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I have iid random variables $D_t \sim \mathcal{N}(\mu, \sigma^2)$ and a constant $Q$.

Define a random variable $Z_t = (Z_{t-1} + Q - D_t)^+$. Essentially $Z$ is the waiting time distribution in a queue with interarrival times distributed according to $D_t$ (negative interarrival times are censored to 0) and constant service times of $Q$.

Now define $X_t = D_t - Q - Z_{t-1} + Z_t$. $X_t$ is the distribution of the idle time in the same queue. As it is well known $\mathbb{E}(X_t) = 1-\rho$, where $\rho$ is the utilization of the queue, which is in our case $Q/\mu$.

I need to know the variance of $X_t$, but I don't how to start.

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    $\begingroup$ You can't have a queue with Normally distributed interarrival times, since, e.g., the arrival time for arrival 1,943,195,902,901 has a nonzero probability of being before the arrival time for arrival 1. $\endgroup$ – jbowman Jun 10 at 15:42
  • $\begingroup$ You are right, thanks. I forgot to mention, that negative interarrival-times are censored to 0. The distributions I consider have a low coefficent of variation, so that does not change much in the analysis and it is very common in my field. I'll edit the question. $\endgroup$ – Chris Jun 10 at 16:03
  • $\begingroup$ Multiple or single server queue? $\endgroup$ – jbowman Jun 10 at 16:53
  • $\begingroup$ Single server queue and FIFO (see e.g. Lindley equation ). Note, that the problem was motivated by an inventory problem: What is the demand at the supplier $X_t$, given that the buyer sources a comitted quantity $Q$ each period. That's e.g. why I have normal distributed interarrival times which is not very common in queuing theory. $\endgroup$ – Chris Jun 10 at 17:21

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