I'm learning queueing theory and just finished Little's Law and Utilization.
If the Earth is interpreted as a system that provides a service for its customers, is it unstable? Let the customers be the humans and let the service be sustaining life of said humans.
Let $\lambda =$ Long-term Average Arrival of Customers
The long-term average arrival of customers (humans) into the system (planet Earth) is the global annual birth-rate.
Currently, the global average birth rate per 1,000 total population is 19.15 births in 2012. For the absolute global average birth rate is $\ 19.15 \cdot 7\times10^6 = 1.34 \times 10^8 \frac{\text{humans}}{\text{year}}$.
(World population in 2012 is $7$ billion $ = 7 \times 10^9 $)
Let $\ \bar x = $Average Time in System a.k.a. Average Life-Expectancy of Human.
Current average life-expectancy (regardless of sex) is$\ 71 \frac{\text{years of service}}{\text{human}}$.
$$\lambda \cdot \bar x = 9.54\times10^9 $$
Let $\ C$ be the capacity of humans Earth can have. What is $\ C$? If $\ C < 9.54\times10^9 $ then Earth, as a system, is unstable. Is $\ C $ the death rate?
P.S.: The utilization factor is $\rho = \dfrac{\lambda \cdot \bar x}{C} $