Electronic Companion – Fluid Models for Overloaded Multi-Class Many-Server Queueing Systems with FCFS Routing

1. Transient Dynamics of the Fluid Model We discuss the transient dynamics of the fluid model, thus elaborating on Conjecture 1 of the main paper. We assume we are given transient routing flow rates for the model and specify dynamics in terms of these flow rates. This section closely follows the discussion of transient dynamics of the single-class single-pool fluid model given in Whitt (2006). We can think of Bi,j(t, y) as the amount of class i fluid in service in service pool j at time t that has been in service for time less than or equal to y, and Qi(t, y) as the amount of class i fluid in queue at time t that has been in queue for time less than or equal to y. As stated in Conjecture 1 in the main paper, we assume that these functions are continuous with respect to y. We further stipulate that they should have densities b and q: