enhancement parameter Xi, ..f(P(X1,...Xi,....Xm))<f(P(X1,....Xi,....Xm)). F(X1;X2,...,Xm;A)=f(X1,X2,...,Xm) Therefore solution P(X1,...,Xi,...,Xm)has lower cost +入·g(X1,X2,,Xm) (14) than P(X1,...Xi,...,Xm).Thus the former assumption that P(X1,X2,...,Xm)is an optimal solution point is untenable, By calculating the following partial derivatives according to which proves that the optimal solution exists on some closed the Lagrange multiplier method,we finally get the optimized lower boundary of the constraint space. recommendation (X1,X2,...,Xm). Therefore,the closed lower boundaries for the constraint space can be expressed with the equation: a晟FX1,X2,,Xm,)=0 g(X1,X2,,Xm)=0 FX1X2....Xm)=0 (15) where g(X1,X2,...,Xm)is a piecewise function that depicts the different closed boundaries.The optimized HA enhance- 最F(X1,X2,Xm,A)=0 ment recommendation is eventually determined by the overall With the optimized HA enhancement recommendation utility function.The utility function for a given resource (X1,X2,...Xm),we can get the enhanced availabilities Ci is associated with two parameters:ni,the original HA (P(C1),P(C2),..,P(Cm)),and the exact HA solutions cluster size of resource Ci(for standalone resources,ni is can be found (e.g.,whether a cluster should be constructed set to 1),and Xi,the enhancement parameter for resource Ci. and the size of that cluster).Assume we need n members Therefore,the utility function for resource Ci can be expressed to support the HA cluster;the availability capability for the as fi(ni,Xi),and the overall cost as: cluster is:   B1 ≤ R1,1 · X1 + ... + R1,m · Xm B2 ≤ R2,1 · X1 + ... + R2,m · Xm ... Bi ≤ Ri,1 · X1 + ... + Ri,m · Xm ... Bn ≤ Rn,1 · X1 + ... + Rn,m · Xm 0 ≤ X1 ≤ ln( 1 P (C1) ) 0 ≤ X2 ≤ ln( 1 P (C2) ) ... 0 ≤ Xm ≤ ln( 1 P (Cm) ) (10) The above constraints form a continuous region for the solutions in the multi-dimensional space S(X1, X2, X3,..., Xm). Let the utility function f be the overall cost of the HA enhancement. Let us prove that the closed lower boundaries of the solution space include the optimal solution for the minimum enhancement cost (i.e., we can achieve the optimal solution for the utility function f subject to the constrained solution space of the closed lower boundaries). Theorem 1: The closed lower boundaries of the solution region in the multi-dimensional space S(X1, X2, X3,..., Xm) include the optimal solution Popt. Proof: Assume there exists an optimal solution point P(X1, X2, ..., Xm) in the constraint space beyond the closed lower boundaries; we need to prove that there exists a solution point that is a better solution than point P, and therefore the optimal solution Popt is located in the closed lower boundaries of the constraint space. We define ⇒Xi as the mapping from point P1(X1, ..., Xi , ..., Xm) to point Pi(X1, ..., X0 i , ..., Xm) in the closed lower boundary Bi along decreasing direction in the Xi dimension: P1(X1, ..., Xi , ..., Xm) ⇒Xi Pi(X1, ..., X0 i , ..., Xm). ∵ 0 < X0 i < Xi and the utility function f always has positive correlation with enhancement parameter Xi , ∴ f(P1(X1, ..., X0 i , ..., Xm)) < f(P1(X1, ..., Xi , ..., Xm)). Therefore solution Pi(X1, ..., X0 i , ..., Xm) has lower cost than P1(X1, ..., Xi , ..., Xm). Thus the former assumption that P(X1, X2, ..., Xm) is an optimal solution point is untenable, which proves that the optimal solution exists on some closed lower boundary of the constraint space. Therefore, the closed lower boundaries for the constraint space can be expressed with the equation: g(X1, X2, ..., Xm) = 0, where g(X1, X2, ..., Xm) is a piecewise function that depicts the different closed boundaries. The optimized HA enhance￾ment recommendation is eventually determined by the overall utility function. The utility function for a given resource Ci is associated with two parameters: ni , the original HA cluster size of resource Ci (for standalone resources, ni is set to 1), and Xi , the enhancement parameter for resource Ci . Therefore, the utility function for resource Ci can be expressed as fi(ni , Xi), and the overall cost as: f(X1, X2, ..., Xm) = f1(n1, X1) + f2(n2, X2) + ... +fm(nm, Xm) = Xm i=1 fi(ni , Xi) (11) For example, the utility function fi(ni , Xi) can be defined as: fi(ni , Xi) = Ei(n 0 i − ni) (12) where n 0 i denotes the cluster size of resource Ci after HA enhancement, and Ei denotes the HA enhancement cost per unit (e.g., it can include the initial fixed cost for purchasing hardware and software, and the annual maintenance cost). The utility function is then determined by the business service providers who want to provide appropriate IT resources to support their business services at an appropriate cost; it may vary according to their demands. We can now calculate n 0 i according to Xi .    P 0 (Ci) = 1 − (1 − P(Ci))n 0 i P 0 (Ci) = P(Ci) · P Ci Xi = ln(P Ci) ⇒ n 0 i = d ln(1 − P(Ci) · e Xi ) ln(1 − P(Ci)) e (13) In the above formula, P 0 (Ci) denotes the enhanced avail￾ability for resource Ci and P(Ci) denotes the availability of one single resource. Therefore the optimized recommendation can be calculated with the utility function subject to the constraint with the equation g(X1, X2, ..., Xm) = 0. By using the Lagrange multiplier method [10], we construct the auxil￾iary function F(X1, X2, ..., Xm, λ) to calculate the optimized recommendation (see Equation 14), where f(X1, X2, ..., Xm) denotes the utility function and g(X1, X2, ..., Xm) denotes the function for the constraint space: F(X1, X2, ..., Xm, λ) = f(X1, X2, ..., Xm) +λ · g(X1, X2, ..., Xm) (14) By calculating the following partial derivatives according to the Lagrange multiplier method, we finally get the optimized recommendation (X1, X2, ..., Xm).    ∂ ∂X1 F(X1, X2, ..., Xm, λ) = 0 ∂ ∂X2 F(X1, X2, ..., Xm, λ) = 0 ... ∂ ∂λ F(X1, X2, ..., Xm, λ) = 0 (15) With the optimized HA enhancement recommendation (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 the size of that cluster). Assume we need n members to support the HA cluster; the availability capability for the cluster is: