A first-passage-time problem for symmetric and similar two-dimensional birth-death processes

A spatial symmetry property of a two-dimensional birth-death process ${\bf X}(t)$ with constant rates is exploited in order to obtain closed-form expressions for first-passage-time densities through straight-lines $x_2=x_1+r$ and for the related taboo transition probabilities. An analogous study is performed on a birth-death process $\widetilde{\bf X}(t)$ with state-dependent rates that is similar to ${\bf X}(t)$ in the sense that the ratio of their transition functions is time independent. Examples of applications to double-ended queues and stochastic neuronal modeling are also provided.