Compound geometric and related distributions

Abstract

In this chapter we consider the two special cases where the number of claims distribution {p n ;n = 0,1,2, …} is a geometric distribution (possibly modified or truncated at 0), and a negative binomial distribution. Compound geometric distributions play an important role in reliability, queueing, regenerative processes, and insurance applications. For example, the equilibrium waiting time in the G/G/1 queue has a compound geometric distribution and so does the maximal aggregate loss of the surplus process under the classical and renewal risk models. For discussions of these applications see Gertsbakh (1984), Kalashnikov (1994b, 1997a), and Panjer and Willmot (1992). Brown (1990) discussed moment-based bounds on the tails of compound geometric distributions. Bounds in the context of G/G/l queues are given in Kingman (1964, 1970), Ross (1974), and Stoyan (1983, p. 83). Similar results in the insurance risk setting are given in Gerber (1973, 1979) and Taylor (1976). Their bounds are largely based on the service time distribution or the claim amount distribution of the underlying aggregate claims process, as opposed to the ‘individual claim amount distribution’ in the present compound geometric context. In the next section, various bounds on the tails of compound geometric distributions are derived using the results in chapter 4. These bounds are based on the individual claim amount df F(y). Section 7.2 gives a detailed analysis when the number of claims distribution is a discrete compound geometric distribution.