Littles Law Derivation
Overview
Following
leon-garcia
{% N(t) = A(t) - D(t) %}
where
- {% A(t) %} - the number of arrivals by time t
- {% D(t) %} - the total number of departures by time t
At at time t where {% N(t) = 0 %}
{% \bar{N}(t) = \frac{1}{t} \sum_{i=1}^{A(t)} T_i %}
where {% \bar{N}(t) %} is the average number of customers
The average arrival rate is
{% \bar{\lambda}(t) = \frac{A(t)}{t} %}
{% \bar{N}(t) = \frac{\bar{\lambda}(t)}{A(t)} \sum_{i=1}^{A(t)} T_i %}
{% \bar{T}(t) = \frac{1}{A(t)} \sum T_i %}
{% \bar{N}(t) = \bar{\lambda}(t) \bar{T}(t) %}
Which in the limit as {% t \rightarrow \infty %} is
{% \mathbb{E}(N) = \lambda \mathbb{E}(T) %}
