## 5 Towards Self-Modeling Sensor Networks  
In large-scale WSNs, any form of centralized control is often undesired if not impossible. Instead, a self-organized paradigm is followed; thus, it is consequent to also perform the modeling in  
Figure 8: Investigation of PMOO vs. TFA under the drifting apart of maximum and minimum scaling curves. Figure 7: Baseline comparison between pay multiplexing only once (PMOO) and total flow analysis (TFA) for different scenarios. a distributed, network-internal manner. The model is supposed to allow the WSN to act accordingly, e.g., by distributed decision on the admission of certain additional sensing tasks. For such a self-modeling with SensorNC, the two basic pieces of modeling information are: the residual service a sensor can offer, i.e., the left-over service curve, and the arrival curve valid for a flow at a certain location in the network. The left-over service curve is the result of an analysis that requires the bound on interfering traffic. Deriving the arrival curves for interfering traffic, on the other hand, requires executing the deconvolution of flow arrivals with (left-over) service curves according to the exact shape of the sink tree. For this reason, general network calculus relies on the global view of the network and thus does not support self-modeling in WSNs. In this section, we present a way how, in a restricted, but practically very relevant setting, the SensorNC computations can be distributed over the WSN's sink tree network such that self-modeling is realized. We achieve self-modeling by rephrasing the SensorNC analysis, in particular the output arrival curve presented in Sections 3.2 and 3.3, as well as the PMOO analysis in Algorithm 2 where these curves are used. In order to do so, we need to restrict the curve shapes for modeling and analysis to the oftentimes used rate-latency service curves \(\beta=\beta_{R,L}\in\mathcal{F}_{\text{RL}}\) and token-bucket arrival curves \(\alpha=\gamma_{r,b}\in\mathcal{F}_{\text{TB}}\). Then, the three network calculus bounds can be computed as follows:  
**Corollary 3**. : _(Performance bounds in SensorNC) Consider a sensor node that offers a service curve \(\beta=\beta_{R,T}\in\mathcal{F}_{RL}\). Assume a flow \(f\) with arrival curve \(\alpha=\gamma_{r,b}\in\mathcal{F}_{TB}\) traversing this node. Then, we obtain the following SensorNC performance bounds:_  
\[\text{output arrival curve:}\ \ \gamma_{r,b^{\prime}}\left(d \right)=\left(\gamma_{r,b}\,\dot{\phi}\,\beta_{R,T}\right)\left(d\right)= \begin{cases}0&\text{if }d=0\\ \gamma_{r,b\,+\,r\,\text{-}\,T}\left(d\right)&\text{otherwise,}\end{cases} \] \[\text{backlog:}\ \ \forall t\in\mathbb{R}^{+}\colon\ B\left(t \right)=b+r\cdot\text{T},\] \[\text{delay:}\ \ \forall t\in\mathbb{R}^{+}\colon\ D\left(t \right)=T+\frac{b}{R}.\]  
Calculating the bound on network-internal traffic at any location uses aggregation and deconvolution. Given the above restriction of curve shapes, the following property holds for the combination of these operations. **Lemma 1**. : _(Distributivity of \(\dot{\phi}\) with respect to \(+\)) For any \(\alpha^{f_{1}},\alpha^{f_{2}}\in\mathcal{F}_{TB}\) and \(\beta\in\mathcal{F}_{RL}\), it holds that:_  
\[\left(\alpha^{f_{1}}+\alpha^{f_{2}}\right)\dot{\phi} = \alpha^{f_{1}}\dot{\phi}\beta+\alpha^{f_{2}}\dot{\phi}\beta.\]  
Proof. : Let \  
Moreover, the composition rule of \(\hat{\phi}\) follows from \(f\osigma g\osigma h=f\osigma(g\otimes h)\)[6] by an argumentation similar to Lemma 1. The above distributivity and the composition rule allow one to compute the network internal output arrival curve of Algorithm 2 with a new SensorNC concatenation theorem. **Theorem 7**. : _(SensorNC concatenation theorem) Consider a sensor node \(s_{i}\) in a sink tree that is crossed by a set of flows \(F=f_{1},\ldots,f_{n}\) with arrival curves \(\alpha^{f_{1}},\ldots,\alpha^{f_{n}}\in\mathcal{F}_{\text{TB}}\).