Abstract
In this paper, we study the problem of finding the number of integer solutions solving z 1 + ⋯ + z k ⩽ w , 1 ⩽ z i ⩽ r , i = 1 , … , k , 1 ⩽ k < f for given f , r , w ∈ N with w ⩾ max ( f , r ) . This problem is naturally from calculating exact distributions of some sooner waiting time random variables of run and frequency quotas in statistics. We present several solutions to the problem and develop an algorithm for the sooner waiting time problems. Numerical results are given to show the efficiency of our algorithm for calculating the exact distributions of the sooner waiting time random variable.
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