Two Desk Tandem Queuing System

In this paper, a two-station tandem queueing system is considered, in which customers must receive service at both stations before leaving the system. When the input process is Poisson distribution and the service processes are negative exponential distributions, the limit distribution of length is given by predecessors applying the theory of Birth and Death Process. Based on that, we give every queuing targets, length, waiting time and so on.However, if the input process is not Poisson distribution or any of the service processes is not negative exponential distribution, the theory of Birth and Death Process is not suitable, such as the three more models studied in this paper. Firstly, the input process is Poisson distribution, the service time of the first station is negative exponential distribution, and the service time of the second station is general distribution. Secondly, the input process is Poisson distribution, and the service times of the two stations are both general distributions. Thirdly, the input process and the service time of the two stations are all general distributions.Be aimed at the three models which mentioned above, we apply the Markov skeleton process theory, which was firstly put forward by professor Hou. Zhenting etc to study the transient and limit distribution of their states. In this thesis, we draw the following conclusions: Firstly, applying the Markov skeleton process approach, we present the equations which satisfy the transient distribution of the length of each model, and prove that its probability distribution is the minimal nonnegative solution of some equations. Then we study the limit distribution of each model. Furthermore we find out a Doob skeleton process of each model, and give the generalized limit distribution by the theory of Doob skeleton process and limit theory.