Suppose we have a hierarchical DCN design as shown in the figure below. Each rack has $20$ hosts and we need high throughput connections on all flows from hosts i$=1,...,20$ in each of racks $1,2,3,4$ to hosts i$=1,..,20$ in each of racks $5,6,7,$ and $8$ respectively. Each host is connected to the TOR with a $1$Gbps link, and the connections from TORs to Tier$-2$ and Tier$-2$ to Tier$-1$ switches are $20$Gbps$.$ Using this architecture:

$($a$)$ What is the end$-$to$-$end throughput for a host i in rack $1$ to the corresponding host i in rack $5$ in the architecture below? Please explain your answer.