This paper presents a problem of optimal stopping and proposes simple rules that are asymptotically optimal in a specific sense. The problem is central to quality control and has applications in reliability theory and other areas. The goal is to detect a change in the distribution of a sequence of independent random variables, where the first m variables have distribution F₀ and the rest have distribution F₁ ≠ F₀. The challenge is to determine the optimal stopping time N to react to the change, minimizing the expected number of observations before reacting, while controlling the frequency of false reactions. The paper introduces a minimax-type criterion for quickness of reaction, defined as the smallest bound on the average number of differently distributed X's observed before reacting, regardless of the behavior of the X's before the change. It shows that the procedure proposed by Page (1954) is asymptotically optimal for this criterion. This procedure involves stopping when the cumulative sum of log-likelihood ratios exceeds a threshold. The procedure is equivalent to a sequential probability ratio test (SPRT) with log-boundaries and is shown to have expected run lengths that can be calculated using Wald's equation. The paper extends the problem to a family of distributions {Fθ, θ ∈ Θ} with an unknown θ, aiming to minimize the expected number of observations before reacting for each θ, subject to a constraint on the expected number of observations under F₀. It shows that asymptotically, one can minimize this for each θ as γ → 0. The results are established by connecting the problem to one-sided sequential testing. The paper also discusses practical applications of these procedures in quality control and reliability theory, and provides examples of how to implement the procedures, including the use of cumulative sum charts and maximum likelihood methods. The paper concludes with a discussion of the theoretical and practical implications of these results.This paper presents a problem of optimal stopping and proposes simple rules that are asymptotically optimal in a specific sense. The problem is central to quality control and has applications in reliability theory and other areas. The goal is to detect a change in the distribution of a sequence of independent random variables, where the first m variables have distribution F₀ and the rest have distribution F₁ ≠ F₀. The challenge is to determine the optimal stopping time N to react to the change, minimizing the expected number of observations before reacting, while controlling the frequency of false reactions. The paper introduces a minimax-type criterion for quickness of reaction, defined as the smallest bound on the average number of differently distributed X's observed before reacting, regardless of the behavior of the X's before the change. It shows that the procedure proposed by Page (1954) is asymptotically optimal for this criterion. This procedure involves stopping when the cumulative sum of log-likelihood ratios exceeds a threshold. The procedure is equivalent to a sequential probability ratio test (SPRT) with log-boundaries and is shown to have expected run lengths that can be calculated using Wald's equation. The paper extends the problem to a family of distributions {Fθ, θ ∈ Θ} with an unknown θ, aiming to minimize the expected number of observations before reacting for each θ, subject to a constraint on the expected number of observations under F₀. It shows that asymptotically, one can minimize this for each θ as γ → 0. The results are established by connecting the problem to one-sided sequential testing. The paper also discusses practical applications of these procedures in quality control and reliability theory, and provides examples of how to implement the procedures, including the use of cumulative sum charts and maximum likelihood methods. The paper concludes with a discussion of the theoretical and practical implications of these results.