Question: Find the distribution for the (random) time it takes an $M/M/1/2$ queuing system with $\lambda = \mu = 1$ to change its state from being full to being empty. ($\lambda, \mu$, arrival rate and serivce rate)

I have a hard time understanding this solution, can anyone help me out?:

Let $X_1, X_2, X_3, . . .$ be independent identically distributed random variables each of which is distributed like the sum of two independent random variables that are exponentially distributed with expected values 1 and 1/2, respectively.

Further, let N be a discrete random variable independent of the $X_n$'s such that $P(N = n) = 2^{−n}$ for $n = 1, 2, 3, . . .$

Then the asked for random time is distributed like $\sum_{n=0}^N X_n$.


Let $Y(t)$ be the queue size at time $t$. Each $X_i$ represents the time to get $2$ transitions starting from the state where $Y(t)=0$. The first transition is to $Y(t)=1$ and occurs at exponential rate $\lambda=1$. The next transition, to either $0$ or $2$, occurs at rate $\lambda+\mu=2$.

Each such two-step transition goes to $Y(t)=2$ with probabilty $1/2$ and returns to $Y(t)=0$ with probability $1/2$. So if $N$ is the number of such two-step transitions until we reach $Y(t)=2$, it has $Geom(1/2)$ distribution.

  • $\begingroup$ I understand what you mean but your explaination is from empty to full right? Does not matter,i understand it. And regarding the geometric distribution its like $P(N=n) = (1/2)(1/2)^{n-1}$. Prob. you succeed on the nth try. Why on earth didnt he write it like that or explain it like you did :) So much easier to understand. I got it now but still pretty hard to get your head around a distribution which is a sum of n variables, X_n, which in turn is a sum of two exponential r.v's and the limit of the sum is in it self a geometric r.v. Extra hard when we havent worked with distr.'s much. Ty btw. $\endgroup$ – JKnecht Jan 8 '16 at 10:04
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    $\begingroup$ @JKnecht It sounds like you understand it right. I'm a little surprised though if you haven't covered much about probability distributions already. Stochastic processes is normally done after basic probability theory, which would cover all the fundamental dist'ns such as geometric, exponential and poisson (as well as a bunch of other useful topics). Anyway, good luck with your studies. $\endgroup$ – Mick A Jan 8 '16 at 11:55
  • $\begingroup$ yeah, we have covered the basic probability theory with all the fundamental distributions but not much at all about when you combine them in this way. $\endgroup$ – JKnecht Jan 8 '16 at 12:06

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