Networks of queues

Consider an open migration process with $c$ stations, in which individuals arrive at station $j$ at rate $v_{j}$, individuals move from $i$ to $j$ at rate $\lambda_{i j} \phi_{i}\left(n_{i}\right)$, and individuals depart from $i$ at rate $\mu_{i} \phi_{i}\left(n_{i}\right)$, where $n_{i}$ denotes the number of individuals currently at station $i$. Show when $\phi_{i}\left(n_{i}\right) = n_{i}$ for all $i$ that the system behaves as though the customers move independently through the network. Identify the explicit form of the stationary distribution, subject to an assumption of irreducibility, and explain a connection with the Bartlett theorem of Problem (8.10.6).

Let $Q$ be an $\mathrm{M}(\lambda ) / \mathrm{M}(\mu ) / s$ queue where $\lambda < s \mu$, and assume $Q$ is in equilibrium. Show that the process of departures is a Poisson process with intensity $\lambda$, and that departures up to time $t$ are independent of the value of $Q(t)$.

Customers arrive in the manner of a Poisson process with intensity $\lambda$ in a shop having two servers. The service times of these servers are independent and exponentially distributed with respective parameters $\mu_{1}$ and $\mu_{2}$. Arriving customers form a single queue, and the person at the head of the queue moves to the first free server. When both servers are free, the next arrival is allocated a server chosen according to one of the following rules:

Consider an $\mathrm{M}(\lambda ) / \mathrm{M}(\mu ) / 1$ queue modified so that on completion of service the customer leaves with probability $\delta$, or rejoins the queue with probability $1-\delta$. Find the distribution of the total time a customer spends being served. Hence show that equilibrium is possible if $\lambda < \delta \mu$, and find the stationary distribution. Show that, in equilibrium, the departure process is Poisson, but if the rejoining customer goes to the end of the queue, the composite arrival process is not Poisson.

Consider an open migration process in equilibrium. If there is no path by which an individual at station $k$ can reach station $j$, show that the stream of individuals moving directly from station $j$ to station $k$ forms a Poisson process.

Show that an open migration process is non-explosive.

Let $X$ be an irreducible, continuous-time Markov chain with generator $\mathbf{G}$ on the state space $T = S \cup \left\{ \infty \right\}$, where $S$ is countable and non-empty. Show that a distribution $\mathbf{\pi }$ on $T$ satisfies $\mathbf{\pi } \mathbf{G} = \mathbf{0}$ if and only if, for $j \in S, \sum_{i \in T} \pi_{i} g_{i j} = 0$.