Appendix for FCD Load Balancing under Switching Costs and Imperfect Observations

I. PROOF OF PROPOSITION 1 Proof: Consider a simple example of n = 2 servers in a super time slot. At t = 2k, assume without loss of generality that X 1 (2k) > X 2 (2k). At the exploration slot t = 2k+1 with exploration probability γ 2k , the expected number of users is thus E [X 1 (2k + 1)] = X 1 (2k)(1 − γ 2k) + X 2 (2k)γ 2k E [X 2 (2k + 1)] = X 2 (2k)(1 − γ 2k) + X 1 (2k)γ 2k. On one hand, each user a who switched from server 1 to server 2 experiences lighter load and switches back with probability f 21 := f (X 1 (2k), X 2 (2k + 1), γ 2k). On the other hand, each user b who switched from server 2 to server 1 experiences heavier load and hence, switches back with probability 1. At t = 2k + 2, the expected load in each server is thus where f 12 = 1. In order to ensure fast convergence to-Nash equilibrium, it is intuitive to have the expected loads in the two servers to be balanced after each round of the algorithm, i.e., setting E[X 1 (2k + 2)] = E[X 2 (2k + 2)], we obtain f 21 = 1 − 1 2γ 2k 1 − X 2 (2k) X 1 (2k). Extend to n-server scenario. In order for rapid convergence, the backtracking probability can be achieved by setting load balance of the next time slot. f ji (X i (2k), X j (2k)) is as follows: f ji = max 1 − n−1 nγ 2k 1 − Xj (2k) Xi(2k) , 0 , X j (2k) < X i (2k) 1, X j (2k) ≥ X i (2k) In the case that the exploration probability meets the condition that 0 ≤ γ 2k ≤