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We derive conditions for recurrence and transience for time-inhomogeneous birth-and-death processes considered as random walks with positively biased drifts. We establish a general result, from which the earlier known particular results by Menshikov and Volkov [‘Urn-related random walk with drift 
$\rho x^\alpha /t^\beta $’, Electron. J. Probab. 13 (2008), 944–960] follow.
