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 sequ ence of periods of linear birth, death, and immigration. This paper pr ovides a general method for calculating the absolute probabilities p(i )(t) = P(X(t) = i) at any stage in this sequence. In general, the diff erent periods of birth, death, and immigration will have different par ameters, 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 represent s photon numbers, transmission through fibers causes attenuation (deat h), and amplification can be described by a linear birth, death, and i mmigration process.