This paper proposes a robust optimization model for probabilistic protection under uncertain capacity demands to minimize the total required capacity against multiple simultaneous failures of physical machines. The proposed model determines both primary and backup virtual machine allocations simultaneously under the probabilistic protection guarantee. To express the uncertainty of capacity demands, we introduce an uncertainty set that considers the upper bound of the total demand and the upper and lower bounds of each demand. The robust optimization technique is applied to the optimization model to deal with two uncertainties: failure event and capacity demand. With this technique, the model is formulated as a mixed integer linear programming (MILP) problem. To solve larger sized problems, a simulated annealing (SA) heuristic is introduced. In SA, we obtain the capacity demands by solving maximum flow problems. Numerical results show that our proposed model reduces the total required capacity compared with the conventional model by determining both primary and backup virtual machine allocations simultaneously. We also compare the results of MILP, SA, and a baseline greedy algorithm. For a larger sized problem, we obtain approximate solutions in a practical time by using SA and the greedy algorithm.

This paper presents robust optimization models for minimizing the required backup capacity while providing probabilistic protection against multiple simultaneous failures of physical machines under uncertain virtual machine capacities in a cloud provider. If random failures occur, the required capacities for virtual machines are allocated to the dedicated backup physical machines, which are determined in advance. We consider two uncertainties: failure event and virtual machine capacity. By adopting a robust optimization technique, we formulate six mixed integer linear programming problems. Numerical results show that for a small size problem, our presented models are applicable to the case that virtual machine capacities are uncertain, and by using these models, we can obtain the optimal solution of the allocation of virtual machines under the uncertainty. A simulated annealing heuristic is presented to solve large size problems. By using this heuristic, an approximate solution is obtained for a large size problem.