In class exercises, Lecture $21$

Problem $8\mathrm{.}14$ book Customers arrive to a single server system in accordance with a Poisson process with rate $.$ Arrivals only enter if the server is free. Each customer is either a type $1$ customer with probability $p$ or a type $2$ customer with probability $1-p.$ The time it takes to serve a type i customer is exponential with rate ${}_i,i=1,2.$ Find the average amount of time an entering customer spends in the system.

Problem $8\mathrm{.}12$ book $($for detailed solution, check the solutions provided for the queuing homework$)$

A group of $m$ customers frequents a single$-$server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon completing service the customer departs the system, but then returns after an exponential time with rate $.$ All service times ar exponentially distributed with rate $.$

$($a$)$ Find the average rate at which customers enter the station.

$($b$)$ Find the average time that a customer spends in the station per visit.