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# Modeling and Analysis of Wireless Communication Systems using Automatic Retransmission Request Protocols

## Giovanidis, Anastasios

The focus of the current thesis is on the modeling, analysis and control of Automatic Retransmission reQuest (ARQ) protocols as part of a wireless communications system. The function of these protocols is the detection and correction of errors. To achieve this the receiver informs the transmitter over the result of packet decoding using a binary control signal ACK/NACK. The NACK triggers a retransmission of the erroneous packet, while an ACK informs the transmitter that the packet has been correctly received and the next packet awaiting in the buffer is prepared. A significant performance measure related to such protocols is the goodput, defined as the rate of correctly transmitted packets over the wireless link. Typically goodput is expressed as the product of scheduled transmission rate times the success probability, which results from renewal-reward theory assuming fixed probability distribution and ergodicity of the fading process. Alternative goodput measures for short term communications are suggested which are more appropriate in case the number of packets to be transmitted is finite. Fixing the success probability values per retransmission, by selecting a priori a retransmission policy, the evolution of an ARQ protocol can be described as a success run. A definition of reliability in communications is provided, which is related to the notion of delay limited capacity. It is proven that an ARQ protocol is reliable if and only if its transition probability matrix is ergodic. Conditions for ergodicity and non-ergodicity result in a categorization of ARQ protocols (and subsequently of power allocation policies per retransmission), into reliable and unreliable. Since in practical communications the ARQ protocols are always truncated and a packet dropping occurs when the maximum number of retransmissions is exceeded, the problem of optimal truncation has been investigated. The method utilizes optimal stopping arguments where the successful transmission of a packet is related to a reward, whereas the delay and power consumption are modeled as generalized costs. The system incurs additionaly a penalty when the packet is dropped. Optimal truncation length has resulted from the sequential analysis which provides a rule combining all the above costs and rewards into a simple inequality. ARQ protocols are of course related to queuing. Incorporating a retransmission protocol at the server of a queue brings additional delay to the buffered packets so that reliability of transmission can be guaranteed. To reduce delay certain packets can be dropped by interrupting the retransmission process. Applying dynamic programming the optimal dropping policy is derived. The decision to drop depends on the system state which is the pair of queue length and current retransmission effort. The resulting policies are optimal in the sense of minimizing the time average of a linear combination of queue length and number of dropped packets. A next step in the analysis is the power control of ARQ protocols in a downlink system. Data destined to a certain number of users are buffered at the base station. Each buffer uses the retransmission protocol to achieve reliablity, while the base station has a specific total power budget to divide among users at each time slot. Considering fixed transmission rate per user and taking interference into account, the stability region of the system is derived. A power allocation policy is shown to achieve this stability region and algorithms to compute the power per user are applied and compared. The work concludes with an investigation of an ad hoc wireless network, where data enter in different source nodes and should be routed through the system nodes to their destination. Errors occur per hop due to fading and interference. Each node is again equiped with an ARQ protocol for error correction. The stability region of the system is derived. Each data flow is related to a utility function and a network utility maximization problem with stability constraints is formulated. Its solution provides the optimal per slot congestion control, routing and power allocation policy to maximize the sum of utilities while keeping all buffers in the system finite. The requirement that the power allocation policies should be implemented in a decentralized manner can be fulfilled if cooperation between nodes is allowed and at the same time each node performs measurements to estimate its interference level. Applying game theory and using the above information, each node can choose an optimal power to transmit.