# Resource Allocation in C-V2X Mode 3 Based on the Exchanged Preference Profiles
## 2 System Model
### V2V Links' Effective Capacity  
As can be observed in Formula (2), the SINR of the V2V link is dynamic in different slots, leading to the instability of the network serviceability. Thus, in this paper, we assume that the traffic generated by the VUE enters an infinite-size first-in-first-out (FIFO) buffer before it is transmitted through the V2V channel, which is shown in Figure 2. Since the vehicles periodically broadcast the CAMs to inform the neighbors of their presence, the  
Figure 1: C-V2X mode 3 communication model.  
generated data traffic rate of the VUE can be regarded as a constant \(\lambda_{k}\) and the source traffic of VUE \(k\) during \([0,t]\) is \(A_{k}(t)=\lambda_{k}t\). We define the actual service traffic as \(S_{k}(t)\), and the achievable service traffic is \(\hat{S}_{k}(t)\). It is obvious that \(S_{k}(t)\) is less than or equal to \(A_{k}(t)\) for any \(t>0\) and \(S_{k}(t)\) is bounded by the minimum value of \(A_{k}(t)\) and \(\hat{S}_{k}(t)\), which can be expressed as  
\[S_{k}(t)=\min\{\hat{S}_{k}(t),A_{k}(t)\} \tag{4}\]  
Figure 3 illustrates the LVP of the V2V link. As shown in Figure 3, \(D_{max}\) represents the maximum tolerable latency of packet transmission, and \(Q_{max}\) is the maximum traffic queue length in the buffer. The minimum traffic curve, which should be served to guarantee a transmission latency of no more than \(D_{max}\), is expressed as \(\psi(t)=\lambda_{k}(t-D_{\max})\). The latency violation events happen when the actual service curve \(S_{k}(t)\) is lower than the curve \(\psi(t)\). Thus, the LVP of the V2V link \(k\) can be expressed as  
\[\Pr\{D_{k}\geq D_{\max}\}=\frac{\int_{t\in\mathcal{V}}dS_{k}(t)}{S_{k}(lT)} \tag{5}\]  
where \(\mathcal{V}\) denotes the part where \(S_{k}(t)\leq\psi(t)\) and \(D_{k}\) is the latency experienced by the VUE \(k\). \(T\) is the length of one time slot, and \(l\in\{0,1,2,\dots\}\).  
Formula (5) reflects that the LVP of a V2V link is closely related to the maximum tolerable latency \(D_{max}\) and the channel's actual cumulative service traffic \(S_{k}(t)\). Thus, to characterize the relations between \(D_{max}\) and \(S_{k}(t)\), according to [11], we introduce the effective capacity conception \(EC_{\xi}(\theta_{k}^{V})\), which provides a measure for the maximum constant source traffic rate supported by a given service rate under the constraint of latency requirement factor \(\theta_{k}^{V}\). The effective capacity \(EC_{\xi}(\theta_{k}^{V})\) of VUE \(k\) can be expressed as  
\[EC_{k}(\theta_{k}^{V})=-\frac{1}{\theta_{k}^{V}T}\ln E\left\{e^{-\theta_{k}^{ V}T\text{W}\theta_{2}(1+\gamma_{k,n}^{V})}\right\} \tag{6}\]  
As proved in [11], the effective capacity \(EC_{\xi}(\theta_{k}^{V})\) is monotonically decreasing with \(\theta_{k}^{V}\in[0,\infty)\). The effective capacity reaches its maximum value when \(\theta_{k}^{V}=0\), which is  
Figure 3: Traffic and service model.  
Figure 2: Queue theory model.  
\[EC_{k}(\theta_{k}^{V}=0)=\mathbb{E}\Big{\{}W\log(1+\gamma_{k,n}^{V})\Big{\}} \tag{7}\]  
Moreover, the effective capacity reaches its minimum value when \(\theta_{k}^{V}=\infty\), i.e., \(EC_{k}(\theta_{k}^{V}=\infty)=0\).  
Particularly, when the source traffic rate \(\lambda_{k}\) is equal to \(EC_{k}(\theta_{k}^{V})\), the LVP of the VUE \(k\) can be expressed as  
\[\Pr\{D_{k}\geq D_{\text{max}}\}=p(\lambda_{k})e^{-\theta_{k}^{V}\lambda_{k}D_ {\text{max}}} \tag{8}\]  
where \(p(\lambda_{k})\) denotes the buffer nonempty probability in the steady queue state, which is  
\[p(\lambda_{k})=\frac{\lambda_{k}}{\mathbb{E}\big{\{}W\text{log}_{2}(1+\gamma_ {k}^{V})\big{\}}} \tag{9}\]