Ferkans — Interactive Telecom Tutor

Generalizing MAN to Multiple Transmitters

The single-server MAN scheme (Chapter 2) sends $\binom{K}{t+1}$ XOR messages in sequence. With $S$ cooperating servers, we can send multiple messages in parallel — one per server — provided the messages don't interfere at the receivers. Zero-forcing across $SL$ aggregate antennas lets us do this.

This section describes the multi-server MAN scheme: the same combinatorial placement as standard MAN, with an adapted delivery phase that exploits both coded multicasting and cooperative MIMO. The rate (in file units) is simply $R_\text{MAN}/S$ — scale the single-server rate by the number of cooperating servers. The DoF matches Theorem 9.1.

Multi-Server MAN Delivery

Complexity: Per

(t + SL)

-subset:

SL

coded streams, each serving

t+1

users. Total messages:

\binom{K}{t+SL}

subsets ×

SL

streams =

SL \binom{K}{t+SL}

. Each message of size

F/\binom{K}{t}

. Total bits:

O(F \cdot SL \binom{K}{t+SL}/\binom{K}{t})

. Delivery time: scales as

K(1-\mu)/(SL(1+t))

channel uses.

Input:

S

servers each with full library; MAN placement with

integer

t

; demand vector

\mathbf{d}

; channels

\{\mathbf{h}_{k,i}\}

.

Output: Transmit signals

\mathbf{x}_1, \ldots, \mathbf{x}_S

.

1. Enumerate

(t + SL)

-subsets

\mathcal{S}' \subseteq [K]

.

2. for each subset

\mathcal{S}'

do

3.

\quad

Partition

\mathcal{S}'

into

SL

groups, each of size

(t+1)

.

4.

\quad

for each sub-group

\mathcal{G}_j

(j = 1 to SL) do

5.

\quad\quad

Form XOR:

\tilde{x}_j = \bigoplus_{k \in \mathcal{G}_j} W_{d_k, \mathcal{S}' \setminus \{k\}}

6.

\quad

end for

7.

\quad

Cooperatively beamform:

[\mathbf{x}_1; \ldots; \mathbf{x}_S] = \mathbf{V} [\tilde{x}_1; \ldots; \tilde{x}_{SL}]

8.

\quad

\mathbf{V}

is the aggregate

SL \times SL

zero-forcing precoder across servers.

9. end for

The scheme requires joint precoding across servers: each server's transmit signal is a function of all users' demands and all servers' channels. This is the "full cooperation" assumption.

Theorem: Multi-Server MAN Achievability

The multi-server MAN scheme with $S$ cooperating $L$ -antenna servers achieves sum-DoF $= \min(t + S L, K)$ , matching the Lampiris-Caire formula with effective $L_\text{eff} = SL$ .

The $S$ servers are treated as a single transmitter with $SL$ antennas by joint precoding. All of Chapter 5's analysis applies verbatim.

Proof

Consolidation

Under full cooperation, the $S$ servers emit a joint signal vector $\mathbf{x}^\text{agg} \in \mathbb{C}^{SL}$ from the perspective of users. This is identical to a single $SL$ -antenna transmitter.

Apply Chapter 5

The cache-aided $SL$ -antenna BC has DoF $= t + SL$ by Theorem 5.1 (Lampiris-Caire).

Per-server rate

The per-server transmit rate is $(t + SL)/S$ DoF per channel use — scaled by $S$ versus the single-server case. $\blacksquare$

Per-Server Rate vs Memory Ratio

Per-server delivery rate under cooperative multi-server MAN. With $S$ cooperating servers, each server's load is $R_\text{MAN}/S$ — the aggregate rate is the same as single-server MAN, but split across $S$ servers.

Parameters

Users K12

Max servers S6

Example: Multi-Server MAN: $K = 4$ , $S = 2$ , $L = 1$ , $t = 1$

Work through the multi-server MAN delivery for $K = 4$ users, $S = 2$ cooperating servers (each with $L = 1$ antenna), $t = 1$ (so $M/N = 1/4$ ), demand $\mathbf{d} = (1, 2, 3, 4)$ .

Solution

Placement

MAN, $t = 1$ : each file split into $\binom{4}{1} = 4$ subfiles $W_{n,\{k\}}$ . User $k$ caches $W_{n,\{k\}}$ .

Effective antennas

$SL = 2$ . Group size for Lampiris-Caire: $t + SL = 3$ .

Delivery groups

$(t+SL) = 3$ -subsets of $[4]$ : $\{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}$ . Four groups; each group $SL = 2$ beams.

Per-group beams

For group $\{1,2,3\}$ : two ZF beams. Beam 1 target $\{1,2\}$ (null user 3): $\tilde x_1 = W_{d_1, \{2\}} \oplus W_{d_2, \{1\}} = W_{1,\{2\}} \oplus W_{2,\{1\}}$ . Beam 2 target $\{1,3\}$ (null user 2): $\tilde x_2 = W_{d_1, \{3\}} \oplus W_{d_3, \{1\}} = W_{1,\{3\}} \oplus W_{3,\{1\}}$ .

Per-server signals

Cooperative precoder: $[\mathbf{x}_1; \mathbf{x}_2] = \mathbf{V} [\tilde x_1; \tilde x_2]$ where $\mathbf{V}$ zero-forces across the two antennas (one per server).

Decoding

Each user in the group receives its targeted beam (the other is nulled); XOR-cancels using cached subfiles.

Rate

4 groups × 2 beams × 1 channel use per beam = 8 beam-channel-uses. Each beam delivers 1 subfile-sized message = $F/4$ bits. Total transmission time: normalized DoF = $t + SL = 3$ ; per-user DoF = 3/4. Compare single-server: DoF = $t + L = 2$ , per-user 0.5. Cooperation gives 1.5× DoF improvement.

⚠️Engineering Note

Multi-Server Cooperation in Practice

Deployed multi-server cooperation schemes:

3GPP CoMP (Rel-11+). Coordinated multi-point transmission; up to 3-cell cooperation typical. Latency budget: ~10 ms for shared data exchange.
5G Joint Transmission (JT-CoMP). Data is simultaneously sent from multiple cells with cooperative precoding. Used in dense deployments.
6G cell-free massive MIMO. Envisioned $S = 10\text{-}100$ cooperating APs; requires dedicated fronthaul infrastructure.

For cache-aided multi-server systems, the inter-server communication requirement is higher than for uncoded CoMP: servers must also coordinate on which MAN subfiles each holds, and on the XOR structure of delivery. Realistic scale: $S = 2-8$ per cluster with full cooperation, time-shared between clusters.

Practical Constraints

•
CoMP Rel-11: up to 3 cooperating cells per cluster
•
Joint Transmission: requires ~1 ms latency for data sharing
•
Cell-free mMIMO (6G vision): 10-100 APs cooperating
•
Cache coordination: aligned placement required across servers

Partial Cooperation

When full cooperation is infeasible, one falls back to partial cooperation:

Cluster cooperation. Group servers into clusters of size $S_c < S$ ; full cooperation within cluster, time-share between clusters. DoF reduces from $t + SL$ to $t + S_c L$ per cluster, averaged.
Non-cooperative coded caching. Each server independently runs MAN. No DoF gain from $S$ ; each server handles $K/S$ users. Rate per server: $R_\text{MAN}(K/S, L, M)/S$ . This is the worst case.

Typical deployment: $S_c = 2-4$ . With $L = 4$ and $S_c = 4$ , cluster DoF = $t + 16$ — substantial even at moderate cluster sizes.

The Multi-Server MAN Scheme