$4\mathrm{.}42.$ Patients arrive at the emergency$|$room of a hospital according to a Poisson process. A patient is classified into one of three types: high priority $($i$.$e$.,$ a life is at stake$),$ medium priority $($i$.$e$.,$ the patient might be in pain but his or her life is not threatened$),$ and low priority. On an average day, there

are approximately $10$ arrivals per hour. One$-$fifth of the arrivals are high priority, three$-$tenths are medium priority, and one$-$half are low priority. All registration matters are handled by a single clerk. The time to register a patient is exponentially distributed with a mean of $5\mathrm{.}5$ min $($regardless of the

classification of the patient$).$ Assume that there is no preemption. What is the average time to complete the registration for each type of patient $($wait in queue plus service time$)?$ What is the average time for an arbitrary patient. Verify that the latter can also be obtained using an M$/$M$/1$ model.