Queueing Systems

, Volume 71, Issue 4, pp 405-444

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

The G t /GI/s t +GI many-server fluid queue

  • Yunan LiuAffiliated withDepartment of Industrial Engineering, North Carolina State University
  • , Ward WhittAffiliated withDepartment of Industrial Engineering and Operations Research, Columbia University Email author 


This paper introduces a deterministic fluid model that approximates the many-server G t /GI/s t +GI queueing model, and determines the time-dependent performance functions. The fluid model has time-varying arrival rate and service capacity, abandonment from queue, and non-exponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixed-point equation for the time-varying rate that fluid enters service and the solution of an ordinary differential equation for the time-varying head-of-line waiting time, during each overloaded interval.

Simulations are conducted to confirm that the algorithm and the approximation are effective.


Queues with time-varying arrivals Nonstationary queues Transient behavior Many-server queues Deterministic fluid model Customer abandonment Non-Markovian queues

Mathematics Subject Classification

60K25 90K22