Queue Nonlinearity

A single server handles jobs that arrive randomly and take a random amount of time to process. If both inter-arrival times and service times follow exponential distributions, this is called an M/M/1 queue, and is the simplest model in queueing theory. Managers often treat utilization linearly: “90% busy is only a little worse than 80% busy.” The M/M/1 formula shows this intuition is badly wrong. The mean number of jobs in the system (waiting and being served) is: $$L = \frac{\rho}{1 - \rho}$$ where $\rho = \lambda / \mu$ is the utilization ratio (arrival rate divided by service rate). The mean time a job spends in the system follows from Little’s Law $L = \lambda W$: $$W = \frac{1}{\mu - \lambda} = \frac{1}{\mu(1 - \rho)}$$ The denominator $(1 - \rho)$ causes both $L$ and $W$ to blow up as $\rho \to 1$. $\rho$ $L = \rho/(1-\rho)$ Marginal $\Delta L$ per 0.1 step 0.50 1.00 — 0.60 1.50 +0.50 0.70 2.33 +0.83 0.80 4.00 +1.67 0.90 9.00 +5.00 Each equal step in $\rho$ produces a larger jump…