# Improving Bundle Routing in a Space DTN by Approximating the Transmission Time of the Reliable LTP
## 3 System Analysis
### Bundle Service Time with Retransmissions  
The bundle service time \(S\) depends on the size of the bundle, the protocol values, and the channel parameter values. Since the bundle reliability is achieved through the retransmission of the lost segments, the number of transmission rounds drives the service time in particular when long propagation delays are involved. Assuming that the bundles travel on a single LTP block, a bundle requires the following transmission rounds on average to be successfully delivered (see [5] for details):\[\kappa(p)\approx\max\biggl{\{}1,\frac{-log_{p}(n)+\gamma}{1-p^{\prime}}\biggr{\}} \tag{3}\]  
where \(p\) is the data segment loss probability, \(p^{\prime}\) the control segment loss probability, and \(\gamma=0.5772\dots\) is the Euler-Mascheroni constant. Typically, the size of the data segments is much larger than the size of the control segments in the LTP, and therefore \(p>p^{\prime}\). Assuming independent bit errors, the value of \(p\) (and \(p^{\prime}\)) can be determined with \(1-(1-p_{e})^{L}=1-e^{Llog(1-p_{e})}\), where \(p_{e}\) is the channel bit error rate (BER) and \(L\) is the segment size, including the lower layer headers. To remove the extra complexity of having to set a large number of protocol parameters, and to better focus the discussion, a simplified expression for the block service time \(S\) than the one given in [5] was adopted. The proposed expression works reasonably well for common parameter values: a 100 kB block and bundle size, 1 Mbps channel rate, 1000 B segment payload, and 100 B header length. The effective bundle service time (\(S_{e}\)), which depends on the value of the packet loss ratio \(p\), can be found using  
\[S_{e}(p)\approx S_{0}+(\kappa(p)-1)(2D+1) \tag{4}\]  
where \(D\) is the one-way propagation delay and \(S_{0}\) is the nominal bundle transmission time over a lossless channel (i.e., \(D\) plus the ratio of the bundle size and the channel rate). The radiation time of the segment retransmission is assumed to be negligible, as the propagation delay tends to dominate the expression. However, even if that assumption does not hold, the error is small, as shown in Figure 1, which depicts the approximated service time (Equation (4)) compared to the actual value. The notations \(S_{e}(p)\) and \(\kappa(p)\) emphasize the dependency of these values on the packet loss ratio. It is worth noting that the parameter \(p\) is observable by the sender, as in the LTP protocol, the receiver sends negative acknowledgment (NACK) messages for the lost segments. It is therefore possible to measure \(p\) instead of calculating it from the channel's BER.  
Figure 1: Visual verification of the proposed simplified expression for the bundle service time compared to the complete model [5]. It is assumed there are a 100 kB bundle and block size, 1 Mbps channel rate, 1000 B payload, and 100 B headers. Results for one-way propagation delays of 0.1 s, 1 s, 10 s, and 100 s are presented.  
While the accurate calculation of the average service time could be used instead, the simplified expression improved the route computation speed.