Ferkans — Interactive Telecom Tutor

The Densification Imperative

The 1000 $\times$ traffic growth forecast for cellular networks cannot be met by spectral efficiency improvements alone — the Shannon limit constrains per-link gains. The dominant lever is network densification: deploying many low-power small cells (picocells, femtocells, relay nodes) within the coverage footprint of existing macro cells. The resulting heterogeneous network (HetNet) presents a fundamentally different analytical challenge: cells differ in transmit power, antenna height, and backhaul capacity, creating an asymmetric interference landscape. The central question is how to associate users to cells — the strongest signal is not always the best choice when load balancing is considered.

Definition:
Heterogeneous Network (HetNet)

A heterogeneous network (HetNet) consists of $K$ tiers of base stations, where tier $k$ has:

BS locations forming an independent PPP $\Phi_k$ with intensity $\lambda_k$ ,
Transmit power $P_k$ ,
Path-loss exponent $\alpha_k$ (often assumed equal across tiers).

A typical user connects to the BS providing the strongest biased received power:

$x^{\star} = \arg\max_{k, x_i \in \Phi_k} B_k P_k \|x_i\|^{-\alpha_k}$

where $B_k \geq 1$ is the cell range expansion (CRE) bias for tier $k$ . Setting $B_k = 1$ for all tiers gives max-received-power association; $B_k > 1$ for small cells offloads users from macro to small cells.

Definition:
Cell Range Expansion (CRE) Bias

The cell range expansion bias $B$ (in linear scale, or $B_{\text{dB}}$ in dB) artificially inflates the perceived signal strength of a small cell during cell selection. A user associates with the small cell if:

$P_{\text{small}} \|x_{\text{small}}\|^{-\alpha} \cdot B > P_{\text{macro}} \|x_{\text{macro}}\|^{-\alpha}$

The CRE-expanded coverage region of a small cell has effective radius:

$R_{\text{eff}} = R_0 \cdot B^{1/\alpha}$

where $R_0$ is the unbiased coverage radius. Typical values in LTE are $B_{\text{dB}} = 0$ -- $15$ dB.

CRE offloads users from congested macro cells to lightly loaded small cells. However, CRE users in the expanded region receive a weaker signal from their serving small cell than from the macro cell, creating a CRE zone where additional interference protection (such as eICIC with Almost Blank Subframes) is needed.

Theorem: Coverage Probability in a Two-Tier HetNet

Consider a two-tier HetNet with macro BSs (PPP $\Phi_1$ , intensity $\lambda_1$ , power $P_1$ ) and small BSs (PPP $\Phi_2$ , intensity $\lambda_2$ , power $P_2$ , bias $B$ ). Under Rayleigh fading with common path-loss exponent $\alpha > 2$ and interference-limited conditions, the association probability to the small-cell tier is:

$A_2 = \frac{\lambda_2 (B P_2 / P_1)^{2/\alpha}} {\lambda_1 + \lambda_2 (B P_2 / P_1)^{2/\alpha}}$

The overall coverage probability is:

$p_c(\tau) = A_1 \cdot p_{c,1}(\tau) + A_2 \cdot p_{c,2}(\tau)$

where $p_{c,k}(\tau)$ is the conditional coverage probability given association with tier $k$ . In the interference-limited regime, each $p_{c,k}(\tau)$ has the same functional form as the single-tier result but with a modified interference field.

The bias $B$ expands the effective footprint of small cells, increasing $A_2$ and offloading traffic from macro cells. With $B = 1$ and $P_2 \ll P_1$ , almost all users associate with macro cells despite the much larger number of small cells. The bias corrects this imbalance by accounting for load: a user may get higher throughput from a weaker small cell that serves fewer users than from a stronger but congested macro cell.

Proof

Association regions via Voronoi

Under max-biased-power association, user at origin connects to tier $k$ if the nearest tier- $k$ BS (at distance $r_k$ ) satisfies $B_k P_k r_k^{-\alpha} > B_j P_j r_j^{-\alpha}$ for all $j \neq k$ .

For tier 2 (small cells with bias $B$ ):

$A_2 = \Pr\!\left[B P_2 r_2^{-\alpha} > P_1 r_1^{-\alpha}\right]$

Since $r_k^2$ is exponential with rate $\pi\lambda_k$ :

$A_2 = \frac{\lambda_2 (B P_2/P_1)^{2/\alpha}} {\lambda_1 + \lambda_2 (B P_2/P_1)^{2/\alpha}}$

Conditional coverage

Conditional on associating with tier $k$ at distance $r$ , the coverage probability follows from the Laplace functional of the aggregate interference from both tiers. The computation parallels the single-tier case (Theorem 21.3) but with the interference field comprising contributions from both $\Phi_1$ and $\Phi_2$ (excluding the serving BS):

$p_{c,k}(\tau) = \int_0^{\infty} \mathcal{L}_{I_1}(\tau r^{\alpha}/P_k) \cdot \mathcal{L}_{I_2}(\tau r^{\alpha}/P_k) \cdot f_{r|k}(r)\,dr$

The total coverage is $p_c = A_1 p_{c,1} + A_2 p_{c,2}$ by the law of total probability. $\blacksquare$

,

HetNet SINR and Cell Association

Visualise the SINR distribution and cell association in a two-tier HetNet. Adjust the number of macro and small cells, the range expansion bias, and the path-loss exponent. Observe how increasing the CRE bias offloads more users to small cells (expanding their Voronoi regions) but degrades the SINR of CRE-zone users who connect to a weaker small cell. The plot shows the SINR CDF for macro-associated and small-cell-associated users separately.

Parameters

Number of macro BSs3

Number of small cells15

CRE bias

B

(dB)6

Path-loss exponent

\alpha

3.5

Example: Small-Cell Offloading with CRE

A two-tier HetNet has macro BSs ( $\lambda_1 = 2$ /km $^2$ , $P_1 = 46$ dBm) and pico BSs ( $\lambda_2 = 10$ /km $^2$ , $P_2 = 30$ dBm). The path-loss exponent is $\alpha = 4$ .

(a) Compute the fraction of users associated with pico cells without CRE ( $B = 0$ dB). (b) Compute the fraction with CRE bias $B = 10$ dB. (c) If each macro cell serves $K_1$ users, estimate the per-user rate improvement from the offloading in (b) vs. (a).

Solution

Power ratio

Power ratio: $P_2/P_1 = 10^{(30-46)/10} = 10^{-1.6} = 0.0251$ .

$(P_2/P_1)^{2/\alpha} = 0.0251^{0.5} = 0.1585$ .

Association without CRE

(a) With $B = 1$ (0 dB):

$A_2 = \frac{10 \times 0.1585}{2 + 10 \times 0.1585} = \frac{1.585}{3.585} = 0.442$

About 44% of users associate with pico cells.

Association with CRE

(b) With $B = 10$ dB ( $B = 10$ in linear):

$(B P_2/P_1)^{2/\alpha} = (10 \times 0.0251)^{0.5} = 0.251^{0.5} = 0.501$

$A_2 = \frac{10 \times 0.501}{2 + 10 \times 0.501} = \frac{5.01}{7.01} = 0.715$

With 10 dB CRE bias, 71.5% of users associate with pico cells.

Rate improvement

(c) Without CRE: macro serves $0.558 K_{\text{tot}}$ users, pico serves $0.442 K_{\text{tot}}$ users. With CRE: macro serves $0.285 K_{\text{tot}}$ , pico serves $0.715 K_{\text{tot}}$ .

Macro-cell per-user rate improvement: resources shared among $0.285/0.558 = 0.51\times$ fewer users, so per-user rate roughly doubles.

Overall network throughput improves because the macro load is redistributed to pico cells with shorter propagation distances. $\blacksquare$

Quick Check

In a two-tier HetNet, what is the primary purpose of the cell range expansion (CRE) bias?

To increase the transmit power of small cells

To offload users from macro to small cells for better load balancing

To reduce interference from macro cells to small cells

To extend the physical coverage of the network to new areas

Correction:

To offload users from macro to small cells for better load balancing

CRE expands the effective coverage area of small cells, causing more users to associate with them. This offloads the congested macro cell, improving per-user throughput even though some CRE-zone users receive weaker signals from their serving small cell.

Common Mistake: CRE Without Interference Protection Degrades Performance

Mistake:

Applying a large CRE bias ( $B > 10$ dB) without coordinated interference protection, expecting automatic throughput improvement from offloading alone.

Correction:

CRE-zone users connect to a small cell whose signal is weaker than the macro interference. Without protection, these users experience negative SINR, effectively creating a coverage hole. 3GPP addresses this with eICIC (enhanced ICIC): the macro cell transmits Almost Blank Subframes (ABS) — subframes with only CRS/PSS/SSS and no data — during which CRE-zone users are scheduled on the small cell. The ABS duty cycle $\beta$ trades macro capacity (reduced by $\beta$ ) for CRE-zone protection. Typical values: $\beta = 25\%$ -- $50\%$ .

⚠️Engineering Note

Backhaul Constraints in HetNet Deployments

The stochastic geometry analysis assumes all BSs have ideal backhaul (infinite capacity, zero latency). In practice, small cells face severe backhaul limitations:

Fibre backhaul: Ideal (1--10 Gbps, < 1 ms latency) but requires costly civil works. Available at only 30--40% of small-cell sites in urban deployments.
Microwave backhaul: 100 Mbps--1 Gbps capacity, 1--5 ms latency. Sensitive to rain fading above 10 GHz.
Non-ideal backhaul: The throughput of a small cell is $\min(R_{\text{radio}}, R_{\text{backhaul}})$ . If $R_{\text{backhaul}} < R_{\text{radio}}$ , the radio capacity gain from densification is wasted.
5G IAB (Integrated Access and Backhaul): Uses the same mmWave spectrum for both access and backhaul, eliminating the need for separate backhaul infrastructure. Time-division between access and backhaul reduces the effective access capacity by 30--50%, which must be accounted for in ASE calculations.

Practical Constraints

•
Fibre: 1-10 Gbps, < 1 ms, available at 30-40% of small-cell sites
•
Microwave: 100 Mbps - 1 Gbps, 1-5 ms latency
•
5G IAB: 30-50% access capacity reduction due to TDM with backhaul

Hexagonal Model vs Stochastic Geometry Comparison

Property	Hexagonal Model	PPP/Stochastic Geometry
BS placement	Regular grid	Random (Poisson)
Closed-form SIR?	Yes (first-tier approx.)	Yes (exact under PPP)
Density dependence	Fixed by design	Coverage invariant to $\lambda$
Multi-tier (HetNets)	Difficult	Natural (independent PPPs)
Realism for macro	Moderate (planned sites)	Lower bound (too random)
Realism for small cells	Poor (irregular)	Good (opportunistic placement)
Key strength	Frequency planning, reuse	ASE scaling, HetNet analysis
Key limitation	Assumes perfect grid	Ignores BS repulsion

Heterogeneous Network (HetNet)

A cellular network comprising multiple tiers of base stations with different transmit powers, coverage areas, and backhaul capabilities (e.g., macro cells, pico cells, femto cells). Stochastic geometry models each tier as an independent PPP, enabling tractable analysis of coverage and rate in dense, irregular deployments.

Why This Matters: Cell-Free Massive MIMO as the Ultimate HetNet

The HetNet framework of this section separates the network into distinct tiers (macro, pico, femto) with independent scheduling. The MIMO book develops an alternative architecture — cell-free massive MIMO — where all access points (APs) coherently serve all users without cell boundaries:

No handover: every user is simultaneously served by all nearby APs
No cell association: users do not "belong" to any single cell
CommIT contributions: Ngo, Caire et al. developed the user-centric scalable framework and fronthaul optimisation
The cell-free architecture eliminates the cell-edge problem that drives the SIR analysis in this chapter

Cell-free massive MIMO can be viewed as the asymptotic limit of a HetNet where all tiers merge into a single cooperative network.

Cell Range Expansion (CRE)

A cell association technique that adds a positive bias $B$ (in dB) to the received power from small cells, expanding their effective coverage area and offloading users from macro cells. CRE users in the expanded zone need interference protection (e.g., eICIC) since their serving signal is weaker than the macro interference.

Related: Heterogeneous Network (HetNet)

HetNets and Small Cells