G/M/1

Consider $\mathrm{G} / \mathrm{M}(\mu ) / 1$, and let $\alpha_{j} = \mathbb {E}\left[(\mu X)^{j} e^{-\mu X} / j!\right]$ where $X$ is a typical interarrival time. Suppose the traffic intensity $\rho$ is less than 1. Show that the equilibrium distribution $\pi$ of the imbedded chain at moments of arrivals satisfies

Look for a solution of the form $\pi_{n} = \theta^{n}$ for some $\theta$, and deduce that the unique stationary distribution is given by $\pi_{j} = (1-\eta ) \eta^{j}$ for $j \geq 0$, where $\eta$ is the smallest positive root of the equation $s = M_{X}(\mu (s-1))$.

Consider a $\mathrm{G} / \mathrm{M}(\mu ) / 1$ queue in equilibrium. Let $\eta$ be the smallest positive root of the equation $x = M_{X}(\mu (x-1))$ where $M_{X}$ is the moment generating function of an interarrival time. Show that the mean number of customers ahead of a new arrival is $\eta (1-\eta )^{-1}$, and the mean waiting time is $\eta \left\{ \mu (1-\eta )\right\}^{-1}$.

Consider $\mathrm{D}(1) / \mathrm{M}(\mu ) / 1$ where $\mu > 1$. Show that the continuous-time queue length $Q(t)$ does not converge in distribution as $t \rightarrow \infty$, even though the imbedded chain at the times of arrivals is ergodic.