Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth–Death Processes with Immigration

Abstract

Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

“…In Figure 5, the transition probabilities p j,n (t|0), given in (36), for the NHDI process with constant death rate and periodic immigration intensity function (25) are plotted as function of t for j = 2, 15 and n = 0, 1, 2 for some fixed choices of parameters. We note that, for large times, the transition probabilities p j,n (t|0) oscillate around the probabilities q n , given in (39), related to the HDI process. For the NHDI process, having λ n (t) = ν(t) and µ n (t) = µ n, with ν(t) given in (25), the probabilities p j,n (t|0) are plotted as function of t for µ = 0.8, ν = 2.0, a = 0.9, Q = 1.0 with n = 0, 1, 2.…”

“…We consider the generalized Polya-death (GPyD) process {N(t), t ≥ t 0 }, obtained by setting ν(t) = ν λ(t) in the NHBDI process. Then, in the GPyD process we assume that the birth and death intensity functions are λ n (t) = λ(t) (n + ν) for n ∈ N 0 and µ n (t) = n µ(t) for n ∈ N, with ν positive real number and λ(t), µ(t) bounded and continuous functions for t ≥ t 0 (cf., for instance, Ohkubo [39]). A special case of GPyD process has been considered in Giorno et al [23] and Di Crescenzo and Nobile [40] to describe a time-inhomogeneous adaptive queuing system, by assuming that λ n (t) = (λ n + α) k(t) for n ∈ N 0 and µ n (t) = n µ k(t) for n ∈ N. Moreover, a time-homogeneous GPyD process, characterized by λ n (t) = λ (n + α) and µ n (t) = n µ, has been used to describe an adaptive queuing system, known as "Model D" with panic-buying and compensatory reaction of service (cf., for instance, Conolly [2], Lenin et al [11] and Giorno and Nobile [18]).…”

“…For the NHDI process, having λ n (t) = ν(t) and µ n (t) = µ n, with ν(t) given in(25), the probabilities p j,n (t|0) are plotted as function of t for µ = 0.8, ν = 2.0, a = 0.9, Q = 1.0 with n = 0, 1, 2. The dashed lines indicate the probabilities q 0 , q 1 , q 2 , given in(39).…”

Reaction diffusion systems and extensions of quantum stochastic processes

“…For example, it provides a convenient method to describe spatially varying birth-death models with time dependent rates, which are difficult to model by other methods. Although non-spatial time dependent models can be approached with Wei-Norman methods [46], extracting exact results for spatial dependent models would then likely require perturbative diagram summation tricks, or possibly exact methods of path integration. For discrete processes without spatial dependency, the machinery also provides utility, producing algebraic reduction techniques which removes the need for path integration.…”

Lie algebraic discussion for affinity based information diffusion in social networks

“…While the application of Wei-Norman method in physical science has a long history [21,25], its application in some simple biological population dynamics, such as SIR and SIS, appears only recently (see e.g. [24,[26][27][28]) due to lack of symmetry in such systems (Hence, it would be much more difficult to construct an appropriate Lie algebra with a low dimension). It is hoped that the approach offered in this study could shed some light on the analytical solution of more complicated (and realistic) social dynamics models.…”