Generalized sooner waiting time problems in a sequence of trinary trials

Let {ξn,n≥1} be a sequence of independent trials with three possible outcomes 0, 1, 2 labeled as failure, success of type I and success of type II, respectively. Suppose that at each time a success of type I (type II) occurs in {ξn,n≥1} a random reward of type I (type II) is received. We obtain distributions of the number of trials until either the sum of consecutive rewards of type I is equal to or exceeds the level k1 or the sum of consecutive rewards of type II is equal to or exceeds the level k2 under two different schemes.