The Metrics

Waiting Time

Little's law converts average counts into average times. W is time in the system, while Wq is time waiting before service, so the time metrics are interpretations of the count metrics. The relationship is exact inside the idealized queueing model, not an empirical claim about an unmeasured real line.

highlighted = computed this step

Little's law

Little's law gives W=1 because L equals lambda times W. Why: average count equals arrival rate times average time in system. Interpretation: W is the mean time a customer spends from arrival until departure inside this steady state model.

L=λW=1L=\lambda W=1
Waiting timeLittle's law and service-time decomposition are exact.W Wq service lambdaWWWqservicelambdaW11/21/21

Wait before service

The mean wait in queue is 1/2. Why: this excludes the service time itself. It measures only the delay before service begins, so it corresponds to the queue portion rather than the whole system visit.

Wq=1/2W_q=1/2
Waiting timeLittle's law and service-time decomposition are exact.W Wq service lambdaWWWqservicelambdaW11/21/21

Add service time

The service-time part is 1/2, and W equals Wq plus that service part. Why: time in system includes waiting and service. The decomposition keeps the operational meanings separate while preserving the same exact total.

system time=queue wait+service time\text{system time}=\text{queue wait}+\text{service time}
Waiting timeLittle's law and service-time decomposition are exact.W Wq service lambdaWWWqservicelambdaW11/21/21

Diagram note

The table recomputes W, Wq, service time, and lambda times W from exact fractions. The equalities are steady-state identities inside the queueing model; real queues still require measured rates before the model is evidence. Pixel positions are rounded for layout; every number shown is exact.

Little’s law closes the time metrics\text{Little's law closes the time metrics}
Waiting timeLittle's law and service-time decomposition are exact.W Wq service lambdaWWWqservicelambdaW11/21/21