Exercises

A source is SR if $R(D_1) + R_\text{enh}(D_2|D_1) = R(D_2)$ for $D_2 < D_1$. Rate adds across layers without redundancy.

Cache base layer (widely useful); enhance on demand from server. Separates storage concerns: base in cache, enhancements in delivery.

All at 4K: $b_{\max} = 10$ Mbps.

$K(1-\mu)/(1+K\mu) = 10 \cdot 0.8/3 = 2.67$.

$R = 10 \cdot 2.67 = 26.7$ Mbps. Uncoded would be $10 \cdot 10 = 100$ Mbps. $\sim 3.7\times$ savings.

$\alpha_\ell^* \propto p_\ell b_\ell$.

$\alpha_1^* = p_1 b_1 / (p_1 b_1 + p_2 b_2)$. $\alpha_2^* = p_2 b_2 / (p_1 b_1 + p_2 b_2)$.

$b_1 = 1$ (base), $b_2 = 5$ (HD). $p_1 = 0.2, p_2 = 0.8$. Weights: $0.2 \cdot 1 = 0.2$ vs $0.8 \cdot 5 = 4$. Split: $0.05$ for base, $0.95$ for HD.

HD dominates because both popular and high-bitrate.

Human vision: SD → HD is a huge perceptual jump. HD → 4K less. 4K → 8K barely visible on typical screens.

QoE $\sim \log(\text{bitrate})$ or similar concave form. Marginal QoE per Mbps decreases.

Optimal cache prioritizes getting many users to HD rather than few users to 4K — matches concavity.

File-level: subfile size = $F/\binom{K}{t}$. For large $K$: tiny subfiles, high header overhead.

Chunk size fixed ($\sim$ 5 MB). Subpacketization applies per chunk: $\sim 5\text{MB}/\binom{K}{t}$.

Chunk-level becomes impractical when $\binom{K}{t} >$ chunk size in bits / packet size. Typically $K < 30$ before chunks become too small.

Bandwidth $B$ serves $K$ users at $B/K$ each. QoE per user: $\propto B/K$ - rebuffer.

Effective bandwidth $B \cdot (1 + K\mu)$. Per-user bitrate $\sim B(1+K\mu)/K$. Massive boost.

Better-provisioned bandwidth → less congestion → fewer stalls. Non-linear gain in rebuffering.

Approximately $(1 + K\mu)$ gain on bitrate component; $\sim$ constant gain on rebuffer penalty. Composite: roughly $(1+K\mu) \times$ QoE.

All users see same channel variance. Rebuffer on congestion.

Multicast delivers to many users in one message. Fewer individual transmissions; less per-request variance.

Central Limit: variance scales as $1/K$ for coded. Rebuffering probability $\propto \sigma$: reduces by $\sqrt{K}$ factor.

Approximately: variance reduction → rebuffering reduction → QoE boost. Compound with bandwidth gain.

Client initiates each HTTP request for each chunk. Server responds per-request.

Server initiates coded broadcast; all users decode their needed content.

Different philosophies. DASH infrastructure (CDN, HTTP/2) is unicast pull.

Options: (1) MBMS parallel layer for multicast pushes; (2) HTTP/2+/3 server-push; (3) client-side cache with server-side coded injection. Each has tradeoffs in protocol complexity.

Add "coded-chunk" descriptor: tells client this chunk can be received via coded multicast. List of coded-delivery opportunities.

On coded-chunk: tune to multicast channel + use cached XOR components to decode. Fallback to unicast if multicast fails.

Side channel (e.g., WebSocket) for multicast metadata. Primary data via MBMS or HTTP/3 push.

Non-aware clients fall back to pure DASH. No disruption.

4K $\approx 15$ Mbps. 2-hour movie: $15 \cdot 7200 = 108$ Gb = 13.5 GB.

$1000 \cdot 13.5 = 13.5$ TB.

$(1 + 1000 \cdot 0.3) = 301$. Effective size $13.5 / 301 \approx 0.045$ TB = 45 GB.

$\sim 300\times$ bandwidth reduction. Enormous at scale.

Cache hit: immediate. MAN-coded delivery on miss. Large library; gain depends on popularity.

Sudden concurrent demand; cache can't be pre-populated. MAN-coded delivery via multicast massively beneficial (single broadcast to all).

Live benefits more from coded: concurrent demand is ideal for multicast. MBMS already deployed for live; coded MBMS is the natural evolution.

5-10 second chunks: easier subpacketization ($\binom{K}{t} \lesssim 10^6$ still fits). But coarse- grained startup latency.

1-2 second chunks: better latency but $\binom{K}{t}$ fits poorly into small chunks. Coded gain reduced.

Chunk size balancing subpacketization feasibility with latency goals. Typical: 2-5 seconds.

Server sends first chunk to each of $K$ users: bandwidth $KB$. Per-user latency $\sim 1/B$.

Server sends first chunk via MAN-coded multicast: bandwidth $KB/(1+K\mu)$. Per-user latency $\sim 1/[B(1+K\mu)]$.

Startup latency improvement: factor $1+K\mu$.

Assumes cache populated. On first-ever watch (cold cache), coded gain not yet available.

Multicast Broadcast Multimedia Services. 5G feature supporting scalable video broadcast.

Natural substrate for MAN-style broadcast. Each user decodes via cached MAN subfiles.

3GPP SA4 is studying coded-caching + MBMS integration. Standards expected 2025-2027.

Live events (stadium, concert): immediate use case. Commercial VoD: longer horizon.

Current analysis assumes concave QoE. Some QoE models (e.g., rebuffering-dominated) are discontinuous. Optimal caching under non-smooth QoE unclear.

Live content: no pre-caching possible. How does coded caching help? Real-time multicast encoding.

Video codecs have inter-chunk dependencies (P, B frames). Coded delivery must respect these. Open scheme design.