An urn model with bernoulli removals and independent additions

A single urn model is considered for which, at each of a discrete set of time values, the balls in the urn are first removed independently with a probability that depends on the time value and then, independently of the number of balls remaining, a random number of new balls are added to the urn. The distribution and moments of the number of balls in the urn at time n are studied as well as the asymptotic behavior as n approaches infinity. Some special cases are considered in detail.