P?n, it can be shown that ?2Cn(Pn)?P n2 > 0. C.4 Proof of Theorem 5.2 The reflection point is P?n = Ln?22(Ln?1)( ?P +An). As Ln ? ?, we have P?n = 0.5( ?P +An). Only one link can operate in the convex region due to constraint (5.17). Since ?P?n?Ln > 0, P?n is an increasing function of Ln. When 1 ?Ln 0, for Ln > 2. The second part can be easily shown by evaluating (5.14), (5.15), and (C.1). 170 Appendix D Proofs in Chapter 6 D.1 Proof of Lemma 6.1 According to the definition of Xm(t) in (6.3), we have Cm(t) = [Xm(t)?Xm(t?1)]/?. From the definition ofCminm (t), the playout buffer is emptied at the end of time slott, i.e., Xm(t) = Dm(t). Therefore, we can derive the minimum required rate as Cminm (t) = max{0,Dm(t)?Xm(t?1)}/?. (D.1) From the feasibility condition (6.4), we have Xm(t?1) ? Dm(t?1). Substituting it into (D.1), we have Cminm (t) ? [Dm(t)?Dm(t?1)]/? ? ?Cminm (t). (D.2) Rate ?Cminm (t) occurs when the playout buffer is empty at both the beginning and end of time slot t, but without buffer overflow during the entire time slot. D.2 Proof of Theorem 6.1 Recall that?minm is the SINR corresponding to the minimum required rateCminm (t). Let ??minm (t) be the SINR corresponding to ?Cminm (t). Since (6.2) is a monotonically increasing function, we have 0 ??minm (t) ? ??minm (t). We now consider the power assignment that achieves rates ?Cminm (t), or, the corresponding SINRs ??minm (t). From (6.7) and (6.8), the minimum SINR constraint is: ?m(t) = G m mPm(t)summationtext knegationslash=mG m k Pk(t) +?m ? ??minm (t), ?m. (D.3) 171 Eqn. (D.3) is a system of linear equations of the power vector vectorP(t), which can be written in the matrix form as: parenleftbigI???minAparenrightbigvectorP(t) followsequal ??minvector?, (D.4) where I is the identity matrix, A is an M ?M matrix with Amk = ?? ? ?? 0, m = k Gmk /Gmm, mnegationslash= k, (D.5) ??min = diag{??min1 (t),??min2 (t),??? ,??minM (t)} is a diagonal matrix, and vector? = [?1/G11,?2/G22,??? , ?M/GMM]T. Define ?min = diag{?min1 (t),?min2 (t),??? ,?minM (t)} and ? = ??min ??min followsequal 0. Assume vectorP is a power assignment that achieves ??minm (t) for all m, which satisfies (D.4). Substituting ??min = ?+?min into (D.4), we have parenleftbigI??minAparenrightbigvectorP followsequal?minvector? +?parenleftBigvector? +AvectorPparenrightBig. Since ?, vector?, A and vectorP all have non-negative elements, we have ? parenleftBig vector? +AvectorP parenrightBig followsequal0, and therefore, parenleftbigI??minAparenrightbigvectorP followsequal?minvector?. That is, vectorP can also achieve ?minm (t) for all m and it satisfies the minimum SINR constraint in (6.8). Once the minimum SINR constraint in (6.8) (i.e., no buffer underflow) is satisfied, the max- imum SINR constraint in (6.8) (i.e., no buffer overflow) can be satisfied since BS m can stop transmission when the playout buffer at user unm is full. 172 Appendix E Proofs in Chapter 7 E.1 Proof of Theorem 7.1 Due to i.i.d. channel gains and noise powers, the random variables ?n(t)/Gn(t)?s are ex- changeable, for all t. Define w(g,?,c) = (2c/Bw ?1)?/g, which is convex and increasing with c, for all g ? 0 and ? ? 0. Let ?(vectorC) = E[?(vectorC)] = E[summationtextTt=1w(g(t),?(t),c(t))]. ?(vectorC) is a symmet- ric, convex and increasing function in vectorC for each fixed vectorG and vector?. According to Proposition 11.B.5 in [13], ?(vectorC) is symmetric, convex and increasing. Following Fact 2.1, the objective function (7.9) is Schur-convex and increasing. E.2 Proof of Corollary 7.2.3 To evaluate the smoothness of a transmission schedule vectorC, the following smoothness utility function can be used: U(vectorC) = Tnsummationdisplay t=1 ([c(t)??c]/Tn), (E.1) where ?c = summationtextTnt=1c(t)/Tn is the average rate. This is a continuous symmetric convex function U : RTn ? R. From Fact 2.1, U is Schur-convex and order preserving. The optimal power transmission schedule vectorC?n satisfies vectorC?n ? vectorCin for all i. Therefore, it also achieves the minimum value for U(?). 173 Appendix F Acronyms AMI Automated Meter Infrastructure BS Base Station CBR Constant Bit Rate CDMA Code Division Multiple Access CPS Cyber-Physical System CSMA Carrier Sense Multiple Access DCC Distribution Control Center DCPC Distributed Constrained Power Control DCT Discrete Cosine Transform DR Demand Response DRER Distributed Renewable Energy Resource DUBMLC Distributed User Benefit Maximization Load Control DVS Dynamic Voltage Scaling ESS Energy Storage System GSEPS General Smooth Electric Power Scheduling FDMA Frequency-Division Multiple Access HAN Home Area Network ICT Information and Communications Technology IDCT Inverse DCT LAN Local Area Network LB Lower Bound LP Linear Programming 174 MB Macro Block MG Microgrid MGCC MG Central Controller PMA Power Minimization Algorithm PMU Phasor Measurement Unit PHEV Plug-in Hybrid Electric Vehicles PLC Power Line Communication QoE Quality of Experience QoS Quality of Service QoSE Quality of Service in Electricity RLT Reformulation-Linearization Technique RTP Real-time Transport Protocol SEPS-DL Smooth Electric Power Scheduling for Deferrable Load SINR Signal to Interference-plus-Noise Ratio SG Smart Grid SST Solid State Transformer SUDP Supply Until Deadline Policy TCP Transmission Control Protocol TDMA Time Division Multiple Access UB Upper Bound UDP User Datagram Protocol UMRP Utility Maximization Real-time Pricing V2G Vehicle-to Gird VBR Variable Bit Rate VPP Virtual Power Plant VSN Visual Sensor Networks WAN Wide Area Network 175 Bibliography [1] N. 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