Exercises

$t = 20 \cdot 0.2 = 4$. $\mathrm{DoF} = \min(4 + 12, 20) = 16$.

$R = 4 \cdot 5 \cdot 0.6/(1 + 2) = 12/3 = 4$ files.

Caching gain $t$ comes from XOR cancellation at users, using the user's cache. Adding more servers doesn't give users more cache; it only lets servers coordinate better at the transmit side (spatial gain). Thus spatial gain scales with $S$ but caching gain stays at $t$.

$t = S M/N = 10 \cdot 0.2 = 2$.

If the same storage were at users ($K = 50$ users, $M = 0.04 N$ per user), then $t = KM/N = 50 \cdot 0.04 = 2$ — same gain. But the location of storage matters for the delivery scheme.

Joint precoding across $S$ servers forms $\mathbf{x}_i = [\mathbf{V}_i] \tilde{\mathbf{x}}$ where $\tilde{\mathbf{x}}$ contains shared data across servers, and $\mathbf{V}_i$ is the i-th server's contribution to the joint precoder. This requires servers to know (i) the same joint data vector and (ii) the cooperative precoding matrix, both within a channel coherence block. At 100s of MHz bandwidth, this is microseconds — dedicated fiber between co-located servers is feasible.

$t = 100 \cdot 0.1 = 10$. DoF = $\min(10 + 40, 100) = 50$.

Each cluster of 4 servers achieves DoF = $10 + 8 = 18$ locally. With 5 clusters time-sharing, aggregate DoF averaged across time = 18 (not multiplied by 5; each cluster serves a fraction of users at a time).

Full cooperation is 50/18 ≈ 2.8x better than 4-server clusters. But full cooperation requires $20 \cdot L = 40$-antenna joint precoding — expensive. Clusters of 4 with $S_c L = 8$ antennas are more practical.

Most 6G architectures target cluster sizes of 2-8 APs. The resulting DoF is substantially above non-cooperative ($t + L = 12$), substantially below full cooperation. Cost-effective middle ground.

$t$ depends on the product $SM_\text{AP}$. For fixed aggregate storage $S \cdot M_\text{AP}$ = constant $B$, $t = B/N$ is invariant.

Smaller $S$ but larger $M_\text{AP}$ means fewer cooperation nodes (less spatial gain $SL$). So for the DoF $= t + SL$, both $t$ (fixed by $B$) and $SL$ (decreasing in $S$ shrinkage) matter.

For fixed $B$, prefer to spread across more APs to maintain spatial DoF. At $B$ = constant, $t$ invariant, and $SL$ increasing in $S$: more APs is strictly better. The practical limit: backhaul cost per AP, not analytical. Operators balance capex (many APs) vs opex (backhaul).

MAN, $t = 1$: each file split into $\binom{6}{1} = 6$ subfiles $W_{n,\{k\}}$. User $k$ caches $W_{n,\{k\}}$ for all files.

$(t + SL) = 3$-subsets of $[6]$: $\binom{6}{3} = 20$ groups.

For each 3-subset $\mathcal{S}'$: send $SL = 2$ beams. Each beam targets a 2-subset of $\mathcal{S}'$ (excluding one user); cooperative precoding nulls the excluded user.

DoF = $t + SL = 3$. Per-user DoF = 3/6 = 0.5.

The aggregate delivery rate of the full MAN scheme is $K(1-\mu)/(1+KM/N)$ files per channel use. Under full cooperation, the $S$ servers transmit this in parallel via joint precoding; each server handles its share of the signal. Per-server rate: $R_\text{MAN}/S$.

Each server has $L$ antennas; total cooperative antennas $SL$. The MAN rate formulation is invariant to the number of cooperating transmitters at fixed total antennas. The rate is divided by $S$ in the per-server view because $S$ servers share the load equally.

$\Lambda = K/K_s = 10$, $M_s = K_s M = 0.2 N$.

$R_\text{ded} = 20 \cdot 0.9/3 = 6$.

$R_\text{shared} = 2 \cdot 10 \cdot 0.8/(1+2) = 6.67$. Wait, larger.

At this $\mu$, shared-cache is strictly worse than dedicated (more rate needed). The break-even happens at different $\mu$ depending on $K, K_s$. Full analysis in Parrinello et al.

For any uncoded placement and demand vector, consider all permutations $\pi$ of $[K]$ and $\sigma$ of $[S]$. Averaging over these yields a symmetric scheme.

The symmetric scheme is precisely the Lampiris-Caire cooperative scheme of §9.3. Its DoF is $t + SL$ by Chapter 5.

Worst-case rate $\geq$ average-case rate $= t + SL$. Hence no uncoded-placement scheme achieves DoF less than $t + SL$. Combined with achievability (Lampiris-Caire), equality holds. $\blacksquare$

Placement subpacketization: $\binom{K}{t}$ per file. Delivery groups: $\binom{K}{t + SL}$. Effective subpacketization incorporates both: approximately $\binom{K}{t+SL-1}$.

For $K = 100$, $t = 5$, $SL = 10$: subfiles = $\binom{100}{15} \sim 10^{17}$. Infeasible at this scale. Polynomial-subpacketization multi-server schemes are a major open area; Chapter 14 discusses PDAs and extensions.

At $K = 20$, $t = 5$, $SL = 5$: subfiles = $\binom{20}{10} = 184756$. Manageable for large files; infeasible for small.

Each cluster's sum-DoF: $\min(t + 4L, K_{\text{cluster}})$, where $K_\text{cluster}$ is the user count served by that cluster.

Two clusters alternate. Each cluster delivers to its $K/2$ users for half the time. Effective per-cluster DoF: $(t + 4L) \cdot 1/2$ on average.

Aggregate across both clusters: $(t + 4L) \cdot 1$ (time-sharing conservation). But users get time-shared DoF, so effective per-user DoF is $1/K$ of the aggregate.

Full cooperation: DoF = $t + 8L$ for all $K$ users simultaneously. Time-shared 2-cluster: effectively DoF $= t + 4L$ at half the time-share, averaged. Full is strictly better but requires more cooperation overhead.

Under demand privacy, no individual server should learn more about $\mathbf{d}$ than what its received signals trivially reveal.

Use common randomness between all servers (pre-distributed via the CPU or offline). Mask demand-dependent quantities with this randomness. Wan-Caire '22 shows this is possible at asymptotic rate cost zero.

The extension preserves DoF = $t + SL$ but requires careful coordination of the shared-randomness protocol across servers. Exact characterization is ongoing CommIT research.

Is there a tighter converse when privacy is a constraint? Not known. The conjecture is that privacy is "free" in the multi-server setting as in single-server.

$\mathrm{DoF} = \min(SM/N + SL, K)$. For $K \geq SM/N + SL$: DoF scales linearly with $S$ as long as caching or spatial contribute.

Per-user rate $\approx \mathrm{DoF}/K \cdot \log_2(1 + \text{eff SNR})$. Effective SNR scales with $SL$ (coherent combining).

Per-user rate scales approximately linearly with $S$ (both in DoF and in effective SNR) up to the $K$-user saturation point. Beyond that, additional APs don't help without more users.

For realistic 6G: $K = 50\text{--}500$, $SL = 100\text{--}500$, caching provides additional $t = 5\text{--}20$ per-user DoF. Total per-user DoF in a well-designed fog-mMIMO: 0.3--1.