# Improving Bundle Routing in a Space DTN by Approximating the Transmission Time of the Reliable LTP
## 3 System Analysis
### End-to-End Flow Response Time  
A DTN is affected by repeated link disruptions, with some lasting a very long time. To account for this characteristic, consider that a queuing system is used to model an arbitrary link \((a,b)\in\mathcal{E}\) that becomes available for bundle transmissions at a rate \((C+V)^{-1}\), where \(C>0\) and \(V>0\) are the average contact and disruption durations, respectively (i.e., the average duration of one period is given by a link up average time \(C\) followed by a link down average time \(D\)).  
The queuing system is modeled by exponentially distributed values for the contact and disruption durations, service time, and interarrival time. Note that this model assumes features that are more general than the ones found in a typical space DTN, since both the contact and disruption durations are modeled as random values instead of being deterministic, which allows drawing conclusions for broader operational contexts. In a later section, the focus will switch to a deterministic DTN in the simulation study.  
The bundle arrivals at this queuing system are not impacted by whether the link is disrupted. Instead, an arrival may occur at any time as determined by the load \(\lambda\), whose value aggregates the flow rates scheduled to pass through this link, including both those originated by the current node and those arriving from other nodes. The buffers are of sufficient capacity for the communication requirements of the system, and therefore, it is possible to assume infinite queue lengths. The bundles are then served according to a first-come-first-serve policy, and therefore, a new bundle arrival can be transmitted immediately only if no other bundle is being transmitted and the contact is active. In all other cases, the bundle joins the transmission buffer.  
Let \(S\) denote the bundle service time, which is the time required to successfully transmit the bundle over the link \((a,b)\). Because a contact may end before the service completes (i.e., disruptions are not restricted to idle queues), the LTP implements a pause-and-resume mechanism that prevents dropping the bundle when such a case occurs and allows resuming the transmission at the next contact time. The average service time is therefore impacted by such transmission interruptions. The extended average service time \(X\) can be found by multiplying the service time without interruptions \(S\) by a constant factor as follows:  
\[X=S\bigg{(}1+\frac{V}{C}\bigg{)}, \tag{1}\]  
with the following constraint needed for stability; that is, the average capacity provided by the contacts must be larger than the total traffic load \(\lambda\), or in terms of the link utilization, one has  
\[\lambda X<\frac{C}{C+V}.\]  
With this result, and by extending the queuing model to a system with server vacations, the following expression can be found for the average response time \(T\) of the bundles over the selected link. The extended service time given in Equation (1) by the interrupted transmissions has been accounted for in the following expression [5]:  
\[T=\frac{C^{2}S+2 CSV+(C+S)V^{2}}{(C+V)(C-S(C+V)\lambda)} \tag{2}\]  
This result shows the role of the average contact and disruption durations, traffic load, and service time in the end-to-end response time performance. The former three parameters are normally known. The contact and disruption times are usually precomputed ahead of the mission, as the contact plan and the load could either be inferred from the intended application or measured. However, the latter parameter is variable, as it strongly depends on the specific channel conditions that determine the segment retransmissions.