t. C1 EGST, (7.1b) (7.1c) (7.1d) C4 0 , (7.1e) (7.1f) where Wk is the weighting factor that can be used to prioritize different QoS requirements of UEs. P is a resource allocation problem that optimizes the offloading transmission time tk and power Pk, as well as local computing chip frequency fk. C states that all the tasks 7 Mobile Edge Computing in 5G Wireless Networks should be completed before the end of the block. Notice that here we omit the processing and transmission time at the server by following [15, 199]. Lk in C2 denotes the minimum data bits for computing for UE k. ER in C3 is the total energy available in UE k. C4 defines the maximum CPU frequency of each UE. The above problem is non-convex since the objective function involves sum-of-ratio max- imization. Also, the coupling of some variables makes the optimization problem even more complicated. To address the coupling problem, let Pk = Pktk. Besides, for notational brevity, denote Rk(Pkstk.,kk) = and = first employ simple transformations and the original problem becomes: (7.2a) s.t. C1 : (7.2b) C2 : E4ST (7.2c) (7.2d) (7.2e) (7.2f) C6 : Ek(Pktk,)k) Ak. (7.2g) Lemma 7.1 For kifk))is the optimal solution of P2, there must exist {2k} such that ({t}, {f*}, {Pk}) satisfies the Karush-Kuhn-Tucker condition of the following problem for Ak= Bk= P3 : max (7.3a) s.t. (7.2c) - (7.2g). (7.3b) Furthermore, {P}) satisfies the following equations for Bk= (7.4) Lemma 7.1 can be proved by taking the derivative of the Lagrange function of P2. Nk is the non-negative multiplier of (7.2b). A detailed proof can be obtained in [92]. Lemma 7.1 implies that the optimal solution of P2 can be obtained by solving the equations of (7.4) among the solutions of P3. The Lagrange function of P3 is (7.5) 7.3 Problem Formulation where Ak, Mk, Ok, Nk, and m are non-negative Lagrange multipliers for the respective con- straints. It can be readily proved that P3 3 is convex for given Nk and Bk, Ak, and satisfies Slater's condition. Thus, strong duality holds between the primal and du