Part $($a$)$

DD Bike Rental Company has $5$ identical bikes that it rents to students at a collegetown shop. Potential bike renters come to this shop at the rate of $3$ per hour, and that customer arrival rate is a negative exponential distribution rate. Historical data shows that a rented bike is kept by the renter for an average of $30$ minutes, and that rent time period has a negative exponential distribution. Whenever all bikes have been rented, any potential renter is promptly directed to another nearby bike rental shop.

Produce an r$-$CTMC that can be utilized by DD Bike Rental Company to track the number of bikes waiting in the shop to be rented during business hours.

Part $($b$)$

Convert your r$-$CTMC of Part $($a$)$ into an equivalent st$-$CTMC $($by drawing suitable CTMC networks$).$