Problem $5$: Consider an M$/$G$/1$ queue that serves three types of jobs: red, blue, and

green. Red jobs arrive according to a Poisson process with rate ${}_R=\frac{1}{5}$ jobs$/$s$,$ blue jobs

arrive according to a Poisson process with rate ${}_B=\frac{1}{3}$ jobs$/$s$,$ whereas green jobs arrive

according to a Poisson process with rate ${}_G=\frac{1}{6}$ jobs$/$s$.$ Red job sizes have mean $1$ and

variance $1,$ blue job sizes have mean $0\mathrm{.}5$ and variance $1,$ whereas green job sizes have

mean $0\mathrm{.}1$ and variance $1.$ All jobs arrive to the same FCFS queue, so that, at any time,

the server might be serving a red job, a blue or a green one, and there might be jobs of

one, two, or all three types in the queue. What is the mean response time $($i$.$e$.,$ mean

system time$)$ of red jobs? What is the mean response time of blue jobs, and what about

the mean system time of green jobs?