# A Comprehensive Worst Case Bounds Analysis of IEEE 802.15.7
## 4 Worst Case Bound Analysis Using Network Calculus
### Service Curve Analysis of IEEE 802.15.7--Star Topology  
Akin to several predecessor standards like the IEEE 802.15.4e [30], for star topology, the PAN Coordinator sends a beacon for every superframe interval for synchronization. The Beacon Interval (\(BI\)) and the Superframe Duration (\(SD\)) can be represented as:  
\[BI=aBaseSD\ \times 2^{BO}opticalclocks \tag{3}\]  
\[for\ 0\leq BO\leq 14\]  
\[SD=aBaseSD\ \times 2^{SO}opticalclocks \tag{4}\]  
\[for\ 0\leq SO\leq BO\leq 14\]  
_aBaseSD_ is the number of optical clocks forming a superframe when the superframe order (\(SO\)) is equal to 0, this value can be given by:  
\[aBaseSD=aBaseSlotDuration\ \times n(T_{s}) \tag{5}\]  
The slot duration can be given as the number of optical clocks forming a superframe slot when the superframe order is equal to 0; this value is fixed to 60 optical clocks when SO = 0. \(n(T_{s})\) is the number of timeslots in the superframe. It is mandatory to include the GTS transmission, the acknowledgements and the CAP region within for the worst case bounds analysis. For the sake of simplicity, we consider one data frame transmission in each a GTS per superframe. As every VLC superframe comprises 16 timeslots, the size of a single timeslot will be:  
\[T_{s}=\ \frac{SD}{16}=\ aBaseSD\ \times 2^{SO-4} \tag{6}\]  
Every timeslot \(T_{s}\) in a superframe comprises \(T_{data}\) and \(T_{idle}\). \(T_{data}\) is the maximum duration used for data transmission in a GTS. \(T_{data}\) in our case this is is the single occupied timeslot in the optical frame. \(T_{idle}\) is the time period that accommodates the acknowledgments and Inter Frame Spacing (\(IFS\)) in the network. VLC has different \(IFS\) such as: long (\(LIFS\)), short (\(SIFS\)), and reduced (\(RIFS\)). These \(IFS\) values vary based on the PHY layer the VLC operates on. The first three PHY layers that are used in outdoor communication have large \(IFS\) times ranging from 40 to 400 optical clocks. However, the latter ones that use discrete light sources have no \(IFS\) at all. As shown in Equation (7), the latency \(T\) is the time for which a burst waits to gets served. This is the difference between the burst arrival at the Beacon Interval and the time at which data are served. In a worst case scenario, the burst can arrive towards the end of the superframe and get served in the beginning of the adjacent superframe. \[T=BI-T_{s} \tag{7}\]  
Figure 5: Network calculus—(_b,r_) model. The overall service provided by the network can be given by the product of the data rate and the time at which the system receives the service. If the network comprises idle time, it should be noted that the \(BI\) must include the optical clocks involved in it as well. In such a case the \(BI\) can even be twice the size of the \(SD\). As the overall service for the VLC network has to be calculated in accordance with a data rate, we multiply the service offered by a constant data rate \"\(C\)\". The service given for the GTS, i.e., the number of bits that has to be sent during a GTS within a time \(t\), is as given by Equation (8),  
\[\beta_{1}= \begin{cases}C\left((t-(BI-T_{s}))\right)^{+},\forall\;0\;\leq t\; \leq BI-\;T_{idle}\\ 0,\;\;\;\;\;\;otherwise\end{cases}  \tag{8}\]  
This value of the service curve can now be derived for \(N\) number of superframes, similar to the equation derived for the service curve for \(n\) superframes of the IEEE 802.15.4 in ref. [31]. When considering \(N\) superframes, we must take the entire service by summing the overall service provided for the data frames. This service curve engulfs all the latencies (\(n(BI-T_{s})\)) across the superframes taken for the VLC network. This service of the \(N_{th}\) superframe can be represented as:  
\[\beta_{N}= \begin{cases}(n-1)CT_{data}+C(t-(n(BI-T_{s})))^{+}\\ \forall 0\leq t\leq(n-1)BI-T_{idle}\\ 0,\;\;\;\;\;otherwise\end{cases}