For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure distribution depends on the number of people in the queue.

First let us assume that at time $t = 0$ we start in a state called $idle$.

The states the system walks through are described below. The forward arrows in the chain represent arrivals and are exponentially distributed with some rate parameter $\lambda$. The backward arrows represent departures but every departure is distributed in an other way with possibly it's own unique state dependent distribution.

My question is how to model this queue mathematically. My intend is to set up a model to analyse properties of this queue (such as it's waiting time distribution, utilisation time, ...).

My first idea was to use a continuous-time markov chain but my departure rates are not expentionally distributed. Any other ideas?

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    $\begingroup$ I don't know much about M/G queues, but I'm guessing a result this general is going to be really hard to derive. Maybe start assuming everything is exponential, then try some simulations with only one of the departure rates non-M in various ways to accumulate some intuition. $\endgroup$
    – BruceET
    Mar 23, 2015 at 20:55

1 Answer 1


Your diagram looks like a variation of a $M/G/n$ system. If service times are exponential (with rates dependent on the number in service) then you have a straightforward birth and death process and the stationary distribution can be computed.

Simplest model is the $M/M/n$, where the service rates are $\mu_i=i\mu$ for $i< n$ and $n\mu$ for $i\geq n$. This is the case when each of the $n$ servers can serve at most one customer at any time, and all have the same service rate $\mu$.


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