# Distribution of the idle time in a queue

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.

• 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. – jbowman Jun 10 at 15:42
• 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. – Chris Jun 10 at 16:03
• Multiple or single server queue? – jbowman Jun 10 at 16:53
• 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. – Chris Jun 10 at 17:21