little's law

Basis

$N$:
$T$
$X$: throughput
$\lambda$:

Little’s Law for Open Systems

For any ergodic open system we have that

$$\mathbb{E}[N] = \lambda \cdot \mathbb{E}[T]$$

where$\mathbb{E}[N]$is the expected number of jobs in the system, $\lambda$ is the average arrival
rate into the system, and $\mathbb{E}[T]$ is the mean time jobs spend in the system.

Little’s Law for Closed Systems

Given any ergodic closed system,

$$N = X \cdot \mathbb{E}[T],$$

where $N$ is a constant equal to the multiprogramming level, $X$ is the throughput (i.e., the rate of completions for the system), and$\mathbb{E}[T]$is the mean time jobs spend in the system.

Little’s Law for Open Systems Restated

Given any system where $\bar{N}^{Time Avg}$ and $X$ exist and where $\lambda = X$, then

$$\bar{N}^{Time Avg} = \lambda \cdot \bar{T}^{Time Avg}.$$