Optimal Control of a Single Queue with Retransmissions: Delay-Dropping Tradeoffs
📝 Abstract
A single queue incorporating a retransmission protocol is investigated, assuming that the sequence of per effort success probabilities in the Automatic Retransmission reQuest (ARQ) chain is a priori defined and no channel state information at the transmitter is available. A Markov Decision Problem with an average cost criterion is formulated where the possible actions are to either continue the retransmission process of an erroneous packet at the next time slot or to drop the packet and move on to the next packet awaiting for transmission. The cost per slot is a linear combination of the current queue length and a penalty term in case dropping is chosen as action. The investigation seeks policies that provide the best possible average packet delay-dropping trade-off for Quality of Service guarantees. An optimal deterministic stationary policy is shown to exist, several structural properties of which are obtained. Based on that, a class of suboptimal <L,K>-policies is introduced. These suggest that it is almost optimal to use a K-truncated ARQ protocol as long as the queue length is lower than L, else send all packets in one shot. The work concludes with an evaluation of the optimal delay-dropping tradeoff using dynamic programming and a comparison between the optimal and suboptimal policies.
💡 Analysis
A single queue incorporating a retransmission protocol is investigated, assuming that the sequence of per effort success probabilities in the Automatic Retransmission reQuest (ARQ) chain is a priori defined and no channel state information at the transmitter is available. A Markov Decision Problem with an average cost criterion is formulated where the possible actions are to either continue the retransmission process of an erroneous packet at the next time slot or to drop the packet and move on to the next packet awaiting for transmission. The cost per slot is a linear combination of the current queue length and a penalty term in case dropping is chosen as action. The investigation seeks policies that provide the best possible average packet delay-dropping trade-off for Quality of Service guarantees. An optimal deterministic stationary policy is shown to exist, several structural properties of which are obtained. Based on that, a class of suboptimal <L,K>-policies is introduced. These suggest that it is almost optimal to use a K-truncated ARQ protocol as long as the queue length is lower than L, else send all packets in one shot. The work concludes with an evaluation of the optimal delay-dropping tradeoff using dynamic programming and a comparison between the optimal and suboptimal policies.
📄 Content
arXiv:0902.3979v1 [cs.MM] 23 Feb 2009 submitted to the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Optimal Control of a Single Queue with Retransmissions: Delay–Dropping Tradeoffs Anastasios Giovanidis, Student Member, IEEE, Gerhard Wunder, Member, IEEE, J¨org B¨uhler, Student Member, IEEE Abstract A single queue incorporating a retransmission protocol is investigated, assuming that the sequence of per effort success probabilities in the Automatic Retransmission reQuest (ARQ) chain is a priori defined and no channel state information at the transmitter is available. A Markov Decision Problem with an average cost criterion is formulated where the possible actions are to either continue the retransmission process of an erroneous packet at the next time slot or to drop the packet and move on to the next packet awaiting for transmission. The cost per slot is a linear combination of the current queue length and a penalty term in case dropping is chosen as action. The investigation seeks policies that provide the best possible average packet delay-dropping trade-off for Quality of Service guarantees. An optimal deterministic stationary policy is shown to exist, several structural properties of which are obtained. Based on that, a class of suboptimal < L, K >-policies is introduced. These suggest that it is almost optimal to use a K-truncated ARQ protocol as long as the queue length is lower than L, else send all packets in one shot. The work concludes with an evaluation of the optimal delay-dropping tradeoff using dynamic programming and a comparison between the optimal and suboptimal policies. Index Terms Automatic Retransmission reQuest Protocols, ARQ, Single Queue, Delay, Dropped Packets, Markov Decision Process, Dynamic Programming 0This work is supported by the German Federal Ministry of Education and Research as part of the ScaleNet project 01BU566. The authors are with the Fraunhofer German-Sino Mobile Communications Lab, Heinrich-Hertz-Institut, Einsteinufer 37, D-10587 Berlin, Germany. Tel/Fax: +493031002-860/-863, e-mail: {giovanidis, wunder, buehler} @ hhi.de and with the Technical University of Berlin, Heinrich Hertz Chair for Mobile Communications, Werner-von-Siemens-Bau (HFT 6), Einsteinufer 25, D-10587 Berlin, Germany. submitted to the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2 I. INTRODUCTION Retransmission protocols are applied in communications over fading channels to achieve reliability. The concept of such protocols is to detect erroneous packets at the receiver and then request retransmission for these data. The information whether the message has been detected as correct or erroneous is sent back to the transmitter via a binary feedback link using the signal NACK to request a new retransmission or ACK to declare that the packet is correctly received. In the latter case, the sender moves on to the first transmission of the next packet waiting in the buffer. In this way error-free communications can be guaranteed. However, reliability naturally comes at a cost. The costs include a waste of resources, e.g. power or spectrum, for the unsuccessful efforts, a reduction in transmission rate when error correction and detection parity bits are added to the original code, as well as a noteworthy increase in packet delay due to multiple channel reuses for the correct transmission of a single message. During the last decades, different types of retransmission protocols have been proposed in the literature (see [1], [2] and references therein) and adopted by standards of mobile networks such as UMTS, WiMAX and 3GPP LTE. These include the simple Stop-and-Wait ARQ (SW-ARQ) protocol, as well as Type-I or Type-II Hybrid-ARQ (HARQ) schemes, where Forward Error Correction codes (FEC) are used together with packet combining to enhance the protocols’ performance, [3], [4], [5], [6], [7]. Several efforts to optimize ARQ protocols, in the sense of reducing necessary retransmission efforts with economy in resources at the same time, can be found in the recent literature. Power control per retransmission to maximize the throughput of ARQ protocols has been investigated in [8]. In [9] the optimal sequence of redundancy per effort is found based on a dynamic programming formulation. The work in [10] investigates the optimal power and rate allocation among SW-ARQ retransmissions, that guarantees average delay or throughput constraints, based on some channel state information (CSI) at the receiver. A trade-off between throughput and energy consumption when the CSI is partially observable is given in [11] through a semi-Markov Decision Process formulation, while delay- and overflow-aware joint rate and power adaptation for type-I and power adaptation for type-II HARQ protocols is provided in [12] and [13], respectively. Rather noteworthy is also the work in [14], where the average delay of a single user wireless communication system, which includes a buffer and incorporates SW-ARQ retransmissions, is optimized by c
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