Question

A two-runway (one runway for landing, one runway for taking off) airport is being designed for propeller-driven aircraft. The time to land an airplane is known to be exponentially distributed, with a mean of 1.5 minutes. If airplane arrivals are assumed to occur at random, what arrival rate can be tolerated if the average wait in the sky is not to exceed 3 minutes? (Hint: A single landing strip airport is modeled by an M/M/1 queue.)

Answer 1

Answer:

$\frac{4}{9}$

Step-by-step explanation:

First, we gather the data:

The model is a single $M/M/1$ landing strip with $\mu = \frac{2}{3}$

This based on the assumption that service times are exponential.

Then, we must find the maximum rate such that $w_{q} \leq 3$

Using the $M/M/1$ table, we find that:

$w_{p} = \frac{\rho }{(\mu (1 -\rho) } \\ \frac{\lambda }{\mu (1 - \lambda } \leq 3$

if and only if

$\lambda\leq \frac{3\mu ^{2} }{(1 + 3\mu) }$

=$\frac{4}{9}$