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Consider an electric scooter rental station in a bustling city center where both the scooters

and potential renters arrive continuously throughout the day. The inter$-$arrival times of the scooters $($for

charging or maintenance$)$ and the customers $($renters$)$ are both exponential, with mean times of $0\mathrm{.}33$ and

$1$ minutes, respectively. The station has $4$ docking ports where scooters can be parked while they await

renters. When all docking ports are full, any arriving scooters must leave without stopping. Similarly,

arriving customers who find no available scooters at the station will leave without renting.

a$.$ Model this system as a CTMC by defining the state and the state space, and drawing the rate

diagram

b$.$ Compute the steady state probabilities.

c$.$ Determine the long$-$run probability that a customer leaves without renting a scooter.

d$.$ Determine the long$-$run probability that there are at least two scooters at the rental station in the

long run.

e$.$ Calculate the long$-$run probability that an arriving scooter has to leave because all docking ports

are occupied.

f$.$ Determine the expected number of scooters at the rental station in the long run.

g$.$ Consider an electric scooter rental station with varying operational costs depending on scooter

availability. The station incurs a fixed cost of $$100$ per hour when there are no scooters

available, due to potential lost revenue and customer dissatisfaction. Conversely, when scooters

are available, the station incurs a cost of $$25$ per scooter per hour, reflecting operational

expenses such as maintenance, electricity for charging, and space utilization. Calculate the

expected cost per hour in the long run for the scooter rental station.