$($Data Center Networking$)$ Consider the data center network with hierarchical topology in Figure $6\mathrm{.}30,$ shown below. Suppose there are $80$ pairs of flows, with ten flows between the first and ninth rack, ten flows between the second and tenth rack, and so on$.$ Further suppose that all links in the network are $10Gbps,$ except for the links between hosts and TOR switches, which are $1$ Gbps$.$

Figure $6\mathrm{.}30$ A data center network with a hierarchical topology

a$)$ Each flow has the same data rate; determine the maximum rate of a flow.

b$)$ Suppose you upgrade your tier$-1$ switches to $4$ switches, each of the $4$ tier$-1$ switches having a link to each of the $4$ tier$-2$ switches $($full$-$mesh$).$ Determine the maximum flow rate for this enhanced topology.