A fluid approximation for large-scale service systems

Abstract

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We also show that the system dynamics greatly simplifies in two special cases: (i) when the service time distribution is exponential (M) and (ii) when the service time distribution is deterministic (D) and the model is stationary. We develop an efficient algorithm to compute all standard performance functions in both cases. In case (i), we establish an asymptotic loss of memory (ALOM) property, i.e., asymptotic independence from the initial conditions as time evolves. We show that the difference in the performance functions with different initial conditions dissipates over time exponentially fast, under regularity conditions. In contrast, in case (ii) we show that ALOM fails dramatically. Instead, although all model parameters are constants, we show that the performance rapidly approaches a periodic steady state (PSS) with a period equal to the service time, whenever the system does not start with the unique stationary distribution. Moreover, the form of the PSS depends on the initial condition. Simulation and a heavy-traffic limit confirm that this anomalous behavior also occurs in the large-scale queueing model.