A Note on Generalization of the Simplest Time-Dependent Discrete Markov Process: Linear Growth Process With Immigration-Emigration

Lineargrowth process with immigration and emigration is the general model in the study of population in biological and ecological systems, and their transient analysis is the most important factor in the understanding of the structural behavior of such systems. The probability-generating function π( z, t ) of the probability distribution {p n ( t )} of the random variable N ( t ) in many queuing situations concerning Birth, Death, Immigration, and Emigration were studied and we find the generalization of the transient solutions of the queuing systems and also studied its particular cases.