Queueing Systems

, Volume 91, Issue 3–4, pp 367–401 | Cite as

On the distributions of infinite server queues with batch arrivals

  • Andrew DawEmail author
  • Jamol Pender


Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.


Batch arrivals Infinite server General service Time-varying 

Mathematics Subject Classification

60K25 90B22 60G50 



We acknowledge the generous support of the National Science Foundation (NSF) for Jamol Pender’s Career Award CMMI # 1751975 and Andrew Daw’s NSF Graduate Research Fellowship under Grant DGE-1650441.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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