Queueing Systems

, Volume 75, Issue 1, pp 65–77 | Cite as

The M/M/1 queue with inventory, lost sale, and general lead times

  • Mohammad Saffari
  • Søren Asmussen
  • Rasoul Haji


We consider an M/M/1 queueing system with inventory under the \((r,Q)\) policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.


Queueing Inventory Stationary distribution Lost sale  Regenerative process 

Mathematics Subject Classification

60K25 90B05 60G07 60G10 



The first author acknowledges the grant from the Iranian Science Ministry which allowed him to spend an extended period in 2009 at Aarhus University, Denmark. He is also very grateful the hospitality shown by the Thiele Center and Department of Mathematical Sciences, Aarhus University, during that time.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Mohammad Saffari
    • 1
  • Søren Asmussen
    • 2
  • Rasoul Haji
    • 3
  1. 1.Department of Industrial EngineeringUniversity of TafreshTafreshIran
  2. 2.Department of Mathematical SciencesAarhus UniversityAarhusDenmark
  3. 3.Department of Industrial EngineeringSharif University of TechnologyTehranIran

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