## 5 Towards Self-Modeling Sensor Networks
For the purpose of their aggregate output arrival curve calculation, the share of service offered to each flow \(f\in F\) from its source to \(s_{i}\) within this aggregate is the concatenation of the service on its path. Then, the entire flow aggregate's output is bounded by_  
\[\overline{\alpha}_{i} = \sum_{f\in F}\left(\alpha^{f}\hat{\osigma}\bigotimes_{j=0}^{L(f _{i})}\beta_{P(f,i)}\right)\]  
_where \(L(f,s)\) is the location (index) of sensor node \(s\) on \(f\)'s path, and in reverse, \(P(f,i)\) denotes the sensor at location (index) \(i\) on \(f\)'s path. Applying Corollary 3 and the composition rule, we can rephrase the equation to:_  
\[\overline{\alpha}_{i} = \gamma_{\overline{r}_{i}\overline{b}_{i}}\]  
_with_  
\[\overline{r}_{i}=\sum_{f\in F}r^{f}\quad\text{and}\quad\overline{b}_{i}=\sum_ {f\in F}\left(b^{f}+r^{f}\cdot\sum_{j=1}^{L(f_{i})}T_{P(f,i)}\right).\]  
Proof. : Can be found in [22]. Using this result in the PMOO analysis of Algorithm 2 requires separating the flow of interest from its interfering traffic. To do so, Theorem 7 can be applied to a the subset of flows \(F^{\prime}\subseteq F\) not including the flow of interest. This reformulated computation's decisive improvement for a self-modeling WSN is the change from a total output arrival curve for an flow aggregate to an aggregate of flow-local results. The computations only make use of individual flow's arrival curves and the service curves of servers they cross. Information about the sink tree network's exact shape, merging locations with other flows or left-over service curves, are not required. Figure 9 illustrates the flow-locality of the computation. Applying the new method from Theorem 7 instead of the conventional one of Section 3.3 has several practical advantages for self-modeling WSNs:  
Complete model with low communication overhead: Flow-locality enables one to overcome the need for an additional protocol such as Deluge [23], distributing the information required to derive network-internal arrival curves and left-over service curves. Each flow's arrival at a server can be calculated hop-by-hop without compromising accuracy. A flow can carry information about its  
Figure 9: Sink tree network with flow aggregation (left) converted to a set of flow-local views by Theorem 7 (right). current arrival bound as payload, pushing the information to all sensors concerned. From all of the flows crossing them, sensor nodes can derive their local delay bound and backlog bound, as well as their left-over service curves. Thus, the network model is updated continuously, i.e., independent of a polling interval, with only a small payload overhead. Local reaction to changes: The flow-locality also affects the recomputation effort in the case of parameter modifications. Using the conventional method, the locality of a modified parameter did not matter much due to the complex setting of flow aggregation locations; a change to a single parameter always invalidated a large amount of the derivation's intermediate results and triggered expensive recomputations, usually of the entire subtree. Theorem 3 prevents such an invalidation from spreading to flows not directly affected by a change, e.g., flows not crossing a sensor that adapted its rate, and thus enables a quick reaction to changes.