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$\textbf{Problem}$ Suppose customers arrive at a system according to the poisson process with rate $\lambda$. Every customer stays in the system for an exp($\mu$) amount of time and then leaves. Customers behave independently of each other. Show that the expected number of customers in the system at time $t$ is $\frac{\lambda}{\mu} (1-e^{-\mu t})$

Hint: Calculate the probability that a customer arrive at time $s(<t)$ is still in the system at time $t$ and apply the result of non-homogeneous Poisson process.

I don't have any clue about the problem.... Any help is appreciated ...

Thank you!

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  • $\begingroup$ Could you clarify -- does every customer stay for a constant period of time (exp($\mu)$) or that period is distributed according to the exponential distribution with parameter $\mu$? $\endgroup$ – dnqxt Apr 6 '19 at 23:09
  • $\begingroup$ A period is distributed according to the exponential distribution with parameter $\mu$! $\endgroup$ – user453447 Apr 7 '19 at 1:59
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The problem describes an $M/M/\infty$ queue.

See, for example, here

https://en.wikipedia.org/wiki/M/M/%E2%88%9E_queue

or the book by Vidyadhar Kulkarni, Modeling and analysis of stochastic systems, 2nd edition.

Some details of exactly the problem above are given in Example 5.17 (p. 173) in the book.

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