## 5 Analysis of 3G/LTE Cost  
For analyzing 3G/LTE cost, a cycling fleet \(F\) consisting of one or more long-thin groups is considered. We describe the assumptions (which are similar to [25; 35]) used in our analytical model and derive the expected amounts of cyclist sensing data \(d_{\rm L}\) uploaded (i.e., \(d_{\rm E}^{\rm UL}\)) and global fleet information \(d_{\rm F}\) downloaded (i.e., \(d_{\rm E}^{\rm DL}\)) via 3G/LTE in a group \(L\). We use a Poisson distribution to approximately model the number of fleet cyclists in any subinterval of the common cycling path \(P\) and thus the distance between any two consecutive fleet cyclists is exponentially distributed. Each fleet cyclist rules on \(P\) with a constant speed and has a fixed transmission range of \(r\). If the distance between two fleet cyclists is equal to or less than \(r\), they can directly communicate with each other. Suppose that the total number of fleet cyclists on \(P\) of length \(C\) is \(N\) and the fleet cyclists on \(P\) are \(b_{1},b_{2},\ldots,\) and \(b_{N}\). The size of \(L\) is decided by two conditions. The first condition is the density \(\lambda\ (=N/C)\) of fleet cyclists on \(P\). The second condition is the distance \(d_{i,i+1}\), for \(i=1,2,\ldots,N-1\), of consecutive fleet cyclists \(b_{i}\) and \(b_{i+1}\). In particular, \(d_{\rm E}^{\rm UL}\) and \(d_{\rm E}^{\rm DL}\) may increase as \(\lambda\) increases or \(d_{i,i+1}\) decreases. A random variable \(D_{i}\) that represents the distance between \(b_{i}\) and \(b_{i+1}\) (for \(i=1,2,\ldots,N-1\)) is exponentially distributed with the following cumulative distribution function:  
\[F_{D_{i}}(d)=\left\{ \begin{array}{ll}1-e^{-\lambda d}&d\geq 0\\ 0&d<0\end{array} \right.. \tag{3}\]Assume that \(L\) consists of \(n\) group cyclists (i.e., exactly \(n-1\) consecutive pairs of fleet cyclists are within \(r\)). It is required that \(D_{i}\leq r\) for \(i=1,2,\ldots,n-1\) and \(D_{n}>r\) if \(n<N\), whereas it is required that \(D_{i}\leq r\) for \(i=1,2,\ldots,n-1\) if \(n=N\). The probability \(P_{n}\) that has \(n\) group cyclists in \(L\) follows:  
\[P_{n}=\left\{ \begin{array}{ll}\prod_{i=1}^{n-1}Pr\{D_{i}\leq r\}\times Pr\{ D_{n}>r\}&n<N\\ \prod_{i=1}^{n-1}Pr\{D_{i}\leq r\}&n=N\end{array} \right.. \tag{4}\]  
\(D_{1},D_{2},\ldots,D_{n-1}\), and \(D_{n}\) are independent and identically distributed random variables. From Equations (3) and (4), by substituting \(1-e^{-\lambda r}\) and \(e^{-\lambda r}\) for \(Pr\{D_{i}\leq r\}\) and \(Pr\{D_{n}>r\}\), respectively, \(P_{n}\) follows:  
\[P_{n}=\left\{ \begin{array}{ll}(1-e^{-\lambda r})^{n-1}e^{-\lambda r}&n<N\\ (1-e^{-\lambda r})^{n-1}&n=N\end{array} \right.. \tag{5}\]  
Therefore, the expected number \(n_{GC}\) of group cyclists in \(L\) is  
\[n_{GC}=\sum_{i=1}^{N}P_{i}\times i. \tag{6}\]  
The expected number \(n_{L}\) of groups in \(F\) is  
\[n_{L}=\frac{N}{n_{GC}}. \tag{7}\]  
In our framework, each group cyclist \(j\) transmits its \(d_{\mathrm{L}}\) to group cyclist \(j-1\) in sequence via ad-hoc communication and then the GH (i.e., group cyclist 1) transmits the aggregated \(d_{\mathrm{L}}\) to the server via 3G/LTE. Let \(d_{\mathrm{H}}\) and \(d_{\mathrm{P}}\) represent the sizes of the packet header and absolute position of \(d_{\mathrm{L}}\), respectively. Let \(\alpha\) be the sensing data aggregation ratio using relative positions instead of absolute positions. The expected \(d_{\mathrm{E}}^{\mathrm{UL}}\) of \(d_{\mathrm{L}}\) uploaded via 3G/LTE for sensing data collection can be modeled by summing the expected size of aggregated \(d_{\mathrm{L}}\) over all possible numbers of group cyclists in \(L\) as follow:  
\[d_{\mathrm{E}}^{\mathrm{UL}}=\sum_{i=1}^{N}P_{i}\times[d_{\mathrm{H}}+d_{ \mathrm{P}}+\alpha\times(i-1)\times d_{\mathrm{P}}]. \tag{8}\]  
After collecting all \(d_{\mathrm{L}}\) in \(F\), the server returns \(d_{\mathrm{F}}\) to the GH.