A note on solutions of a functional equation arising in a queuing model for a LAN gateway

We discuss some issues concerning solutions of the functional equation (M(x,y)-xy)P(x,y)=(1-y)(M(x,0)+r^1ξ2xy)P(x,0)+(1-x)(M(0,y)+r^2ξ1xy)P(0,y)-(1-x)(1-y)M(0,0)P(0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (M(x,y)-xy)P(x,y)=&\;(1-y)(M(x,0)+\widehat{r}_{1}\xi_{2}xy)P(x,0) \\ &+(1-x)(M(0,y)+\widehat{r}_{2}\xi_{1}xy)P(0,y)\\ &-(1-x)(1-y)M(0,0)P(0,0) \end{aligned}$$\end{document}in the class of analytic functions P mapping D¯2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{D}^2}$$\end{document}(D¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{D}}$$\end{document} stands for the closure of the unit disc D in the complex plane C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}) into C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}. Here rj,sj∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r_j,s_j\in (0,1)}$$\end{document} for j=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j=1,2}$$\end{document} are fixed, ξj=rjsj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\xi_j=r_js_j}$$\end{document}, q^=1-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{q}=1-q}$$\end{document} for every q∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q \in \mathbb{R}}$$\end{document} and M(x,y)=(r^1+r1s^1y+ξ1xy)(r^2+r2s^2x+ξ2xy).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(x,y)=(\widehat{r}_{1}+r_{1} \widehat{s}_{1}y+\xi _{1}xy)(\widehat{r}_{2}+r_{2} \widehat{s}_{2}x+\xi _{2}xy).$$\end{document}The equation arises in a two-dimensional queueing model for a LAN gateway.