A horologist $($watchmaker$)$ is typically able to service a mechanical watch in about $120$ minutes $($exponential distribution$).$ Assuming an $8$ hour work$-$day, the watchmaker averages $3$ customers arriving at his shop every day following a poisson distribution. The watchmaker just purchased a new machine that will polish the dial and crystal reducing his service time by $20$ minutes per watch. How much will the line for a watch waiting to be serviced be reduced by the watchmaker using this new machine, assuming no increase in customer demand?

$2\mathrm{.}25$ watches waiting for service

The number waiting to be serviced will be reduced by about $1$:$2$ watches

The number waiting to be serviced will be reduced by about $0\mathrm{.}4$ watches

The number waiting to be serviced will be reduced by about $1\mathrm{.}45$ watches