, Volume 59, Issue 2, pp 135-156
Date: 06 Aug 2008

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

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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.

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