Exercises

$t = 4 \cdot 0.25 = 1$. Downlink term: $20 \cdot 0.75/(4 \cdot 2 \cdot 2) = 15/16 = 0.9375$. Fronthaul term: $20 \cdot 0.75/(4 \cdot 2) = 15/8 = 1.875$. Sum: $2.8125$. NDT $= \max(1, 2.8125) = 2.8125$.

NDT is normalized to $T_\text{ref}$, the delivery time with infinite fronthaul and no cache needed. Any real architecture cannot be faster than the infinite-fronthaul baseline. NDT = 1 is achieved when $\mu = 1$ (full library cached at each EN) or when $C \to \infty$ (pure cloud-RAN without cache bottleneck, DoF = $N_\text{EN} L_\text{EN}$).

NDT $\to K/(N_\text{EN} L_\text{EN})$ — the per-user MU-MIMO DoF formula from Chapter 5 with $t = 0$. The architecture reduces to a cooperative $N_\text{EN} L_\text{EN}$-antenna transmitter serving $K$ users.

In MAN: $t = K M/N$. Caches at users; users benefit directly from their own cache contents. In C-RAN: $t = N_\text{EN} M/N$. Caches at ENs; users benefit only indirectly through EN-to-user downlink. The number of cache "locations" is $N_\text{EN}$, not $K$. Typically $N_\text{EN} \ll K$, so the aggregate cache gain in C-RAN is smaller than in MAN for the same per-cache size.

$-\partial C/\partial \mu = N_\text{EN} C^2 / (K (1-\mu)^2) = 2 \cdot 4/(10 \cdot 0.64) = 1.25$. Interpretation: adding 0.1 to $\mu$ saves 0.125 in $C$ along the iso-NDT contour.

For each user's demand $d_k$, the cloud determines which subfiles are not cached at the nearest EN. These must be transferred via fronthaul. Total per-EN fronthaul: $K(1-\mu)/ N_\text{EN}$ files per delivery round.

After all ENs have the requested content, they cooperatively transmit to users via Lampiris-Caire: DoF = $t + N_\text{EN} L_\text{EN}$. Delivery time: $K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t))$.

The phases run sequentially (fronthaul + downlink); total time in units of $T_\text{ref}$ gives the stated NDT bound.

$\mathcal{L} = c_M \mu N + c_C C + \lambda [f(\mu, C) - \Delta^*]$ where $f$ is the NDT formula. KKT: $c_M N = -\lambda \partial f/\partial \mu$; $c_C = -\lambda \partial f/\partial C$.

Dividing: $c_M N/c_C = (\partial f/\partial \mu)/(\partial f/\partial C)$. The ratio of marginal NDT sensitivities equals the cost ratio. Solve explicitly using the NDT formula.

Cheap cache ($c_M \ll c_C$): push $\mu$ high, $C$ low. Cheap fronthaul: opposite. The optimum slides along the iso-NDT contour based on cost ratios.

At $(0.01, 10)$: $\partial C/\partial \mu = 2 \cdot 100/(10 \cdot (0.99)^2) = 20.4$. Very steep — adding cache saves lots of fronthaul here. At $(0.9, 0.1)$: $\partial C/\partial \mu = 2 \cdot 0.01/ (10 \cdot 0.01) = 0.2$. Very shallow — fronthaul trivially saved per unit of cache.

Cache-heavy end has low substitution rate; operators already have ample cache, marginal cache isn't very useful. Cache-poor end has high substitution rate: every bit of cache saves lots of fronthaul. If cache is cheap, add lots at the cache-poor end.

$\Delta = K/(N_\text{EN} L_\text{EN}) + K/(N_\text{EN} C)$.

First term = $K/(\text{effective MIMO DoF})$. Second term = $K/(N_\text{EN} C)$, the fronthaul's "DoF budget." Total delay = sum of delays at the two bottlenecks.

In a non-cached C-RAN, the cloud generates messages; fronthaul transports them; ENs radiate them. Both legs have DoF-like capacities; total delay is the sum of the leg delays.

$\mu = 0$: Downlink NDT contribution = $K/(N_\text{EN} L_\text{EN}) = 40/16 = 2.5$. $\mu = 0.1$: $t = 0.4$; $K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t)) = 36/(16 \cdot 1.4) = 1.61$. Reduction: 36%.

Coded caching contributes a multiplicative factor $(1+t)$ on the downlink term, reducing NDT substantially. The CommIT NDT extension (vs. uncoded) is what makes this gain realizable.

Non-cached content $K(1-\mu)$ files must cross the fronthaul in each delivery round. Fronthaul aggregate capacity: $N_\text{EN} C$ files/use. Time to transfer: $K(1-\mu)/(N_\text{EN} C)$.

All $K$ files (post-fronthaul) must be delivered to $K$ users via MIMO BC with $N_\text{EN} L_\text{EN}$ effective antennas. DoF = $N_\text{EN} L_\text{EN}$; time = $K/(N_\text{EN} L_\text{EN})$ normalized to $K(1-\mu)/(N_\text{EN} L_\text{EN})$ in the NDT convention (only non-cached content needs downlink transmission in the achievable protocol; cached content is delivered directly).

The two cuts are sequential; total time ≥ sum. Hence $\Delta \geq$ stated expression. $\blacksquare$

$\Delta_\text{ach} = K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t)) + K(1-\mu)/(N_\text{EN} C)$.

$\Delta_\text{cs} = K(1-\mu)/(N_\text{EN} L_\text{EN}) + K(1-\mu)/(N_\text{EN} C)$.

Difference = $K(1-\mu)[1/(N_\text{EN} L_\text{EN}) - 1/(N_\text{EN} L_\text{EN}(1+t))] = K(1-\mu) t/(N_\text{EN} L_\text{EN}(1+t))$. Nonzero whenever $t > 0$.

The cut-set bound is loose when caching gain is active (${t \geq 1}$). The true NDT is below $\Delta_\text{cs}$ but the upper bound $\Delta_\text{ach}$ is usually achievable. The gap is the coded multicast contribution.

Let $R_c, R_u$ be cacheable and uncacheable rates per user. Cacheable mode achieves downlink NDT = $K(1-\mu)/(N_\text{EN} L_\text{EN}(1+t))$; fronthaul = $K(1-\mu)/(N_\text{EN} C)$. Uncacheable mode: same architecture but $t = 0$; downlink NDT = $K/(N_\text{EN} L_\text{EN})$; fronthaul = $K/(N_\text{EN} C)$.

Total delivery time = $\theta \Delta_c + (1 - \theta) \Delta_u$ for fraction $\theta \in [0, 1]$ of cacheable content. The achievable $(\Delta_c, \Delta_u)$ pairs form a pentagon-like region analogous to JLEC 2019's GDoF region.

Park-Caire-Simeone 2020+ show separation is GDoF-optimal in the C-RAN setting. The NDT formulation extends the JLEC separation theorem to architectures with fronthaul constraints.

Exact finite-SNR tradeoffs and NDT-optimal joint schemes are open. The separation is a clean achievability and conjecturally tight; converse proofs for general $(\mu, C)$ regimes remain research-active.

Privacy requires that the fronthaul message does not reveal $\mathbf{d}$ to any single EN. Use shared randomness between cloud and users (like secret sharing) to obscure demands.

Shared randomness comes at zero rate cost (Wan-Caire 2022). NDT under demand privacy matches the non-private NDT for most operating regimes.

The cost is in coordination: each user must receive its share of the shared randomness. Communication-complexity cost may appear in finite-$F$ analyses; the NDT (asymptotic) remains invariant.

The precise characterization of NDT with combined fronthaul, caching, and privacy constraints is ongoing CommIT research.

Let EN $i$ have cache $M_i$, fronthaul $C_i$, $L_i$ antennas. Aggregate gain: $t_\text{total} = \sum_i M_i/N$. Effective antenna count: $\sum_i L_i$ (cooperative).

Downlink: $K(1-\mu_\text{avg})/(\sum_i L_i (1 + t_\text{total}))$; fronthaul: $K(1-\mu_\text{avg})/(\sum_i C_i)$; sum for heterogeneous NDT.

Heterogeneity can be exploited or hurt fairness. Larger caches at some ENs could serve nearby users faster; careful scheduling required.

Heterogeneous-EN NDT is less well-studied than the symmetric case. Open problems include optimal cache placement under spatial heterogeneity and user demand correlation.