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 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.