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The motion of polymer molecules along channels is essential for many biological processes such as DNA and RNA transport across nuclear pores, viral infection, gene therapy etc. Recent experiments allow the process to be investigated with single molecule precision. In such experiments generally DNA or RNA molecule is driven through a tiny hole of diameter few nanometers (compatible to the diameter of the molecule) by external electrical field, and it presence is detected by the resistance. In our work we are trying to investigate simple, but exactly solvable models for the process.