# Cooperative Crossing Cache Placement in Cache-Enabled Device to Device-Aided Cellular Networks
## 3 Offloading Probability Analysis
### Offloading Probability in D2D Mode  
If the requested content is not stored locally or the requesting user has no cache ability, the content is searched for in the cache of nearby devices within a D2D distance \(R\), in which at least one D2D transmitter is available to serve the requesting user while guaranteeing the successful content transmission within a certain time \(\tau\). The offloading probability in D2D mode is given by  
Figure 1: Illustration of system model. \[P_{2}=\sum\limits_{k=1}^{K}\ p_{k}\times P_{k}\times(1-q_{k})\mathbb{P}_{u,n}\times P (SIR_{k}^{d2d})\geq\xi \tag{3}\]  
where \(P_{k}\) denotes the probability of establishing a D2D link when the user requests content \(k\) and \(\mathbb{P}_{u,n}\) is the coverage probability of D2D transmitters that can serve a requesting user, where at least one of them is available. More precisely,  
\[ \begin{array}{ll}\mathbb{P}_{u,n}&=\sum\limits_{n=0}^{\infty}\Pr(N_{k}=n) \left(1-\frac{n}{i=1}\ (1-\Omega_{i})\right)\\ &=\sum\limits_{n=1}^{\infty}\Pr(N_{k}=n)\big{(}1-\big{(}1-(1-e^{-\lambda_{t} \tau})\big{)}\big{)}^{n}\\ &=1-\sum\limits_{n=1}^{\infty}\Pr(N_{k}=n)e^{-n\lambda_{t}\tau}\end{array}  \tag{4}\]  
where \(\Pr\ (N_{k}=n)=e^{-\lambda^{\prime}_{u,k}\tau R^{2}\frac{(\lambda^{\prime}_{u,k} \tau R^{2})^{n}}{n! }}\) is the probability that there are \(n\) available D2D transmitters with content \(k\) cached within a D2D distance \(R\), and \(\Phi^{t}_{u,k}\) denotes the set of all available D2D transmitters with density \(\lambda^{\prime}_{u,k}=\lambda_{u,k}\)\(\Omega\). Thus, (4) can be rewritten as:  
\[\mathbb{P}_{u,n}\ =1-e^{-\pi\lambda^{\prime}_{u,k}R^{2}(1-e^{-\lambda_{t}\tau})}+ e^{-\pi\lambda^{\prime}_{u,k}R^{2}} \tag{5}\]  
\(P(SIR_{k}^{d2d}\geq\xi)\) is the probability of successful content transmission in D2D communication, which can be achieved as follows:  
\[ \begin{array}{ll}P(SIR_{k}^{d2d}&(r)\geq\xi)=P\left[\frac{P_{u,k}br^{-\alpha }}{\sum\limits_{i\in\{\Phi^{t}_{u,k}\cup\Phi^{t}_{u}\}\setminus\{0\}}P_{u}h_{ i}r_{i}-\alpha+\sum\limits_{i\in\Phi_{b}}P_{b}h_{i}r_{i}-\alpha}\geq\xi\right]\\ &=P\left[h\geq\frac{\xi^{\prime}\alpha}{P_{u}}\left(\sum\limits_{i\in\{\Phi^{t }_{u,k}\cup\Phi^{t}_{u}\}\setminus\{0\}}P_{u}h_{i}r_{i}-\alpha+\sum\limits_{i \in\Phi_{b}}P_{b}h_{i}r_{i}-\alpha\right)\right]\\ &\stackrel{{(a)}}{{=}}E_{r_{k}}\big{[}\mathcal{L}_{I_{u}}(r^{ \alpha}\xi P_{u}^{-1})\mathcal{L}_{I_{b}}(r^{\alpha}\xi P_{u}^{-1})\big{]}\\ &=\int\limits_{0}^{R}\mathcal{L}_{I_{u}}(r^{\alpha}\xi P_{u}^{-1})\mathcal{L}_ {I_{b}}(r^{\alpha}\xi P_{u}^{-1})f_{r_{k}}(r)dr,\end{array}  \tag{6}\]  
where \(\Phi^{t}_{u}\) denotes the set of D2D transmitters without content \(k\) cached and \(\xi=e^{\frac{\xi\ln 2}{\pi\
u}}-1\); (a) is due to the complementary cumulative distribution function of \(h\), which follows an exponential distribution with unit mean. Further, \(\mathcal{L}_{I_{u}}(r^{\alpha}\xi P_{u}^{-1})\) and \(\mathcal{L}_{I_{b}}(r^{\alpha}\xi P_{u}^{-1})\) are the Laplace transforms of the interference from all content-cached devices except the corresponding D2D transmitter \(i_{0}\) and from all the BSs, respectively. The probability density function (PDF) of the distance between a typical user and the corresponding D2D transmitter, when content \(k\) is requested, is \(f_{r_{k}}(r)=\frac{2\pi r\lambda^{\prime}_{u,k}}{P_{k}}\exp\left(-\pi r^{2}( \frac{P_{b}}{P_{u}})^{2/\alpha}\lambda_{b}+\lambda^{\prime}_{u,k})\right)\), and \(P_{k}=2\pi r\lambda^{\prime}_{u,k}\int_{0}^{R}r\exp(-\pi r^{2}[(\frac{P_{b}}{P_ {u}})^{2/\alpha}\lambda_{b}+\lambda^{\prime}_{u,k}])dr\), which are given in the proof of [2]. We have \(\mathcal{L}_{I_{u}}(r^{\alpha}\xi P_{u}^{-1})\), shown below in (7), where (a) results from the assumption of the exponential distribution of \(h_{i}\) with unit mean and follows Theorem 2 in [33]. Note that this step arises from the interferer devices with and without content \(k\) cached.