**2. Main tools**

In the following, we will briefly introduce our main tools: (a) the matrix-analytic approach and the theory of Markov-modulated fluid flows, initiated by a series of papers by Ahn and Ramaswami [34–37] and Ramaswami [33] and (b) an application of the optional sampling theorem (OST) to the multi-dimensional martingale of Asmussen and Kella [31].

Here and in the following, we use bold symbols to denote vectors and blackboard to denote matrices. To be consistent, for any matrix , we shall denote its elements by ð Þ *ij* or by ½ � *ij* and reserve the notation *ij* for the sub-matrix of with row indices in the set ℑ*<sup>i</sup>* and column indices in the set ℑ*j*. Moreover, for *n*-vector **x**, we use Δ**<sup>X</sup>** for a diagonal matrix Δ**<sup>x</sup>** ¼ *diag x*ð Þ 1, *x*2, … , *xn :* We further use *E* and *Ei* to represent expectation and conditional expectation operators, respectively. (**E**) represents a matrix (a vector) of expectations. We denote by **e** a column vector with ones, i.e. **e** ¼ ð Þ 1, 1, … , 1 *<sup>T</sup>*, by **<sup>e</sup>***<sup>i</sup>* a row vector with the *<sup>i</sup>*th component equal to 1 and all the other components 0, by the identity matrix, by **0** the zero matrix (all with the appropriate dimensions); finally, by **1**f g *<sup>A</sup>* the indicator of an event *A*.
