Final answer:
To calculate the fraction of incoming cars turned away at a parking garage, we need to apply queueing theory principles, specifically using the Erlang B formula for an M/M/c queueing system. The expected number of cars in the garage at any time depends on the arrival and service rates. The detailed calculations would determine the probability the garage is at full capacity, which equals the fraction of cars turned away.
Step-by-step explanation:
The student's question pertains to the fraction of incoming cars that are turned away at a parking garage operating under a Poisson process. We are given that cars arrive at a rate of 60 per hour and that the garage can hold 75 cars. Each car stays for an exponentially distributed time with a mean of 45 minutes.
To find the fraction of cars turned away in the steady state, we need to calculate the expected number of cars in the garage. The arrival rate (λ) is 60 cars per hour and the service rate (μ), since each car stays 45 minutes on average, is 1.33 (or 4/3) cars served per hour per space.
Using the formula for the expected number of customers (E[N]) in an M/M/c system (Poisson arrivals, exponential service times, c servers), where c is the number of spaces in this case: E[N] = λ/μ. Since we are interested in the fraction turned away, we need to consider the probability that all 75 spaces are full (system is at full capacity).
The exact calculations require the use of the Erlang B formula or a similar queueing theory model and are beyond the scope of a simple answer. However, queueing theory will give the fraction of time the system is full, and since arrivals are Poisson, this is also the fraction of incoming cars that will be turned away.