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Šiame darbe nagrinėta tam tikra asimptotiškai homogeninė Markovo grandinė ir rasta jos stacionaraus skirstinio uodegos asimptotika. Nagrinėta grandinė negali būti ištirta šiuo metu žinomais metodais, todėl darbas turi praktinę reikšmę. Spręstas uždavinys aktualus sunkių uodegų analizėje. / In this work we have investigated some asymptotically homogeneous Markov chain and found asymptotics of the stationary distribution tail. To our best knowledge, considered chain cannot be investigated by means of existing methods, hence obtained results have practical value. Solved problem is relevant in heavy tail analysis.

This dissertation is concerned with the tail asymptotics of queueing networks with subexponential service time distributions. Our objective is to investigate the tail characteristics of key performance measures such as cycle times and waiting times on a variety of queueing models which may arise in many applications such as communication and manufacturing systems. First, we focus on a general class of closed feedforward fork and join queueing networks under the assumption that the service time distribution of at least one station is subexponential. Our goal is to derive the tail asymptotics of transient cycle times and waiting times. Furthermore, we argue that under certain conditions the asymptotic tail distributions remain the same for stationary cycle times and waiting times. Finally, we provide numerical experiments in order to understand how fast the convergence of tail probabilities of cycle times and waiting times is to their asymptotic counter parts. Next, we consider closed tandem queues with finite buffers between stations. We assume that at least one station has a subexponential service time distribution. We analyze this system under communication blocking and manufacturing blocking rules. We are interested in the tail asymptotics of transient cycle times and waiting times. Furthermore, we study under which conditions on system parameters a stationary regime exists and the transient results can be generalized to stationary counter parts. Finally, we provide numerical examples to understand the convergence behavior of the tail asymptotics of transient cycle times and waiting times. Finally, we study open tandem queueing networks with subexponential service time distributions. We assume that number of customers in front of the first station is infinite and there is infinite room for finished customers after the last station but the size of the buffer between two consecutive stations is finite. Using (max,+) linear recursions, we investigate the tail asymptotics of transient response times and waiting times under both communication blocking and manufacturing blocking schemes. We also discuss under which conditions these results can be generalized to the tail asymptotics of stationary response times and waiting times. Finally, we provide numerical examples to investigate the convergence of the tail probabilities of transient response times and waiting times to their asymptotic counter parts.

An ever increasing demand for instant and reliable information on modern communication networks forces codewords to operate in a non-asymptotic regime. To achieve reliability for imperfect channels in this regime, codewords need to be retransmitted from receiver to the transmit buffer, aided by a fast feedback mechanism. Large occupancy of this buffer results in longer communication delays. Therefore, codewords need to be designed carefully to reduce transmit queue-length and thus the delay experienced in this buffer. We first study the consequences of physical layer decisions on the transmit buffer occupancy. We develop an analytical framework to relate physical layer channel to the transmit buffer occupancy. We compute the optimal code-rate for finite-length codewords operating over a correlated channel, under certain communication service guarantees. We show that channel memory has a significant impact on this optimal code-rate. Next, we study the delay in small ad-hoc networks. In particular, we find out what rates can be supported on a small network, when each flow has a certain end-to-end service guarantee. To this end, service guarantee at each intermediate link is characterized. These results are applied to study the potential benefits of setting up a network suitable for network coding in multicast. In particular, we quantify the gains of network coding over classic routing for service provisioned multicast communication over butterfly networks. In the wireless setting, we study the trade-off between communications gains achieved by network coding and the cost to set-up a network enabling network coding. In particular, we show existence of scenarios where one should not attempt to create a network suitable for coding. Insights obtained from these studies are applied to design a distributed rate control algorithm in a large network. This algorithm maximizes sum-utility of all flows, while satisfying per-flow end-to-end service guarantees. We introduce a notion of effective-capacity per communication link that captures the service requirements of flows sharing this link. Each link maintains a price and effective-capacity, and each flow maintains rate and dissatisfaction. Flows and links update their respective variables locally, and we show that their decisions drive the system to an optimal point. We implemented our algorithm on a network simulator and studied its convergence behavior on few networks of practical interest.