Before revision:  Unfairness is common for some clients and they may always be regarded as outliers, so we also consider the fairness problem, and give randomized algorithms that embody lottery models, so that for each client j with its submitted parameter e_j , the expected number of connections for it is at least e_j.
After revision:  Unfairness is common for some clients and they may always be regarded as outliers, so we also consider the fairness problem, and give randomized algorithms that embody lottery models, so that for each client j with its submitted parameter e_j has a value e_j, and we need to stochastically open a set of facilities such that the expected number of connections for it is at least e_j.