Problem $2.$ A Scatternet is a number of interconnected piconets that supports communication between more than $M$ devices, where the devices participating in different piconets can relay data through bridges that connect them.

In a situation where $32$ Bluetooth devices are present and need to connect to each other. For the sake of this problem, assume that FS delay is fixed to $10ms,RB$ delay to $400ms,$ Connectivity lists are sent in DM$1$ packets, and the "Connected" notifications are sent instantly with no time delay. Also assume that all devices are turned on at $t=0s.$

Remember that for a master to form a piconet, it pages slave $S_i,$ then $S_{i+1}$ and so on$.$

Given the above, provide the following:

a$.$ What is the value of $M?$

b$.$ How many coordinator$($s$)$ are there in the Scatternet?

c$.$ What is the minimum number of masters needed to fully connect this Scatternet?

d$.$ What is the number of bridges?

e$.$ What is the size of each piconet

Assuming the best case scenario, how long will it take $(s$ tarting at $t=0s)$ to:

a$.$ Elect the coordinator?

b$.$ Page all masters in the scatternet?

c$.$ Notify all masters their connectivity lists?

d$.$ Terminate the protocol?

Notes:

Assuming the best case scenario, meaning that the duration of the formation should be minimal. This happens when only nodes with same #of votes confront each other without colliding with others. You should elaborate and explain.

A confrontation period $=FS$ delay $+RB$ delay $+FS$ delay

Paging each device takes $40\mathrm{.}625ms=2\mathrm{.}5ms$

Connectivity lists are sent in DM$1$ packets.

The bridges would be the first nodes to be contacted by their masters.

For the sake of this problem, the "connected notification" sending time is assumed negligible, also the time needed to break the temporary piconet is also assumed negligible.