$.$ A digital printing studio offers personalised merchandise such as custom$-$printed t$-$shirts, mugs, and

phone cases. They have recently introduced a state$-$of$-$the$-$art printing machine called the PrintMaster, capable of

swiftly producing high$-$quality custom designs on various items.

The studio has launched an online platform where customers can submit their design order. Once an order is

submitted, it enters a queue to be processed by the PrintMaster. A status tracker is displayed, initially showing

Pending until the printing process begins. While the PrintMaster is processing the design, the status updates to

Printing and once the item is ready for pickup or delivery, it changes to Ready.

The design orders are uploaded to the platform according to a Poisson process, with an average of $24$ orders per

day. The time taken to print each request follows an exponential distribution, with an average processing time of

$42$ minutes. The studio operates continuously with no breaks and has been running long enough for steady state

properties to apply. With these assumptions, the entire process can be modelled as an M$/$M$/1$ queue.

$($a$)$ Write down the arrival rate, service rate, and traffic intensity for this queue. Make sure to express the

arrival rate and service rate in terms of the same units.

$($b$)$ What is the long$-$term average number of requests in the queue?

$($c$)$ Determine the average time taken for the status to change from Pending to Ready.

$($d$)$ Once a request begins being printed, what is the probability that it takes $30$ minutes or less to finish being

printed?