Gaussian skewness approximation for dynamic rate multi-server queues with abandonment

William A. Massey, Jamol Pender

Research output: Contribution to journalArticlepeer-review

43 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 30:149-201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567-588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.

Original languageEnglish (US)
Pages (from-to)243-277
Number of pages35
JournalQueueing Systems
Issue number2-4
StatePublished - Nov 2013

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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