# American Institute of Mathematical Sciences

July  2011, 7(3): 607-622. doi: 10.3934/jimo.2011.7.607

## Optimal design and analysis of a two-hop relay network under Rayleigh fading for packet delay minimization

 1 Department of Mathematical Sciences and Telecommunication Engineering Program, Korea Advanced Institute of Science and Technology, Daejeon, South Korea 2 Department of Mathematical Sciences and Telecommunication Engineering Program, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, South Korea

Received  September 2010 Revised  May 2011 Published  June 2011

In this paper, we consider a wireless network consisting of a source node, a destination node and multiple relay nodes under Rayleigh fading. Cooperative diversity in the network is achieved by selecting an opportunistic relay node with the best channel condition to the destination node. We focus on the packet level performance in this paper and analyze the average packet delay of the network for various modulation and coding schemes. We assume that the arrival process of packets follows a Markov modulated Poisson process (MMPP). To derive the average packet delay, we first derive the distribution of the number of packets that are successfully transmitted from the source node to the destination node via a relay node. Using the distribution obtained above, a queueing process of the M/G/1 type is developed to model the queue at the source node. The average packet delay is then obtained from the stationary distribution of the queueing process of the source node by applying Little's Lemma. Based on our results on the average packet delay, the optimal modulation and coding scheme for given network parameters is determined to minimize the average packet delay. We validate our analytic model through simulation. The detailed relations between the average packet delay and network parameters such as average signal to noise ratios (SNRs) between nodes, the number of relay nodes and packet arrival rate, are investigated through numerical studies based on our analytic model as well as simulation studies. From our numerical results, we conclude that the optimal modulation and coding scheme that minimizes the average packet delay depends not only on SNRs of channels between nodes but also on the arrival rate of packets at the data link layer.
Citation: Hong Il Cho, Gang Uk Hwang. Optimal design and analysis of a two-hop relay network under Rayleigh fading for packet delay minimization. Journal of Industrial and Management Optimization, 2011, 7 (3) : 607-622. doi: 10.3934/jimo.2011.7.607
##### References:
 [1] A. Bletsas, H. Shin and M. Z. Win, Cooperative communication with outage-optimal opportunistic relaying, IEEE Trans. on Wireless Commun., 6 (2007), 3450-3460. doi: 10.1109/TWC.2007.06020050. [2] J. Cai, A. S. Alfa, P. Ren, X. Shen and J. W. Mark, Packet level performance analysis in wireless user-relaying networks, IEEE Trans. on Wireless Commun., 7 (2008), 5336-5345. doi: 10.1109/T-WC.2008.070960. [3] G. Casella and R. L. Berger, "Statistical Inference," 2nd edition, Duxbury, 2001. [4] W. Chen, L. Dai, K. B. Letaief and Z. Cao, A unified cross-layer framework for resource allocation in cooperative networks, IEEE Trans. on Wireless Commun., 7 (2008), 3000-3012. doi: 10.1109/TWC.2008.060831. [5] T. M. Cover and A. E. Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, 25 (1979), 572-584. doi: 10.1109/TIT.1979.1056084. [6] H. Heffes and D. M. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. on Sel. Areas in Commun., 4 (1986), 856-868. doi: 10.1109/JSAC.1986.1146393. [7] J. N. Laneman and G. W. Wornell, Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks, IEEE Trans. Inform. Theory, 49 (2003), 2415-2425. doi: 10.1109/TIT.2003.817829. [8] Q. Liu, S. Zhou and G. B. Giannakis, Queueing with adaptive modulation and coding over wireless links: Cross-layer analysis and design, IEEE Trans. on Wireless Commun., 4 (2005). [9] C. K. Lo, R. W. Heath and S. Vishwanath, The impact of channel feedback on opportunistic relay selection for hybrid-ARQ in wireless networks, IEEE Trans. on Veh. Tech., 58 (2009), 1255-1268. [10] M. F. Neuts, "Structured Stochastic Matrices of $M$/$G$/$1$ type and their Applications," Probability: Pure and Applied, 5, Marcel Dekker, Inc., New York, 1989. [11] T. S. Rappaport, "Wireless Communications: Principle and Practice," 2nd edition, Pentice Hall PTR., 2002. [12] B. Rong and A. Ephremides, Protocol-level cooperation in wireless networks: Stable throughput and delay analysis, in "Proc. the 7th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)," Seoul, Korea, 2009. [13] A. Sendonaris, E. Erkip and B. Aazhang, User cooperation diversity - Part I: System description, IEEE Trans. on Commun., 51 (2003), 1927-1938. doi: 10.1109/TCOMM.2003.818096. [14] H. Y. Wei and R. D. Gitlin, Two-hop-relay architecture for next-generation WWAN/WLAN integration, IEEE Wireless Commun., 11 (2004), 24-30. doi: 10.1109/MWC.2004.1295734. [15] S. Xu and T. Saadawi, Does the IEEE 802.11 MAC protocol work well in multihop wireless Ad Hoc networks?, IEEE Commun. Mag., 39 (2001), 130-137. doi: 10.1109/35.925681. [16] K. Zheng, Y. Wang, L. Lei and W. Wang, Cross-layer queueing analysis on multihop relaying networks with adaptive modulation and coding, IET Commun., 4 (2010), 295-302. doi: 10.1049/iet-com.2009.0380.

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##### References:
 [1] A. Bletsas, H. Shin and M. Z. Win, Cooperative communication with outage-optimal opportunistic relaying, IEEE Trans. on Wireless Commun., 6 (2007), 3450-3460. doi: 10.1109/TWC.2007.06020050. [2] J. Cai, A. S. Alfa, P. Ren, X. Shen and J. W. Mark, Packet level performance analysis in wireless user-relaying networks, IEEE Trans. on Wireless Commun., 7 (2008), 5336-5345. doi: 10.1109/T-WC.2008.070960. [3] G. Casella and R. L. Berger, "Statistical Inference," 2nd edition, Duxbury, 2001. [4] W. Chen, L. Dai, K. B. Letaief and Z. Cao, A unified cross-layer framework for resource allocation in cooperative networks, IEEE Trans. on Wireless Commun., 7 (2008), 3000-3012. doi: 10.1109/TWC.2008.060831. [5] T. M. Cover and A. E. Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, 25 (1979), 572-584. doi: 10.1109/TIT.1979.1056084. [6] H. Heffes and D. M. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. on Sel. Areas in Commun., 4 (1986), 856-868. doi: 10.1109/JSAC.1986.1146393. [7] J. N. Laneman and G. W. Wornell, Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks, IEEE Trans. Inform. Theory, 49 (2003), 2415-2425. doi: 10.1109/TIT.2003.817829. [8] Q. Liu, S. Zhou and G. B. Giannakis, Queueing with adaptive modulation and coding over wireless links: Cross-layer analysis and design, IEEE Trans. on Wireless Commun., 4 (2005). [9] C. K. Lo, R. W. Heath and S. Vishwanath, The impact of channel feedback on opportunistic relay selection for hybrid-ARQ in wireless networks, IEEE Trans. on Veh. Tech., 58 (2009), 1255-1268. [10] M. F. Neuts, "Structured Stochastic Matrices of $M$/$G$/$1$ type and their Applications," Probability: Pure and Applied, 5, Marcel Dekker, Inc., New York, 1989. [11] T. S. Rappaport, "Wireless Communications: Principle and Practice," 2nd edition, Pentice Hall PTR., 2002. [12] B. Rong and A. Ephremides, Protocol-level cooperation in wireless networks: Stable throughput and delay analysis, in "Proc. the 7th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)," Seoul, Korea, 2009. [13] A. Sendonaris, E. Erkip and B. Aazhang, User cooperation diversity - Part I: System description, IEEE Trans. on Commun., 51 (2003), 1927-1938. doi: 10.1109/TCOMM.2003.818096. [14] H. Y. Wei and R. D. Gitlin, Two-hop-relay architecture for next-generation WWAN/WLAN integration, IEEE Wireless Commun., 11 (2004), 24-30. doi: 10.1109/MWC.2004.1295734. [15] S. Xu and T. Saadawi, Does the IEEE 802.11 MAC protocol work well in multihop wireless Ad Hoc networks?, IEEE Commun. Mag., 39 (2001), 130-137. doi: 10.1109/35.925681. [16] K. Zheng, Y. Wang, L. Lei and W. Wang, Cross-layer queueing analysis on multihop relaying networks with adaptive modulation and coding, IET Commun., 4 (2010), 295-302. doi: 10.1049/iet-com.2009.0380.
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