Grant: Discovery Grants Program - Individual

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.