Table 1 Summary of main symbols and functions used throughout the paper
Symbol/function | Definition |
|---|---|
2F1(·,·,·,·) | Gauss hypergeometric function |
\(\mathcal {K} = \left \{ {1,2} \right \}\) | Tier set: tier 1 is related to macro BSs, and tier 2 is related to small cell BSs |
\(\tilde j = \left \{ {k \in \mathcal {K}:k \ne j} \right \}\) | Complementary tier, i.e., \(\tilde 1~=~2\) and \(\tilde 2~=~1\) |
Φ(j),λ(j) | PPP and its density related to the locations of macro (j = 1) and small cell BSs (j = 2) |
λ MT | Density of the PPP of MTs’ positions |
Φ,λ | PPP and its density related to the locations of all BSs |
t ( j) | Association weight for tier j |
i0,p0,ε,pmax | Interference threshold, target receive power, partial compensation factor, and maximum transmit power |
τ,α | Path loss slope and path loss exponent |
MT0,MTi | Position of the probe MT and position of a generic MT, e.g., an interfering MT |
Ψ ( k) | PPP of interfering MTs’s locations |
\(R_{x,(q)}^{(j)}\) | Distance (including shadowing) between location x and the qth nearest BS from tier j |
\(\phantom {\dot {i}\!}R_{{\text {MT}}_{\mathrm {i}}},U_{{\text {MT}}_{\mathrm {i}}},D_{{\text {MT}}_{\mathrm {i}}}\) | Distances (including shadowing) between MTi and its serving BS, its most interfered BS and the probe BS |
\(\phantom {\dot {i}\!}H_{{\text {MT}}_{\mathrm {i}}}\) | Power gain of the multi-path fading which is exponentially distributed |
pMT(r)=p0(τr)αε | Transmit power for a given distance towards the serving BS for active MTs. Muted MTs has 0 transmit power |
\(\sigma _{n}^{2},I\) | Noise power and aggregate interference according to Assumption 1 |
\(\mathcal {X}_{{\text {MT}}_{i}}^{(j)}\) | Event defined as MTi is associated with tier j |
\(\mathcal {Q}_{{\text {MT}}_{i}}^{(m)}\) | Event defined as the most interfered BS of MTi belongs to tier m |
\(\mathcal {X}_{{\text {MT}}_{i}}^{(j,m)}\) | Event defined as MTi is associated with tier j and the most interfered BS of MTi belongs to tier m |
\(\mathcal {A}_{{\text {MT}}_{i}}\) | Event defined as MTi is active, i.e., non-muted |
\(\overline {\mathcal {A}_{{\text {MT}}_{i}}}\) | Event defined as MTi is muted |
\(\mathcal {O}_{{\text {MT}}_{i}}^{(j,k)}\) | Event defined as the interfering MTi of tierk receives higher weighted average power |
from its serving BS than from the probe BS that belong to tier j | |
\(\mathcal {Z}_{{\text {MT}}_{i}}\) | Event defined as the interfering MTi causes a level of interference less than i0 to the probe BS |
fX(·) | PDF (Probability Density Function) of random variable X |
\(\bar F_{X} (\cdot)\) | CCDF (Complementary Cumulative Distribution Function) of random variable X |
\({\mathcal {L}}_{X} (\cdot)\) | Laplace transform of random variable X |
f′(x0),f′′(x0) | First and second derivatives of function f(x) evaluated at x=x0 |
\(\mathbb {E}\left [\cdot \right ],\Pr \left (\cdot \right),{\bf {1}}\left (\cdot \right)\) | Expectation operator, probability measure and indicator function |
Γ(z), Γ(a,z) | Euler gamma function and incomplete gamma function |