We consider a mixed boundary-value problem for the Poisson equation in a two-level junction $Ω_ε,$ which is the union of a domain $Ω_0$ and a large number $3N$ of thin cylinders. The thin cylinders are divided into two levels depending on their length and the boundary conditions given on their lateral surfaces. In addition, the thin cylinders from each level are $ε$-periodically alternated. The nonuniform Neumann conditions and the uniform Dirichlet conditions are given respectively on the lateral surfaces of the thin cylinders from the first level and the second level. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct the asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space $H^1 (Ω_ε)$ as $ε → 0 (N → ∞).$ 

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