A note on solutions of a functional equation arising in a queuing model for a LAN gateway
Janusz Brzdȩk, El‐sayed El‐hady, Wolfgang Förg-Rob, Zbigniew Leśniak · Aequationes Mathematicae · 2016
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.