Sequences of linear birth, death, and immigration processes with applications to cascades of optical amplifiers

Consider a population having size X(t) at time t that undergoes a sequence of periods of linear birth, death, and immigration. This paper provides a general method for calculating the absolute probabilities p(i)(t) = P(X(t) = i) at any stage in this sequence. In general, the different periods of birth, death, and immigration will have different parameters, although some special cases are also considered where certain parameters are common to all periods. Applications of this model can be found in fiber optics when considering a general cascade of erbium-doped fiber amplifiers. In this application, population size represents photon numbers, transmission through fibers causes attenuation (death), and amplification can be described by a linear birth, death, and immigration process.