# Low Delay Inter-Packet Coding in Vehicular Networks
## 2 Preliminaries
### Performance Metric  
Our goal is to improve the reliability of communications by introducing redundancy into the transmitted data. When applying this commonly used approach to V2R communications, we meet specific problems which make it difficult to evaluate the system performance.  
First, the packet losses cannot be modeled as a random stationary process. For a given vehicle and a given RSU, we distinguish poor communications zones (entry zone and exit zone) when the distance between the receiver and the transmitter is large, and so-called production zone with relatively good transmission conditions [7].  
As a side effect of non-stationary channel conditions, the conventional criteria such as bit error rate (BER) or frame (word or packet) error rate (FER) do not make sense since these quantities change over time. The average overall error probability over the session is determined by the worst-case values.  
To formalize the problem of improving reliability of V2R communications, we use the following approach. Let \(e_{t}\) denote the probability of the packet loss at time moment \(t\), for \(t=1,...,T\) and where \(T\) is the total number of packets transmitted during one communication session. Assume that a message consisting of \(K\) packets should be transmitted, and the transmission is considered as successful only if all \(K\) packets are delivered to the receiver. To improve reliability, \(N-K\) redundant packets are added to form a block of length \(N\geq K\) packets, in such a way that, if the number of lost packets is \(\mu\leq N-K\), then \(K\) information packets can still be reconstructed by the decoder. More precisely, if \(\mu\) is less than the minimum distance of the code, then lost packets can always be correctly estimated, otherwise, only some of the lost packets can be correctly estimated.  
Let \(\xi_{t}\) denote the indicator function of the successful delivery of the data packet number \(t\) in the coded system, with \(t=1,...,T^{\prime}\), and where \(T^{\prime}\leq T\) is the total number of transmitted information packets during the session of length \(T\) packets. By definition, \(\xi_{t}\) is equal to 1 or 0 in case of success or failure of delivering packet number \(t\), respectively. In particular, \(\Pr(\xi_{t}=1)=1-e_{t}\) is the probability of successful delivery of the \(t\)th packet. If the message consists of \(K\) packets then the success in delivering the entire message can written in terms of \(\xi_{t}\) as  
\[\left(\sum_{\tau=t}^{t+K-1}\xi_{\tau}\right)=K\]  
For a sequence of the \(T^{\prime}\) transmitted packets, the average probability of message delivering should be computed as average over \(t=1,...,T^{\prime}-K+1\).  
The _successful delivery function_ (SDF) is defined as  
\[P_{\mathrm{s}}(K)\triangleq\frac{1}{T^{\prime}-K+1}\sum_{t=1}^{T^{\prime}-K+1} \mathrm{Pr}\Big{\{}\Big{(}\sum_{\tau=t}^{t+K-1}\xi_{\tau}\Big{)}=K\Big{\}}. \tag{1}\]  
This function characterizes the average (over session) probability of successful delivery of length \(K\) messages, as a function of \(K\).  
In the particular case of uncoded system, function \(P_{\mathrm{s}}(1)\) equals the fraction of successfully delivered packets, and uncoded transmission is optimal. In general, \(P_{\mathrm{s}}(K)\) decreases as \(K\) grows. This successful delivery function (SDF) shall be our performance measure.  
In our analysis, we assume that for coverage zones of RSUs the channel parameter \(\epsilon_{t}\) changes slowly with \(t\) and can be approximated as a constant value for a single block of length \(N\). This allows us to compare coding systems using different erasure correcting codes by simulating them on the BEC with a fixed symbol erasure probability.  
In addition, we compare codes using simulation on the M-BEC described by the Gilbert-Eliott model whose parameters do not change during one block of erasure correcting code. For the final evaluation, to take into account specific of drive-thru scenario, we measure SDF by simulation in real-life network conditions.