A queueing model and a set of orthogonal polynomials

A single serving queueing model is studied where potential customers are discouraged at the rate λn = λqn, 0 < q < 1, n is the queue length. The serving rate is μn = 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 λn = λqn(1 + bqn), μn = μ(1 − qn)(1 + bqn) 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.

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