Yoni Nazarathy and Gideon Weiss. The Asymptotic Variance Rate of the Output Process of Finite Capacity Queues

Abstract. It is well known that the output process of many infinite capacity birth-death queues is a Poisson process. Thus for such queues the output rate equals the asymptotic variance rate (the asymptotic rate of increase of the variance function). The situation is quite different for finite capacity systems: Here it is known that in most cases the output process is not a renewal process but rather a more complex Markov Arrival Process (MAP).

In this case we develop a formula for the asymptotic variance rate of the form lambda* + R where lambda* is the output rate and R is an expression which may be easily evaluated. The proof relies on a relationship between a class of MAPs and Markov Modulated Poisson Process (MMPP) and uses results of W. Whitt with regards to the asymptotic variance rate of MMPPs that have a birth-death structure.  

For the M/M/1/K queue, 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), R/lambda* is minimal. The situation is similar for the M/M/c/K queue and the Erlang loss system. Here also, balancing the system parameters results in a pronounced decrease in the asymptotic variance rate.  We call this phenomenon, BRAVO (the acronym stands for Balancing Reduces Asymptotic Variance of Outputs) and end with some numerical results related to BRAVO that leave some open questions.