The asymptotic behavior of a divergent linear birth and death process
- 1 June 1976
- journal article
- research article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 8 (02) , 315-338
- https://doi.org/10.1017/s0001867800042166
Abstract
We consider a birth and death process with Q-matrix of rates q m,m + 1 = mβ, q m,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.Keywords
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