**In minisymposium: Matrix Equations**

Fluid queues, also known as first-order fluid processes, are two-dimensional stochastic processes $\{X(t), \varphi(t)\}_{t \geq 0}$ where $X$ (the level) takes real values and $\varphi$ (the phase) takes a finite number of distinct values. The process $\{\varphi(t)\}$ is a continuous-time Markov chain, it controls the process $\{X(t)\}$ as follows: during intervals where $\varphi$ is constant, and equal to $i$, $X$ varies linearly at speed $c_i$.

Fluid queues have long been used as models of dams, they became very popular in the 1980s to analyse buffer behaviour in telecommunication networks, and they have more recently found applications in risk theory and finance.

Two matrices are key to the analysis of many properties of fluid queues, denoted as $\Psi(s)$ and $\widehat\Psi(s)$ and defined as follows. Let $\theta = \inf\{t >0: X(t)=0\}$ be the first return time of the fluid to level 0; $\Psi_{ij}(s)$ is the Laplace transform of $\theta$, constrained on $\varphi(\theta)=j$, conditionally given that $\varphi(0)=i$, with $c_i >0$ and $c_j < 0$; $\widehat\Psi(s)$ corresponds to pairs of phases such that $c_i < 0$ and $c_j >0$. For $s \geq 0$, the matrices $\Psi(s)$ and $\widehat\Psi(s)$ are minimal nonnegative solutions of two Riccati equations.

I shall explain why these matrices are important to the analysis of fluid queues, and give a high-level probabilistic derivation of the Riccati equations.