```{exercise} Let $N$ be a Poisson process on $[0, \infty)$. Fix $T >0$. Show that $R(t) = N(T-t)$ is a Poisson process on $[0, T]$. Hint: appeal to appropriate construction of Poisson processes. ``` ```{exercise} **By brute force**, show that for a $M(\lambda)/M(\mu)/1$ queue that is started with at stationarity, so that the number of items in the system at time any time $t$ has distribution $$\mathbb{P}(Q(t) = n) = \rho^n(1-\rho)$$ has the property that an arrival waits time $W$ until being served, where $W$ has law $$ \mathbb{P}(W \leq x) = 1- \rho e^{-x(\mu - \lambda)}.$$ ``` ```{exercise} We say that a continuous-time Markov chain with generator-(Q matrix) $G$ is **reversible** with respect to the probability measure $\pi$ if $$ \pi_i g_{ij} = \pi_j g_{ji}$$ for all states $i,j$. * Show that $\pi$ must be stationary. * Show the the continuous-time Markov chain corresponding to a $M(\lambda)/M(\mu)/1$ queue is reversible with respect to its stationary distribution. ```

* Version: `r format(Sys.time(), '%d %B %Y')` * [Rmd Source](https://tsoo-math.github.io/ucl/QHW8.1.Rmd)