Dynamic rate Erlang-A queues

William A. Massey, Jamol Pender

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large-scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al. (Queueing Syst 30(1):149–201, 1998) for Markovian service networks. These asymptotics were inspired by the Halfin and Whitt (Oper Res 29(3):567–588, 1981) scaling for multi-server queues. In this paper, we provide a review and an in-depth analysis of the Erlang-A queueing model. We prove new results about cumulant moments of the Erlang-A queue, the transient behavior of the Erlang-A limit cycle, new fluid limits for the delay time of a virtual customer, and optimal static staffing policies for healthcare systems. We combine tools from queueing theory, ordinary differential equations, complex analysis, cumulant moments, orthogonal polynomials, and dynamic optimization to obtain new insights about this fundamental queueing model.

Original languageEnglish (US)
Pages (from-to)127-164
Number of pages38
JournalQueueing Systems
Issue number1-2
StatePublished - Jun 1 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


  • Abandonment
  • Asymptotics
  • Cumulant moments
  • Dynamical systems
  • Fluid and diffusion limits
  • Hermite polynomials
  • Multi-server queues
  • Skewness
  • Time-inhomogeneous Markov processes
  • Time-varying rates


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