Queue Formation

See the first post in this series for the basics of queueing theory. We can combine arrivals (Poisson at rate $\lambda$) with a server (exponential service at rate $\mu$) into a complete queue. The system is stable because $\rho = \lambda/\mu < 1$, which means that on average, the server handles more work than arrives. Our first question is, how long is the queue? The surprising answer is that even when the server has plenty of spare capacity, customers wait. The mean number of customers in the system (both waiting and being served) is: $$L = \frac{\rho}{1 - \rho}$$ The table below gives some representative values: $\rho$ $L$ 0.1 0.11 0.5 1.00 0.8 4.00 0.9 9.00 When $\rho = 0.5$, half the server’s capacity is idle, but there is on average one customer in the system at any moment. That customer either had to wait for a previous customer, or is currently being served. The queue is never consistently empty, even at moderate load. The formula also explains why simple queues are so…