## 4 Network Model
### MS Mobility Model (\(m=1\))  
Intuitively, increasing the sink velocity \(v\) will improve the system efficiency, since in a unit time interval the mobile sink can contact more sensors and gather more information throughout the sensor field. We should carefully choose this parameter, explained as follows. On the one hand, the higher the MS velocity is, the higher the probability for sensors to meet MSs. On the other hand, when MSs are moving too fast across the effective communication region of sensors, there may not be a sufficiently long session interval for the sensor and sink to successfully exchange one potentially long packet. In other words, with the increase of MS velocity, the \"outage probability\" of packet transmission will arise. Therefore, finding a proper value for sink velocity must be a tradeoff between minimizing the sensor-MS meeting latency and minimizing the outage probability. Suppose the network consists of \(m\) MSs and \(n\) sensors in a disk of unit size (of radius \(1/\sqrt{\pi}\)). Both the sink and sensor operate with transmission range of \(r\). The mobility pattern of the mobile sinks \(\text{M}_{i}\) (\(i=1\),..., \(m\)) is according to \"Random Direction Mobility Model\" [22], however, with a constant velocity \(v\). The sink's trajectory is a sequence of epochs, and during each epoch the moving speed v of \(\text{MS}_{i}\) is invariant and the moving direction of \(\text{MS}_{i}\) over the disk is uniform and independent of its position. Denote \(\sigma_{i}\) as the epoch duration of \(\text{MS}_{i}\), which is measured as the time interval between \(\text{MS}_{i}\)'s starting and finishing points. \(\text{Q}_{i}\) is an exponentially distributed random variable, and the distributions of different \(\sigma_{i}\) (i = 1,...,\(m\)) are independent and identically-distributed (iid) random variables with common average of \(\overline{\sigma}\). Consequently the epoch length of different \(\text{L}_{i}\)s are also iid random variables, sharing the same average of \(\text{L}=\overline{\sigma}\times v\). \begin{table}
\begin{tabular}{|c|c|} \hline
**Sink Type** & **Coordinate** \\ \hline
1 (SS) or 0 (MS) & \((\mathcal{X}^{\text{j}}_{\text{MS}-\text{EN}_{i}^{\text{j}}},\mathcal{Y}^{ \text{j}}_{\text{MS}-\text{EN}_{i}^{\text{j}}})\) or \\  & \((\mathcal{X}^{\text{SS}-\text{EN}_{i}^{\text{j}}},\mathcal{Y}^{\text{SS}-\text {EN}_{i}^{\text{j}}}_{\text{MS}-\text{EN}_{i}^{\text{j}}})\) \\ \hline \end{tabular}
\end{table} 
Table 2: Response (queuing) message. Figure 4: FIFO SS-cache. Assume a stationary distribution of MSs, then the probabilities of independent MSs approaching a certain static sensor from different directions are equal. Specifically, the meeting of one static sensor N\({}_{\rm j}\) (j = 1,..., n) and one mobile sink MS\({}_{\rm i}\) is defined as MS\({}_{\rm i}\) covers N\({}_{\rm j}\) during an epoch. Since MS\({}_{\rm i}\)will cover an area of size \(\rm{\uppi r^{2}+2r\times L_{\rm i},k}\) (Figure 5) during the k-th epoch, then the number of epochs X\({}_{\rm i}\) needed till the first sensor-MS meeting is geometrically distributed with average of \(\rm{\frac{1}{p}=\frac{1}{\pi r^{2}+2r\times L}}\) (Theorem 3.1 of [22]), with the cumulative density function (cdf) as:  
\[\rm{F_{X_{i}}(x)=\sum_{x_{k}\preceq x}p(1-p)^{k-1}} \tag{20}\]  
where, p is a service probability. In the case of multiple mobile sinks, the sensor-MS meeting delay should be calculated as the delay when the first sensor-MS meeting occurs. Thus the number of epochs X needed should be the minimum of all X\({}_{i}\) (i = 1,..., \(m\)), with the cdf as:  
\[\rm{F_{x}(x)=1-[1-F_{x_{i}}(x)]^{m}\,\approx\sum_{x_{k}\preceq x}mp(1-p)^{k-1}} \tag{21}\]  
Denote \(\rm{\overline{X}}\) as the average of X, then the expected sensor sink meeting delay will be:  
\[\rm{D_{1}=\overline{X}\cdot\frac{\Gamma}{\
u}} \tag{22}\]  
If we increase the radio transmission range \(r\), or increase the number of MSs \(m\), or increase the sink velocity \(\
u\), the sensor sink meeting delay can be reduced.