NDT Upper and Lower Bounds

Achievability and Converse

Having defined NDT, we now analyze the achievable bounds. The Sengupta-Tandon-Simeone 2017 paper established the first tight characterization for several regimes; subsequent CommIT work has refined the bounds for coded placement and mixed traffic.

This section presents the achievable upper bound (Lampiris-Caire- style scheme adapted to C-RAN) and the information-theoretic converse. For many regimes, the bounds coincide; for others, the gap is a factor of 2 or less.

Theorem: NDT Achievability (Upper Bound)

For the cache-aided cloud-RAN with integer t=NENM/Nt = N_\text{EN} M/N, the following NDT is achievable: Δach(M,CF)  =  max⁡ ⁣(1,K(1−μ)NENLEN(1+t)+K(1−μ)NENCF).\Delta_\text{ach}(M, C_F) \;=\; \max\!\left(1, \frac{K(1-\mu)}{N_\text{EN} L_\text{EN}(1 + t)} + \frac{K(1-\mu)}{N_\text{EN} C_F}\right). The scheme combines MAN-style placement at ENs with cooperative Lampiris-Caire delivery on the downlink.

Extend the Lampiris-Caire scheme from Chapter 5: the NENN_\text{EN} cooperating ENs act as a single transmitter with NENLENN_\text{EN} L_\text{EN} antennas. Aggregate caching gain t=NENM/Nt = N_\text{EN} M/N provides an additional coded multicast boost. The downlink term K(1−μ)/(NENLEN(1+t))K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t)) captures the Lampiris- Caire DoF; the fronthaul term captures the cloud-to-EN transfer.

Proof
Placement

Each EN caches a subset of subfiles such that aggregate coverage is NENμN_\text{EN} \mu file copies. Specifically, run MAN with NENN_\text{EN} "users" at caching level μ\mu, but treat each MAN-subfile as replicated across ENs in a suitable pattern.

Fronthaul phase

For non-cached content, cloud sends coded-XOR messages to ENs via fronthaul. Fronthaul rate required: K(1−μ)/NENK(1-\mu)/N_\text{EN} files per channel use per EN.

Downlink phase

ENs act as cooperating Leff=NENLENL_\text{eff} = N_\text{EN} L_\text{EN} antenna transmitters. Apply Lampiris-Caire: DoF = t+Lefft + L_\text{eff}. Delivery time: K(1−μ)/(NEN(1+t)LEN)K(1-\mu)/(N_\text{EN}(1+t) L_\text{EN}).

Total NDT

Sum: Δ=max⁡(1,fronthaul term+downlink term)\Delta = \max(1, \text{fronthaul term} + \text{downlink term}). ■\blacksquare

Theorem: NDT Converse (Lower Bound)

For the cache-aided cloud-RAN, any achievable NDT satisfies Δ(M,CF)≥max⁡ ⁣(1,K(1−μ)NENLEN+K(1−μ)NENCF).\Delta(M, C_F) \geq \max\!\left(1, \frac{K(1-\mu)}{N_\text{EN} L_\text{EN}} + \frac{K(1-\mu)}{N_\text{EN} C_F}\right). The bound matches the upper bound when tt = 0 (no coded caching gain). For t≥1t \geq 1, the achievable upper bound is strictly tighter — the coded caching gain contributes.

Cut-set argument on the cloud-EN and EN-user boundaries. Each has a capacity; the total delivery time is at least the sum. The converse does not fully exploit the coded caching structure; the gap to achievability is the caching gain.

Proof
Cut-set argument

Two cuts: (i) cloud →\to all ENs (bottleneck: fronthaul); (ii) all ENs →\to all users (bottleneck: downlink). Both cuts must carry the non-cached content, of size K(1−μ)K(1-\mu) files.

Per-cut time

Cut (i) time: K(1−μ)/(NENC)K(1-\mu)/(N_\text{EN} C). Cut (ii) time: K(1−μ)/(NENLEN)K(1-\mu)/(N_\text{EN} L_\text{EN}).

Sum

Delivery time ≥\geq sum of per-cut times (they run in sequence in the simplest protocol). Hence Δ≥max⁡(1,…)\Delta \geq \max(1, \ldots).

Tightness

Tight when t=0t = 0. For t≥1t \geq 1, the converse does not account for the Lampiris-Caire DoF gain; tighter converses (not in 2017 paper) close the gap partially.

Cloud-Edge Delivery Split

Fraction of delivery served from cloud (via fronthaul) vs from edge (via cache) as a function of fronthaul capacity CC. At low CC, the edge dominates; at high CC, the cloud takes over. The transition reflects the cache-fronthaul tradeoff.

Parameters
8
4
0.3

Example: High-Cache Regime

As cache μ→1\mu \to 1 (full library at each EN), show that NDT →1\to 1 regardless of fronthaul CC.

Solution
Substitute $\mu = 1$

Upper bound: Δ=max⁡(1,K⋅0/(NENLEN(1+t))+K⋅0/(NENC))=max⁡(1,0)=1\Delta = \max(1, K \cdot 0/(N_\text{EN} L_\text{EN}(1+t)) + K \cdot 0/(N_\text{EN} C)) = \max(1, 0) = 1.

Interpretation

With full cache, no fronthaul is needed, and the downlink serves users via cooperative MU-MIMO. NDT = 1 is the baseline, achieved by pure MU-MIMO when fronthaul is unnecessary.

Diminishing returns

Above some threshold μ∗\mu^*, further cache increases do not improve NDT. The architectural insight: oversize cache is wasted once NDT saturates at 1. Optimal μ\mu depends on CC.

Example: High-Fronthaul Regime

As C→∞C \to \infty (abundant fronthaul), show that NDT →K(1−μ)/(NENLEN(1+t))\to K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t)), the Lampiris-Caire DoF limit.

Solution
Substitute $C \to \infty$

Fronthaul term: K(1−μ)/(NENC)→0K(1-\mu)/(N_\text{EN} C) \to 0. Downlink term dominates.

Result

Δ→max⁡(1,K(1−μ)/(NENLEN(1+t)))\Delta \to \max(1, K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t))). For small μ\mu and moderate NEN,LENN_\text{EN}, L_\text{EN}, this is > 1; the downlink is the bottleneck.

Interpretation

At infinite fronthaul, the architecture reduces to Chapter 5's single-transmitter MIMO BC with effective antennas NENLENN_\text{EN} L_\text{EN}. Cache still matters, but the cloud-EN handoff is free.

Key Takeaway

NDT bounds guide architectural design. Upper bound (achievable): Lampiris-Caire-style scheme adapted to C-RAN. Lower bound (converse): cut-set argument. Tight at several operating points; in general the gap is ≤2\leq 2. For deployment planning, the achievable bound is what matters — it tells you what latency your design can hit.

⚠️Engineering Note

NDT in 5G NR Deployments

Typical 5G NR parameters and their NDT implications:

  1. Small cell C-RAN: NEN=4N_\text{EN} = 4, LEN=4L_\text{EN} = 4, K=50K = 50 users, C=10C = 10 files/use (abundant fronthaul at small scale). With μ=0.2\mu = 0.2, t=0.8t = 0.8: NDT ≈K(1−μ)/(4⋅4⋅1.8)+K(1−μ)/(4⋅10)=40⋅0.8/28.8+40⋅0.8/40=1.11+0.8=1.91\approx K(1-\mu)/(4 \cdot 4 \cdot 1.8) + K(1-\mu)/(4 \cdot 10) = 40 \cdot 0.8/28.8 + 40 \cdot 0.8/40 = 1.11 + 0.8 = 1.91.
  2. Macro cell C-RAN: NEN=1N_\text{EN} = 1 (single-site), no aggregate caching gain; NDT reduces to conventional MU-MIMO. Need more ENs for NDT benefit.
  3. Fog / Open RAN: NEN=20+N_\text{EN} = 20\text{+}, distributed caches; aggregate tt can reach 10+. NDT approaches 1 for moderate fronthaul.

The NDT framework provides a principled way to size the cache and fronthaul at deployment time. Production 5G operators increasingly use it in their design tooling.

Practical Constraints
  • •

    5G NR small-cell C-RAN: N_EN = 2-8 per cluster

  • •

    Fronthaul: 25-100 Gbps per EN; translates to C = 10-50 files/use at typical SNR

  • •

    RU cache: 10-100 GB; translates to μ = 0.01-0.1 for 1-10 TB library

  • •

    Target latency: <10ms downlink for typical 5G eMBB (NDT < 5 at baseline TrefT_\text{ref})

Common Mistake: Fronthaul Units: Files-per-Use vs Absolute

Mistake:

Confusing the symbolic fronthaul CC (files per channel use) with absolute fronthaul bandwidth (Gbps).

Correction:

In the NDT framework, CC is in files per channel use at asymptotic SNR. This normalizes the fronthaul rate to the downlink rate. Conversion: if fronthaul is RFR_F Gbps and each file is FF bits at downlink symbol rate WsW_s, then C=RF⋅109/(F⋅Ws)C = R_F \cdot 10^9 / (F \cdot W_s) files/use.

A 10 Gbps fronthaul delivering 1 GB files over a 100 MBaud downlink: C=1010/(1010⋅108)=10−8C = 10^{10}/(10^{10} \cdot 10^8) = 10^{-8} files/use. But at 100 kBaud downlink: C=10−5C = 10^{-5}. These are wildly different regimes; read the assumptions carefully.

The Cache-Fronthaul TradeoffChapter Summary