Mathematical Methods of Operations Research

, Volume 87, Issue 1, pp 139–168 | Cite as

An M/PH/K queue with constant impatient time

  • Qi-Ming HeEmail author
  • Hao Zhang
  • Qingqing Ye
Original Article


This paper is concerned with an M/PH/K queue with customer abandonment, constant impatient time, and many servers. By combining the method developed in Choi et al. (Math Oper Res 29:309–325, 2004) and Kim and Kim (Perform Eval 83–84:1–15, 2015) and the state space reduction method introduced in Ramaswami (Stoch Models 1:393–417, 1985), the paper develops an efficient algorithm for computing performance measures for the queueing system of interest. The paper shows a number of properties associated with matrices used in the development of the algorithm, which make it possible for the algorithm, under certain conditions, to handle systems with up to one hundred servers. The paper also obtains analytical properties of performance measures that are useful in gaining insight into the queueing system of interest.


Queueing systems Markov process Matrix-analytic methods Impatient customers 

Mathematics Subject Classification

Primary: 60K25 Secondary: 90-08 



The authors would like to thank three anonymous reviewers and the associate editor for their insightful comments and suggestions on this paper. The authors would also like to thank Dr. Stan Dimitrov for sharing computing resource with us.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of WaterlooWaterlooCanada
  2. 2.Department of Management EngineeringWuhan UniversityWuhanChina
  3. 3.Department of Statistics and Financial MathematicsNanjing University of Science and TechnologyNanjingChina

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