# A Novel Multi-Server Federated Learning Framework in Vehicular Edge Computing
## 5 Transmission Latency Analysis
### Ms-Fl  
In our MS-FL framework, the total transmission latency in each communication round denoted by \(\rm\tau_{MS-FL}(t)\) comprises three key components: the local uploading time from clients to their regional FL servers, the downloading time from regional servers to clients, and the inter-server (RSU-to-RSU) transmission time. Owing to the significantly higher communication capabilities of RSU-to-RSU links compared to client-to-RSU links, the overall latency is predominantly determined by the maximum latency among the clients. Nonetheless, the inter-server latency is also accounted for in the analysis. * Client-to-RSU Communication: For each client i in round t, the uploading time to its associated server s is given by the following: \[\rm\tau_{i,s}^{U}(t)=\frac{q_{i}}{b_{i,s}\rm\tau_{i,s}^{U}(t)},\] (16) where \(\rm q_{i}\) is the size of the data (in bits) to be uploaded by client i, and \(\rm\tau_{i,s}^{U}(t)\) is the achievable uplink rate (in bits/s/Hz) from client i to server s at round t. \(\rm b_{i,s}\) is the uplink bandwidth (in Hz) allocated to client i by server s.  
As in [18; 19], we use [PERSON]'s formula [30] to model the uplink rate under noise-limited conditions. The uplink rate is given by the following:  
\[\tau_{\text{i,s}}^{\text{U}}(\text{t})=\log_{2}{(1+\frac{\text{P}_{ \text{i,s}}^{\text{U}}\big{|}g_{\text{i,s}}^{\text{U}}(\text{t})\big{|}^{2}}{ \mu^{2}})}, \tag{17}\]  
where \(\text{P}_{\text{i,s}}^{\text{U}}\) represents the uplink transmit power and \(g_{\text{i,s}}^{\text{U}}(\text{t})\) denotes the uplink channel gain at time \(t\). The noise power is \(\mu^{2}\). Similarly, the downloading time from server s to client i is modeled as follows:  
\[\tau_{\text{i,s}}^{\text{D}}(\text{t})=\frac{\text{q}_{\text{i,s}}}{\text{b}_{ \text{i,s}}\tau_{\text{i,s}}^{\text{D}}(\text{t})}, \tag{18}\]  
where \(\text{q}_{\text{i,s}}\) is the size (in bits) of the data that client \(i\) downloads from server s, \(\text{b}_{\text{i,s}}\) is the downlink bandwidth allocated by server s to client i, and \(\tau_{\text{i,s}}^{\text{D}}(\text{t})\) is the achievable downlink rate (in bits/s/Hz) from server s to client i at round t, given by the following:  
\[\tau_{\text{i,s}}^{\text{D}}(\text{t})=\log_{2}{(1+\frac{\text{P}_{ \text{i,s}}^{\text{D}}\big{|}g_{\text{i,s}}^{\text{D}}(\text{t})\big{|}^{2}} {\mu^{2}})}, \tag{19}\]  
where \(\text{P}_{\text{i,s}}^{\text{D}}\) is the downlink transmit power and \(\text{g}_{\text{i,s}}^{\text{D}}(\text{t})\) is the downlink channel gain at time t.  
* RSU-to-RSU Communication: Although RSUs typically use high-speed backhaul links [31], we include the following general mode for completeness. The latency to exchange a model between two servers at round t is as follows: \[\tau_{\text{RSU-RSU}}(\text{t})=\frac{\text{q}_{\text{RSU}}}{\text{b}_{\text{s, s}}\tau_{\text{RSU-RSU}}(\text{t})},\] (20) where \(\text{q}_{\text{RSU}}\) is the size of the data (e.g., model parameters) exchanged between servers, \(\text{b}_{\text{s,s}}\) is the bandwidth reserved for RSU-to-RSU communication, and \(\tau_{\text{RSU-RSU}}(\text{t})\) is the effective transmission rate between RSUs at time t, given by the following: \[\tau_{\text{RSU-RSU}}(\text{t})=\log_{2}{(1+\frac{\text{P}_{ \text{RSU-RSU}}\big{|}g_{\text{RSU-RSU}}(\text{t})\big{|}^{2}}{\mu^{2}})},\] (21) with \(\text{P}_{\text{RSU-RSU}}\) as the RSU transmit power, \(\text{g}_{\text{RSU-RSU}}(\text{t})\) as the instantaneous channel gain, and \(\mu^{2}\) as the noise power. * Overall Transmission Latency: Assuming that training must proceed for \(T_{MS-FL}\) rounds to reach the target performance, the total transmission latency is simply the sum of the per-round latencies: \[\tau_{\text{MS-FL}}^{\text{Total}}(\text{t})=\sum_{\text{t}=1}^{T_{\text{MS- FL}}}\tau_{\text{MS-FL}}(\text{t}),\] (22) where at each communication round t, the dominant latency component is the maximum of (i) all client-to-server upload delays, (ii) all server-to-client download delays, and (iii) all RSU-to-RSU transfer delays.