I.G. Karpov, G.N. Nurutdinov

Mathematical models of double differences of phase-energy spectrum and energy spectrum are analyzed. Areas of their applicability We consider a discrete Markov random process with continuous time, called the birth and death process. It may be a positive jumps and negative jumps. At present this process in various forms is widely used in the theory of queuing systems. The solution of the Kolmogorov equations for state probabilities of a Markov birth and death process is known only for some special cases in the transition regime, as well as for the steady state when t → ∞. The purpose of this paper  using the method of separation of variables to obtain solutions of the Kolmogorov equations for state probabilities of a Markov birth and death process with continuous time, as having a finite number of states and a countable number of states. In this case the intensity of birth and death depend linear functions. The solution of the Kolmogorov equations is performed for the transient and steady state when t → ∞. On the basis of expansions in orthogonal polynomials of discrete, we obtain solutions of the Kolmogorov equations for the unconditional probabilities of states of discrete Markov processes, birth and death in the transition regime, as well as. The intensity of birth and death depend linear functions, for the processes of pure breeding and pure death processes, solutions of the Kolmogorov equations for the unconditional probabilities of the states in the transitional regime. We show that the distribution for the birth and death processes can be distinguished by the magnitude and sign of the coefficient.