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A note on solutions of a functional equation arising in a queuing model for a LAN gateway. (English) Zbl 1345.30030

Summary: We discuss some issues concerning solutions of the functional equation \[ \begin{aligned} (M(x,y)-xy)P(x,y)=&\;(1-y)(M(x,0)+\widehat{r}_{1}\xi_{2}xy)P(x,0) \\ &+(1-x)(M(0,y)+\widehat{r}_{2}\xi_{1}xy)P(0,y)\\ &-(1-x)(1-y)M(0,0)P(0,0) \end{aligned} \] in the class of analytic functions \(P\) mapping \({\overline{D}^2}\) (\({\overline{D}}\) stands for the closure of the unit disc \(D\) in the complex plane \(\mathbb{C}\)) into \(\mathbb{C}\). Here \(r_j,s_j\in (0,1)\) for \(j=1,2\) are fixed, \(\xi_j=r_js_j\), \(\widehat{q}=1-q\) for every \(q \in \mathbb{R}\) and \[ M(x,y)=(\widehat{r}_{1}+r_{1} \widehat{s}_{1}y+\xi_{1}xy)(\widehat{r}_{2}+r_{2} \widehat{s}_{2}x+\xi_{2}xy). \] The equation arises in a two-dimensional queueing model for a LAN gateway.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30E25 Boundary value problems in the complex plane
39B32 Functional equations for complex functions
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