Problem 6 - Queuing Birth Death Process A gas station has 2 pumps and 1 waiting spot. Customers arrive at a rate of 20 per hour. The service rate of a pump is 10 per hour. Customer balk (leave the system) if the waiting spot and the pumps are full. All processing times and inter-arrival times have an exponential distribution. How many customers are lost per hour? Define the states of this system. Draw the transition diagram. What is the probability of each state in the system? How many customers balk per hour? A balking customer costs $1 in lost profit. The waiting slot can be converted into a third pump at the cost of 20,000 . The station is open 24 hours per day 365 days per year. The arrival rate is constant for all days and times. What is the payback period for this investment (10 points)? Problem 2 - Customers arrive at a rate of 5 per hour. They are serviced by 1 person at a rate of 7 per hour. What is the average number of customers in the line? Assume exponential'arrivals and service times. Problem 3 - Develop a markov chain problem to describe a credit card? Develop numbers and estimated the value of the typical credit card user? Use your Markov chain to discuss what factors make a good customer. Hint, use the balance as the state with monthly transitions (payment period). Problem 4 - Write a Markov Chain that has a transient state but no absorbing state Problem 5 -Queuing Jackson Network A simple assembly line has 3 stations. Station 1 has service rate of 5 per hour, station 2 has a service rate of 6 per hour and station 3 has a service rate of 12 per hour. Jobs arrive at the rate of 4 per hour. 10% of the parts leave station 3 and return to station 1 for rework. Reworked parts require the same amount of time as normal parts. Assume the processing time and inter-arrival times are exponential distribution. Model this system as a Jackson network. Please compute the average number of items at each station and the total average in the system ( 5 points)? What is the maximum rework percent that this system can support? Simulate a system with deterministic processing time? What is the difference in number in the system