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

From: Joint congestion control and resource allocation for energy-efficient transmission in 5G heterogeneous networks

\({\mathcal {S}}\) 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