The problem determining and characterizing efficient sequential estimation procedures in linear birth-and-death processes with immigration is considered. Basing on the likelihood function in the case of random observation time some equations of WALD type and an inequality of RAO-CRAMR-WOLFOWITZ type are obtained. Sequential procedures with minimal variance estimators are characterized by linear equations in the sufficient statistic. Using this fact efficiently estimable functions and the associated estimators are determined. With respect to increasing sequences of stopping times the efficient estimators are strongly consistent and asymptotically normal.