Some international pharmaceutical companies have succeeded in producing vaccines against COVID-19. Countries all over the world have aimed to obtain these vaccines with minimum cost. We consider a set of K-independent Markovian waiting lists. Each list contains a set of countries, where each one of them has an exponential service time and a Poisson arrival process. These companies differ in some characteristics such as the vaccine production cost and the speed of the required quantity delivery. We present a new detection model that helps in providing an appropriate decision to choose a suitable company. Moreover, the concept of balking and the retention of reneged countries is taken into consideration under the quality control process of each waiting list. Under steady state, we face an interesting and difficult discrete stochastic optimization problem. Its solution gives an optimal distribution of the searching effort, which is bounded by a known probability distribution. A simulation study has been derived to get the minimum value of the paid cost random values. The highest service rate, the total expected profit of each queuing system, and the optimum performance measures, which depend on this cost, have been obtained to show the effectiveness of this model.

中文翻译：

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检测合适的制药公司以在质量控制过程中以最低成本获得合适的 COVID-19 疫苗

一些国际制药公司已成功生产出针对 COVID-19 的疫苗。世界各国都致力于以最低成本获得这些疫苗。我们考虑一组K- 独立的马尔可夫等候名单。每个列表包含一组国家，其中每个国家都有指数服务时间和泊松到达过程。这些公司在某些特征上有所不同，例如疫苗生产成本和所需数量交付的速度。我们提出了一种新的检测模型，有助于提供适当的决策来选择合适的公司。此外，在每个候补名单的质量控制过程中都考虑了退缩和保留弃权国家的概念。在稳态下，我们面临一个有趣且困难的离散随机优化问题。它的解决方案给出了搜索工作的最佳分布，该分布受已知概率分布的限制。一个模拟研究已经被推导出来获得支付成本随机值的最小值。获得了最高服务率、每个排队系统的总预期利润以及取决于此成本的最佳性能度量，以证明该模型的有效性。