Well, this is a hard question ! Queueing theory was originated to answer actual problems (performance valuations of telephone switching systems). This situation is still unchanged at present days. As long as many people want to use limited resources effectively, problems for queueing theory are never exhausted. In fact, the problems become more and more complicated, and there remain many unsolved problems. Queueing theory is a tool to solve those problems. Our study is to make and improve such a tool. About 90 years have passed since queueing theory was born. So, the theory has been sophisticated, and demands more and more mathematics. However, fortunately, queueing theory is always asked to answer actual problems. The theory tries to provide right answers for them. We are not sure to answer your questions, but we can say that queueing theory is intended to be useful.
Queueing theory was originated for performance evaluations of telephone switching systems. Now a day, telephone networks have changed to huge networks of telecommunication and information systems. According to those changes, queueing theory has contributed to performance evaluations of those networks. The theory is also used to evaluate performances of computer systems and their networks. Production systems in factories are another important applications.
You may use standard models of queueing theory in your applications. You may also use software packages for peperformance evaluations, based on queueing theory. However, queueing models include random components, and it is very important how to incorporate such random elements in modeling the systems. It is not easy to explain this modeling technique. We recommend you to simulate your system of interest on a computer. To this end, you need to consider mathematical modeling of the system. There are a lot of software packages for simulations, but those package may use their own assumptions, which may be black boxed. If you can afford a time, there is nothing better than to program your simulation by yourself. We now can use high speed computers and well organized programing languages at hand, so simulations are very easy. After the simulations, you may consider to apply queueing theory. Then you may need to do numerical computations or further simulations.
Nothing is required to understand queueing models. However, it may be required some knowledge of probability theory so as to understand probabilistic assumptions of the models and results of queueing theory. It would be better to know about Markov processes as well. Sufficient knowledge on these subjects is about at the level of the first or second year of university.
If you want to study queueing theory, you may need to understand elementary probability theory in addition to analysis and linear algebra at the level of the first year of university. If you want to study advanced topics, you are recommended to learn naive set theory, elementary topology and measure theory. An advanced book of probability theory usually includes elementary set theory and measure theory. So you may start with such a probability book. Then, you need to lean stochastic processes including Markov processes in applied probability books. Some knowledge on function spaces and operators on them would be helpful as well.