Abstract

This paper proposes an efficient framework for deterministic service guarantees in slot-based optical networks. The framework uses the combination of a control plane and a data plane to solve the complex problem of capacity allocation, slot-matching and traffic scheduling. The control plane implements admission control and capacity allocation to source-destination node pairs and the data plane handles traffic aggregation, buffering, and scheduling. We propose an efficient algorithm in the control plane for slot collision-resolution. In the data plane, we present a comprehensive aggregation and scheduling mechanism that realizes Service Curves assurance. We use the Time-Domain Wavelength Interleaved Network (TWIN) architecture for the proof of concept and conduct extensive simulations to assess the performance of the algorithm and scheduling mechanism.

© 2006 Optical Society of America

Full Article  |  PDF Article
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References

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  1. S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
    [Crossref]
  2. C. Qiao and M. Yoo, “Optical Burst Switching (OBS) - A new Paradigm for an Optical Internet,” J. High Speed Nets. 8, 69–84 Jan. (1999).
  3. I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
    [Crossref]
  4. C. Nuzman and I. Widjaja, “Time-domain wavelength interleaved networking with wavelength reuse,” in Proc. IEEE INFOCOM’06, Mar. (2006).
  5. N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
    [Crossref]
  6. B. Li and Y. Qin, “Traffic scheduling in a Photonic Packet Switching System with QoS Guarantee,” J. Lightwave Technol. 16, 2281–2295 (1998).
    [Crossref]
  7. M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
    [Crossref]
  8. Maode Ma and M. Hamdi, “Providing deterministic quality-of-service guarantees on WDM Optical Networks,” IEEE J. Sel. Areas Commun. 18, 2072–2083 (2000).
    [Crossref]
  9. B. Teitelbaum et al., QBone Architecture (v1.0), Internet 2 QoS Working Group Draft, Aug. 1999
  10. H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
    [Crossref]
  11. K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
    [Crossref]
  12. M. Chen and T. -S. Yum, “A conflict-free protocol for Optical WDMA Networks,” in Proc. IEEE GLOBECOM, Dec. (1991).
  13. I. Saniee and I. Widjaja, “Simplified layering and flexible bandwidth with TWIN,” in Proc. Workshop on Future Directions in Network Architecture, SIGComm, (2004).
  14. R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,” IEEE J. Sel. Areas Commun. 13, 1048–1056, Aug. (1995).
    [Crossref]
  15. J. -Y. Le Boudec and P. Thiran, Network Calculus - A Theory of deterministic queueing systems for the Internet, (Lecture Notes in Computer Science, Springer2001).
  16. R. L. Cruz and C. M. Okino, “Service guarantees for window flow control,” in Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Oct. (1996).
    [PubMed]
  17. H. Sariowan, “A Service-curve approach to performance guarantees in integrated-service networks,” Ph.D. thesis, Dept of Electrical & Computer Engineering, UCSD, June (1996).
  18. J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).
  19. J. Zheng, V. O. K. Li, and X. Yuan, “An adaptive flow classification Algorithm for IP switching,” in Proc. IEEE GLOBECOM, Nov. (1999).
  20. W. Wang and C. C. Shen, “An adaptive flow classification scheme for data-driven label switching networks,” in Proc. IEEE ICC, June (2001).
  21. K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).
  22. C. K. Siew and M. H. Er, “A new multiservice provisioning mechanism with service curves assurance for per-class scheduling delay guarantees” in press for IEE Proc. Communications. Available at URL: http://www.icis.ntu.edu.sg/our_institute/staff/cksiew/revised_COM_2005_0246-uncorrected%20final%20draft.pdf

2004 (2)

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).

2003 (2)

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

2001 (1)

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
[Crossref]

2000 (3)

Maode Ma and M. Hamdi, “Providing deterministic quality-of-service guarantees on WDM Optical Networks,” IEEE J. Sel. Areas Commun. 18, 2072–2083 (2000).
[Crossref]

N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
[Crossref]

S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
[Crossref]

1999 (1)

C. Qiao and M. Yoo, “Optical Burst Switching (OBS) - A new Paradigm for an Optical Internet,” J. High Speed Nets. 8, 69–84 Jan. (1999).

1998 (1)

1995 (1)

R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,” IEEE J. Sel. Areas Commun. 13, 1048–1056, Aug. (1995).
[Crossref]

Baba, K.

K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).

Bambos, N.

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

Chapin, S. J.

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

Chen, M.

M. Chen and T. -S. Yum, “A conflict-free protocol for Optical WDMA Networks,” in Proc. IEEE GLOBECOM, Dec. (1991).

Cruz, R. L.

R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,” IEEE J. Sel. Areas Commun. 13, 1048–1056, Aug. (1995).
[Crossref]

R. L. Cruz and C. M. Okino, “Service guarantees for window flow control,” in Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Oct. (1996).
[PubMed]

Dixit, S.

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
[Crossref]

S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
[Crossref]

Er, M. H.

C. K. Siew and M. H. Er, “A new multiservice provisioning mechanism with service curves assurance for per-class scheduling delay guarantees” in press for IEE Proc. Communications. Available at URL: http://www.icis.ntu.edu.sg/our_institute/staff/cksiew/revised_COM_2005_0246-uncorrected%20final%20draft.pdf

Giles, R.

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

Golmie, N.

N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
[Crossref]

Hamdi, M.

Maode Ma and M. Hamdi, “Providing deterministic quality-of-service guarantees on WDM Optical Networks,” IEEE J. Sel. Areas Commun. 18, 2072–2083 (2000).
[Crossref]

Hollick, M.

J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).

Hurley, P.

J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).

Hwang, J. S.

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

Kumaran, K.

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

Le Boudec, J. -Y.

J. -Y. Le Boudec and P. Thiran, Network Calculus - A Theory of deterministic queueing systems for the Internet, (Lecture Notes in Computer Science, Springer2001).

Li, B.

Li, V. O. K.

J. Zheng, V. O. K. Li, and X. Yuan, “An adaptive flow classification Algorithm for IP switching,” in Proc. IEEE GLOBECOM, Nov. (1999).

Ma, Maode

Maode Ma and M. Hamdi, “Providing deterministic quality-of-service guarantees on WDM Optical Networks,” IEEE J. Sel. Areas Commun. 18, 2072–2083 (2000).
[Crossref]

Mantar, H. A.

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

Mitra, D.

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

Mukherjee, B.

S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
[Crossref]

Ndousse, T.

N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
[Crossref]

Nuzman, C.

C. Nuzman and I. Widjaja, “Time-domain wavelength interleaved networking with wavelength reuse,” in Proc. IEEE INFOCOM’06, Mar. (2006).

Okino, C. M.

R. L. Cruz and C. M. Okino, “Service guarantees for window flow control,” in Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Oct. (1996).
[PubMed]

Okumus, I. T.

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

Qiao, C.

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
[Crossref]

C. Qiao and M. Yoo, “Optical Burst Switching (OBS) - A new Paradigm for an Optical Internet,” J. High Speed Nets. 8, 69–84 Jan. (1999).

Qin, Y.

Ross, K.

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

Saniee, I.

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

I. Saniee and I. Widjaja, “Simplified layering and flexible bandwidth with TWIN,” in Proc. Workshop on Future Directions in Network Architecture, SIGComm, (2004).

Sariowan, H.

H. Sariowan, “A Service-curve approach to performance guarantees in integrated-service networks,” Ph.D. thesis, Dept of Electrical & Computer Engineering, UCSD, June (1996).

Schmitt, J.

J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).

Shen, C. C.

W. Wang and C. C. Shen, “An adaptive flow classification scheme for data-driven label switching networks,” in Proc. IEEE ICC, June (2001).

Siew, C. K.

C. K. Siew and M. H. Er, “A new multiservice provisioning mechanism with service curves assurance for per-class scheduling delay guarantees” in press for IEE Proc. Communications. Available at URL: http://www.icis.ntu.edu.sg/our_institute/staff/cksiew/revised_COM_2005_0246-uncorrected%20final%20draft.pdf

Steinmetz, R.

J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).

Su, D.

N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
[Crossref]

Teitelbaum, B.

B. Teitelbaum et al., QBone Architecture (v1.0), Internet 2 QoS Working Group Draft, Aug. 1999

Thiran, P.

J. -Y. Le Boudec and P. Thiran, Network Calculus - A Theory of deterministic queueing systems for the Internet, (Lecture Notes in Computer Science, Springer2001).

Wang, W.

W. Wang and C. C. Shen, “An adaptive flow classification scheme for data-driven label switching networks,” in Proc. IEEE ICC, June (2001).

Widjaja, I.

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

I. Saniee and I. Widjaja, “Simplified layering and flexible bandwidth with TWIN,” in Proc. Workshop on Future Directions in Network Architecture, SIGComm, (2004).

C. Nuzman and I. Widjaja, “Time-domain wavelength interleaved networking with wavelength reuse,” in Proc. IEEE INFOCOM’06, Mar. (2006).

Yamaoka, K.

K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).

Yao, S.

S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
[Crossref]

Yasukawa, K.

K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).

Yoo, M.

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
[Crossref]

C. Qiao and M. Yoo, “Optical Burst Switching (OBS) - A new Paradigm for an Optical Internet,” J. High Speed Nets. 8, 69–84 Jan. (1999).

Yuan, X.

J. Zheng, V. O. K. Li, and X. Yuan, “An adaptive flow classification Algorithm for IP switching,” in Proc. IEEE GLOBECOM, Nov. (1999).

Yum, T. -S.

M. Chen and T. -S. Yum, “A conflict-free protocol for Optical WDMA Networks,” in Proc. IEEE GLOBECOM, Dec. (1991).

Zheng, J.

J. Zheng, V. O. K. Li, and X. Yuan, “An adaptive flow classification Algorithm for IP switching,” in Proc. IEEE GLOBECOM, Nov. (1999).

ICICE Trans. Commun. (1)

K. Yasukawa, K. Baba, and K. Yamaoka, “Dynamic class assignment for stream flows considering characteristics of non-stream flow classes,” ICICE Trans. Commun. 11, 3242–3254, Dec. (2004).

IEEE Commun. Mag. (4)

S. Yao, S. Dixit, and B. Mukherjee, “Advances in Photonic Packet Switching: an overview,” IEEE Commun. Mag. 38, 84–94, Feb. (2000).
[Crossref]

I. Widjaja, I. Saniee, R. Giles, and D. Mitra, “Light core and intelligent edge for a flexible, thin-layered and cost-effective optical transport network,” IEEE Commun. Mag. 41, 30–36, (2003).
[Crossref]

N. Golmie, T. Ndousse, and D. Su, “A differentiated optical service model for WDM Networks,” IEEE Commun. Mag. 38, 68–73, Feb. (2000).
[Crossref]

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next generation optical Internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).
[Crossref]

IEEE J. Sel. Areas Commun. (4)

Maode Ma and M. Hamdi, “Providing deterministic quality-of-service guarantees on WDM Optical Networks,” IEEE J. Sel. Areas Commun. 18, 2072–2083 (2000).
[Crossref]

H. A. Mantar, J. S. Hwang, I. T. Okumus, and S. J. Chapin, “A scalable model for interbandwidth broker resource reservation and provisioning,” IEEE J. Sel. Areas Commun. 22, 2019–2034 (2004).
[Crossref]

K. Ross, N. Bambos, K. Kumaran, I. Saniee, and I. Widjaja, “Scheduling bursts in Time-Domain Wavelength Interleaved Networks,” IEEE J. Sel. Areas Commun. 21, 1441–1451 (2003).
[Crossref]

R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,” IEEE J. Sel. Areas Commun. 13, 1048–1056, Aug. (1995).
[Crossref]

J. High Speed Nets. (1)

C. Qiao and M. Yoo, “Optical Burst Switching (OBS) - A new Paradigm for an Optical Internet,” J. High Speed Nets. 8, 69–84 Jan. (1999).

J. Lightwave Technol. (1)

Other (11)

B. Teitelbaum et al., QBone Architecture (v1.0), Internet 2 QoS Working Group Draft, Aug. 1999

C. Nuzman and I. Widjaja, “Time-domain wavelength interleaved networking with wavelength reuse,” in Proc. IEEE INFOCOM’06, Mar. (2006).

J. -Y. Le Boudec and P. Thiran, Network Calculus - A Theory of deterministic queueing systems for the Internet, (Lecture Notes in Computer Science, Springer2001).

R. L. Cruz and C. M. Okino, “Service guarantees for window flow control,” in Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Oct. (1996).
[PubMed]

H. Sariowan, “A Service-curve approach to performance guarantees in integrated-service networks,” Ph.D. thesis, Dept of Electrical & Computer Engineering, UCSD, June (1996).

J. Schmitt, P. Hurley, M. Hollick, and R. Steinmetz, “Per-flow guarantees under class-based priority queueing,” in Proc. IEEE GLOBECOM, Nov. (2003).

J. Zheng, V. O. K. Li, and X. Yuan, “An adaptive flow classification Algorithm for IP switching,” in Proc. IEEE GLOBECOM, Nov. (1999).

W. Wang and C. C. Shen, “An adaptive flow classification scheme for data-driven label switching networks,” in Proc. IEEE ICC, June (2001).

M. Chen and T. -S. Yum, “A conflict-free protocol for Optical WDMA Networks,” in Proc. IEEE GLOBECOM, Dec. (1991).

I. Saniee and I. Widjaja, “Simplified layering and flexible bandwidth with TWIN,” in Proc. Workshop on Future Directions in Network Architecture, SIGComm, (2004).

C. K. Siew and M. H. Er, “A new multiservice provisioning mechanism with service curves assurance for per-class scheduling delay guarantees” in press for IEE Proc. Communications. Available at URL: http://www.icis.ntu.edu.sg/our_institute/staff/cksiew/revised_COM_2005_0246-uncorrected%20final%20draft.pdf

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Figures (8)

Fig. 1.
Fig. 1.

Traffic flow to nodes G and F in a TWIN network

Fig. 2.
Fig. 2.

TWIN network with a bandwidth broker controller

Fig. 3.
Fig. 3.

Slot transmission in a period

Fig. 4.
Fig. 4.

The arrival curve and service curve of an aggregate flow I

Fig. 5.
Fig. 5.

The class-based traffic assembly and scheduling process

Fig. 6.
Fig. 6.

Flow classifications with M QoS classes and a best effort class (class 0)

Fig. 7.
Fig. 7.

Comparison of service rates required for two arrival curves 1(t):(Σj ,ρj ) and 2(t):(Mj ,pj ,Σj ,ρj ).

Fig. 8.
Fig. 8.

Delay distribution with three classes of traffic

Tables (3)

Tables Icon

Table 1. Network utilization in TWIN networks with different number of nodes

Tables Icon

Table 2. Network utilization in a 200-node TWIN network for 2 slots allocation

Tables Icon

Table 3. Network utilization in a 200-nodes TWIN network for random of 1 to 4 slots allocation

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

δ ij mod ( N 1 ) = p ( 0 p < N 1 , p is a constant )
Φ = k = 1 N 1 C N 1 k Str ( N 1 , k ) k ! k ( N 1 ) N
S I ( t ) = k = 1 w I b u ( t z I k τ ) , where u ( t ) = { 0 , t < 0 1 , t 0
D k d i D k + 1 for 1 k M .
S ¯ j ( t D h ) + k = 1 M i F k S ¯ i ( t D k ) k = 1 w b u ( t k τ ) and
S j ( t ) + i F h S i ( t ) α h w b , for 0 t τ and 1 h M
S ¯ M ( t D M ) S ( t ) and
i F M S i ( t ) α M w b , for 0 t τ
k = 1 M i F k S ¯ i ( t D k ) k = 1 w b u ( t k τ ) and
i F h S i ( t ) α k w b , for 0 t τ and 1 k M
σ j + k = h M i F k σ i k D h w b τ ( 1 k = h + 1 M α k ) and
ρ j + i F h ρ i h α h w b τ
S ¯ M ( t ) S M ( t + D M )
σ j + i F M σ i M D M w b τ and
ρ j + i F M ρ i M α M w b τ
σ j + k = M 1 M i F k σ i k D M 1 w b τ ( 1 α M ) and
ρ j + i F M 1 ρ i M 1 α M 1 w b τ

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