Queueing Systems

, Volume 54, Issue 1, pp 55–78 | Cite as

M/M/1 Queueing systems with inventory

  • Maike Schwarz
  • Cornelia Sauer
  • Hans Daduna
  • Rafal Kulik
  • Ryszard Szekli


We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system.


Queueing systems Inventory systems Performance analysis Inventory policy Queue lengths Service level Inventory level Stationary distribution Convex ordering 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Maike Schwarz
    • 1
  • Cornelia Sauer
    • 1
  • Hans Daduna
    • 1
  • Rafal Kulik
    • 2
  • Ryszard Szekli
    • 2
  1. 1.Department of Mathematics, Center of Mathematical Statistics and Stochastic ProcessesUniversity of HamburgHamburgGermany
  2. 2.University of Wroclaw, Mathematical InstituteWroclawPoland

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