$2\mathrm{.}19.$ At $t=0$ customer A places a request for service and finds all $m$

servers busy and $n$ other customers waiting for service in an $\frac{M}{M}\frac{?}{m}$

queueing system. All customers wait as long as necessary for service,

waiting customers are served in order of arri val, and no new requests

for service are permitted after $t=0.$ Service times are assumed to be

mutually independent, identical, exponentially distributed random

variables, each with mean duration $\frac{1}{}.$

$($a$)$ Find the expected length of time customer A spends waiting for

service in the queue.

$($b$)$ Find the expected length of time from the arrival of customer A

at $t=0$ until the system becomes completely empty $($all customers

complete service$).$

$($c$)$ Let $x$ be the order of completion of service of customer A; that

is$,x=k$ if $A$ is the $kth$ customer to complete service after

$t=O.$ Find $P[x=k](k=1,2,$dots,$m+n+I).$

$($d$)$ Find the probability that customer A completes service before

the custo mer immediately ahead of him in the queue.

$($e$)$ Let tilde$(w)$ be the amount of time customer A waits for service. Find

$P[tilde(w)>x].$In a make$-$to$-$order system, when a customer places a request for a product or service with a producer, A$.$ a work order is created. B$.$ an assembly chart is created. C$.$ an engineering change notice is created. D$.$ a route sheet is created.