A queueing model and a set of orthogonal polynomials

A single serving queueing model is studied where potential customers are discouraged at the rate λ_n = λqⁿ, 0 < q < 1, n is the queue length. The serving rate is μ_n = μ(1 ? qⁿ), n = 0, 1,…. The spectral function is computed and the corresponding set of orthogonal polynomials is studied in detail. The slightly more general model with $\text{λ}{}_{\mathrm{n}}\text{=}\text{λq}{}^{\mathrm{n}}\text{(1 + bq}{}^{\mathrm{n}}\text{)}\text{, μ}{}_{\mathrm{n}}\text{=}\text{μ(1 ? q}{}^{\mathrm{n}}\text{)}\text{(1 + bq}{}^{\mathrm{n}}\text{)}$ and the analogous orthogonal polynomials are also investigated. In both cases a method developed by Pollaczek is used which has been used very successfully to study new sets of orthogonal polynomials by Askey and Ismail.