| Research Abstract |
A main subject of this project is to develop a theoretical method and useful tools for studying asymptotic tail behaviors of the stationary distributions that appear in queueing networks, and to apply them to various models to see their performance. Except for special cases, so called product form networks, this asymptotic problem is known to be very hard. They are usually studied by the large deviation theory. However, this theory is limited in use for queueing networks. Furthermore, it only provides the orders of their decays. In this research project, we are interested in more detailed information, in particular, the geometric decay under the light tail assumptions on service time distributions.
We start to conjecture asymptotic behaviors of typical queueing networks such as the generalized Jackson networks. These conjectures are partially verified by ourselves and some others. However, they are generally very hard to verify. So, we first formulate the decay rate problem using reflec
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ted Markov additive processes. We term this a Markov additive approach, hi this formulation, we choose the characteristic of interest as an additive component and put all the other information into background states. Since the network states are multidimensional, this characteristic takes values along a given direction so that it is one-dimensional. Usually, the characteristic is nonnegative and has complicated state transitions around the origin, while it has certain uniform additive structure when it is away from the origin. Hence, we can formulate them as a Markov additive process in many cases.
In this way, we have the reflected Markov additive process. Since we put all the information except for the characteristics of interest, the background state space is usually infinite. This is a difficult aspect different from the corresponding processes studied in the queueing literature. The latter usually assume the finite background state spaces. We overcome this difficulty using the Wiener-Hope factorization on a Markov additive process. We then derive sufficient conditions for the stationary tail. probabilities of the characteristic to asymptotically decay with a geometric term, and identify a prefactor of the term. Here, we twist the stationary distribution of the reflected Markov additive process provided it exists, and apply the Markov renewal theorem to get the geometric decay rate and the corresponding prefactor.
We apply these results to queues and their networks. In particular, we found interesting asymptotic behaviors on three models, a two node Jackson network with a truncated buffer, two parallel queue in which arriving customers choose the shortest queue, a finite buffer system in which arrival and service times are controlled by a finite state Markov chain. For example, using basic model parameters such as arrival and service rates and routing probabilities, we characterize the limiting decay rate of the stationary tail probability of the unlimited queue when the truncation level of the other queue gets large. This reveals unexpected behaviors of the limiting decay rate. We also studies fluid queues and their networks. These studies have contributed to develop the Markov additive approach. Less
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