Starting at time t=0, cars arrive at a car wash according to a Poisson process with rate of λ cars/hour. At any given moment, the car wash is either free or occupied. The car wash is initially free at time t=0. If a car arrives at the car wash when it is free, the car is serviced immediately. Service lasts for 1/4 hours, during which the car wash is occupied. If a car arrives at the car wash when it is occupied, the car is denied service and it leaves the car wash.
1. Write down the PMF pN(k) of N, the number of cars arriving at the car wash between times 0 and 3, in terms of λ and k.
To write down the probability mass function (PMF) pN(k) of N, the number of cars arriving at the car wash between times 0 and 3, we need to consider the Poisson distribution.
The Poisson distribution is a probability distribution that describes the number of events that occur in a fixed interval of time or space, given the average rate of occurrence.
In this case, the number of cars arriving at the car wash follows a Poisson process with a rate of λ cars per hour. Since we want to find the number of cars arriving within 3 hours, we need to adjust the rate accordingly.
Let's denote the adjusted rate as λ' (lambda prime). The adjusted rate is given by the product of the original rate λ and the duration of the interval, which is 3 hours.
λ' = λ * 3
Now, we can use the formula for the Poisson distribution to calculate the PMF pN(k):
pN(k) = (e^(-λ') * λ'^k) / k!
Here, e is the base of the natural logarithm (approximately 2.71828), and k! represents the factorial of k.
Substituting the value of λ' into the formula, we get:
pN(k) = (e^(-λ * 3) * (λ * 3)^k) / k!
So, the PMF of N, the number of cars arriving at the car wash between times 0 and 3, in terms of λ and k is given by:
pN(k) = (e^(-λ * 3) * (λ * 3)^k) / k!