5.4 Immigration

Suppose the population also receives immigrants as a Poisson process with rate \(\nu \), independently of its size. With \(n\) present, the size rises by one with probability \((n\lambda + \nu )h + o(h)\) and falls by one with probability \(n\mu h + o(h)\), so \[P_n'(t) = -(n\lambda + \nu + n\mu )P_n(t) + \left [(n - 1)\lambda + \nu \right ]P_{n-1}(t) + (n + 1)\mu P_{n+1}(t) .\] The mean \(m(t)\) satisfies \(m'(t) = (\lambda - \mu )m(t) + \nu \). If \(\lambda <\mu \) it tends to \(\frac {\nu }{\mu - \lambda }\): immigration keeps a population alive that would otherwise die out, at a level where births and immigration balance deaths. In the case \(\lambda = 0\) (immigration and death only) the stationary distribution is Poisson with mean \(\frac \nu \mu \); this is the same model as the infinite-server queue of example 5.5.4.

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