<p>It can be proved that $M/GI/\infty$ queues for any general service time distribution is quasi-reversible, that is the departure process , $D(-\infty,t] $ is Poisson and is independent of the state of the system at time $t$, that is $X(t)$.
A very nice proof of this can be found in this paper : <a href="http://www.columbia.edu/~ww2040/physics.pdf" rel="nofollow">http://www.columbia.edu/~ww2040/physics.pdf</a> </p>

CC BY-SA 3.0
2015-04-16T17:49:23.440Z
2015-04-16T17:49:23.440Z
rajatsen91
<p>Hey guys I am studying queueing theory and I am trying to understand this problem. Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In other words the service time consists of a mixture of $K$ deterministic service times. I am trying to understand if the departure process of the model is Poisson? Does anyone have any ideas? Thank you in advance ! </p>

<queueing-theory>
M/G/$\infty$ queue with mixture of deterministic service times
SpawnKilleR
# M/G/$\infty$ queue with mixture of deterministic service times

(Asked by: username_0 on 2015-04-15)

Hey guys I am studying queueing theory and I am trying to understand this problem. Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu\_i$ with probabbility $p\_i$, and $\sum\_{i=1}^kp\_i=1$ and $\sum\_{i=1}^kp\_i/\mu\_i=1/\mu$. In other words the service time consists of a mixture of $K$ deterministic service times. I am trying to understand if the departure process of the model is Poisson? Does anyone have any ideas? Thank you in advance !

**Tags:** queueing-theory


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## 1 answers

### Answer by username_1 on 2015-04-16. Score: 0

It can be proved that $M/GI/\infty$ queues for any general service time distribution is quasi-reversible, that is the departure process , $D(-\infty,t] $ is Poisson and is independent of the state of the system at time $t$, that is $X(t)$.
A very nice proof of this can be found in this paper : <http://www.columbia.edu/~ww2040/physics.pdf>


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