Abstract

The access network has remained the bottleneck in efforts to deliver bandwidth-intensive new-generation applications and services to subscribers. In the wired access network, the gigabit Ethernet passive optical network (GEPON) is a promising technology for relieving this bottleneck, while its counterpart in the wireless access network is worldwide interoperability for microwave access (WiMAX). A converged quadruplet-service-enabled (video, voice, data, and mobility) network, which takes full advantage of the strengths and weaknesses of each of these promising technologies, has been proposed. Besides, research and Internet measurements have revealed that actual Ethernet and wireless data traffic are self-similar and long-range dependent. Therefore, we review the quality of service (QoS) architecture for integrating WiMAX and GEPON access networks that we proposed in previous work. Then, we present an analysis of the queuing behavior of the QoS architecture under self-similar and long-range-dependent data traffic conditions and derive closed-form expressions of the expected waiting time in queue (queuing delay) and the packet loss rate per QoS traffic class. This work brings novelty in terms of presenting performance analysis of the proposed QoS-aware integrated architecture under realistic load conditions and facilitates the provisioning of tightly bound QoS parameters to end users of the converged access network.

© 2009 Optical Society of America

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  1. B. O. Obele, M. H. Kang, “Fixed mobile convergence: a self-aware QoS architecture for converging WiMAX and GEPON access networks,” in The 2nd Int. Conf. on Next Generation Mobile Applications, Services and Technologies, 2008. NGMAST '08, Cardiff, Wales, Sept. 16–19, 2008, pp. 411–418.
  2. G. Shen, R. S. Tucker, C. J. Chae, “Fixed mobile convergence architectures for broadband access: integration of EPON and WiMAX,” IEEE Commun. Mag., vol. 45, no. 8, pp. 44–50, Aug. 2007.
    [CrossRef]
  3. Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.
  4. Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.
  5. Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.
  6. W. E. Leland, M. S. Taqqu, W. Willinger, D. Wilson, “On the self-similar nature of Ethernet traffic,” in IEEE/ACM Trans. Netw., vol. 2, no. 1, pp. 1–15, Feb. 1994.
    [CrossRef]
  7. W. Willinger, M. S. Taqqu, A. Erramilli, “A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks,” in Stochastic Networks: Theory and Applications, F. P. Kelly, S. Zachary, and I. Ziedins, eds., pp. 339–366, New York, NY: Oxford Univ. Press, 1996.
  8. C. Williamson, “Internet traffic measurement,” in IEEE Internet Comput., vol. 5, no. 6, pp. 70–74, Nov. /Dec. 2001.
    [CrossRef]
  9. T. Karagiannis, M. Molle, M. Faloutsos, “Long-range dependence: ten years of Internet traffic modeling,” in IEEE Internet Comput., vol. 8, no. 5, pp. 57–64, Sept.–Oct. 2004.
    [CrossRef]
  10. M. E. Crovella, A. Bestavros, “Self-similarity in World Wide Web traffic: evidence and possible causes,” in IEEE/ACM Trans. Netw., vol. 5, no. 6, pp. 835–846, Dec. 1997.
    [CrossRef]
  11. W. Odom, M. J. Cavanaugh, “Cisco QOS Exam Certification Guide,” 2nd ed., Indianapolis, IN: Cisco, 2004, pp. 1–310.
  12. M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
    [CrossRef]
  13. M. Iftikhar, B. Landfeldt, M. Caglar, “Traffic engineering and QoS control between wireless DiffServ domains using PQ and LLQ,” in Proc. 5th ACM Int. Workshop on Mobility Management and Wireless Access, Chania, Crete Island, Greece, Oct. 22, 2007, pp. 120–129.
  14. M. Iftikhar, B. Landfeldt, M. Caglar, “An analytical model based on G/M/1 with self-similar input to provide end-to-end QoS in 3G networks,” in Proc. 4th ACM Int. Workshop on Mobility Management and Wireless Access, Terromolinos, Spain, Oct. 2, 2006, pp. 180–189.
    [CrossRef]
  15. L. Kleinrock, Queuing Systems, Volume 2: Computer Applications, New York, NY: Wiley-Interscience, 1976.
  16. M. Çağlar, “A long-range dependent workload model for packet data traffic,” Math. Op. Res., vol. 29, no. 1, pp. 92–105, Feb. 2004.
    [CrossRef]
  17. D. Gross, J. Shortle, M. Fischer, D. Masi, “Difficulties in simulating queues with Pareto service,” in Proc. Winter Simulation Conf., 2002, San Diego, CA, Dec. 8–11, 2002, vol. 1, pp. 407–415.
  18. T. Nakashima, T. Sueyoshi, “Analysis of queuing property for self-similar traffic,” in 22nd Int. Conf. on Advanced Information Networking and Applications, 2008, Okinawa, Japan, March 25–28, 2008, pp. 241–248.
    [CrossRef]
  19. H. Fei, B. Yu, “Performance evaluation of wireless mesh networks with self-similar traffic,” in Int. Conf. on Wireless Communications, Networking and Mobile Computing, 2007. WiCom 2007, Shanghai, China, Sept. 21–25, 2007, pp. 1697–1700.
  20. P. Ulanovs, E. Petersons, “Modeling methods of self-similar traffic for network performance evaluation,” in Scientific Proceedings of Riga Technical University. Series 7. Telecommunications and Electronics, 2002, pp. 40–49.
  21. M. Fras, J. Mohorko, “Estimating the parameters of measured self similar traffic for modeling in OPNET,” in 14th Int. Workshop on Systems, Signals and Image Processing, 2007 and 6th EURASIP Conf. Focused on Speech and Image Processing, Multimedia Communications and Services, Maribor, Slovenia, June 27–30, 2007, pp. 78–81.
  22. Z. Sahinoglu, S. Tekinay, “On multimedia networks: self-similar traffic and network performance,” IEEE Commun. Mag., vol. 31, no. 1, pp. 48–52, Jan. 1999.
    [CrossRef]
  23. M. Crovella, A. Bestavros, “Explaining World Wide Web traffic self-similarity,” Tech. Rep. TR-95–015, Computer Science Department, Boston University, Boston MA, Aug. 1995.
  24. M. W. Garrett, W. Willinger, “Analysis, modeling and generation of self-similar VBR video traffic,” ACM SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp. 269–280, Oct. 1994.
    [CrossRef]
  25. K. Park, G. T. Kim, M. E. Crovella, “On the relationship between file sizes, transport protocols and self-similar network traffic,” in 4th Int. Conf. on Network Protocols (ICNP'96), Columbus, OH, Oct. 29–Nov. 1, 1996, pp. 171–180.
  26. T. Tuan, K. Park, “Performance evaluation of multiple time scale TCP under self-similar traffic conditions,” Tech. Rep. CSD-TR-99–040, Department of Computer Sciences, Purdue University, 1999, http://citeseer.ist.psu.edu/article/tuan99performance.html.

2008 (1)

M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
[CrossRef]

2007 (1)

G. Shen, R. S. Tucker, C. J. Chae, “Fixed mobile convergence architectures for broadband access: integration of EPON and WiMAX,” IEEE Commun. Mag., vol. 45, no. 8, pp. 44–50, Aug. 2007.
[CrossRef]

2004 (2)

T. Karagiannis, M. Molle, M. Faloutsos, “Long-range dependence: ten years of Internet traffic modeling,” in IEEE Internet Comput., vol. 8, no. 5, pp. 57–64, Sept.–Oct. 2004.
[CrossRef]

M. Çağlar, “A long-range dependent workload model for packet data traffic,” Math. Op. Res., vol. 29, no. 1, pp. 92–105, Feb. 2004.
[CrossRef]

2001 (1)

C. Williamson, “Internet traffic measurement,” in IEEE Internet Comput., vol. 5, no. 6, pp. 70–74, Nov. /Dec. 2001.
[CrossRef]

1999 (1)

Z. Sahinoglu, S. Tekinay, “On multimedia networks: self-similar traffic and network performance,” IEEE Commun. Mag., vol. 31, no. 1, pp. 48–52, Jan. 1999.
[CrossRef]

1997 (1)

M. E. Crovella, A. Bestavros, “Self-similarity in World Wide Web traffic: evidence and possible causes,” in IEEE/ACM Trans. Netw., vol. 5, no. 6, pp. 835–846, Dec. 1997.
[CrossRef]

1994 (2)

W. E. Leland, M. S. Taqqu, W. Willinger, D. Wilson, “On the self-similar nature of Ethernet traffic,” in IEEE/ACM Trans. Netw., vol. 2, no. 1, pp. 1–15, Feb. 1994.
[CrossRef]

M. W. Garrett, W. Willinger, “Analysis, modeling and generation of self-similar VBR video traffic,” ACM SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp. 269–280, Oct. 1994.
[CrossRef]

Ansari, N.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.

Bestavros, A.

M. E. Crovella, A. Bestavros, “Self-similarity in World Wide Web traffic: evidence and possible causes,” in IEEE/ACM Trans. Netw., vol. 5, no. 6, pp. 835–846, Dec. 1997.
[CrossRef]

M. Crovella, A. Bestavros, “Explaining World Wide Web traffic self-similarity,” Tech. Rep. TR-95–015, Computer Science Department, Boston University, Boston MA, Aug. 1995.

Caglar, M.

M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
[CrossRef]

M. Iftikhar, B. Landfeldt, M. Caglar, “Traffic engineering and QoS control between wireless DiffServ domains using PQ and LLQ,” in Proc. 5th ACM Int. Workshop on Mobility Management and Wireless Access, Chania, Crete Island, Greece, Oct. 22, 2007, pp. 120–129.

M. Iftikhar, B. Landfeldt, M. Caglar, “An analytical model based on G/M/1 with self-similar input to provide end-to-end QoS in 3G networks,” in Proc. 4th ACM Int. Workshop on Mobility Management and Wireless Access, Terromolinos, Spain, Oct. 2, 2006, pp. 180–189.
[CrossRef]

Çaglar, M.

M. Çağlar, “A long-range dependent workload model for packet data traffic,” Math. Op. Res., vol. 29, no. 1, pp. 92–105, Feb. 2004.
[CrossRef]

Cavanaugh, M. J.

W. Odom, M. J. Cavanaugh, “Cisco QOS Exam Certification Guide,” 2nd ed., Indianapolis, IN: Cisco, 2004, pp. 1–310.

Chae, C. J.

G. Shen, R. S. Tucker, C. J. Chae, “Fixed mobile convergence architectures for broadband access: integration of EPON and WiMAX,” IEEE Commun. Mag., vol. 45, no. 8, pp. 44–50, Aug. 2007.
[CrossRef]

Crovella, M.

M. Crovella, A. Bestavros, “Explaining World Wide Web traffic self-similarity,” Tech. Rep. TR-95–015, Computer Science Department, Boston University, Boston MA, Aug. 1995.

Crovella, M. E.

M. E. Crovella, A. Bestavros, “Self-similarity in World Wide Web traffic: evidence and possible causes,” in IEEE/ACM Trans. Netw., vol. 5, no. 6, pp. 835–846, Dec. 1997.
[CrossRef]

K. Park, G. T. Kim, M. E. Crovella, “On the relationship between file sizes, transport protocols and self-similar network traffic,” in 4th Int. Conf. on Network Protocols (ICNP'96), Columbus, OH, Oct. 29–Nov. 1, 1996, pp. 171–180.

Cvijetic, M.

Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.

Erramilli, A.

W. Willinger, M. S. Taqqu, A. Erramilli, “A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks,” in Stochastic Networks: Theory and Applications, F. P. Kelly, S. Zachary, and I. Ziedins, eds., pp. 339–366, New York, NY: Oxford Univ. Press, 1996.

Faloutsos, M.

T. Karagiannis, M. Molle, M. Faloutsos, “Long-range dependence: ten years of Internet traffic modeling,” in IEEE Internet Comput., vol. 8, no. 5, pp. 57–64, Sept.–Oct. 2004.
[CrossRef]

Fei, H.

H. Fei, B. Yu, “Performance evaluation of wireless mesh networks with self-similar traffic,” in Int. Conf. on Wireless Communications, Networking and Mobile Computing, 2007. WiCom 2007, Shanghai, China, Sept. 21–25, 2007, pp. 1697–1700.

Fischer, M.

D. Gross, J. Shortle, M. Fischer, D. Masi, “Difficulties in simulating queues with Pareto service,” in Proc. Winter Simulation Conf., 2002, San Diego, CA, Dec. 8–11, 2002, vol. 1, pp. 407–415.

Fras, M.

M. Fras, J. Mohorko, “Estimating the parameters of measured self similar traffic for modeling in OPNET,” in 14th Int. Workshop on Systems, Signals and Image Processing, 2007 and 6th EURASIP Conf. Focused on Speech and Image Processing, Multimedia Communications and Services, Maribor, Slovenia, June 27–30, 2007, pp. 78–81.

Garrett, M. W.

M. W. Garrett, W. Willinger, “Analysis, modeling and generation of self-similar VBR video traffic,” ACM SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp. 269–280, Oct. 1994.
[CrossRef]

Gross, D.

D. Gross, J. Shortle, M. Fischer, D. Masi, “Difficulties in simulating queues with Pareto service,” in Proc. Winter Simulation Conf., 2002, San Diego, CA, Dec. 8–11, 2002, vol. 1, pp. 407–415.

Iftikhar, M.

M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
[CrossRef]

M. Iftikhar, B. Landfeldt, M. Caglar, “An analytical model based on G/M/1 with self-similar input to provide end-to-end QoS in 3G networks,” in Proc. 4th ACM Int. Workshop on Mobility Management and Wireless Access, Terromolinos, Spain, Oct. 2, 2006, pp. 180–189.
[CrossRef]

M. Iftikhar, B. Landfeldt, M. Caglar, “Traffic engineering and QoS control between wireless DiffServ domains using PQ and LLQ,” in Proc. 5th ACM Int. Workshop on Mobility Management and Wireless Access, Chania, Crete Island, Greece, Oct. 22, 2007, pp. 120–129.

Kang, M. H.

B. O. Obele, M. H. Kang, “Fixed mobile convergence: a self-aware QoS architecture for converging WiMAX and GEPON access networks,” in The 2nd Int. Conf. on Next Generation Mobile Applications, Services and Technologies, 2008. NGMAST '08, Cardiff, Wales, Sept. 16–19, 2008, pp. 411–418.

Karagiannis, T.

T. Karagiannis, M. Molle, M. Faloutsos, “Long-range dependence: ten years of Internet traffic modeling,” in IEEE Internet Comput., vol. 8, no. 5, pp. 57–64, Sept.–Oct. 2004.
[CrossRef]

Kim, G. T.

K. Park, G. T. Kim, M. E. Crovella, “On the relationship between file sizes, transport protocols and self-similar network traffic,” in 4th Int. Conf. on Network Protocols (ICNP'96), Columbus, OH, Oct. 29–Nov. 1, 1996, pp. 171–180.

Kleinrock, L.

L. Kleinrock, Queuing Systems, Volume 2: Computer Applications, New York, NY: Wiley-Interscience, 1976.

Landfeldt, B.

M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
[CrossRef]

M. Iftikhar, B. Landfeldt, M. Caglar, “Traffic engineering and QoS control between wireless DiffServ domains using PQ and LLQ,” in Proc. 5th ACM Int. Workshop on Mobility Management and Wireless Access, Chania, Crete Island, Greece, Oct. 22, 2007, pp. 120–129.

M. Iftikhar, B. Landfeldt, M. Caglar, “An analytical model based on G/M/1 with self-similar input to provide end-to-end QoS in 3G networks,” in Proc. 4th ACM Int. Workshop on Mobility Management and Wireless Access, Terromolinos, Spain, Oct. 2, 2006, pp. 180–189.
[CrossRef]

Leland, W. E.

W. E. Leland, M. S. Taqqu, W. Willinger, D. Wilson, “On the self-similar nature of Ethernet traffic,” in IEEE/ACM Trans. Netw., vol. 2, no. 1, pp. 1–15, Feb. 1994.
[CrossRef]

Luo, Y.

Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.

Masi, D.

D. Gross, J. Shortle, M. Fischer, D. Masi, “Difficulties in simulating queues with Pareto service,” in Proc. Winter Simulation Conf., 2002, San Diego, CA, Dec. 8–11, 2002, vol. 1, pp. 407–415.

Mohorko, J.

M. Fras, J. Mohorko, “Estimating the parameters of measured self similar traffic for modeling in OPNET,” in 14th Int. Workshop on Systems, Signals and Image Processing, 2007 and 6th EURASIP Conf. Focused on Speech and Image Processing, Multimedia Communications and Services, Maribor, Slovenia, June 27–30, 2007, pp. 78–81.

Molle, M.

T. Karagiannis, M. Molle, M. Faloutsos, “Long-range dependence: ten years of Internet traffic modeling,” in IEEE Internet Comput., vol. 8, no. 5, pp. 57–64, Sept.–Oct. 2004.
[CrossRef]

Nakamura, S.

Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.

Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Nakashima, T.

T. Nakashima, T. Sueyoshi, “Analysis of queuing property for self-similar traffic,” in 22nd Int. Conf. on Advanced Information Networking and Applications, 2008, Okinawa, Japan, March 25–28, 2008, pp. 241–248.
[CrossRef]

Obele, B. O.

B. O. Obele, M. H. Kang, “Fixed mobile convergence: a self-aware QoS architecture for converging WiMAX and GEPON access networks,” in The 2nd Int. Conf. on Next Generation Mobile Applications, Services and Technologies, 2008. NGMAST '08, Cardiff, Wales, Sept. 16–19, 2008, pp. 411–418.

Odom, W.

W. Odom, M. J. Cavanaugh, “Cisco QOS Exam Certification Guide,” 2nd ed., Indianapolis, IN: Cisco, 2004, pp. 1–310.

Park, K.

T. Tuan, K. Park, “Performance evaluation of multiple time scale TCP under self-similar traffic conditions,” Tech. Rep. CSD-TR-99–040, Department of Computer Sciences, Purdue University, 1999, http://citeseer.ist.psu.edu/article/tuan99performance.html.

K. Park, G. T. Kim, M. E. Crovella, “On the relationship between file sizes, transport protocols and self-similar network traffic,” in 4th Int. Conf. on Network Protocols (ICNP'96), Columbus, OH, Oct. 29–Nov. 1, 1996, pp. 171–180.

Petersons, E.

P. Ulanovs, E. Petersons, “Modeling methods of self-similar traffic for network performance evaluation,” in Scientific Proceedings of Riga Technical University. Series 7. Telecommunications and Electronics, 2002, pp. 40–49.

Sahinoglu, Z.

Z. Sahinoglu, S. Tekinay, “On multimedia networks: self-similar traffic and network performance,” IEEE Commun. Mag., vol. 31, no. 1, pp. 48–52, Jan. 1999.
[CrossRef]

Shen, G.

G. Shen, R. S. Tucker, C. J. Chae, “Fixed mobile convergence architectures for broadband access: integration of EPON and WiMAX,” IEEE Commun. Mag., vol. 45, no. 8, pp. 44–50, Aug. 2007.
[CrossRef]

Shortle, J.

D. Gross, J. Shortle, M. Fischer, D. Masi, “Difficulties in simulating queues with Pareto service,” in Proc. Winter Simulation Conf., 2002, San Diego, CA, Dec. 8–11, 2002, vol. 1, pp. 407–415.

Singh, T.

M. Iftikhar, T. Singh, B. Landfeldt, M. Caglar, “Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority,” Comput. Commun., vol. 31, no. 5, pp. 1012–1027, March 2008.
[CrossRef]

Suemura, Y.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Sueyoshi, T.

T. Nakashima, T. Sueyoshi, “Analysis of queuing property for self-similar traffic,” in 22nd Int. Conf. on Advanced Information Networking and Applications, 2008, Okinawa, Japan, March 25–28, 2008, pp. 241–248.
[CrossRef]

Taqqu, M. S.

W. E. Leland, M. S. Taqqu, W. Willinger, D. Wilson, “On the self-similar nature of Ethernet traffic,” in IEEE/ACM Trans. Netw., vol. 2, no. 1, pp. 1–15, Feb. 1994.
[CrossRef]

W. Willinger, M. S. Taqqu, A. Erramilli, “A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks,” in Stochastic Networks: Theory and Applications, F. P. Kelly, S. Zachary, and I. Ziedins, eds., pp. 339–366, New York, NY: Oxford Univ. Press, 1996.

Tekinay, S.

Z. Sahinoglu, S. Tekinay, “On multimedia networks: self-similar traffic and network performance,” IEEE Commun. Mag., vol. 31, no. 1, pp. 48–52, Jan. 1999.
[CrossRef]

Tuan, T.

T. Tuan, K. Park, “Performance evaluation of multiple time scale TCP under self-similar traffic conditions,” Tech. Rep. CSD-TR-99–040, Department of Computer Sciences, Purdue University, 1999, http://citeseer.ist.psu.edu/article/tuan99performance.html.

Tucker, R. S.

G. Shen, R. S. Tucker, C. J. Chae, “Fixed mobile convergence architectures for broadband access: integration of EPON and WiMAX,” IEEE Commun. Mag., vol. 45, no. 8, pp. 44–50, Aug. 2007.
[CrossRef]

Ulanovs, P.

P. Ulanovs, E. Petersons, “Modeling methods of self-similar traffic for network performance evaluation,” in Scientific Proceedings of Riga Technical University. Series 7. Telecommunications and Electronics, 2002, pp. 40–49.

Wang, T.

Y. Luo, S. Yin, T. Wang, Y. Suemura, S. Nakamura, N. Ansari, M. Cvijetic, “QoS-aware scheduling over hybrid optical wireless networks,” in Nat. Fiber Optic Engineers Conf., Anaheim, CA, March 25, 2007, OSA Technical Digest Series (CD), Washington, DC: Optical Society of America, 2007, paper NThB1.

Y. Luo, N. Ansari, T. Wang, M. Cvijetic, S. Nakamura, “A QoS architecture of integrating GEPON and WiMAX in the access network,” in 2006 IEEE Sarnoff Symp., Princeton, NJ, March 27–28, 2006, pp. 1–4.

Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.

Weinstein, S.

Y. Luo, T. Wang, S. Weinstein, M. Cvijetic, S. Nakamura, “Integrating optical and wireless services in the access network,” in Optical Fiber Communication Conf. and Exposition and The Nat. Fiber Opt. Engineers Conf., Anaheim, CA, March 5, 2006, Technical Digest (CD), Washington, DC: Optical Society of America, 2006, paper NThG1.

Williamson, C.

C. Williamson, “Internet traffic measurement,” in IEEE Internet Comput., vol. 5, no. 6, pp. 70–74, Nov. /Dec. 2001.
[CrossRef]

Willinger, W.

W. E. Leland, M. S. Taqqu, W. Willinger, D. Wilson, “On the self-similar nature of Ethernet traffic,” in IEEE/ACM Trans. Netw., vol. 2, no. 1, pp. 1–15, Feb. 1994.
[CrossRef]

M. W. Garrett, W. Willinger, “Analysis, modeling and generation of self-similar VBR video traffic,” ACM SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp. 269–280, Oct. 1994.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

WiMAX–GEPON convergence architecture.

Fig. 2
Fig. 2

Converged ONU-BS architecture.

Fig. 3
Fig. 3

(a) Inbound queuing delay versus Hurst parameter; (b) outbound queuing delay versus Hurst parameter

Fig. 4
Fig. 4

Numerical and simulation results for (a) inbound queuing delay and (b) outbound queuing delay.

Fig. 5
Fig. 5

(a) Inbound queue size versus Hurst parameter; (b) outbound queue size versus Hurst parameter.

Fig. 6
Fig. 6

(a) Inbound packet loss rate versus Hurst parameter; (b) outbound packet loss rate versus Hurst parameter.

Fig. 7
Fig. 7

(a) Inbound packet loss rate versus Hurst parameter (closer view of I-UGS and I-rtPS queues); (b) outbound packet loss rate versus Hurst parameter (closer view of O-UGS and O-rtPS queues).

Fig. 8
Fig. 8

Numerical and simulation results for (a) inbound packet loss rate, (b) outbound packet loss rate.

Equations (29)

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S = { ( in 1 , in 2 , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A P , InSv , d , G ) :
A P { A 1 , A 2 , A 3 , A 4 } ,
d { d in , d out } ,
InSv { s 1 in , s 2 in , s 3 in , s 4 in , s 1 out , s 2 out , s 3 out , s 4 out , I } ,
G { 0 , 1 } , in 1 , in 2 , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 Z + } ,
f T i j ( t ) = f j 0 ( t ) F ¯ T i 0 ( t ) ,
f T i j ( t ) = f j 0 ( t ) F ¯ T i 0 ( t ) ,
f T i j ( t ) = f j 0 ( t ) F ¯ k 0 ( t ) F ¯ l 0 ( t ) ,
f T i i ( t ) = f T i ( t ) F ¯ j 0 ( t ) F ¯ k 0 ( t ) F ¯ l 0 ( t ) ,
A P , o { A 1 , A 2 , A 3 , A 4 } ; d , q { d in , d out } ; G , r { 0 , 1 } ;
InSv , q { s 1 in , s 2 in , s 3 in , s 4 in , s 1 out , s 2 out , s 3 out , s 4 out } .
P [ X n + 1 = ( 0 , in 2 k , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A 2 , s 2 in , d out , 0 ) | X n = ( in 1 , in 2 , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A 1 , s 1 in , d in , 0 ) ] = P [ in 1 + 1 served from in 1 , k served from in 2 , no grant during the transition T 12 and a type - 2 packet is in service during this interarrival time ] ,
P [ X n + 1 = ( 0 , in 2 k , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A 2 , s 2 in , d out , 0 ) | X n = ( in 1 , in 2 , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A 1 , s 1 in , d in , 0 ) ] = F ¯ G 0 ( t ) 0 0 t t x f s 2 in ( s ) f s 1 in in 1 + 1 + s 2 in k ( x ) f T 12 ( t ) d s d x d t .
P [ X n + 1 = ( in 1 , in 2 , in 3 , in 4 , 0 , 0 , out 3 k , out 4 , A 1 , s 3 out , d in , 1 ) | X n = ( in 1 , in 2 , in 3 , in 4 , out 1 , out 2 , out 3 , out 4 , A 1 , s 1 out , d out , 1 ) ] = P [ out 1 + 1 served from out 1 , out 2 served , k served from out 3 , a type- 3 outbound is in service , a grant is active ] = F G 1 ( t ) 0 0 t t x f s 3 out ( s ) f s 1 out out 1 + 1 + s 2 out out 2 + s 3 out k ( x ) f T 11 ( t ) d s d x d t ,
E [ W 1 ] = j 1 = 1 J 1 1 j 2 = 0 J 2 j 1 μ 1 π ( j 1 , j 2 , a 1 , s 1 ) + j 1 = 0 J 1 1 j 2 = 1 J 2 ( j 1 μ 1 + 1 μ 2 ) π ( j 1 , j 2 , a 1 , s 2 ) ,
E [ W 2 ] = j 1 = 1 J 1 j 2 = 0 J 2 1 ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , a 2 , s 1 ) + j 1 = 0 J 1 j 2 = 1 J 2 1 ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , a 2 , s 2 ) + c 1 W 2 μ 1 ,
( J 1 , J 2 , J 3 , J 4 , J G , A P , s P , G ) : A P { A 1 , A 2 , A 3 , A 4 , } ,
s P { s 1 , s 2 , s 3 , s 4 , } , G { 0 , 1 } ,
E [ W P e 2 e ] = W ONUBS i OSC + W OSC OLT + E [ W P out ] .
PLR 1 in = i = 0 J 4 j = 0 J 3 k = 0 J 2 l = 0 J out m = 1 4 n = 0 , 1 π ( J 1 , k , j , i , l , A 1 , s m , n ) ,
PLR 1 out = i = 0 J 4 j = 0 J 3 k = 0 J 2 l = 0 J in m = 1 4 n = 0 , 1 π ( J 1 , k , j , i , l , A 1 , s m , n ) ,
E [ W 1 in ] = j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j out = 1 J out ( j 1 μ 1 + j out μ out ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s out , 1 ) + j 1 = 1 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s 1 , 0 ) + j 1 = 0 J 1 1 j 2 = 1 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + 1 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s 2 , 0 ) + j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 1 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + 1 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s 3 , 0 ) + j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 1 J 4 j out = 0 J out ( j 1 μ 1 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s 4 , 0 ) ,
E [ W 1 out ] = j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + j in μ in ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 1 , s in , 0 ) + j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + 1 μ in ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 1 , s in , 1 ) + j 1 = 1 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 1 , s 1 , 1 ) + j 1 = 0 J 1 1 j 2 = 1 J 2 j 3 = 0 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + 1 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 1 , s 2 , 1 ) + j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 1 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + 1 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 1 , s 3 , 1 ) + j 1 = 0 J 1 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 1 J 4 j in = 0 J in ( j 1 μ 1 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 1 , s 4 , 1 ) ,
E [ W 2 in ] = j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j out = 1 J out ( j 1 μ 1 + j 2 μ 2 + j out μ out ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s out , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s 1 , 0 ) + j 1 = 0 J 1 j 2 = 1 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s 2 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 1 J 3 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + 1 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s 3 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 1 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s 4 , 0 ) + c 1 E [ W 2 in ] μ 1 ,
E [ W 2 out ] = j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + j in μ in ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 2 , s in , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + 1 μ in ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 2 , s in , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 2 , s 1 , 1 ) + j 1 = 0 J 1 j 2 = 1 J 2 1 j 3 = 0 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 2 , s 2 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 1 J 3 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + 1 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 2 , s 3 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 1 j 3 = 0 J 3 j 4 = 1 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 2 , s 4 , 1 ) + c 1 E [ W 2 out ] μ 1 ,
E [ W 3 in ] = j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j out = 1 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j out μ out ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s out , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s 1 , 0 ) + j 1 = 0 J 1 j 2 = 1 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s 2 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 1 J 3 1 j 4 = 0 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + 1 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s 3 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 1 J 4 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s 4 , 0 ) + ( c 1 μ 1 + c 2 μ 2 ) E [ W 3 in ] ,
E [ W 3 out ] = j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j in μ in ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 3 , s in , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + 1 μ in ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 3 , s in , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 3 , s 1 , 1 ) + j 1 = 0 J 1 j 2 = 1 J 2 j 3 = 0 J 3 1 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 3 , s 2 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 1 J 3 1 j 4 = 0 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 3 , s 3 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 1 j 4 = 1 J 4 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + 1 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 3 , s 4 , 1 ) + ( c 1 μ 1 + c 2 μ 2 ) E [ W 3 out ] ,
E [ W 4 in ] = j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j out = 1 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 + j out μ out ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s out , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s 1 , 0 ) + j 1 = 0 J 1 j 2 = 1 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s 2 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 1 J 3 j 4 = 0 J 4 1 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s 3 , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 1 J 4 1 j out = 0 J out ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s 4 , 0 ) + ( c 1 μ 1 + c 2 μ 2 + c 3 μ 3 ) E [ W 4 in ] ,
E [ W 4 out ] = j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 + j in μ in ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 4 , s in , 0 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j in = 1 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 + 1 μ in ) π ( j 1 , j 2 , j 3 , j 4 , j out , A 4 , s in , 1 ) + j 1 = 1 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 4 , s 1 , 1 ) + j 1 = 0 J 1 j 2 = 1 J 2 j 3 = 0 J 3 j 4 = 0 J 4 1 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 4 , s 2 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 1 J 3 j 4 = 0 J 4 1 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 4 , s 3 , 1 ) + j 1 = 0 J 1 j 2 = 0 J 2 j 3 = 0 J 3 j 4 = 1 J 4 1 j in = 0 J in ( j 1 μ 1 + j 2 μ 2 + j 3 μ 3 + j 4 μ 4 ) π ( j 1 , j 2 , j 3 , j 4 , j in , A 4 , s 4 , 1 ) + ( c 1 μ 1 + c 2 μ 2 + c 3 μ 3 ) E [ W 4 out ] .