In a hierarchical network, multiplexing is used at various levels of hierarchy to carry end$-$user data, which is all digitized voice $($that is$,$ phone lines$).$ The sound of a voice is sampled $8000$ times each second by an Analog$-$to$-$Digital$-$Converter in the telephone, each sample being converted to an $8$ bit number. $5000$ of these digitized voice data streams are TDM multiplexed onto a Level$-1$ network, whose links are just fast enough to carry all this data. The multiplexing is carried out on a per sample basis; that is one sample from one voice data stream is succeeded by one sample from a different voice stream, etc. The data from $54$ such Level$-1$ networks are multiplexed into a Level$-2$ network using FDM; each Level$-1$ network is given its own frequency channel. The $54$ Level$-1$ channels are seen to take up only $90\%$ of the usable frequency bandwidth of the links of the Level$2$ network; the extra capacity of the Level$-2$ links is left free for future expansion. Ignore all overheads connected with headers etc.

$($a$)$ What is the bitrate capability of a Level$-1$ network link? $($Answer in Mbps$,$ assuming $1$Mbps $=1000000$ bps$)$

L$1$ bitrate: $8000$ samples $/$ second x $8$ bits $=64000$ bps$.$

The capacity for L$1$ is $5000$ streams so $5000$ x $64000$ bps $=320$ Mbps

$($b$)$ What would be the bitrate capability of a Level$-2$ link if all its bandwidth were entirely used for a single data channel?

If the bitrate capability of L$1$ is $320$ Mbps and it takes $54$ L$1$ networks to use $90\%$ of L$2$s bandwidth, then the bitrate capability of a L$2$ link is $320$ Mbps x $54/0\mathrm{.}9=19200$ Mbps

$($c$)$ Due to a malfunction, the Level$-2$ network drops all data injected into it for a duration of $5$s$.$ This causes a small disruption in each phone conversation whose data got lost. What is the minimum number of conversations which got disrupted?

$19200$ Mbps x $5$ microseconds $=96$ bits $96$ bits $/8=12$ conversations.

$($d$)$ With same conditions, what is the maximum number of disrupted conversations?