Many-Server Fluid Queue with Abandonment

We consider the Gt/Mt/st + GIt many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model, having general arrival process (the Gt), Markovian service (the Mt) and customer abandonment with mutually independent times to abandon with a general distribution (the +GIt) when there are many servers (the st) and the system experiences periods of overload. We establish an asymptotic loss of memory (ALOM) property under regularity conditions, i.e., asymptotic independence from the initial conditions as time t evolves. We show that the difference in the performance functions dissipates over time exponentially fast, under the regularity conditions. We apply ALOM to show that the stationary G/M/s+GI fluid queue converges to steady state and the periodic Gt/Mt/st + GIt fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.