Fluid Models for Multiserver Queues with Abandonments

Deterministic fluid models are developed to provide simple first-order performance descriptions for multiserver queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multiserver queues with a large number of servers and nonexponential service-time and time-to-abandon distributions. The first fluid model serves as an approximation for the G/GI/s+GI queueing model, which has a general stationary arrival process with arrival rate λ, independent and identically distributed (IID) service times with a general distribution, s servers and IID abandon times with a general distribution. The fluid model is useful in the overloaded regime, where λ > s, which is often realistic because only a small amount of abandonment can keep the system stable. Numerical experiments, using simulation for M/GI/s+GI models and exact numerical algorithms for M/M/s+M models, show that the fluid model provides useful approximations for steady-state performance measures when the system is heavily loaded. The fluid model accurately shows that steady-state performance depends strongly upon the time-to-abandon distribution beyond its mean, but not upon the service-time distribution beyond its mean. The second fluid model is a discrete-time fluid model, which serves as an approximation for the G_t (n)/GI/s+GI queueing model, having a state-dependent and time-dependent arrival process. The discrete-time framework is exploited to prove that properly scaled queueing processes in the queueing model converge to fluid functions as s → ∞. The discrete-time framework is also convenient for calculating the time-dependent fluid performance descriptions.