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

Yoni Nazarathy, Gideon Weiss

Research output: Contribution to journalArticlepeer-review


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 *+σvi 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, σvi\λ* 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.

Original languageEnglish
Pages (from-to)135-156
Number of pages22
JournalQueueing Systems
Issue number2
StatePublished - Jun 2008

Bibliographical note

Funding Information:
Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


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