Birth and death process with one-side bounded jumps in random environment

2014/07/12 by Huaming Wang, Wang, Hua-Ming
Business, Management and Accounting · Decision Sciences · Mathematics · #60J80 #60K37 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1407.3385

openalex publication_date 2014/07/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let ω=(ω_i)_{i∈\mathbb Z}=(μ^L_i,...,μ¹_i,λ_i)_{i∈ \mathbb Z}, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with L≥1 a positive integer. We study birth and death process N_t which, given the environment ω, waits at a state n an exponentially distributed time with parameter λ_n+∑_l=1^Lμ^l_n and then jumps to n-i with probability μⁱ_n/(λ_n+∑_l=1^Lμ^l_n), i=1,...,L or to n+1 with probability λ_n/(λ_n+∑_l=1^Lμ^l_n). A sufficient condition for the existence, a criterion for recurrence, and a law of large numbers of the process N_t are presented. We show that the first passage time T₁\overset\mathscr D=ξ_0,1+∑_{i≤ -1}∑_k=1^U_i,1ξ_i,k+∑_{i≤ -1}∑_{k= 1}^U_i,1+...+U_i,Lξ_i+1,k, where (U_i,1,...,U_i,L)_i≤0 is an L-type branching process in random environment and, given ω, ξ_i,k, ξ_i,k, i≤ 0, k≥ 1 are mutually independent random variables such that P_ω(ξ_i,k≥ t)=e^-(λ_i+∑_l=1^Lμ^l_i)t, t≥ 0. This fact enables us to give an explicit velocity of the law of large numbers.

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