Closed migration process (${\sum }_j{n}_j=N$)

A closed migration process is a continuous-time Markov chain of populations moving between colonies with a conserved total number of individuals. A colony containing $k$ individuals has departure rate ${\varphi }_j(k)$, with ${\varphi }_j(0)=0$, and each departure chooses its next colony using a fixed routing matrix. Its colony populations can represent stages of a service system, including an idle stage.

// Target: probability-and-statistics.bigb

Represent $N$ telephone lines by a closed migration process with free-line, operator and automated-service colonies. Accepted arrivals have rate $\nu$ when a free line exists, $c$ operators complete at rate $\lambda \min (k,c)$, and $i$ automated calls complete at rate $\mu i$. Define $D_c(k)={\prod }_{h=1}^k\min (h,c)$ andThe product-form stationary distribution of a closed migration process and Poisson arrivals see time averages give the displayed blocking probability. This extends a single-operator model without treating the independent operators as a server with rate $c\lambda$ when fewer than $c$ calls are present.

// Target: probability-and-statistics.bigb

In a one-operator closed migration process representing a finite-line switchboard, let $K$ count calls waiting for or receiving the operator's connection service and let $I$ count connected calls. At a stationary distribution, the expected drift of $K$ is $\nu \mathrm{Pr}\{K+I<N\}-\lambda \mathrm{Pr}\{K>0\}=0$, and that of $I$ is $\lambda \mathrm{Pr}\{K>0\}-\mu \mathbb{E}[I]=0$. Poisson arrivals see time averages identifies the full-state probability with $P_{\mathrm{loss}}$. Thus the accepted arrival rate, operator throughput and completed-call rate coincide. This is a useful independent check on the product-form stationary distribution of a closed migration process.

For an irreducible closed migration process with finite population $N$, positive departure rates and routing matrix $P$, choose positive traffic weights satisfying $\theta =\theta P$. On ${\sum }_j{n}_j=N$, its stationary distribution has the displayed form. For a predecessor $m=n+e_j-e_k$, its weighted transition into $n$ is $\pi (n)({\theta }_j/{\theta }_k)P_{jk}{\varphi }_k(n_k)$. Summing over $j$ gives $\pi (n){\varphi }_k(n_k)$ by the traffic equation, and summing over $k$ proves global balance for a continuous-time Markov chain. The distribution need not satisfy detailed balance.

// Target: probability-and-statistics.bigb