G/G/1

Show that, for a $G / G / 1$ queue, the starting times of the busy periods of the server constitute a renewal process.

Consider a $\mathrm{G} / \mathrm{M}(\mu ) / 1$ queue in equilibrium, together with the dual (unstable) $\mathrm{M}(\mu ) / \mathrm{G} / 1$ queue. Show that the idle periods of the latter queue are exponentially distributed. Use the theory of duality of queues to deduce for the former queue that: (a) the waiting-time distribution is a mixture of an exponential distribution and an atom at zero, and (b) the equilibrium queue length is geometric.

Consider $\mathrm{G} / \mathrm{M}(\mu ) / 1$, and let $G$ be the distribution function of $S-X$ where $S$ and $X$ are typical (independent) service and interarrival times. Show that the Wiener-Hopf equation

for the limiting waiting-time distribution $F$ is satisfied by $F(x) = 1-\eta e^{-\mu (1-\eta ) x}, x \geq 0$. Here, $\eta$ is the smallest positive root of the equation $x = M_{X}(\mu (x-1))$, where $M_{X}$ is the moment generating function of $X$.