## Abstract

We assess the impact of utilizing electrical cross-connects for the intermediate grooming and 3R regeneration of wavelength paths in a hybrid hierarchical optical path network. Simulation results prove that they offer a significant cost reduction. We also investigate the dependencies of network cost on network parameters including optically-transparent reach, electrical switch port cost, and waveband capacity. It is demonstrated that it is critical to choose the waveband capacity properly in order to minimize total network cost.

©2011 Optical Society of America

## 1. Introduction

Recent advances in WDM technologies and related optical technologies have significantly increased network capacities. In order to meet the rapid growth of Internet traffic driven by the penetration of broadband accesses including xDSL and FTTx, single layer optical path networks based on optical cross-connects (OXCs) and ROADMs [1] and wavelength path routing have been studied, and a large number of ROADM ring networks have been deployed in North America and Japan. However, OXC deployment remains limited due to the high cost of the large-scale optical switches required. The need for cost-effective and low power consumption networks that can support the ever-increasing traffic is becoming more and more critical.

An attractive approach to cost-effectively enhancing optical switch capacity is the development of hierarchical optical path networks that employ hierarchical optical cross-connects (HOXCs). HOXCs are capable of switching optical signals at different granularities: wavelengths and wavebands (bundles of wavelengths) [2–5]. Their efficiency, indicated by the cost and port count reductions achieved by hierarchical optical path networks, has been verified [6–9]. In works [6, 7], authors investigated the efficiency of waveband switching networks compared to wavelength-routed networks, and demonstrated that their effectiveness depends on the network and traffic conditions. Cost evaluations of hierarchical optical path networks in terms of optical switch port and fiber requirements are given in [8, 9]. The results demonstrated that the degree of network cost reduction depends on the network design algorithms applied. Reference [10] introduced various HOXC architectures for hierarchical optical path networks, and clarified that necessary optical switch scales can be reduced by employing waveband switching.

In HOXC architectures, the optical cross-connect switch scale required for switching waveband paths can be very small [10, 11], and that for the wavelength paths cross-connect (WXC) part is dominant. WXC size is determined by waveband add/drop ratio, the ratio of number of added/dropped waveband paths to that of outgoing/incoming waveband paths. Bounding the waveband add/drop ratio is, therefore, the direct way of effectively reducing the WXC portion [10, 11]. Research on a network design algorithm that incorporates a waveband add/drop ratio restriction [11] revealed that even if the add/drop ratio is held to a small value, say 0.3, the increase in required network resources such as optical switch ports and fibers can be very small.

Another possible approach to realizing HOXCs with current technologies is utilizing electrical switches for handling wavelength paths. Subsequent works confirmed the effectiveness of hybrid hierarchical optical path networks that incorporate both optical cross-connects for higher order optical paths, waveband paths, and electrical cross-connects (EXCs) for lower order optical paths, wavelength paths [12–16]; the key is suppressing the electrical cross-connect portion. Although transparent optical switches offer various merits in terms of large throughput, bit rate and protocol independence, electrical switches can provide wavelength conversion and 3R regeneration functions. The 3R functions are inevitable when network size is larger than the transparent optical reach, and they can additionally provide wavelength conversion as elaborated below. This can be achieved by exploiting the mature commercial technologies that underlie SDH/SONET and OTN cross-connects. We have already proposed an efficient hybrid-hierarchical optical cross-connect (hybrid-HOXC) architecture [16] that separates the wavelength adding/dropping and grooming functions at the WXC layer, which significantly reduces switch scale. For intermediate grooming of wavelength paths, therefore, the proposed hybrid-HOXC can utilize small scale EXCs. Due to the necessity of OE/EO converters at the inputs and outputs of the electrical switching matrices, the EXCs can easily provide wavelength conversion and 3R regeneration. Thus, hybrid hierarchical optical path networks can perform 3R regeneration and wavelength conversion in combination with intermediate grooming of wavelength paths at EXCs. This idea will help to enhance the value of the hybrid-HOXCs, particularly when they are applied to large-scale (long reach) networks that require 3R regeneration. This paper highlights this point and clarifies the effectiveness of the hybrid-HOXCs for such optical networks.

The quality of the transmitted optical signals is degraded by noise and signal distortion, i.e. physical impairments; they accumulate along the optical paths and consequently limit the optical transparent transmission reach, the maximum distance that an optical signal can be transmitted without 3R regeneration. The impairment-aware design problem recently has received attention for the routing and wavelength assignment (RWA) of transparent optical networks [17–21]. The impairment-aware design problem including RWA and 3R placement problems is known to be NP-complete [17]. Thus these studies commonly establish optical paths one-by-one, i.e. they employ the sequential heuristic approach, so as to satisfy the impairment constraints. In hierarchical optical path networks, the difficulty of the design problem is greatly increased since the grooming issue of wavelength paths into waveband paths must be taken into account.

This paper is the first, as far as the authors know, to investigate the impairment-limited design problem in translucent hierarchical optical path networks employing hybrid-HOXCs which combine electrical cross-connects for wavelength path level cross-connection and 3R regeneration with optical waveband cross-connects (BXCs). To design large-scale translucent optical path networks and make full use of the advantages of hybrid-HOXCs, we propose a heuristic algorithm that considers the optical transparent reach limit and encourages the establishment of highly utilized waveband paths by grooming residual sparse wavelength paths while considering electrical port cost. Simulations prove that the hybrid-HOXC based networks designed by the proposed algorithm offer a substantial cost reduction compared to the corresponding single layer optical path network. We also analyze the effectiveness of the hybrid-hierarchical optical path network with respect to the EXC port cost and waveband capacity.

## 2. Impairment-aware design for hybrid hierarchical optical path network

We assume the hybrid-hierarchical optical cross-connect, shown in Fig. 1 , that can handle different optical path granularities: wavelength paths and waveband paths. This hybrid-HOXC architecture consists of 2 switching layers; waveband and wavelength path switching layers. The waveband switching layer is realized by an optical waveband cross-connect to route waveband paths. The wavelength switching layer is separated into two functional parts for adding/dropping and grooming operations with wavelength path granularity. This configuration minimizes the required switch scale. The wavelength path adding/dropping operations can be done using supplemental optical devices (i.e. WSSs) or they can be integrated into the switch portions that are required to attain colorless/directionless capabilities, which is the same as in a single layer OXC (not discussed further). An EXC is implemented for grooming wavelength paths. As mentioned before, the EXC can offer wavelength conversion and 3R regeneration. As a result, no separate expensive 3R regenerator is implemented in the hybrid-hierarchical optical path network. Moreover, the EXC may also offer sub-lambda (Virtual Container of ODU) level switching capabilities when it is necessary, however, this is not discussed here for simplicity.

In addition, optical signals in optical networks traverse optical fiber links and node systems that consist of many passive and/or active optical components. The physical impairments accumulate along the optical path and restrict the maximum transparent transmission reach (the maximum distance or hop count) that an optical signal can traverse without requiring 3R regeneration. In order to overcome the impairments, 3R regeneration is necessary to clean up the optical signals. Therefore, the maximum reach of the optical paths without regeneration must be considered by routing and wavelength assignment algorithms. The signal quality needs to be acceptable for any situation such as protection/restoration operations. In the translucent hybrid-hierarchical optical path networks employing hybrid-HOXCs, because the regeneration operation is processed simultaneously with waveband grooming operation at the EXCs of intermediate nodes (see Fig. 2 ), all grooming wavelength paths are processed by OEO conversion at the input/output interfaces of the EXC and the optical transparent reach of a waveband path is also exactly that of the wavelength paths accommodated within the waveband path. Hence, only the limitation of the optical transparent reach of waveband paths needs to be considered. This is an advantage over transparent hierarchical optical path networks that require impairment considerations not only for waveband paths but also for wavelength paths (wavelength paths may traverse multiple concatenated wavebands). Practically, the impairment constraint can be introduced in the form of a maximum number of physical hops that an optical signal can travel before requiring 3R regeneration for the worst impairment case. This paper adopts this approach.

The total network cost of a hybrid-hierarchical optical path network includes node cost and link cost. The node cost is mainly dominated by the cost of EXC ports required for grooming wavelength paths and that of BXC ports for originating/terminating and routing waveband paths, while the link cost is the cost of all established fibers and amplifiers. In order to minimize the total network cost of the hybrid-hierarchical optical path network, design strategies that maximally reduce both node cost and link cost are required. The work of [9] evaluated the effectiveness of waveband switching with respect to the important network parameters such as waveband capacity, waveband utilization ratio, and average hop count of established waveband paths for hierarchical optical path networks. It proved that the cost efficiency of hierarchical optical path networks is enhanced by increasing the average hop count of waveband paths or improving the waveband utilization ratio. Therefore, in the impairment-aware design of hybrid-hierarchical optical path networks, even though EXC layer operations can help to increase the network resource utilization efficiency and as a result reduce link cost, limiting the maximum optical transparent reach of waveband paths may depress the efficiency of waveband switching and substantially increase the required node cost. Hence, the network design algorithm must consider the optical transparent reach limit of waveband paths while efficiently dealing with the tradeoff between link cost and node cost to minimize the total network cost.

## 3. Network design algorithm considering the optical transparent reach constraint

#### 3.1 Preliminary

We assume that each fiber can carry up to *B* wavebands and each waveband consists of *W* wavelengths. We denote the given physical topology by G(V, A), where V is the set of all nodes and A is the set of all arcs. Matrix **C** is the set of all traffic demands, where each element, C* _{s,d}*, is the number of wavelength paths requested between node pair (

*s, d*). Hereafter, the optical transparent reach limit is represented by the hop count limitation, denoted by

*K*; it is set constant for all optical paths in the network for notational simplicity, however, if it is necessary, the generalization to the variable limit case is straightforward. The design goal is to minimize the total network cost required to accommodate all traffic demands subject to the given optical transparent reach limit.

In the hierarchical optical path network, the cost reduction can be attained by reducing EXC operation along with improving the waveband utilization efficiency by encouraging the use of free wavelengths within already established waveband paths. We need to evaluate the degree of the cost reduction obtained by EXC cut-through operations. Let cost* _{λ}*(

*u, v*) $:{V}^{2}\to {R}_{+}$,

*R*is the set of all nonnegative numbers, be the required port cost of implementing a wavelength path accommodated as a series of sequentially connected 1-hop waveband paths placed on the shortest route between

_{+}*u*and

*v*. Here, we assume that one of the shortest paths between source and destination will be assigned to the wavelength path. Similarly, we define cost

*(*

_{wb}*u,v; s,d*) $:{V}^{2}\times {V}^{2}\to {R}_{+}$ as the necessary port cost of routing the wavelength path between

*u*and

*v*as a part of the waveband path that directly connects nodes (

*s, d*). The cost functions of cost

*(*

_{λ}*u, v*) and cost

*(*

_{wb}*u,v;s,d*) are obtained as follows,

$\begin{array}{l}{\text{cost}}_{\lambda}(u,v)=2{C}_{EXC}(hop(u,v)-1)+\frac{2}{W}({C}_{BXC\_UNI}+{C}_{BXC\_NNI})hop(u,v)\text{and}\\ \begin{array}{c}{\text{cost}}_{wb}(u,v;s,d)=2{C}_{EXC}(hop(u,s)+hop(d,v))+\frac{2}{W}\{{C}_{BXC\_UNI}(hop(u,s)+hop(d,v)+1)\\ +{C}_{BXC\_NNI}(hop(u,s)+hop(s,d)+hop(d,v))\}\end{array}\end{array}$ where *hop(u, v)* is the minimum hop count between node pair (*u, v*), *C _{EXC}* the EXC port cost, and

*C*and

_{BXC_UNI}*C*are, respectively, the add/drop waveband port cost and the by-pass waveband port cost of the BXC. Based on the port cost functions of cost

_{BXC_NNI}*(*

_{λ}*u,v*) and cost

*(*

_{wb}*u,v; s,d*), we then have the following relative cost reduction obtained by employing waveband switching in the network,

$\text{gain}(s,d;u,v)=\frac{{\text{cost}}_{\lambda}(u,v)-{\text{cost}}_{wb}(u,v;s,d)}{{\text{cost}}_{\lambda}(u,v)}$

#### 3.2 Proposed design algorithm

In order to minimize the total network cost while satisfying the optical reach restriction and fully utilizing the advantages of the hybrid-HOXC nodes, the direct establishment of highly utilized waveband paths that do not exceed *K* hops is encouraged, whereas grooming operations will be applied only to accommodate the sparse remaining wavelength paths that cannot be efficiently carried by any end-to-end waveband to improve the utilization of waveband paths, since electrical EXC ports are relatively expensive. Our proposed algorithm consists of 2 major steps. The first step is routing and assignment of waveband paths; it searches for and establishes waveband paths that are equal to or shorter than *K* hops and can be effectively filled up with wavelength paths. The second step is to accommodate all sparse remaining wavelength paths by applying an auxiliary full-mesh virtual wavelength path graph based on established waveband paths with spare capacities or new direct end-to-end unexceeded *K* hops waveband paths. In both steps, to find the route of waveband/wavelength paths, a proposed un-exceeded *K*-hops shortest path algorithm is applied (see Appendix). The developed design algorithm is briefly described as follows:

*<Hybrid-Hierarchical Optical Path Network Design Considering the Optical Transparent Reach Limit>*

**Step 0- Selection of parameter values**Determine the following proper values:

**Step 1- Routing and Assignment of Waveband Paths with the optical transparent reach limit***a) Searching for a highly utilized waveband path*

$f(s,d)={\displaystyle \sum _{\begin{array}{l}(u,v)\in N(s,d)\\ \text{gain}(s,d;u,v)>0\end{array}}\text{gain}(s,d;u,v)}$

in which N(s, d) is the set of candidate wavelength paths that can be groomed and carried by a waveband path connecting s and d; N(s, d) satisfies

$f(s,d)={\displaystyle \sum _{\begin{array}{l}(u,v)\in N(s,d)\\ gain(s,d;u,v)>0\end{array}}gain(s,d;u,v)}$

and ∀ (*u, v*)$\in $*N*(*s, d*): *hop(u,s)* + *hop(s,d)* + *hop(d,v)* *≤ hop(u,v)* + *h _{lim}* .

- • If such a node pair does not exist, go to Step 2. Otherwise, for all
*(s*$\in ${(_{i}, d_{i})*u, v*)$\in $*N(s, d) |*gain*(s, d; u, v)>0*}, assign the traffic demand of node pair (*s*,_{i}*d*) on to concatenated node pairs (_{i}*s*,_{i}*s*) if*s*, (_{i}≠s*s*,*d*) and (*d*,*d*) if_{i}*d*._{i}≠d - • Define an auxiliary multi-layer waveband graph of the network as,

${G}^{Band}=\left\{{G}_{b}^{Band}=\left(V,{A}_{b}\right)\text{\hspace{0.17em}}|\text{\hspace{0.05em}}1\le b\le B\right\}$

in which each graph layer ${G}_{b}^{Band}=(V,{A}_{b})$is the corresponding network graph for the *b*^{th} waveband, *Band _{b}* (

*1≤b≤B*). Arc set

*A*of

_{b}*G*is derived from all edges in

_{b}^{Band}*G*with modified arc weight ${w}_{b}^{Band}:{V}^{2}\to {R}_{+}$, where arc weight${w}_{b}^{Band}$ is based on the number of unoccupied wavebands in established fibers on the arc, denoted by

*f*(

_{b}*f*). The weighting function for arc set

_{b}≥0*A*is defined by,

_{b}${w}_{b}^{Band}({v}_{i},{v}_{j})=\{\begin{array}{l}\left(1-{f}_{b}\epsilon \right)\left\{2{C}_{BXC\_NNI}+\frac{{C}_{fiber}({v}_{i},{v}_{j})}{B}\right\}\text{\hspace{1em}}\text{if}\text{\hspace{0.17em}}{f}_{b}>\text{0}\\ (1+\Delta )\left\{2{C}_{BXC\_NNI}+\frac{{C}_{fiber}({v}_{i},{v}_{j})}{B}\right\}\text{\hspace{1em}}\text{else}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{f}_{b}=0\end{array},$

where *C _{fiber}*(

*v*,

_{i}*v*) is the link cost connecting

_{j}*v*and

_{i}*v*including fibers and amplifiers; a constant $\Delta \le {({\mathrm{max}}_{({v}_{i},{v}_{j})\in V\times V}\{hop({v}_{i},{v}_{j})\})}^{-1}$ and another very small constant $\epsilon <{\left(\mathrm{max}\left\{\text{number}\text{of}\text{fibers}\text{at}\text{a}\text{node}\right\}\right)}^{-1}$ are introduced in order to encourage the use of existing fibers as much as possible while minimizing the hop count.

_{j}- • Find the shortest waveband path that does not exceed
*K*hops,*r*, on each layer of the auxiliary multi-layer waveband graph_{b}*G*(_{b}^{Band}*b = 1,..,B*) and then select the shortest route*r*(_{b0}*b*is the waveband index) from among the found waveband routes:_{0}*r*= min{_{b0}*r*|_{b}*1 ≤ b ≤ B*}. - • Establish the corresponding end-to-end waveband path from
*s*to*d*on the route*r*and assign waveband index_{b0}*b*to carry the total traffic demand_{0}*sum(s, d)*required in Step 1.a.*c) Repeat Step 1 until no such waveband path remains.***Step 2- Grooming Sparse Remaining Wavelength Paths***a)Construct an auxiliary full-mesh virtual wavelength path graph G*^{λ}= (V, A^{λ})Auxiliary full-mesh virtual wavelength graph

*G*is based on established waveband paths with spare capacities or new direct end-to-end waveband paths that do not exceed^{λ}*K*hops.

- • Arc weight
*w*of G^{λ}, where ${w}^{\lambda}:{V}^{2}\to {R}_{+}$, is given by^{λ}

${w}^{\lambda}({v}_{i},\text{\hspace{0.17em}}{v}_{j})=\mathrm{min}\left\{{C}_{existed\text{\hspace{0.17em}}wb}({v}_{i},\text{\hspace{0.17em}}{v}_{j}),\text{\hspace{0.17em}}{C}_{e2e-wb}({v}_{i},\text{\hspace{0.17em}}{v}_{j})\right\},$

where *C _{e2e-wb}(v_{j}, v_{j})* is the minimum cost of routing a wavelength by employing a new end-to-end un-exceeded-

*K*-hop waveband path directly connecting

*v*and

_{i}*v*(if no end-to-end un-exceeded-

_{j}*K*-hop waveband path between

*v*and

_{i}*v*is available,

_{j}*C*is set to infinity), and

_{e2e-wb}(v_{j}, v_{j})*C*the cost of using an established waveband path with free capacity between

_{existed wb}(v_{j}, v_{j}),*v*and

_{i}*v*, is given by

_{j}${C}_{existed\text{\hspace{0.17em}}wb}({v}_{i},\text{\hspace{0.17em}}{v}_{j})=\{\begin{array}{c}\begin{array}{l}2{C}_{EXC}\text{\hspace{1em}}\text{\hspace{1em}}\text{if}\text{\hspace{0.17em}}\text{there}\text{exists}\text{an}\text{established}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{waveband}\text{with}\text{free}\text{wavelength(s)}\\ \end{array}\\ \infty \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{otherwise}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\end{array},$

so as to encourage the use of existing waveband paths and fibers in order to improve the link utilization and attain cost reduction.

*b) Routing and wavelength assignment of each remaining wavelength path*- • For each remaining wavelength path request, in descending order of the hop count between its source and destination, apply Dijkstra’s algorithm to find the shortest path based on the virtual wavelength path graph and then, assign the request to the shortest path thus found.
- • Update graph G
and repeat Step 2.b until all requested traffic demands are satisfied.^{λ}

## 4. Numerical experiments

We adopt the following parameters for the numerical simulations:

- • Uniformly and randomly distributed traffic demand is represented as the average number of wavelength paths requested between each node pair.
- • Capacity of fiber:
*B*x*W*wavelengths per fiber; each fiber carries*B*wavebands and each waveband accommodates*W*wavelengths. - • No waveband conversion.
- • Parameter
*β*is defined as the ratio of the cost of an electrical switch port (including OE/EO) to that of an optical switch port (*β = C*)._{EXC}/C_{BXC_NNI} - • Parameter
*K*stands for the optical transparent reach limit of waveband/wavelength paths.

We applied the proposed algorithm with all possible threshold values *X _{wb$\in $}*{

*1/W*,

*2/W*, …,

*W/W*}; the threshold value that minimizes the total network cost is then selected. The obtained network costs are normalized by that of the comparable single-layer optical path network utilizing OXCs with the same restriction on optical transparent reach. The network costs are evaluated with 10 different traffic patterns for every traffic demand and each threshold

*X*; their ensemble averages were then plotted.

_{wb}#### 4.1 Network cost evaluation results

We evaluated the efficiency of the proposed algorithm for translucent hierarchical optical path networks employing hybrid-HOXCs with different network topologies including the pan-European network with 26 nodes and 51 links (COST266) and 5x5 regular mesh networks. For comparison, we applied the conventional algorithm based on the traffic demand expression in a Cartesian product space (TDEC) [9], which is known to be one of the most efficient heuristic design algorithms for hierarchical optical path networks, and an end-to-end algorithm (End-to-End) [7, 9], which establishes waveband paths that directly connect the source and the destination to accommodate the traffic demand between those source and destination nodes; i.e. wavelength paths are not groomed at intermediate nodes. Because both of the algorithms originally do not consider the optical reach limit, we assume that they apply the method of breaking down waveband paths longer than K hops and utilizing regeneration operations at every far-end node K hops along the waveband paths to satisfy the optical transparent reach constraint. The TDEC and End-to-End algorithms with this modification to satisfy the optical reach limit constraint are denoted as TDEC3R and End-to-End3R, respectively. We also assume that a fiber can carry up to 8 wavebands (*B*=8); the waveband capacity is also 8 (*W*=8).

Figure 4
shows the network cost comparison for the 5x5 regular mesh network with the optical transparent reach *K* of 3 and the EXC port cost ratio *β* of 3. The results indicate that the translucent hierarchical optical path network offers lower cost than the corresponding single-layer optical path network (normalized cost is less than 1). The cost reduction can reach 50% and strengthens as the traffic demand increases for all applied algorithms. Moreover, the graphs also show that our proposed algorithm outperforms other conventional algorithms (TDEC3R and End-to-End3R) and always offers the best cost reduction for all traffic demand values.

Figure 5
illustrates the normalized network costs yielded by the proposed algorithm, TDEC3R and OBXC3R algorithms for the COST266 network with following parameter values; *B*=*W*=8; the optical transparent reach *K*=2 and *β*=1. The graphs also show that the cost reduction of the translucent hierarchical optical path network increases as the traffic demand increases, and can reach 60%. In this network topology, except for the very small traffic area (i.e. less than 0.4), the proposed algorithm offers the lowest cost.

#### 4.2 Impact of network parameters

In this part, we investigate the dependencies of the total network cost on important network parameters such as optical transparent reach, EXC port cost and waveband capacity for the COST266 network. Fiber capacity is 8 wavebands (*B*=8) and each waveband consists of 8 wavelengths (*W*=8).

### a) Optical transparent reach

The normalized network cost with different optical transparent reaches (*K*=2 and 5) and EXC port cost ratios (*β* =1 and 5) with respect to the traffic demand, is shown in Fig. 6
. The results demonstrate that, in the small traffic demand area, less than 3 for *β*=5 and less than 6 for *β*=1, the obtained cost reduction of the translucent hierarchical optical path network (compared to the corresponding single-layer optical path network with the same restriction on optical reach) increases with the optical reach (larger *K*). This traffic area is limited and strongly depends on the EXC port cost ratio because in the translucent hierarchical optical path network, as the required grooming cost, which is mainly determined by *β*, is reasonable, establishing short waveband paths (smaller *K*) encourages grooming at the EXCs, and as a result, it offers higher link utilization, which can help reduce the total cost. However, when the grooming cost is high and the traffic demand is large, longer optical transparent reach permits further exploitation of waveband switching to decrease the number of necessary switch ports, and consequently offers greater cost reduction.

### b) EXC port cost

In translucent hierarchical optical path networks, EXC port cost is the main factor determining the efficiency of grooming operations. Grooming wavelength paths can strengthen waveband utilization and as a result, reduce link cost, however, the cost of EXC ports can increase node cost. Figure 7
illustrates the dependence of the hybrid hierarchical optical path network cost on the EXC port cost (represented as the EXC port cost ratio, *β*) at the traffic demand of 4 with different values of the optical transparent reach (*K*=2, 3, 4 and 5) for the COST266 network. The total network cost increases as *β* becomes large, but its growth rate drops more rapidly with longer optical transparent reach. The reason is that using short waveband paths encourages grooming and helps to improve the link utilization and consequently reduces the total cost if the cost of grooming (EXC port cost) is not high. However, permitting longer waveband paths allows the waveband switching advantages to be better exploited to reduce the total number of switch ports in the network needed. This is especially important when EXC ports are expensive.

### c) Waveband capacity

The theoretical analysis in [9] of the impact of waveband capacity *W* on the total port count, which mainly determines the total network cost of a hierarchical optical path network, proved that the hierarchical optical path network offers greater cost reduction with larger *W* when traffic demands are not small. In this part, we fix the fiber capacity in terms of wavelength number as a constant (*B*x*W*=64) and the waveband capacity, *W*, was varied (2, 4, 8, 16 and 32). The normalized COST 266 network costs with *β*=2 and *K*=3 with respect to different waveband capacity values (*W*=2, 4, 8, 16 and 32), where the average traffic demand between node pairs is changed, are shown in Fig. 8
.

It is shown that, depending on the traffic demand, the proper waveband capacity must be selected to minimize network cost. For the traffic demands of 2 and 4, the optimal waveband capacity value is 8 while the waveband capacity of 16 is the best for larger traffic demand (traffic demand of 8). Because of the fixed fiber capacity, selecting a smaller waveband capacity can help improve the waveband utilization efficiency but at the cost of increasing the number of waveband paths (and hence waveband ports) necessary as well as the number of expensive EXC ports required for grooming operations to carry the same given traffic demand, and as a result, the total network cost is increased. On the other hand, large waveband capacity such as 32 reduces the network cost less significantly. Even though the waveband utilization efficiency is weakened, since the given traffic demands are not great enough to occupy the huge waveband paths.

## 5. Conclusions

The value of the hybrid-HOXC can be significant when it is applied to large-scale networks that require 3R regeneration. Translucent hierarchical optical path networks implementing hybrid-HOXCs can incorporate 3R regeneration in waveband grooming operations; it is processed simultaneously at the EXCs of intermediate nodes. In order to minimize the total network cost under the physical impairment constraint, we have developed a translucent hierarchical optical path network design algorithm that considers the optical transparent reach constraint. Simulations proved that substantial cost reductions can be achieved by implementing the proposed hybrid network. The proposed algorithm is especially effective for large networks and large traffic demands, and strongly depends on the cost of EXC ports. We also investigated the dependence of the network cost on the optical transparent reach and the selection of waveband capacity. It was demonstrated that it is critical to choose the waveband capacity properly to minimize total network cost.

## Appendix

We propose a dynamic programming algorithm that can find the shortest path whose hop count does not exceed a given threshold, *K*, in a non-negative graph. The algorithm is briefly described as follows:

**Input**

- • Given non-negative graph G(V, A) where V={
*v*, …,_{1}*v*} is the set of vertices and A={(_{N}*v*) | a(_{i}, v_{j}*v*) ≥0} is the set of arcs in which a(_{i}, v_{j}*v*) is the weight of the arc between node_{i}, v_{j}*v*and node_{i}*v*; a(_{j}*v*) = ∞ if there is no physical connection between node_{i}, v_{j}*v*and node_{i}*v*._{j} - • Source and destination node pair: (
*s*,*d*) $\in $VxV - • Hop count limit:
*K*

**Output**

- • r(
*s*,*d*) = the shortest route from*s*to*d*that is less than*K*hops - •
*D*(_{K}*s, d*) = Minimum value of the shortest path from*s*to*d*

**Variables**

- • d[
*k*][*v*] = minimum cost of the*k*-hop shortest path connecting node pair (*s,v*) in which*v*$\in $V; and*k*is the hop count of the route (*k $\in ${1,.., K}*). - • prev[
*k*][*v*] = the preceding node of*v*on the shortest route from the source*s*to*v*that has exactly*k*hops. - • Selected[
*v*] = Selection status of node*v*(= 1 iff*v*was selected; = 0 if not)

**The proposed algorithm:**

**End while**

## Acknowledgment

This work was partly supported by Japan NEDO Green IT Project and JSPS KAKENHI (23246072).

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