# Modelling Birth-Death Processes using Normal Distribution for Duration of Events

Suppose I'm modelling a queue using birth-death processes.

Arrival of Events: The arrival of events can be modelled using a Poisson process with parameter $\lambda$. Given a short time interval $\delta$, the probability of arrival of a new event is approximately $\lambda\delta$.

Duration of Events: The duration of events can be modelled using an exponential distribution with parameter $\mu$. If there are $i$ events at a particular time, then the probability of an event ending in a short time interval $\delta$ is about $i\delta\mu$.

An illustration of this Markov chain is shown below. The state of this chain is the number of customer in the queue.

To calculate the steady-state probabilities, I can now write balance equations of the form ${\lambda\delta\pi_{i}}={i\delta\mu\pi_{i+1}}$, which gives me a recurrent relation, using which I can calculate the steady state probability for any $i$.

However, say that the duration of events are now modelled using a Normal distribution with mean $\mu$ and standard deviation $\sigma$. How would I write the balance equations now that the memoryless property no longer holds? Or are Markov chains no longer valid since the Markov property is violated? How would I then go about solving such problems to obtain steady state probabilities?

• It's not clear how you're going to model durations using the normal law, which may well be negative. – zhoraster May 15 '17 at 8:34
• Say if you fix the positive issue by picking an appropriate positive distribution instead of normal to model it, you are actually having a G/G/1 queue. – BGM May 15 '17 at 9:36
• @BGM Apart from exponential distributions, what other distributions can I use to model duration? – Mohideen Imran Khan May 15 '17 at 12:27