Annals of Operations Research

, Volume 274, Issue 1–2, pp 75–100 | Cite as

Exact analysis for multiserver queueing systems with cross selling

  • Mor Armony
  • Efrat Perel
  • Nir PerelEmail author
  • Uri Yechiali
Original Research


Exact analysis of a multi-server Markovian queueing system with cross selling in steady-state is presented. Cross selling attempt is initiated at the end of a customer’s service every time the number of customers in the system is below a threshold. Both probability generating functions (PGFs) and matrix geometric methods are employed. The relation between the methods is revealed by explicitly calculating the entries of the matrix geometric rate-matrix R. Those entries are expressed in terms of the roots of a determinant of a matrix related to the set of linear equations involving the PGFs. This is a further step towards understanding of the analytical relationship between the two methods. Numerical results are presented, showing the effect of the cross selling intensity and of the threshold level on the systems performance measures. Finally, for a given set of parameters, the optimal threshold level is determined.


Cross-selling Probability generating functions Matrix geometric 

Mathematics Subject Classification

60K25 90B22 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Stern School of BusinessNew York UniversityNew YorkUSA
  2. 2.School of Industrial Engineering and ManagementAfeka College of EngineeringTel-AvivIsrael
  3. 3.School of Industrial Engineering and ManagementShenkar - Engineering, Design and ArtRamat GanIsrael
  4. 4.Department of Statistics and Operations Research, School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael

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