Please Dont give me answer from ChatGPT because i will review it$.$ Thanks! Consider a simple barber shop with one $(1)$ "service chair" and two $(2)$ "waiting chairs."

Customers to this barber shop arrive with exponential arrival events depending on the state

of the system. If the system is empty, customers arrive at rate $.$ If there is a customer in

the chair, but no one in the waiting chairs $($a total of one customer in the system$),$ the arrival

rate slows to $0\mathrm{.}8.$ If there is one customer in service and one waiting $($a total of two

customers in the system$),$ the arrival rate slows further to $0\mathrm{.}6.$ Once there are three

customers in the system, an incoming customer decides to turn away,

The barber also changes his work rate according to the system state. Service times are also

exponential. If there is one customer in the chair and no customer waiting, the barber works

at rate $.$ If there is one customer waiting $($a total of two customers in the system$),$ the

barber speeds up to rate $1\mathrm{.}1$; if there are two waiting $($a total of three customers in the

system$),$ he works at rate $1\mathrm{.}2.$

If $$ is estimated to be $3$ customers per hour, and $$ is estimated at $4$ customers per hour;

a$)$ Draw and label a state space diagram, showing all states and the rates of transition

between states.

b$)$ Write a complete set of balance equations

$-$c$)$ Solve your system of equations to determine the long run probability of the system

being in each possible state

d$)$ What is the average number of jobs in the system at any point in time?

e$)$ Assume that with time, the barber ages and as a result, his service rate reduces from $4$

customers per hour to $3\mathrm{.}5$ customers per hour. What impact would this change have

on the system's state probabilities?