Queueing Theory: Difference between Time in Queue and Waiting Time?

Question

I'm looking at the formulas here: http://www.cs.auckland.ac.nz/courses/compsci742s2c/lectures/p-q.pdf

It has one for r-th percentile of waiting time ($r$% of customers wait less than said time), and another one for r-th percentile of time in queue. But if the customers are in the queue, then aren't they waiting?

The formulas:

1. $\pi_w(r) = E(w)\log_e\frac{100}{100 - r}$
2. $\pi_q(r) = E(w)\log_e\frac{100\rho}{100 - r}$

Answer 1

The definitions earlier in the document are:

• time in queue $q$
• time in system (queue+server) $w = q + s$

so $\pi_w(r) > \pi_q(r)$, as suggested by the formulae if the traffic intensity $\rho <1$, necessary if the queue is expected to clear.