When defining stochastic process, I am given the following example:
Customers arrive one at a time to a service facility and queue and wait for attention by the one server who attends to each customer in turn. The fundamental sources of randomness here are times between successive arrivals, and the service-time demand of each customer. Quantities $X_n$ of interest could be the number of waiting customers found by the $n$th arriving customer, or the number left by the $n$th departing customer. These are discrete-state processes. A continuous-state process could be the waiting time (in the queue until commencement of service) of the $n$th arriving customer.
Is this continuous-state example even correct? It seems to me that "the waiting time (in the queue until commencement of service) of the $n$th arriving customer" means that the time is continuous, and the state-space is discrete (since, I'm assuming, $n$ is an integer $\ge 0$). Therefore, it seems to me that this is a discrete-state, continuous-time process – not a "continuous-state" process.