Analytic and computational analysis of models arising in telecommunications and transportation

离散和连续时间排队在许多发生拥堵的地区(如银行、医院、机场、交通路口、高速公路收费站、电信、计算机网络)都使用。建立了此类排队系统的数学模型。我们广泛的科学目标将是开发新的模型或改进现有的模型，就像我们过去所做的那样。我们过去开发的强大的寻根技术现在可以用来寻找此类问题的解，特别是当等待空间是无限的，到达或服务是相关的。**因此，我们建议的总体主旨将是识别、扩展和统一与离散和连续时间排队模型有关的科学知识，以便于计算和应用。正如我们在过去所做的那样，这项工作有望改进排队理论(同样是离散和连续时间模型)和其他随机过程中许多问题的理论和计算分析。与过去一样，预计对这种模型和其他类似模型的计算方面的讨论将使该理论对实践者有用。*意识到我们过去推广的根方法可以以简单可靠的方式给出各种随机模型的解，我们的长期目标是展示根方法如何帮助开发此类模型的分析和计算方面(无论是在离散时间还是连续时间)。这些模型出现在应用数学的各个领域，如排队、可靠性、库存控制和精算学。我们现在开始展示如何在涉及马尔可夫到达和/或马尔可夫服务过程的过程中使用根方法。正如我们过去所做的那样，预计我们现在可以通过展示如何将根方法用于涉及马尔科夫到达和/或马尔可夫服务过程的更复杂的过程，来回应KlienRock和Neuts等知名研究人员表达的关切。我们已经在我们最近的几篇论文(见投稿中的第6和11号)中展示了这种方法如何用于解决涉及马尔可夫服务/到达的问题。它旨在将该方法应用于涉及批量队列的更复杂的模型。还计划将这种方法的效率与正在使用的一些其他方法进行比较。**除了上述拟议的工作外，在不久的将来，我们打算寻找一个不同的方向，并将我们的寻根方法应用于精算学中出现的情况，并处理诸如风险过程等情况，这些过程涉及找到破产时间的分布、破产前的盈余和破产时的赤字。在过去，已经提出了各种方法来研究这些分布的概率性质。我们的方法将试图给出分析上明确的和计算上稳定的结果。**在研究上述建议的同时，我们将像过去一样，促进知识和培训高素质的人员，他们将做好充分准备，培训下一代研究人员/教师，或使用他们所学到的工具在电信、交通、可靠性和精算学等领域应用。

Both the discrete- and continuous-time queues are used in many areas (such as banks, hospitals, airports, traffic intersections, highway toll-booths, telecommunications, computer networks) wherever congestions occur. Mathematical models are developed for such queueing systems. Our broad scientific goal will be to develop new models or improve on the existing models as we have done in the past. The powerful root-finding technique that we have developed in the past can now be used to finding solutions of such problems, particularly when the waiting space is infinite and arrivals or services are correlated.**Thus, the general thrust of our proposal will be to identify, extend and unify scientific knowledge pertaining to both discrete- and continuous-time queueing models with a view toward computations and hence applications. As we have done in the past, the work is expected to improve theoretical and computational analyses of many problems in queueing theory (again both discrete- and continuous-time models) and other stochastic processes. As in the past, it is expected that the discussion of computational aspects of such and other similar models will make the theory useful for practitioners. ***Realizing that the use of roots' method that we have propagated in the past can give solutions to various stochastic models in a simple and reliable way, our long-term objective is to show how the roots' method can help in developing analytic and computational aspects of such models (both in discrete and continuous times). These models arise in various areas of applied mathematics such as queueing, reliability, inventory control and actuarial science. We are now beginning to show as to how the roots' method can be used in processes involving Markovian Arrival and/or Markovian Services Processes. As we have done in the past, it is expected that we can now respond to concerns expressed by various well-known researchers such as Klienrock and Neuts by showing that how the roots' method can be used in more complicated processes involving Markovian Arrival and/or Markovian Services Processes. We have already shown in a couple of our recent papers (see number 6 and 11 in Contributions) that how this method can be used in solving problems involving Markovian service/arrivals. It is intended to apply the method to more sophisticated models involving bulk queues. It is also planned to compare the efficiency of this method with some of the other methods that are being used. **In addition to the above proposed work, in the near future, we intend to look into a different direction and apply our root-finding method to cases that arise in actuarial science and deal with situations such as risk processes which involve finding the distributions of the time to ruin, the surplus before the time of ruin, and the deficit at the time of ruin. In the past, various approaches have been proposed to study the probabilistic properties of these distributions. Our approach will attempt to give analytically explicit and computationally stable results.**While working on the above proposal, we will be advancing the knowledge and training, as we have done in the past, highly qualified personnel who will be well-prepared to train the future generation of researchers/teachers or use the tools they have learned to apply in areas such as telecommunications, transportation, reliability and actuarial science.