Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

TL;DR

This paper introduces MISOCP formulations to convexify queueing formulas, enabling efficient solutions to complex stochastic location problems with congestion, and demonstrates their broad applicability and superior performance over existing methods.

Contribution

It develops novel MISOCP models for queueing metrics and applies them to large-scale congestion-aware location problems, outperforming current approaches.

Findings

Efficiently solves large benchmark problems with new MISOCP formulations.

Outperforms existing methods in solving congestion-related location problems.

Demonstrates broad applicability to queue-based optimization problems.

Abstract

Mixed-Integer Second-Order Cone Programs (MISOCPs) form a nice class of mixed-inter convex programs, which can be solved very efficiently due to the recent advances in optimization solvers. Our paper bridges the gap between modeling a class of optimization problems and using MISOCP solvers. It is shown how various performance metrics of M/G/1 queues can be molded by different MISOCPs. To motivate our method practically, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service networks. Four different MISOCPs are developed and compared on sets of benchmark test problems. The new formulations efficiently solve large-size test problems, which cannot be solved by the best existing method. Then, the general applicability of our method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality.