Exercises

In a D2D caching network, both uncoded (Ch 10) and coded (MAN-style, Ch 11) delivery achieve per-user throughput $\Theta(M/N)$. The scaling orders coincide; the coded scheme wins only in the constant factor. Spatial reuse and coded multicasting do not cumulate asymptotically.

Ratio $= (1 + 10 \cdot 0.25)/(1 + 0.25) = 3.5/1.25 = 2.8$. Coded is ~2.8× faster than uncoded.

Both mechanisms save airtime: spatial reuse gives concurrent links; coded multicast gives more content per link. But airtime is a shared resource — you can't both have $n$ concurrent transmissions AND pack $K_g$ files into each transmission at the scaling-order level. The mechanisms compete; the per-user throughput order is capped at $\Theta(M/N)$.

$t = 10 \cdot 0.3 = 3$ (integer). Subpacketization: $\binom{10}{3} = 120$ subfiles per file. Feasible for large files (>100 MB); marginal for small files.

Practical $K_g$: 5-30 users per cluster. Balances: (i) Coded gain constant $(1 + K_g M/N)$, (ii) Subpacketization $\binom{K_g}{t}$, (iii) Coordination overhead across users. Values beyond 30 hit subpacketization walls; below 5 give negligible coded gain.

MAN: $(K_g - t)/(t+1) = K_g(1-\mu)/(1+K_g \mu)$ file-units. Uncoded: $K_g (1-\mu)$ file-units. Ratio: $1/(1+K_g \mu)$.

Per-user rate = (1 / delivery cost) × hit-probability. Uncoded per-user: $\mu/(K_g(1-\mu))$ per airtime unit. Coded per-user: $\mu (1 + K_g \mu)/(K_g(1-\mu))$ per airtime unit. Ratio: $1 + K_g \mu$.

$K_g = 10$: $\binom{10}{1} = 10$. Fine. $K_g = 20$: $\binom{20}{2} = 190$. Fine. $K_g = 30$: $\binom{30}{3} = 4060$. Fine. $K_g = 40$: $\binom{40}{4} \approx 9 \times 10^4$. Over.

Max $K_g \approx 35$ at $\mu = 0.1$. Beyond, subpacketization infeasible. For realistic file sizes (~1 GB), subfile size below ~0.1 MB gets header-dominated.

Per-user $\Theta(M/N)$, aggregate $\Theta(n M/N)$. Linear in $n$.

Single BS serves $n$ users; MAN rate formula gives aggregate $K(1-\mu)/(1+K\mu) \to (1-\mu)/\mu$ as $K \to \infty$. Per-user: vanishing $O(1/n)$.

D2D aggregate scales linearly in $n$; cellular saturates at a constant. At any moderately large $n$, D2D is asymptotically better.

Each user sends 1-2 discovery messages + listens for neighbors: ~5 ms. 1000 users: aggregate ~5 s (if serial) or ~5 ms (if parallel on separate channels).

Forming 100-user clusters: graph partitioning; ~100 ms with distributed algorithm.

End-to-end cluster setup: ~100 ms. Acceptable for non-real-time content; marginal for interactive (<10 ms requirement).

Uncoded constant (Ji-Caire-Molisch): $c_{\text{uncoded}} \approx 0.5 \cdot c_{\text{link}} = 50$ Mbps × $\mu$. Coded multiplier: $(1 + 2)/(1 + 0.1) \approx 2.7$. Per-user throughput: $50 \cdot 0.1 \cdot 2.7 \approx 13.5$ Mbps. Aggregate: 10000 × 13.5 = 135 Gbps.

Cellular at 10 Gbps / 10000 = 1 Mbps per user. Coded D2D is ~13× better per user.

Consider any user $k$. Its caches + neighbors' caches provide side information of aggregate size $O(M \cdot \log n)$ (few neighbors, each with $M$ files). Its demand is one file ($F$ bits).

Per-round: user $k$ needs $F$ bits delivered. Of these, $M/N \cdot F$ can come from local caches (cache hit) + some from neighbor transmissions. Scheduling latency for neighbor-sourced content: $\Theta(1/r^2)$ slots per user. Together: $T_n \leq M/N + O(1/\sqrt{n \log n}) = \Theta(M/N)$ for large $n$.

Achievability (Ji-Caire-Molisch 2016) matches. No scaling improvement possible with coding. $\blacksquare$

Consider heterogeneous demand: some clusters have Zipf demand (concentration $\alpha = 1$), others uniform ($\alpha = 0$).

Apply coded delivery in high-concentration clusters (MAN gain large). Apply uncoded in low-concentration clusters (coded gain small). Average per-user across all clusters.

Marginal improvement over uniform-coded: few-percent. Implementation complexity added is not always worth it. Typically one chooses a single policy for all clusters.

Optimal heterogeneous scheme is an open question. CommIT research (Ji-Tulino-Llorca-Caire 2017) treats Zipf demand uniformly; per-cluster adaptation is not explicit.

With SINR target $\gamma$, effective interference radius scales as $r_{\text{eff}} \propto \sqrt{\gamma / (n - 1)}$ for path-loss 2. Extends to $r^{2/\alpha}$ for general $\alpha$.

The scaling order $\Theta(M/N)$ is preserved. Constants change: SINR model gives slightly worse constants due to interference-aware rate allocation.

The CommIT JCM analysis is robust to interference model choice; scaling order is model-independent for reasonable $\alpha$.

Each D2D link has capacity $L \log(1 + \text{SNR})$ with $L$ antennas. Link capacity multiplies by $L$.

Within a cluster, users could cooperate as $K_g$-transmitter $K_g$-receiver network. DoF up to $L \cdot K_g$ per cluster.

Per-user throughput: $\Theta(L M/N)$. Scales with $L$ just like in the shared-link case. Result matches Lampiris-Caire extended to distributed transmitters.

This is an active area; see CommIT fog massive MIMO papers that combine D2D + multi-antenna devices.

1. Caching multiplies usable capacity by (1 + t) where t = (aggregate cache)/(library).
2. Spatial resources (antennas, D2D links) give concurrent DoF.
3. Caching + spatial can combine additively (DoF = t + L) or non-cumulatingly (D2D scaling) depending on architecture.
4. Converse bounds come from cut-set + entropy arguments; achievability combines combinatorial placement with scheme-specific delivery.

All results take the form: $\text{per-user metric} = f(\text{cache ratio}, \text{network structure}, \text{interference model})$. The architectural choice determines whether gains are additive, multiplicative, or non-cumulating.

A fully unified framework (including Zipf demand, heterogeneity, privacy) is a major research direction. CommIT group's ongoing work approaches this in specific sub-settings.