Multiple queues with simultaneous arrivals

Queuing theory problems have been the topic of deep research owing to the fact that so many difficulties are in existence and their significance in real life cases can not be ignored. Those problems can be observed in numerous sectors such as telecommunications, airlines, logistics, hospitals, computing, production and inventory. Besides, speed is the key word in today's world because population is almost at the peak, thus demands or requests must be met as much as possible. However, our world has limited sources that is why there has to be some delays and queues. Additionally, game theory is one of the most important topics and it comes into prominence due to increasing competition in the world. There are lots of organizations which dwell in aforementioned sectors and they need to compete with each other to maximize their benefits. Just as in queuing theory, application of game theory spans the huge part of real life problems involving so much burden. So, there are abundance of works which dive into the distinct branches of game theory. In this study, both queueing theory and game theory are taken into consideration. We include the concept of game analysis, server rate optimization, multiple queues, loss systems and simultaneous arrivals at the same time whereas the studies in literature just focus on some of them. In our first case, we apply a game theoretic approach to two loss queuing systems under specific assumptions. With the deployment of server rate optimization we reach Nash equilibrium points. We also provide some analytical derivations and validate them using simulations. In our second case, we deal with one loss system with an uncapacitated queue involving quasi birth death process. We find the steady state probabilities employing two different computation techniques and calculate the expected profit for each queue in the system.