In [1], stochastic discrete resource allocation problems were considered which are hard due to the combinatorial explosion of the feasible allocation search spare, as well as the absence of closed-form expressions for the cost function of interest. An ordinal optimization algorithm for solving a class of such problems was then shown to converge in probability to the global optimum. In this paper, we show that this result can he strengthened to almost sure convergence, under some additional mild conditions, and we determine the associated rate of convergence. In the case of regenerative systems, we further show that the algorithm converges exponentially fast.