| Summary: | Parallel queueing networks have advantage over single server queueing networks, because when some servers simultaneously serve the customers in the line, the efficiency increases. Therefore, in the real world parallel queueing servers such as computer networks and multiple parallel processors, have become common. Since then many scientists have been studying the analysis of parallel queueing networks to give the exact practical models for the real world queueing problems.
One of the topics in parallel queueing networks is the two-dimensional random walk, which recently have been studied by many scientists.
The formulation for a random walk model in the first quadrant has been already studied by Fayolle, Malyshev and Iasnogorodski [19].
In this thesis I extend the formulation of a general random walk model to the half plane, including the first and fourth quadrants, and by using kernel method and Tauberian-like Theorem I investigate the exact tail asymptotic behaviour of the joint stationary distribution of the generating functions. In addition, I apply the results of the formulation of a general random walk model in the half plane to the Generalized-JSQ model, which is a queueing system with two parallel servers that have three streams of arrivals, two of which are dedicated to each servers, and the third one joins the shorter queue. Suppose that arrivals are independent Poisson processes, and service times have identical exponential distributions. Although this queueing model has been already studied by Zhao and Grassmann [75], and M. Miyazawa, [56],
in this thesis I will use a different method named kernel method to investigate the exact tail asymptotic behaviour of the generating functions. The kernel method is simpler and faster than other
methods, since in this method we are not dealing with the explicit expressions in terms of generating functions, but we only discuss the dominant singularity and its location.
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