Bounding errors introduced by clustering of customers in closed product-form queuing networks

Product-form queuing network models have been widely used to model systems with shared resources such as computer systems (both centralized and distributed), communication networks, and flexible manufacturing systems. Closed multichain product-form networks are inherently more difficult to analyze than open networks, due to the effect of normalization. Results in workload characterization for closed networks in the literature are often for networks having special structures and only specific performance measures have been considered. In this article, we derive certain properties (insensitivity of conditional state probability distributions and fractional-linearity of Markov reward functions) for a broad class of closed multichain product-form networks. These properties are derived using the most basic now balance conditions of product-form networks. Then we show how these basic properties can be applied in obtaining error bounds when similar customers are clustered together to speed up computation.