# A Comprehensive Worst Case Bounds Analysis of IEEE 802.15.7
## 4 Worst Case Bound Analysis Using Network Calculus  
Network calculus is a mathematical method that is well adapted to controlled traffic sources and provides upper bounds on delays for traffic flows. In this work we will focus on the star topology based VLC network as it supports time-sensitive communication. We focus on the star topology because it supports both the contention based and guaranteed communication. Topologies like peer-to-peer support only contention based communication. The network infrastructure supporting broadcast topologies are not very deterministic as the number of slots in the CFP downlink is not clearly defined.  
Network calculus is a theoretical formulation based on deterministic queuing systems. This formulation helps in understanding and determining structural properties like schedulability, delay bounds, and buffering requirements for specific bounds of traffic in an integrated network. This makes network calculus a tailored fit for analyzing and determining the worst-case bounds of the IEEE 802.15.7.  
**Arrival curves:** To define the bounds of any network, we first have to determine the guarantees to the specific data flows. These guarantees can be defined by the so-called arrival curves. Given a wide-sense increasing function \(\alpha\) defined for a \(t\geq 0\), we say that a flow-through \(s\) with input \(R\) is constrained by \(\alpha\) for all \(s\leq t\): Considering that the flow \(R\) has an arrival curve of \(\alpha\), then the bound is be given by:  
\[R(t)-R(s)\leq\alpha(t-s) \tag{1}\]  
The accumulation of all the arrivals for the respective flow of \(R(t)\) can be given by \(\alpha(t)\). It is also equally important to learn the service provided by the system for the respective traffic flow. The _service curves_ help in determining this bound.  
**Service curves:** Depending on resource reservations in the network, the _Service Curve_ can be defined as a means of expressing resource availability. Service curves can be categorized into strict, weakly strict, or even variable capacity. To guarantee a strict QoS, it is necessary to specify some minimal service bound for the server.  
For a system \(S\), and a flow-through \(s\) with input \(R\) and an output function \(R*\), \(S\) offers to the flow, a service curve \(\beta\), if and only if \(\beta\) is wide-sense increasing, \(\beta(0)=0\) and \(R*\geq R\otimes\beta\).  
As shown in Figure 5, the envelope of the entire service provided \(\beta(t)\) will determine the overall service curve of the system. For a cumulative arrival function \(R(t)\), there exists an arrival curve \(\alpha(t)=b+r\cdot t\), where \(b\), \(r\), \(t\) are the burst rate, the data rate, and the time interval, respectively. A minimum service curve \(\beta\) is guaranteed to the arrival function of \(R(t)\). The maximum delay bound for the arrival \(\alpha(t)=b+r\cdot t\) that receives a rate-latency service curve \(\beta(t-T)^{+}\) where \(R\) is the bandwidth and \(T\) is the maximum latency can be given by:  
\[D_{max}=\ \frac{b}{R}\ +T \tag{2}\]  
The variance between the \((b,r)\) curve and the realistic model is adequate for a periodic traffic, which is often the case in Wireless Sensor Networks. Figure 5 depicts the basic \((b,r)\) model with the arrival and service curves, and the delay bound (expressed as maximum delay).  
Figure 4: Frame structure—broadcast.