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There are many ways we can apply Little's Law in analyzing networks.
Using Little's Law with a Queue and a Server
Let's first consider its application to the queue and server of a system. N is the expected number of customers in the system, and T is the expected time each customer spends in the system.
Thus, Little's Law can be applied directly as:
Using Little's Law with Just a Queue
Next, let's consider its application to the queue only. Nq is the expected number of customers in the queue, and W is the expected time a customer spends waiting in the queue.
Thus, we can apply Little's Law scoped at the queue as:
Computing the Average Customers Being Serviced
The average service time is the inverse of the service rate, or 1/μ. Thus:
So, multiplying all terms by λ, we get:
which implies:
where:
Note that ρ <= 1, since the arrival rate λ should be less than or equal to the service rate μ in a stable system.
Example: Route Through a Network
This has a queueing module that can be expressed as:
What is the average delay through the network?
The arrival to system 1 in arrivals per second is λ = λ1 + λ2 + λ3 = 3 arrivals/s. Thus, for system 1, the rate in is equal to the rate out, so the arrival rate to system 2 is λ = 3 arrivals/s. Similarly, the arrival rate to system 3 is λ = 3 arrivals/s.
The time spent in the entire system is T = T1 + T2 + T3. We can solve for each subsystem's time:
Thus, T = T1 + T2 + T3 = 0.883 + 0.367 + 0.867 = 2.067 seconds.