Ferkans — Interactive Telecom Tutor

Outage Is Not an Academic Abstraction

Real wireless systems encode at a fixed rate $R$ ; when the channel falls into a deep fade below the rate's SNR requirement, the message is lost (an outage event). Commercial systems set rate conservatively: the $\epsilon$ -outage capacity $C_\epsilon$ , with $\epsilon$ a target outage probability (typically $10^{-3}$ for video, $10^{-6}$ for URLLC).

Coded caching's multicast rate on a fading channel is governed by the worst-user outage rate — the rate that every user in the multicast group can decode with probability $1 - \epsilon$ . This section analyzes the outage-limited rate of cache-aided fading systems and quantifies the operational improvement from caching.

Definition:
$\epsilon$ -Outage Capacity of a Multicast Link

For a block-fading multicast link to $K$ users with i.i.d. fading $|h_k|^2 \sim \text{Exp}(1)$ (Rayleigh), the $\epsilon$ -outage multicast capacity is $C_\epsilon^{(K)} \;=\; \sup\{R : \Pr(\min_k \log_2(1 + |h_k|^2 \text{SNR}) < R) \leq \epsilon\}.$ For i.i.d. fading, $\min_k |h_k|^2$ has distribution $1 - (1 - F(x))^K = 1 - e^{-Kx}$ , so $C_\epsilon^{(K)} \;=\; \log_2(1 - \frac{\ln(1-\epsilon)}{K} \cdot \text{SNR}) \;\approx\; \log_2\!\left(\frac{\epsilon \text{SNR}}{K}\right) \text{ for small } \epsilon.$ The $1/K$ factor captures the multicast penalty: more receivers means a worse worst-case fade.

The penalty $\ln(1-\epsilon)/K \approx -\epsilon/K$ shrinks the effective SNR by a factor of $K$ . For $K = 100$ users and $\epsilon = 10^{-3}$ , the effective SNR is $10^{-3}/100 = 10^{-5}$ times the mean SNR — a 50 dB penalty relative to single-user. Multicasting is brutal on fading channels.

Theorem: Outage-Limited Rate of Cache-Aided Multicast

For the cache-aided fading BC with $K$ users, $L$ antennas, memory ratio $\mu$ , and target outage $\epsilon$ , the achievable per-user rate under the Lampiris-Caire scheme with outage-limited multicast is $R_\epsilon^{(K)}(\mu) \;=\; \frac{(t + L) C_\epsilon^{(t+L)}}{K},$ where $C_\epsilon^{(t+L)}$ is the $\epsilon$ -outage rate of a multicast to a $(t+L)$ -group. The key improvement: the multicast group size shrinks from $K$ to $t + L$ .

Vanilla MAN multicasts one XOR at a time to $t+1$ users, so outage is over a $(t+1)$ -group. The Lampiris-Caire scheme groups users into $(t+L)$ -subsets; each delivery group's multicast is over this smaller set, improving outage performance. Larger groups have worse outage; the scheme balances group size against total DoF.

Proof

Per-group outage

Each Lampiris-Caire delivery group has $t + L$ users. Outage rate per group: $C_\epsilon^{(t+L)}$ .

Per-group DoF

Within a group, the scheme achieves DoF $= L$ (via ZF) plus caching gain $t$ (via XOR). Per-user DoF within group: $(t + L)/(t + L) = 1$ .

Total throughput

Summing over all $\binom{K}{t+L}$ groups and normalizing by delivery time: per-user rate $= (t+L) C_\epsilon^{(t+L)}/K$ .

Compare to pure MAN

Pure MAN (L = 1) achieves $R = (t+1) C_\epsilon^{(t+1)}/K$ . Multi-antenna improvement: factor $(t+L)/(t+1)$ in DoF times the outage-capacity ratio $C_\epsilon^{(t+L)}/C_\epsilon^{(t+1)}$ . Sometimes the two factors cancel; the group-size dependence is subtle. $\blacksquare$

Outage-Limited Per-User Rate vs Memory Ratio

Per-user outage-limited rate for cache-aided Rayleigh fading. Blue: MAN outage-limited at the chosen $\epsilon$ ; red dashed: no-caching baseline. The caching gain translates directly into higher effective outage rate. Notice how the curve's shape depends on both the coded-caching gain and the multicast group-size penalty.

Parameters

Users K10

Outage ε0.01

Mean SNR (dB)10

Example: Video Streaming at 5% Outage

A 5G-NR video streaming service targets 5% outage for smooth playback. Per-user mean SNR $\rho = 10$ dB, $K = 30$ simultaneous users, $L = 4$ antennas. Compare rate with and without $\mu = 0.2$ cache.

Solution

No caching

MU-MIMO. Per-user rate at $\epsilon = 0.05$ : $C_{0.05}^{(30)} = \log_2(1 + (-\ln 0.95/30) \cdot 10) \approx \log_2(1 + 0.017) \approx 0.025$ bits/use. Per user: $\approx 0.025$ bits.

With caching

$t = 30 \cdot 0.2 = 6$ . Group size $t + L = 10$ . $C_{0.05}^{(10)} \approx \log_2(1 + (-\ln 0.95/10) \cdot 10) \approx \log_2(1 + 0.05) \approx 0.072$ bits/use. Per user: $(t+L) \cdot 0.072/K = 10 \cdot 0.072/30 = 0.024$ bits/use.

Comparison

Roughly similar in this regime — but at larger $\mu$ or lower SNR, caching dominates. The outage-rate analysis is delicate because the group-size penalty can offset the caching gain at small group sizes. Engineering implication: cache sizing should account for target outage, not just asymptotic DoF.

Key Takeaway

Outage analysis reveals that caching gain depends on the interaction of group size and fading statistics. In good-channel / low-outage regimes, the $t + L$ DoF dominates. In deep-outage regimes, the worst-user penalty $\log K$ within a group can eat into the gain. System design must account for both effects.

Adaptive Rate Selection

In practice, outage-limited operation is a conservative design choice. Modern systems use adaptive MCS (modulation and coding scheme): measure the channel, pick the highest rate that the worst user can decode. This effectively replaces the fixed outage-rate with a channel-adaptive rate that tracks conditions.

For cache-aided multicast, adaptive MCS must be done at the group level — the slowest user in each $(t+L)$ -group bounds the group's rate. Group formation becomes an optimization: cluster users with similar channel qualities to minimize within-group variation. This connects naturally to Chapter 6's user-grouping discussion.

Common Mistake: DoF ≠ Outage Rate

Mistake:

Quoting the Lampiris-Caire DoF $= t + L$ as the realizable rate of a deployed system operating under outage constraints.

Correction:

DoF is a high-SNR quantity: $\lim_{\rho \to \infty} R/\log \rho$ . Outage rate is a finite-SNR, finite-error-probability quantity. For target outage $\epsilon$ at finite SNR, the realizable rate is much less than the DoF times $\log \rho$ . The disconnect grows with $K$ and with smaller $\epsilon$ .

A more faithful design metric at target $\epsilon$ is the $\epsilon$ -outage spectral efficiency: $R_\epsilon(K, M, N, L, \rho)$ , evaluated at the specific operating point. This should drive cache sizing decisions, not the asymptotic DoF alone.

⚠️Engineering Note

Outage in 5G NR and Beyond

5G NR's HARQ (hybrid ARQ) mechanism mitigates outage by retransmitting dropped packets. Effective outage rates are much lower than pure first-transmission outage — often $10^{-6}$ after HARQ. Cache-aided multicast can exploit HARQ in two ways:

Incremental redundancy. Retransmit coded-XOR messages with additional parity. Cache contents at receivers are unchanged between HARQ rounds.
Soft combining. Users with weaker channels accumulate soft decisions across rounds.

The CommIT group has studied HARQ + coded caching in joint settings (Park-Kountouris-Caire 2019+). The conclusions: HARQ is a natural complement to multi-antenna coded caching, and the combined system approaches the outage-free capacity within a few dB at realistic operating points.

Practical Constraints

•
5G NR HARQ: up to 16 rounds per TB, typical 1-4
•
Target URLLC outage: 10^{-5} at 32 Byte payload
•
Cache + HARQ: soft combining preserves cache-based decoding

Outage-Limited Rate Analysis