A Recursive Approximation Technique for a Combined Source Queueing Model

The model studied in this paper captures the combined effects of finite and infinite source traffic-often used to model interactive and batch traffic, respectively-when they contend for a single server resource. The finite source traffic is modeled by heterogeneous finite sources, the infinite source traffic by a stationary Poisson process, and the single server is assumed to have exponentially distributed service times with distinct service rates for the different customer types. All customers share a common queue and are serviced in FIFO order. A special case of this model where the finite sources are identical combines two fundamental and widely used models (the repairman and models) in a natural manner. Regardless of the homogeneous or heterogeneous nature of the finite sources, the combined source model is not product form due to the realistic assumption that service rates are distinct for different customer types (batch and interactive traffic typically have different CPU processing requirements). In this paper, we show how to recursively calculate all mean quantities of interest in an approximate but quite accurate manner for the general heterogeneous model. The accuracy of the recursive technique is established in part by contrasting the approximate solution to simulation results for a wide parameter range, and in part by studying the asymptotic behavior of the approximation.