separate decoding function for each user depending on the network topology. Hence, the decoding function for user $U_k, k\in [K],$ is $$\begin{aligned}
    \mu^k_{\left(\{\mathcal{Z}_k\}_{k=1}^{K}\right)} :[N]^K \times [2^{\lfloor FR_{\pi^k(1)} \rfloor}] \times \cdots \times [2^{\lfloor FR_{\pi^k(\rho)} \rfloor}] \times [2^{\lfloor FM_U \rfloor}] \rightarrow [2^{\lfloor F \rfloor}],\end{aligned}$$ where $\pi^k(1),\ldots, \pi^k(\rho)$ denote the $\rho$ servers in set $\mathcal{Z}_k$, maps the demand vector $\mathbf{d}$, the received signals $X_{\mathcal{Z}_k}$ from the servers in $\mathcal{Z}_k$, and the local cache content $V_k$ to the estimate $\hat{W}_{d_k},$ i.e., $$\begin{aligned}
    \hat{W}_{d_k} \triangleq \mu^k_{\left(\{\mathcal{Z}_k\}_{k=1}^{K}\right)} (\mathbf{d},X_{\mathcal{Z}_k},V_k)\end{aligned}$$ The probability of error for this scheme, for a fixed topology, is defined as $$\begin{aligned}
    \max_{\mathbf{d}\in [N]^K} \max_{k\in [K]} \mathrm{Pr}(\hat{W}_{d_k} \neq W_{d_k}).\end{aligned}$$ We remark here that the storage and caching functions $\sigma_p$ and $\phi_k$ do not depend on the network topology, while the encoding and decoding functions do.

The tuple $(M_S,M_U,R_1,\ldots,R_P)$ is said to be achievable if for every $\epsilon >0$ and large enough file size $F$ there exists a $(M_S,M_U,R_1,\ldots,R_P)$ caching scheme with probability of error less than $\epsilon$.

Our goal is to minimize the delivery latency, which is the time by which all the user requests can be satisfied. Among other parameters, delivery latency also depends on the operation of the SBSs. If each SBS transmits 