Sojourn Time

How long does a customer actually spend in the system? The previous scenario measured $L$, the mean number of customers in the system at any moment. This scenario measures $W$, the mean time a single customer spends from arrival to departure. This is called the sojourn time, residence time, or response time, and has two components: $W_q$: time spent waiting in the queue because the server is busy. $W_s$: time spent in service while the server is working on this customer. $$W = W_q + W_s$$ The Surprising Finding For an M/M/1 queue, the mean sojourn time is: $$W = \frac{1}{\mu(1 - \rho)}$$ This blows up as $\rho \to 1$, just like $L$. But the split between waiting and service shifts dramatically as load increases. $\rho$ $W_q$ (wait) $W_s$ (service) $W$ (total) 0.1 0.11 1.00 1.11 0.5 1.00 1.00 2.00 0.9 9.00 1.00 10.00 The mean service time $W_s = 1/\mu = 1.0$ is constant: the server always takes the same average time to serve one customer. All the extra delay at high load is pure…