The asymptotic behaviour of the cumulative mean of a reward process 𝒵ρ, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E𝒵ρ(t) = C0 + C1t + o(1), t → ∞, where C0 and C1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given for the mean of the first passage time from a state i to a state j of the dual process, in terms of the means of passage times of the original process.

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.