Extending Ward Whitt's pioneering work "Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] 2006," this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded G/G I /n + G I queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.

To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a nonexponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space D D , assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed. (FCLT's) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables. From [14,41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Introduction. Heavy-traffic (HT) functional central limit theorems