Table 1 A list of important symbols used in the problem formulation

\({\mathcal {S}}\)

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

Z_u(t),H_u(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

R_u(t),μ_u(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

η_EE,ηEEreq

Energy efficiency and its requirement