Exercise 8 Consider a gas station that has 1 gas pump, 1 automated car washing machine,...

 

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Exercise 8 Consider a gas station that has 1 gas pump, 1 automated car washing machine, and 1 air pump. Each facility has an infinite queue. All service times are independent and have an exponential distribution with a mean of 5 minutes at the gas pump, 10 minutes at the washing machine, and 4 minute at the air pump. On the average, 3 cars per hour comes to the gas pump to fuel from the outside of the gas station. The arrival process to the gas pump follows a Poisson process. After fueling, 1/3 of the cars go to the car washing machine, 1/3 of the cars go to the air pump, and the rest of the cars leave the gas station. Some cars directly come to the washing machine without fueling. Except for the cars from the gas pump, the inter arrival time of the cars to the washing machine has exponential distribution with mean 30 minutes. The washed cars leave the gas station with probability 2/3, and the rest of the cars to go to the air pump. There are cars come to the gas station only to pump the flat tires. This process follows Poisson process with mean rate of 1 per hour. Every cars finishing air pumping leaves the gas station. 1. Find the total arrival rate at each of the facilities. 2. Provide the steady-state distribution of the number of customers at the air pump. 3. Find the probability density function of the waiting time, fw (w), at the air pump (including service time). The waiting time is measured in hours. 4. Find the expected waiting time (including service times) at each of the facilities. (unit: hours) 5. Find the expected total waiting time (including service times) for a customer. (unit: hours) Exercise 8 Consider a gas station that has 1 gas pump, 1 automated car washing machine, and 1 air pump. Each facility has an infinite queue. All service times are independent and have an exponential distribution with a mean of 5 minutes at the gas pump, 10 minutes at the washing machine, and 4 minute at the air pump. On the average, 3 cars per hour comes to the gas pump to fuel from the outside of the gas station. The arrival process to the gas pump follows a Poisson process. After fueling, 1/3 of the cars go to the car washing machine, 1/3 of the cars go to the air pump, and the rest of the cars leave the gas station. Some cars directly come to the washing machine without fueling. Except for the cars from the gas pump, the inter arrival time of the cars to the washing machine has exponential distribution with mean 30 minutes. The washed cars leave the gas station with probability 2/3, and the rest of the cars to go to the air pump. There are cars come to the gas station only to pump the flat tires. This process follows Poisson process with mean rate of 1 per hour. Every cars finishing air pumping leaves the gas station. 1. Find the total arrival rate at each of the facilities. 2. Provide the steady-state distribution of the number of customers at the air pump. 3. Find the probability density function of the waiting time, fw (w), at the air pump (including service time). The waiting time is measured in hours. 4. Find the expected waiting time (including service times) at each of the facilities. (unit: hours) 5. Find the expected total waiting time (including service times) for a customer. (unit: hours)

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