Question $1($inspired by H&L$,17\mathrm{.}2-2).$

Newell and Jeff are the two barbers in a barber shop they own and operate. They provide three chairs for customers who are waiting to begin a haircut, which means that, overall, the number of customers in the shop varies between $0$ and $5.($In other words, if Newell and Jeff are

each serving a customer and there are three customers waiting in the provided chairs, then no other customer can arrive to the shop. Effectively, this means that a customer who might try to "arrive" would immediately balk, i$.$e$.,$ not enter the shop. Another arrival cannot occur until one of the haircuts is complete and one of the customers moves from a provided waiting chair to get their haircut, at which time one of the provided waiting chairs is empty and can accept a new arrival.$)$ For each nin$\{0,1,2,3,4,5\},$ the probability $P_n$ that exactly $n$ customers are in

the shop is: $P_0=\frac{1}{20},P_1=\frac{3}{20},P_2=\frac{6}{20},P_3=\frac{6}{20},P_4=\frac{3}{20},$ and $P_5=\frac{1}{20}$

a$)$ Calculate $L$ and state how you would describe the meaning of $L$ to Newell and Jeff.

b$)$ Calculate $L_q$ and state how you would describe the meaning of $L_q$ to Newell and Jeff.

c$)$ Determine the long$-$run expected number of customers being served.

d$)$ Given that an average of $4$ customers per hour arrive and stay to receive a haircut, determine $W$ and $W_q.$ Describe these two quantities in terms meaningful to Newell and Jeff.

e$)$ Given that Newell and Jeff are equally fast in giving haircuts, what is the average duration of a haircut?