Queueing Systems

, 59:135 | Cite as

The asymptotic variance rate of the output process of finite capacity birth-death queues

  • Yoni Nazarathy
  • Gideon Weiss


We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ *+∑v i where λ * is the rate of outputs and v i are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of v i are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity.

In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal, \(\frac{\sum v_{i}}{\lambda^{*}}\) is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced.


Queueing theory Loss systems M/M/1/K MAP Asymptotic variance rate BRAVO 

Mathematics Subject Classification (2000)

60J27 60K25 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of StatisticsThe University of HaifaMount CarmelIsrael

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