Consider the optimal routing problem discussed in Lecture $3$ and shown in the figure below. We have a

system with two M$/$M$/1$ queues.

The total arrival rate to the system is $\backslash$lambda jobs$/$sec and each job is routed to server $1$ with probability p

and routed to server $2$ with probability $1$ p$.$ Assuming that $1=2$ jobs$/$sec and $2=1$ jobs$/$sec$,$ we

have shown in the lectures that mean response time $($for any job$)$ is given by

E$[$T$]=$ p

$2\backslash$lambda p $+$

$1$ p

$1\backslash$lambda $(1$ p$)$

$(1)$

For any $\backslash$lambda in $(0,1),$ find the routing probability p that minimizes the mean response time E$[$T$]$ given in

$(1)$ numerically or analytically, and then plot the optimal p value as a function of $\backslash$lambda $.$ What is the behavior

of the optimal p value as $\backslash$lambda approaches $1$ and give an intuitive explanation of the results you observe