Table 2 Table of notations

\(\mathcal {S}=\left \{ s_{1},...,s_{m}\right \}\)

Set of small cells in the whole network, \(m\in \mathbb {N}^{*}\), and \(\mathbb {N}^{*}\) represents the set of positive integers.

\(\mathcal {B}=\left \{ b_{1},...b_{n}\right \}\)

Set of PRBs in the system resource pool, \(n\in \mathbb {N}^{*}\).

\(\mathcal {U}=\left \{u_{1},...,u_{o}\right \}\)

Set of users (UE) of a small cell, \(o\in \mathbb {N}^{*}\).

R _i,j,k

The rate user u _k obtained based on PRB b _j in small cell s _i .

W

Bandwidth of per PRBs.

Sinr_i,j,k

SINR u _k received based on PRBs b _j in small cell s _i .

g _i,j,k

Channel gain.

p

Transmitted power of small cell.

n ₀

Noise power.

\(\mathcal {BS}_{i}=\left \{ b_{1}s_{i},b_{2}s_{i},..., \right \}\)

The set of available PRBs in small cell s _i .

R _k

A constant to indicate the rate demand of user u _k .

n

Total number of PRB in the system resource pool in initial state.

\(n^{'}_{i}\)

Total number of PRBs in the system resource pool before each pre-allocation, \(n^{'}_{i}\leqslant n\).

i

{i|1≤i≤m}, m is the total number of small cells.

\(n^{'}_{i}-Y_{i}\)

The remainder PRBs in the system resource pool after pre-allocation for s _i .

I

An interference matrix of small cells.

Y _i

The number of allocated PRBs in small cell s _i .

\(\mathcal {S}^{'}=\left \{ s_{1}^{'},...,s_{m}^{'} \right \}\)

The set of small cells in the whole network in descending order based on the degree of vertex in

interference graph.

\(\mathcal {BS}_{i}^{'}\)

The set of PRBs in small cell s _i after global pre-allocation.

\(\mathcal {B}^{'}\)

Total number of PRBs in the system resource pool before resource optimization.

\(\mathcal {B}^{''}\)

Total number of PRBs in the system resource pool after resource optimization.

\(\mathcal {BZ}_{s_{i}}\)

The idle PRB list belonging to adjacent cells of small cell s _i .

\(\mathcal {Z}_{s_{i}^{''}}\)

The adjacent cell set of small cell \(s_{i}^{''}\).

O _downstream

System downlink throughput.

\(O_{\text {downstream}}^{'}\)

System downlink throughput after each resource coordination.

t _delay

Average processing time caused by PRB switching.

V _j

The value of PRB b _j .

\(\mathcal {S}^{''}=\left \{ s_{1}^{''}, s_{2}^{''},...\right \}\)

The set of small cells with new PRB demands.

\(\mathcal {BS}_{l}^{'}\)

The set of PRBs in ascending order based on the value in the adjacent cell s _l of small cell s _l .

\(O_{\text {downstream}}^{'}\)

System downlink output after each resource coordination.

w _j,l

The order of PRB b _j in \(\mathcal {BS}^{'}_{l}\).

L _l

The rate loss of the adjacent cell s _l of small cell s _i .

\(\mathcal {TS}_{l}\)

The set of transferred PRBs in small cell s _l .