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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.

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The example is correct.

There are two sets to consider:

  • The set of values that each $X_n$ can take
  • The index set (i.e. the values that $n$ can take)

The state-space is the first one. In this example, this is the positive real line, so it is a "continuous-state" process. The fact that $X_n$ models a "time" concept (waiting time) is irrelevant to whether the process is a "discrete-time" or "continuous-time" process.

Stochastic processes that are referred to as "continuous-time" are always referring to the index set, not the state-space. In this case the index set is discrete, so it would be "discrete-time".

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  • $\begingroup$ I think you're correct, but I'm not sure that your comments regarding $X_n$ modelling a time concept being irrelevant are correct. It seems to me that the fact that $X_n$ models "time" absolutely matters: time is continuous, and so, if $X_n$, the state-space, is time, then this is exactly what defines the stochastic process to be a continuous-state process. Am I mistaken? $\endgroup$ – The Pointer Mar 22 at 5:01
  • $\begingroup$ I edited to clarify; I meant irrelevant for determining whether the process was "discrete-time" or "continuous-time", which seemed to be the confusion in the question (i.e. saying that because the state was a continuous "time" then the process is "continuous-time", which is not the case). $\endgroup$ – Chris Haug Mar 22 at 12:03
  • $\begingroup$ Interesting; that actually was what I (naively, it seems) thought. Thanks for clarifying this. $\endgroup$ – The Pointer Mar 22 at 12:10

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