My question is the following.
An $M/M/1$ queue has arrival rate $\nu$ and service rate $\mu$, where $\rho = \nu/\mu < 1$. Show that the sojourn time (ie the queueing time plus the service time) of a typical customer is exponentially distributed with parameter $\mu - \nu$. (Call this time $T$.)
Now, we're looking at a 'typical' customer, so we consider the chain in equilibrium (eg justified by the PASTA property). Now, I am able to solve this question. For example, one simply observes that if a customer joins a queue of size $j$, then she waits $\Gamma(j+1,\mu)$. Plus in the numbers, including the cdf for $\Gamma(j+1,\mu)$, and bash out some algebraic manipulation.
I'm interested in a more refined approach, however. The cdf for $\Gamma(j+1,\mu)$ isn't so nice, I feel. Of course, since $j$ is an integer here, we can just see it as a sum of Poisson probabilities -- alternatively, one could derive the waiting time when the queue has size $j$ like this.
I'd like to prove this lemma by showing that the time $T$ has the memoryless property. We then need only calculate the expectation. The expectation of $\Gamma(\alpha,\beta) = \alpha/\beta$ for any $\alpha$ and $\beta$, so it's fairly straightforward to show that $E(T) = 1/(\mu - \nu)$. So my issue is the following.
Show directly that the time $T$ has the memoryless property.