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Forward-Looking WDM Network Reconfiguration with Per-Link Congestion Control

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Abstract

We study reconfigurations of wavelength-routed Wavelength Division Multiplexing (WDM) networks in response to lightpath demand changes, with the objective of servicing more lightpath demands without additional network resources from a long-term network operation point of view. For the reconfiguration problem under study, we assume WDM network operators are provided with lightpath demands in batches. With limited network resources, our problem has two unique challenges: balancing network resource allocations between current and future lightpath demands, and modeling future lightpath demands. The first challenge implies making tradeoffs between accepting as many current immediate lightpath demands as possible and reserving a certain amount of network resources for near future predicted lightpath demands. The second challenge implies modeling future predicted lightpath demands, which are not exactly known or certain as the current lightpath demands. Our proposed model allows a natural separation between the operation of the optical layer and the user traffic layer (predominantly the IP-layer), while supporting their interactions, for which we propose a new formulation for per-link congestion control, associated with a mathematical solution procedure. Our simulation results reveal that by properly controlling resource allocations in the current session using our proposed mechanism, rejections in future sessions are greatly reduced.

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Acknowledgments

Dr. Jing Wu acknowledges the research support from the State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, China.

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Authors and Affiliations

  1. Communications Research Centre (CRC) Canada, Ottawa, ON, K2H 8S2, Canada

    Jing Wu & Michel Savoie

  2. School of Information Technology and Engineering, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    James Zhang & Gregor von Bochmann

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Correspondence to Jing Wu.

Appendix: Solution Methodology

Appendix: Solution Methodology

By properly relaxing some constraints using Lagrange multipliers, we will derive in this section the DP (Dual Problem), which can be decomposed into subproblems that can be solved independently. A schematic depiction of the overall algorithm is given in Fig. 15.

Fig. 15
figure 15

Schematic depiction of the overall algorithm

Full size image

1.1 The LR Solution Procedure

We first use the Lagrange multipliers ξ ijc , λ ic , π ij to relax respectively the wavelength channel capacity constraints (b), and link congestion constraints (c). This leads to the following Lagrangian DP (Dual Problem):

$$ \mathop {\max }\limits_{\xi ,\lambda ,\pi \ge 0} \,q = \mathop {\,\min }\limits_{{{\rm A},{\rm B},\Updelta ,\Upphi ,\Upgamma }} \left\{ {\sum\limits_{(s,d)} {\left[ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right)} \right] + \sum\limits_{(i,j)} {G_{ij} (\gamma_{ij} ) + \sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {\xi_{ijc} \left( {\sum\limits_{(s,d)} {\sum\limits_{{0 < n \le H_{sd} }} {\delta_{ijc}^{sdn} } } - 1} \right)} } } } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\lambda_{ic} \left( {\sum\limits_{(s,d)} {\sum\limits_{{0 < n \le H_{sd} }} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\varphi_{i,ca}^{sdn} } } } } - F_{ic} } \right)} } + \sum\limits_{(i,j)} {\pi_{ij} \left( {\sum\limits_{(s,d)} {\sum\limits_{{0 < n \le H_{sd} }} {\sum\limits_{{0 \le c < n_{ij} }} {\delta_{ijc}^{sdn} } } } - \gamma_{ij} W} \right)} } \right\}, $$

subject to the constraints (a), (d), and (e), where ξ, λ, π are respectively the vectors of Lagrange multipliers {ξ ijc }, {λ ic }, {π ij }.

After regrouping the relevant terms, the dual function leads to the following problem:

$$ \mathop {\,\min }\limits_{{{\rm A},{\rm B},\Updelta ,\Upphi ,\Upgamma }} \left\{ {\sum\limits_{(s,d)} {\left[ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ] + \sum\limits_{{(i,j)}} {\sum\limits_{{0 < c \le n_{{ij}} }} {\delta _{{ijc}}^{{sdn}} \left( {\xi _{{ijc}} + \pi _{{ij}} } \right)} } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } } \right)} \right] - \sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {\xi_{ijc} } } - \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\lambda_{ic} } } F_{jc} + \sum\limits_{(i,j)} {\left( {G_{ij} (\gamma_{ij} ) - W\gamma_{ij} \pi_{ij} } \right)} } } \right\}. $$
(7)

By using the fact that \( \delta_{ijc}^{sdn} = \alpha_{sdn} \delta_{ijc}^{sdn} \) and \( \varphi_{i,ca}^{sdn} = \alpha_{sdn} \varphi_{i,ca}^{sdn} \), we can rewrite (7) as:\( \mathop {\,\min }\limits_{{{\rm A},{\rm B},\Updelta ,\Upphi ,\Upgamma }} \left\{ {\sum\limits_{(s,d)} {\left[ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right) + \sum\limits_{{0 < n \le H_{sd} }} {\alpha_{sdn} \left( {_{sdn} \sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {(\xi_{ijc} + \pi_{ij} )\delta_{ijc}^{sdn} } } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } \right)} } \right]} + \sum\limits_{(i,j)} {\left( {G_{ij} (\gamma_{ij} ) - W\gamma_{ij} \pi_{ij} } \right)} - \sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {\xi_{ijc} - \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\lambda_{ic} } } F_{jc} } } } \right\} \)Since the last two terms are independent of the decision variables (i.e., A, B, Δ, Φ, and Γ), the problem can be further simplified as:

$$ \begin{aligned} \mathop {\,\min }\limits_{{{\rm A},{\rm B},\Updelta ,\Upphi }} \left\{ {\sum\limits_{(s,d)} {\left[ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right) + \sum\limits_{{0 < n \le H_{sd} }} {\alpha_{sdn} \left( {\sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {(\xi_{ijc} + \pi_{ij} )\delta_{ijc}^{sdn} + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } } } \right)} } \right]} } \right\} + \mathop {\min }\limits_{\Upgamma } \left\{ {\sum\limits_{(i,j)} {\left( {G_{ij} (\gamma_{ij} ) - W\gamma_{ij} \pi_{ij} } \right)} } \right\} \\ & = \sum\limits_{(s,d)} {\mathop {\min }\limits_{{{\rm A}_{sd} }} \left\{ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right) + \sum\limits_{{0 < n \le H_{sd} }} {\left[ {\alpha_{sdn} \cdot \mathop {\min }\limits_{{\beta_{sdn} ,\Updelta_{sdn} ,\Upphi_{sdn} }} \left( {\sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {(\xi_{ijc} + \pi_{ij} )\delta_{ijc}^{sdn} } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } } \right)} \right]} } \right\} + \sum\limits_{(i,j)} {\mathop {\min }\limits_{{\gamma_{ij} }} \left\{ {G_{ij} (\gamma_{ij} ) - W\gamma_{ij} \pi_{ij} } \right\}} ,} \\ \end{aligned} $$
(8)

which we shall refer to as the RP (Relaxed Problem).

1.2 A.2. Decomposed Subproblems

We can see that the RP is composed of two minimization subproblem sets. The first subproblem set RWSS (Routing and Wavelength-assignment Subproblem Set) is

$$ \sum\limits_{(s,d)} {\mathop {\min }\limits_{{{\rm A}_{sd} }} \left\{ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right) + \sum\limits_{{0 < n \le H_{sd} }} {\left[ {\alpha_{sdn} \cdot \mathop {\min }\limits_{{\beta_{sdn} ,\Updelta_{sdn} ,\Upphi_{sdn} }} \left( {\sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {(\xi_{ijc} + \pi_{ij} )\delta_{ijc}^{sdn} } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } } \right)} \right]} } \right\}} , $$
(9)

subject to the constraints (a), (d), and (e). RWSS can be decomposed into source–destination-level sub-problems (denoted as SDS sd ), each corresponding to one (s, d):

$$ {\text{SDS}}_{\text{sd}} = \mathop {\min }\limits_{{{\rm A}_{sd} }} \left\{ {P_{sd} \left( {\sum\limits_{{0 < n \le H_{sd} }} {[1 - \alpha_{sdn} ]} } \right) + \sum\limits_{{0 < n \le H_{sd} }} {\alpha_{sdn} \cdot I_{\text{sdn}} } } \right\}, $$
(10)

where I sdn (corresponding to every s sdn ) is defined as

$$ I_{\text{sdn}} = \mathop {\,\min }\limits_{{\beta_{sdn} ,\Updelta_{sdn} ,\Upphi_{sdn} }} \left\{ {\sum\limits_{(i,j)} {\sum\limits_{{0 < c \le n_{ij} }} {(\xi_{ijc} + \pi_{ij} )\delta_{ijc}^{sdn} } + \sum\limits_{{i \in \mathcal{V}}} {\sum\limits_{0 \le c < W} {\sum\limits_{{j \in \mathcal{V}}} {\sum\limits_{{0 \le a < n_{ij} }} {\lambda_{ic} \varphi_{i,ca}^{sdn} } } } } } } \right\}, $$
(11)

subject to the constraints (a), (d), and (e). In RWSS, there are altogether Z lightpath-level subproblems (I sdn ’s) independent of each other.

In the subproblem set CGSS (Congestion Subproblem Set), there are E independent subγproblems, each corresponding to one network link:

$$ \sum\limits_{(i,j)} {\mathop {\min }\limits_{{0 \le \gamma_{ij} \le 1}} \{ G_{ij} (\gamma_{ij} ) - \gamma_{ij} W\pi_{ij} \} } $$
(12)

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Wu, J., Zhang, J., von Bochmann, G. et al. Forward-Looking WDM Network Reconfiguration with Per-Link Congestion Control. J Netw Syst Manage 20, 6–33 (2012). https://doi.org/10.1007/s10922-011-9212-2

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