Research On The Several Classes Of Complex Queueing Systems

In this Ph.D. thesis, we investigate several classes of complex queueing systems which include G-queueing systems, retrial queueing systems, vacation queueing systems, discrete-time queueing systems, Markov arrival queueing systems and their mixture. This Ph.D. thesis is organized as follows.Chapter 1 is preface. The historical background, the subject, the recent develop-ment of queueing theory and the main results and innovative contributions of this thesis are introduced.In Chapter 2, we analyze an M/G/1 retrial queue with negative customers and non-exhaustive vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. It is assumed that the server has arbitrary repair time and vacation time distributions. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of classical M/G/1 queue. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we derived the solutions for queueing measures. Furthermore, we analyze the system reliability by regarding a new queueing system where the failure states of the server are assumed to be absorbent states. We also analyze the busy period of the system. Some special cases of interest are discussed and some known results have been derived. The effects of various parameters on the system performance are analyzed numerically.In Chapter 3, we discuss an MX/G/1 retrial queue with impatient customers and feedback under N-policy vacation schedule subject to disastrous failures at which times all customers in the system are lost. The necessary and sufficient condition for the stability of the system is derived. Applying Foster's criterion and Kaplan's condition, we study the ergodicity of the embedded Markov chain. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we obtain the steady-state distributions of the server state and the number of customers in the orbit along with various performance measures. In addition, we analyze some reliability problems. Further, some special cases of interest are discussed.In Chapter 4, we consider an M/G/1 retrial G-queue with preemptive resume prior-ity and collisions under linear retrial policy subject to the server breakdowns and delayed repairs. A breakdown at the busy server is represented by the arrival of a negative cus-tomer which causes the the customer being in service to be lost. The stability condition of the system is derived. Using generating function technique, the steady-state distri-butions of the server state and the number of customers in the orbit are obtained along with some interesting and important performance measures. The stochastic decompo-sition property is investigated. Further, some special cases of interest are discussed. Finally, numerical illustrations are provided.In Chapter 5, we deal with a discrete-time Geo/G/1 retrial queue with negative customers and general retrial times. Negative customers will make the customer being in service lost but has no effect to the orbit. We analyze the Markov chain underlying the considered queueing system. By applying the supplementary variables method, we derive Kolmogorov equations. By taking generating functions for these equations, we obtain the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stability condition of the system is derived. Further, the special case of no negative customers is discussed. Finally, some numerical examples are provided to illustrate the impact of several parameters on some crucial performance characteristics of the system.In Chapter 6, we study the MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment. The arrival of a negative customer with equal probability goes to any busy server to remove the customer being in service. Two different types of multi-dimensional Markov chains describing the behavior of the system based on state space arrangements are investigated. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated. Main performance measures are obtained. Illustrative numerical examples are presented.In Chapter 7, we investigate an BMAP/G/1 queues with negative customers, sec-ond optional service and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival pro- cess (MAP) respectively. After completion of the essential service of a customer, it may go for a second phase of service. The arrival of a negative customer removes the customer being in service. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method, the censoring technique and the RG-factorization, we obtain the queue length distributions. We obtain the mean of the busy period based on the renewal theory.