$${\mathcal {S}}$$ Set of base stations (BSs), with index $$s \in \left \{1,..., \mathbf {s} \stackrel {\triangle }{=} |{\mathcal {S}}| \right \}$$ $${\mathcal {P}}$$ Set of power levels (PLs), with index $$p \in \left \{1,..., \mathbf {p} \stackrel {\triangle }{=} |{\mathcal {P}}|\right \}$$ $${\mathcal {U}}$$ Set of user equipments (UEs), with index $$u \in \left \{1,..., \mathbf {u} \stackrel {\triangle }{=} |{\mathcal {U}}|\right \}$$ $${\mathcal {C}}$$ Set of component carriers (CCs), with index $$c \in \left \{1,..., \mathbf {c} \stackrel {\triangle }{=} |{\mathcal {C}}|\right \}$$ $${\mathcal {B}}$$ Set of resource blocks (RBs) per CC, with index $$b \in \left \{1,..., \mathbf {b} \stackrel {\triangle }{=} |{\mathcal {B}}|\right \}$$ $${\mathcal {L}}$$ Set of modulation and coding schemes (MCSs) per RB, with index $$l \in \left \{1,..., \mathbf {l} \stackrel {\triangle }{=} |{\mathcal {L}}| \right \}$$ $${\mathcal {N}}_{s}$$ Set of neighboring cells of cell s, which may interfere with s when given the same RBs f s Maximum number of CCs able to be used by cell s k u Maximum number of CCs able to be assigned to UE u W Balance weight Ψ u,c,b,s,p Index of the highest-rate MCS used by UE u on RB b of CC c at PL p of cell s r l Achieved transmission rate of an RB on MCS l $$\underline {e}_{{u,c,b,l,s,p}}$$ Binary variable showing if RB b of CC c on MCS l at PL p of cell s is assigned to UE u $$v(\underline {e})$$ Achieved transmission rate with the allocation indicated by $$\underline {e}$$ $$\hat {\underline {e}}, \hat {\underline {e}}_{1},..., \hat {\underline {e}}_{10}$$ and $$\tilde {\underline {e}}_{4}, \tilde {\underline {e}}_{5}$$ Binary variables similar to $$\underline {e}$$ with certain elements to be varied while fixing the others, for the different scenarios shown in Section 3.2 and Eqs. (5)–(12), (14), (16), and (18)–(21) $$\hat {\underline {y}}_{1},..., \hat {\underline {y}}_{5}$$ Auxiliary variables for the different scenarios shown in Eqs. (13), (15), (17), and (18)–(21) $$P_{\text {tot}}(t), \overline {P}_{\text {tot}}, P_{\text {max}}$$ Total power consumption at time t, its limit of the time-average expectation, and the maximum transmit power $$R_{\text {tot}}(t), \overline {R}_{\text {tot}}$$ Total data rate at time t, and its limit of the time-average expectation $$Q(t), \overline {Q}$$ Data queue length at time t, and its limit of the time-average expectation Zu(t),Hu(t) Virtual queue lengths at time t corresponding to (27-C5) and (27-C6), respectively, for UE u $$A_{u}(t), A_{u}^{\text {max}}, \lambda _{u}$$ Traffic arrival of UE u at time t, its maximum value allowed, and arrival rate of UE u Ru(t),μu(t) Admitted traffic of UE u, and link (service) rate of UE u, at time t $$\overline {r}_{u}$$ Time-average throughput of UE u ηEE,ηEEreq Energy efficiency and its requirement