In a simple homogencous birth-and-death process with λ and μ as the constant birth and death rates, tespectively, let Xt denote the population size at time t and Y(t) = ∮t0X(u) du. The results obtained include among others the following: (i) An emplicit formula for the characteristic function of the joint distribution of X(t) and Yt). (ii) It is shown that, if t↑∞ while λ ≤ μ, jkthe limitaing distribution of Y(t) is aweighted mean of certain X2-distributions. If λ > μ, then Y(t)↑∞ with probability 1-μ/lambda;. Given that Y(t)↑∞, its conditional limiting distribution is again a mixture X2-distributions. (iii) As t↑∞, the stochastic limit of [R(t) = X(t)/μ'1(X(t)). S9t) = Y(t/(Y(t)] exists. Let it be denoted by [R,S]. It is shown that if λ < μ, S is a mixture of certain X2-distributions, with R = 0 A.S., whereas if λ ≤ μ, R = S A.S. Furthermore, it is shown that when λ ≠ μ, S(i)↑ S as t ↑ ∞, bogth in mean square as well as with probability one. (iv)The expression for the characteristic function of the conditional distribution of Y(t1) given that X(t↑0 as t ↑ ∞, is given, with t1 being any fixed moment of time.