Erlang and Jackson Network

This is a note for reviewing the MAP554 and some points about network.

M/M/1, M/M/$\infty$, birth and death

The basic model of queue theory. M/M/1 has just one server and has an invariant measure like geometric law,
$$
\pi(n) = p^n(1-p), p = \frac{\lambda}{\mu}
$$
M/M/$\infty$ has infinite server and Poisson law
$$
\pi(n) = e^{-p} \frac{p^n}{n!}, p = \frac{\lambda}{\mu}
$$
where $\lambda$ is the rate of arrival and $\mu$ the rate of waiting. A more general case can be done like change of power.

Erlang network:

This is just an application for truncated technique. That is if we have already a network with reversible invariant measure, we can generate a new by changing the power of that part. That is  

\begin{eqnarray*}
\tilde{q}(x,y) &=& C q(x,y), \forall x \in \mathcal{A}, y \in \mathcal{S} - \mathcal{A}\\
\tilde{q}(x,y) &=& q(x,y) , \text{ otherwise }
\end{eqnarray*}
Then the new invariant measure becomes
\begin{eqnarray*}
\tilde{\pi}(x) &=& K\pi(x) , \forall x \in \mathcal{A} \\
\tilde{\pi}(y) &=& KC\pi(y) , \forall y \in \mathcal{S} -\mathcal{A}\\
K &=& \frac{1}{\pi(\mathcal{A}) + \pi(\mathcal{S} - \mathcal{A})}
\end{eqnarray*}
The application is that we make $C = 0$ then the network is defined in just the part $\mathcal{A}$. For example, in the network of route with restriction $\mathcal{R}$, we can just do the case without restriction to get $\pi$, which is just the case of several M/M/1 independent, then we do restriction and normalization.
$$
\tilde{\pi}(x) = K\pi(x) , \forall x \in \mathcal{A}, K = \frac{1}{\sum_{x \in \mathcal{R}} \pi(x)}
$$

Jackson network

A more general model of network is like that. Each station has rate $\lambda_i$ of arrival and $\phi_i(n_i)\mu_i$ rate to tackling the service. Here $\phi_i(n_i)$ can be considered as the power of server, in the case M/M/1 it is always 1 and M/M/$\infty$ it is always $\phi_i(n_i) = n_i$. However, the difference is that after each service of station $i$, it has possibility $r_{ij}$ to go to the station j

The key is to find a equivalent $\tilde{\lambda}_i$ which satisfies that
$$
\tilde{\lambda}_i = \lambda_i + \sum_j \tilde{\lambda}_j r_{ji}
$$
then the station looks like independent and has the invariant measure
$$
\pi(n) = \Pi_i \frac{\tilde{p}_i^{n_i}}{\Pi_{m=1}^{n_i}\phi_i(m)}, \tilde{p}_i = \frac{\tilde{\lambda}_i}{\mu_i}
$$