Abstract
We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.
Highlights
Birth–death processes are continuous-time Markov chains on the state space of non-negative integers, in which only transitions to adjacent states are allowed
We focus on a general linear time-inhomogeneous birth and death process with immigration { N (t), t ≥ t0 }, with time-varying intensity functions
We obtain the transition probabilities for the NHBDI process N (t) when the process moves starting from the zero state
Summary
Birth–death processes are continuous-time Markov chains on the state space of non-negative integers, in which only transitions to adjacent states are allowed These processes have been used as models in populations growth and in queuing systems and in many other fields of both theoretical and applied interest (cf., for instance, Bailey [1], Conolly [2], Feldman [3], Iosifescu and Tautu [4], Medhi [5], Ricciardi [6] and Thieme [7]). In growth models, immigration’s effects may occur, due to the circumstance that the population is not isolated In these cases, it is necessary take into account birth–death processes with a reflecting condition in the zero state (see, for instance, Di Crescenzo et al [8], Crawford and Suchard [9], Giorno and Nobile [10], Lenin et al [11] and Tavaré [12]). Various numerical computations were performed with MATHEMATICA to analyze the role played from the parameters, by devoting special attention to the case of periodic immigration intensity functions
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