A spatial symmetry property of a two-dimensional birth–death process 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 with state-dependent rates that is similar to 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.
We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.