Bubble tea is made by blending a tea base with milk, fruit and fruit juices, then adding the tapioca

pearls, commonly called Boba. A famous bubble tea chain opens a new branch, York Bubble Tea

$($YBT$),$ at the university campus. Due to its unique style and variety of flavour, it is very famous

among the students. The customer population is fairly large. Following a Poisson distribution, $21$

customers will arrive at the YBT every hour on average. Currently, the YBT operates with a one$-$lane

operation run by an individual staff member or a team of two to serve customers, who take the order,

get the payment, and prepare the order on a first$-$come, first$-$serve $($FCFS$)$ basis. Individual staff can

serve $25$ customers per hour, whereas a team of two staff can take $80$ seconds to serve a customer,

following an exponential distribution.

$($a$)$ What is the average time in minutes a customer spends at the YBT in the following two

scenarios: when an individual staff is serving and when a team of two staff working together is

serving?

$($b$)$ If a team of two staff work together, what is the probability of more than three customers

waiting to be served?

The management of YBT noticed that the number of customers increased during rush hour, and

consequently, the waiting times increased. YBT wants to manage the system faster. They decided

that the average waiting time in the line should not be more than three minutes, and hence opened

two additional counters, increasing the number of counters to serve customers to three. An individual

staff will serve each counter. On average, $60$ customers will arrive $($during rush hours$)$ at the YBT

every hour following a Poisson distribution. They wait in a single waiting line to place the order, pay

and get the order where the queue discipline is FCFS$.$ Service times at each counter are identical and

distributed exponentially; each individual staff serves an average of $25$ customers per hour.

$($c$)$ What is the probability that at least one customer will be waiting in the line to be served?

$($d$)$ What is the average length of the waiting line with three serving counters? Is YBT achieving

their goal of having the average waiting time in the line less than three minutes? $($Do they need

to add another counter? A simple Yes$/$No answer will not get any mark$/$s$)$

$($e$)$ What is the probability that a customer will arrive and be served immediately?

Show all calculations$/$results to at$-$least $4$places after the decimal point$($expect when all succeeding digits are $0)$