Question

A gas station with only one gas pump employs the following policy: If a customer has to wait to buy the gasoline, the price is $3.50 per gallon. If she does not have to wait to buy the gasoline, the price is $4.00 per gallon. Customers arrive according to a Poisson process with a mean rate of 20 per hour. Service times at the pump have an exponential distribution with a mean of 2 minutes. Arriving customers always wait until they can eventually buy gasoline. Determine the expected price (in $) of gasoline per gallon. Group of answer choices

Answer 1

The expected price of gasoline per gallon is $3.638.

Step-by-step explanation:

To determine the expected price of gasoline per gallon, we need to calculate the average price for each scenario. Let's start by finding the average wait time for customers. Since the arrivals follow a Poisson process with a mean rate of 20 per hour, the average time between arrivals is 1/20 hours or 3 minutes. With the service times having an exponential distribution with a mean of 2 minutes, the average wait time is the average time between arrivals minus the average service time, which is 3 - 2 = 1 minute.

Next, we calculate the probability of a customer having to wait. Using the formula for the probability of at least one customer in a time interval, we have 1 - e^(-λt), where λ is the arrival rate and t is the time interval. Plugging in λ = 20 per hour and t = 1/60 hours (1 minute), we find the probability of a customer having to wait is 1 - e^(-20/60) = 0.1813.

Finally, we can calculate the expected price per gallon by multiplying the probabilities of each scenario by their respective prices and summing the results. The expected price is (0.1813 × $3.50) + (0.8187 × $4.00) = $3.638. Therefore, the expected price of gasoline per gallon is $3.638.

Answer 2

Step-by-step explanation:

The arrival rate (λ) = 20 customers per hour. Since the service times at the pump have an exponential distribution with a mean of 2 minutes, therefore the service rate (μ) = 60 / 2 = 30 customers per hour.

The probability of the no customers being in the system(P₀) is given as:

`P_0=1-(\lambda)/(\mu) =1-(20)/(30)=1-0.67=0.33`

If no customer is in the system we can sell gasoline for $4 /gallon to the next customer. The expected price p of gasoline is given by:

`P=P_0*4+(1-P_0)3.5=0.33*4+(1-0.33)3.5=1.32+2.345=3.665`

P = $3.665 per gallon