Markov denumerable process and queue theory

Abstract

In this thesis, we study a modified Markovian batch-arrival and bulk- service queue including finite states for dependent control. We first consider the stopped batch-arrival and bulk-service queue process Q∗, which is the process with the restriction of the state-dependent control. After we obtain the expression of the Q∗-resolvent, the extinction probability and the mean extinction time are explored. Then, we apply a decomposition theorem to resume the stopped queue process back to our initial queueing model, that is to find the expression of Q-resolvent. After that, the criteria for the recurrence and ergodicity are also explored, and then, the generating function of equilibrium distribution is obtained. Additionally, the Laplace transform of the mean queue length is presented. The hitting time behaviors including the hitting probability and the hitting time distribution are also established. Furthermore, the busy period distribution is also obtained by the expression of Laplace transform. To conclude the discussion of the queue properties, a special case that m = 3 for our queueing model is discussed. Furthermore, we consider the decay parameter and decay properties of our initial queue process. First of all, similarly we consider the case of the stopped queue process Q∗. Based on this q-matrix, the exact value of the decay parameter λC is obtained theoretically. Then, we apply this result back to our initial queue model and find the decay parameter of our initial queueing model. More specifically, we prove that the decay parameter can be expressed accurately. After that, under the assumption of transient Q, the criteria for λC -recurrence are established. For λC -positive recurrent examples, the generating function of the λC-invariant measure and vector are explored. Finally, a simple example is provided to end this thesis