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Ethernet's success in local area networks (LANs) is fueling the efforts to extend its reach to cover metro and long-haul networks. This new Ethernet is refereed to as Carrier Ethernet. Among the various transport infrastructures for realizing Carrier Ethernet, wavelength-division multiplexing (WDM) optical network is a strong candidate for this purpose. Optical transmission rates per channel are increasing from 10 to 40 Gb/s and even 100 Gb/s, and they can also coexist in the same fiber. Along with the flexibility associated with such a network with mixed-line rates (MLR), signal-related constraints at high rates become a challenge for cost-efficient routing. Among these issues is the maximum nonregenerated optical distance that a signal can travel before its quality degrades or maximum transmission range (TR). TR is rate-dependent: The higher the rate, the shorter the range. While high-rate pipes may require signal regeneration to restore the signal's quality, they support more traffic and, hence, can save resources. We study the problem of cost-efficient routing of multi-bit-rate (1/10/40/100 Gb/s) Ethernet tunnels using MLR over a carrier's WDM optical network with signal-transmission-range constraints. We studied the effect of TR for mixed-rate signals (10/40/100 Gb/s) on the network's cost to determine the optimal TR of each bit rate. We present an analytical model based on a mixed-integer linear program (MILP) to determine the optimal TR of a small network. Since MILP has scalability constraints that makes it hard or sometimes impossible to solve for real network topologies, we propose a graph-based solution that constructs a mixed-line-rate auxiliary (MLR-AUX) graph to capture the network's heterogeneity and a weight-assignment approach that allows the routing to be cost-efficient. Our algorithms were tested on a U.S. nationwide network topology. We found that it is possible to reduce the network's cost by using short TR and that the optimal TR depends - - strongly on traffic characteristics and on the TR values of different bit-rate signals.