How to estimate service and arrival RATES

We are talking about rates, meaning number of customer served/arriving per time unit
Therefore,
(1) If each customer is served in 4 minutes, the service rate (µ) is:
$µ =$ customers per minute

(2) If the arrival rate (λ) is one customer every 5 minutes:
$λ =$ customers per minute

How to estimate service and arrival TIMES:

The average service time is $w = E(S) = µ^{-1} = 1/µ$ , where $E(S) =$ .

W = 1/µ = 1/0.20 = 1/1/5 = 5 minutes per customer

For arrival time one customer arrives every 4 minutes or 1/λ.

1/λ = 1/0.25 = 1/1/4 = every 4 minutes one customer arrives

How do we enter these numbers in Little’s Law?

The average number of customers in the system at any time is equal to the arrival rate (λ) by the service time (w). In general, for a single server queue, the long-run server utilization in a single-server queue is equal to the average arrival rate (λ) divided by the service rate (µ).
For c identical servers in parallel, each server works at rate µ customers per unit and the long-run average server utilization is defined by the ratio of arrival rate over the number of servers c by the service rate µ.

	Single Server (M/M/1)	Double Server (M/M/2)
Average number of customers in system $(L)$	$L = λ * w$	$L = λ * w$
Utilization (ρ)	$ρ = λ/µ$	$ρ = λ/cµ$

For our example:

	Single Server (M/M/1)	Double Server (M/M/c)
Average number of customers in system (L) (customers per minute per server)	$L = 4 * 0.20 = 0.8$	$L = 4 * 0.20 = 0.8$
Utilization (ρ) (% time busy per server)	$ρ = 4/5 = 0.8 = 80%$	$ρ = 4/(2*5) = 0.4 = 40%$