Open migration process

An open migration process is a continuous-time Markov chain of colony populations with external Poisson processes, state-dependent departure rates, and fixed probabilistic routing that eventually exits the system. Its traffic equations of an open migration process determine the total arrival rates, and its equilibrium has a product-form stationary distribution of an open migration process when the normalizing sums are finite.

For a stationary open migration process with local total departure rate ${\varphi }_i(n_i)$ and traffic rates ${\alpha }_i$, adjacent product-weight ratios give the displayed reversed parameters. The traffic equations of an open migration process make each reversed routing row sum to one, while total external arrival rates agree in the two directions. Thus the reversed process has the same local departure functions and is again an open migration process.

If no directed routing path returns from $k$ to $j$, the set of colonies reachable from $k$ has no outgoing edge to its complement. The complement is an autonomous open migration process, and the marked $j\to k$ transition is an exit from it. In time reversal of an open migration process, that exit is a state-independent Poisson immigration channel of rate ${\alpha }_j{p}_{jk}$. Reversing a stationary two-sided Poisson process preserves its law, proving the forward stream is Poisson. This argument does not assert Poisson internal streams for networks with feedback.

For positive traffic rates in single-server colonies, a total service budget $F>{\sum }_j{\alpha }_j$ minimizes the stationary mean population when the spare rates ${\varphi }_j-{\alpha }_j$ are proportional to $\sqrt{{\alpha }_j}$. The Cauchy-Schwarz inequality proves the optimum directly.

For an open migration process, colony $j$ with population $k$ has total departure rate ${\varphi }_j(k)$. Its stationary weight is $b_j(k)={\alpha }_j^k/{\prod }_{h=1}^k{\varphi }_j(h)$. The colony weights factor independently and normalize when ${\sum }_{k\ge 0}{b}_j(k)<\infty$ for every colony. Global balance for a continuous-time Markov chain proves the formula even when routing is not reversible.

With row vectors, external arrival rates $\nu$ and transient routing matrix $P$, the total arrival rates satisfy $\alpha =\nu +\alpha P=\nu (I-P{)}^{-1}$. These include all internal migrations and do not depend on colony service speeds.