The Repeated Solicitation Model

This paper presents a probabilistic analysis of what we call the "repeated solicitation model". To give a specific context, suppose B is a direct marketing company with a list of S sales prospects. At epoch 1, B sends a solicitation to every prospect on the list, and elicits X(1) replies. The company deletes the respondents from the list, and at epoch 2 sends a solicitation to the other prospects, of whom X(2) respond, and so on. This continues until an epoch n such that X(n) = 0, which we call epoch T, and then B makes no further solicitations. We seek (a) the probability distribution of T; (b) the distribution of the total number of respondents; (c) the expected total number of solicitations. All three quantities are explicitly computed, assuming that (i) prospects' response times are independent, and (ii) S is Poisson distributed.