Suppose process times on a machine are exponentially distributed with a mean of $10$ minutes.

A job has currently been running for $90$ minutes. What is the expected time until completion?

Compute the capacity in parts per hour of the following:

$($a$)$ A station with three machines operating in parallel with $20-$minute process times at each

station.

$($b$)$ A balanced line with single$-$machine stations, all with average processing times of $15$

minutes.

$($c$)$ A four$-$station line with single$-$machine stations, where the average processing times are

$15,20,10,12$ minutes, respectively for stations $1,2,3,4.$

$($d$)$ A four$-$station line with multimachine stations, where the number of $($parallel$)$ machines

at stations $1,2,3,4$ is $2,6,10,3,$ respectively. The average processing times at stations

$1,2,3,4$ are $10,24,40,18$ minutes, respectively.

$($e$)$ A three$-$station line with ample equipment $($i$.$e$.,$ such that operators are never prevented

from processing a job by a lack of equipment$)$ staffed by six operators who are identical

with regard to average processing times and require $10,15,$ and $5$ minutes, respectively,

on stations $1,2,3.$

$($f$)$ The same line as in the above case except that station $2$ consists of only two parallel

machines. All other stations still have ample capacity.