Consider a network shown in Figure above. A Poisson stream of packets has arrival rate $\backslash$lambda packets$/$sec and

exponentially distributed packet lengths of average value L bits. The network can send these packets along

two different routes to their common destination. Each route consists of one node, that is$,$ of one buffer

equipped with a transmitter. The two nodes use transmitters with different rates. Consequently, the two

nodes $1$ and $2$ are modeled as queues with exponential service times with different rates. For each node

j$($j$=1,2),\backslash$lambda $\_($j$)$ designates the average rate of packets going through the node. Packets are sent to node $1$ with

probability p independently of one another and to node $2$ otherwise, $\backslash$mu $\_($j$)$ is the average service rate of that

node $($Assume $\backslash$mu $\_($j$)$ is Poisson and $\backslash$mu $\_(1)+\backslash$mu $\_(2)=\backslash$mu packets$/$sec$.$ Both nodes have infinite queues$).$

a$.$ Find the average delay per packet on each route. Sketch a curve showing average packet delay versus

p$.$

b$.$ Find the value of p that minimizes the average delay per packet in the network in terms of given system

parameters.

c$.$ Find the value of p that provides the same average delay on each route in terms of given system

parameters.