Consider a large system that is composed of five separate $\backslash ($ M $/$ M $/1/\backslash$infty $\backslash )$ servers serving five different applications, each application has $\backslash (\backslash$lambda$=3\backslash$mathrm$\{$pkt$\}/\backslash$mathrm$\{$s$\}\backslash )$ in packet generation rate and each server has $\backslash (\backslash$mu$=6\backslash )\backslash (\backslash$mathrm$\{$pkt$\}/\backslash$mathrm$\{$s$\}\backslash )$ in service rate.

a$)$ Which is better in terms of packet delay, having a dedicated server for each application with $\backslash (\backslash$mu$=6\backslash )$ per server or having a single powerful server with $\backslash (5\backslash$mu $\backslash )$ service rate serving all of the five applications simultaneously? Compute a proof for your answer $($Hint: compute the packet delay $\backslash (\backslash$mathrm$\{$E$\}[\backslash$mathrm$\{$T$\}]\backslash )$ for both cases and compare them$).$

b$)$ Assume that the separate five servers have been combined into a single powerful server with service rate $\backslash (5\backslash$mu $\backslash ).$ Compute the probability that there are $6$ packets in the system.