An Analysis Of The Queueing System With Impatient Waiting

In this thesis, the queueing systems with impatient waiting were systems studied: the M/G/1 queue with multiple adaptive vacation, server set-up times and impatient waiting and the M/M/c/K queue with impatient waiting and the three threshold policy. These are fully new vacation queue models. In the first queue model, the necessary and sufficient condition were provided in which the queueing system is positive recurrent, and the generating function of the steady-state queue length and the steady-state waiting time distribution. Importantly, their stochastic decomposition results were also given, at the same time, show the generating function of the busy period, vacation period, set-up period and idle period. At last, the probability which the server was in various states in the steady-state situation was shown. In the second model, using the matrix analytic method, the stable distribution of the queue length, waiting number and the waiting time of customers entered into the system were given.The whole thesis includes three chapters. In the first chapter, the development history of the queue theory, the vacation queue theory, and their applications in the up-to-date technical fields were shown. Meanwhile, the main methods were summed up which are used to study the queue system. In the second chapter, the model one was studied. With regarding customer immediate leaving and retarding numbers customers in the system as top of observation, the Markov chain imbedded in which the customer arrived and its transition probability matrix were given. Then the stochastic decomposition theorem of the steady-state queue length and waiting time were discussed, the probability decomposition problem of the additional length and the additional delay were derived. Further, the expressions of the generating function of the busy period, vacation period, vacation period and idle period were given, also the ratio in various states in the steady-state situation were calculated. At last, the two special examples were given and the above results were tested, and the results in the work of Tian Nai-shuo were generalized.In the third chapter, the model two was studied. With the quasi-birth and death process and the matrix analytic method, the rate matrix, the queue length stable distribution , and the distribution of waiting number when the servers are all busy and the waiting time of customers entered into the system were given. In the end, the two special examples were given and the above results were text again. These conclusion generalized the studies of George Zhang(2005) and point out the research direction in the vacation queueing system in the future.