### Abstract

We consider a problem motivated by the desire to provide flexible, rate-based, quality of service guarantees for packets sent over input queued switches and switch networks. Our focus is solving a type of online traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of transmissions in which only whole packets are sent. We prove worst-case bounds on the additional buffer use, which we call backlog, that results from using such an approximation. We first consider the N × N, input queued, crossbar switch. Our main result is an online packet-scheduling algorithm using no speedup that guarantees backlog at most (N + 1)^{2}/4 packets at each input port and each output port. Upper bounds on worst-case backlog have been proved for the case of constant fluid schedules, such as the N^{2} - 2N + 2 bound of Chang, Chen, and Huang (INFOCOM, 2000). Our main result for the crossbar switch is the first, to our knowledge, to bound backlog in terms of switch size N for arbitrary, time-varying fluid schedules, without using speedup. Our main result for Banyan networks is an exact characterization of the speedup required to maintain bounded backlog, in terms of polytopes derived from the network topology.

Original language | English (US) |
---|---|

Pages (from-to) | 1374-1386 |

Number of pages | 13 |

Journal | IEEE/ACM Transactions on Networking |

Volume | 14 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Combinatorics
- Graph theory
- Network calculus
- Packet-switching
- Scheduling

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Hardware and Architecture
- Information Systems

### Cite this

*IEEE/ACM Transactions on Networking*,

*14*(6), 1374-1386. https://doi.org/10.1109/TNET.2006.886320

**Approximating fluid schedules in crossbar packet-switches and Banyan networks.** / Rosenblum, Michael Aaron; Caramanis, Constantine; Goemans, Michel X.; Tarokh, Vahid.

Research output: Contribution to journal › Article

*IEEE/ACM Transactions on Networking*, vol. 14, no. 6, pp. 1374-1386. https://doi.org/10.1109/TNET.2006.886320

}

TY - JOUR

T1 - Approximating fluid schedules in crossbar packet-switches and Banyan networks

AU - Rosenblum, Michael Aaron

AU - Caramanis, Constantine

AU - Goemans, Michel X.

AU - Tarokh, Vahid

PY - 2006/12

Y1 - 2006/12

N2 - We consider a problem motivated by the desire to provide flexible, rate-based, quality of service guarantees for packets sent over input queued switches and switch networks. Our focus is solving a type of online traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of transmissions in which only whole packets are sent. We prove worst-case bounds on the additional buffer use, which we call backlog, that results from using such an approximation. We first consider the N × N, input queued, crossbar switch. Our main result is an online packet-scheduling algorithm using no speedup that guarantees backlog at most (N + 1)2/4 packets at each input port and each output port. Upper bounds on worst-case backlog have been proved for the case of constant fluid schedules, such as the N2 - 2N + 2 bound of Chang, Chen, and Huang (INFOCOM, 2000). Our main result for the crossbar switch is the first, to our knowledge, to bound backlog in terms of switch size N for arbitrary, time-varying fluid schedules, without using speedup. Our main result for Banyan networks is an exact characterization of the speedup required to maintain bounded backlog, in terms of polytopes derived from the network topology.

AB - We consider a problem motivated by the desire to provide flexible, rate-based, quality of service guarantees for packets sent over input queued switches and switch networks. Our focus is solving a type of online traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of transmissions in which only whole packets are sent. We prove worst-case bounds on the additional buffer use, which we call backlog, that results from using such an approximation. We first consider the N × N, input queued, crossbar switch. Our main result is an online packet-scheduling algorithm using no speedup that guarantees backlog at most (N + 1)2/4 packets at each input port and each output port. Upper bounds on worst-case backlog have been proved for the case of constant fluid schedules, such as the N2 - 2N + 2 bound of Chang, Chen, and Huang (INFOCOM, 2000). Our main result for the crossbar switch is the first, to our knowledge, to bound backlog in terms of switch size N for arbitrary, time-varying fluid schedules, without using speedup. Our main result for Banyan networks is an exact characterization of the speedup required to maintain bounded backlog, in terms of polytopes derived from the network topology.

KW - Combinatorics

KW - Graph theory

KW - Network calculus

KW - Packet-switching

KW - Scheduling

UR - http://www.scopus.com/inward/record.url?scp=33947211736&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947211736&partnerID=8YFLogxK

U2 - 10.1109/TNET.2006.886320

DO - 10.1109/TNET.2006.886320

M3 - Article

AN - SCOPUS:33947211736

VL - 14

SP - 1374

EP - 1386

JO - IEEE/ACM Transactions on Networking

JF - IEEE/ACM Transactions on Networking

SN - 1063-6692

IS - 6

ER -