Unbalanced Job Approximation using Taylor Series Expansion and Review of Performance Bounds

TL;DR

This paper introduces the Unbalanced Job Approximation (UJA), a low-cost method using Taylor series to estimate queueing network throughput, and reviews various performance bounds including the Balanced Job Bound (BJB).

Contribution

It presents the UJA method for approximating throughput in queueing networks and reviews existing bounds, extending applicability to multiple job classes and complex network configurations.

Findings

UJA with one term matches BJB throughput.

Accuracy improves with more Taylor series terms.

UJA reduces computational cost in large QNs.

Abstract

Unbalanced Job Approximation - UJA is a family of low-cost formulas to obtain the throughput of Queueing Networks - QNs with fixed rate servers using Taylor series expansion of job loadings with respect to the mean loading. UJA with one term yields the same throughput as optimistic Balanced Job Bound - BJB, which at some point exceeds the maximum asymptotic throughput. The accuracy of the estimated throughput increases with more terms in the Taylor series. UJA can be used in parametric studies by reducing the cost of solving large QNs by aggregating stations into a single Flow-Equivalent Service Center - FESCs defined by its throughput characteristic. While UJA has been extended to two classes it may be applied to more classes by job class aggregation. BJB has been extended to QNs with delay servers and multiple jobs classes by Eager and Sevcik, throughput bounds by Eager and Sevcik, Kriz, Proportional Bound - PB and Prop. Approximation Bound - PAM by Hsieh and Lam and Geometric Bound - GB by Casale et al. are reviewed.