## Abstract

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation penalty is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a “local search”, where we use the problem structure in a novel manner. The second is a one-coordinate “global search”, used in coordinate pivot iteration. We then merge these two heuristics into a unified “local–global” heuristic, and numerically illustrate the effectiveness of this heuristic.

Original language | English |
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Pages (from-to) | 71-102 |

Number of pages | 32 |

Journal | Mathematical Methods of Operations Research |

Volume | 86 |

Issue number | 1 |

DOIs | |

State | Published - 1 Aug 2017 |

Externally published | Yes |

### Bibliographical note

Funding Information:We thank Hai Vu and Moshe Haviv for useful discussions and advice. We are grateful to two anonymous reviewers for their helpful comments. We thank The Australia-Israel Scientific Exchange Foundation (AISEF) for supporting Liron Ravner’s visit to The University of Queensland. Yoni Nazarathy’s research is supported by ARC Grants DE130100291 and DP130100156.

Publisher Copyright:

© 2017, Springer-Verlag Berlin Heidelberg.

## Keywords

- Global optimization
- Road traffic
- Scheduling

## ASJC Scopus subject areas

- Software
- Mathematics (all)
- Management Science and Operations Research