Yoni Nazarathy and Gideon Weiss, The University of Haifa.

We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form lambda* + sum v_i where lambda* 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 a rather surprising phenomena: When the system is balanced, i.e. the arrival and service rates are equal, 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.

[home]