We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

中文翻译：

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应用在操作和精算模型中的复合分布的对数凹度

我们确定，如果 (i) 和中的随机项数具有对数凹概率质量函数 (pmf)，则独立同分布 (iid) 随机量的随机和具有对数凹累积分布函数 (cdf) (ii) iid 项的分布具有非递增的密度函数（连续时）或非递增的 pmf（离散时）。我们使用标准精算风险模型和替换模型来说明这个结果的有用性。我们应用这个基本结果来确定在随机时间间隔内观察到的复合更新过程具有对数凹 cdf，如果观察时间间隔和间隔-更新时间分布具有对数凹密度，而复合分布具有递减的密度或 pmf。我们使用第二个结果来为各种库存模型建立所谓的 (s,S) 策略的最优性，这些模型的缺货成本系数为 [$/unit]，显着概括了需求和提前期流程的条件，结合这些模型中的成本结构。我们还确定了我们的结果对计算最优策略参数的各种算法方法的影响。