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I am considering a queuing model of the form $M/M/\infty$, you find properties of this queue here: http://en.wikipedia.org/wiki/M/M/%E2%88%9E_queue

I am interested in the average busy period of this model, i.e the average time interval during which at least one server is busy. I am a little bit confused here how this should like with an infinite amount of servers.

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Wikipedia gives the formula $$\frac{1}{\lambda}\sum_{i\gt c} \frac{c!}{i!}\left( \frac{\lambda}{\mu} \right)^{i-c}$$ for the length of time the process spends above a fixed level $c$, starting timing from the instant the process transitions to state $c+1$.

You are asking for the case $c=0$, which gives $$\frac{1}{\lambda}\sum_{i \gt 0} \frac{1}{i!}\left( \frac{\lambda}{\mu} \right)^{i} = \frac{1}{\lambda}(e^{\lambda/ \mu}-1)$$

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  • $\begingroup$ This should be divided by $\mu$. $\endgroup$
    – Did
    Mar 4, 2014 at 8:17
  • $\begingroup$ It seems you are correct and Wikipedia is wrong: if both the rates $\lambda$ and $\mu$ double then the expected times should halve. $\endgroup$
    – Henry
    Mar 4, 2014 at 8:22
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    $\begingroup$ I have corrected Wikipedia to match equation 3.1 of Guillemin, Fabrice M.; Mazumdar, Ravi R.; Simonian, Alain D. (1996). "On Heavy Traffic Approximations for Transient Characteristics of M/M/∞ Queues". Journal of Applied Probability (Applied Probability Trust) 33 (2): 490–506. $\endgroup$
    – Henry
    Mar 4, 2014 at 8:30
  • $\begingroup$ Thanks but is there a short proof for this formula? The paper you mentioned is not very helpful here $\endgroup$
    – Alkibiades
    Mar 4, 2014 at 13:45

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