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  1. E. Kuumola, J. Resing and J. Virtamo, Joint Distribution of Instantaneous and Averaged Queue Length in an M/M/1/K System, in Proceedings of the 15th Teletraffic Congress Specialist Seminar ``Internet Traffic Engineering and Traffic Management'', pp. 58-67, 2002, Wurzburg, Germany (pdf)(bib)
    Abstract: We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary differential equations is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three different numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the efficiency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situation which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.