Thesis (M.Sc.) - Cairo University - Institute of Statistical Studies and Research - Department of Mathematical Statistics

Stochastic modelling is the application of probability theory to the description and analysis of real world phenomena. One of the most important domains in stochastic modelling is the field of queueing theory. Queueing theory has many applications in telecommunications, manufacturing process, computer networks and even real life situations. The problem of analyzing complicated queueing models is receiving considerable attention in the last decades. Vacation queues with impatient customers are one of these models. The existing of more than one server with different service rates adds a challenging problem to the analysis of the queueing system. There are many ways used in analyzing such queueing systems. One of the most important methods is the matrix-geometric method which is special case from the matrix-analytic method. In this thesis, we presented and summarized the two methods and gave some examples to show that these methods are efficient and easy to use when dealing with complicated queueing models rather than traditional methods. Moreover, we introduced a two heterogeneous servers queueing system with multiple vacation in which the vacation duration of each server is exponentially distributed. When all servers are on vacation, customers are impatient if their waiting times exceed a constant value. Our model is represented as an M/G/1-type Markov chain. To derive the stationary distribution of the system we employed the matrix-analytic method. The stationary distribution of the model was explicitly obtained by considering the transition structure of the corresponding markov chain

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