### Abstract

The random graph model of parallel computation introduced by Gelenbe et al. depends on three parameters: n, the number of tasks (vertices); F, the common distribution of T_{1},…, T_{n}, the task processing times, and p = p_{n}, the probability for a given i < j that task i must be completed before task j is started. The total processing time is R_{n}, the maximum sum of T_{i}'s along directed paths of the graph. We study the large n behavior of R_{n} when np_{n} grows sublinearly but superlogarithmically, the regime where the longest directed path contains about enp_{n} tasks. For an exponential (mean one) F, we prove that R_{n} is about 4np_{n}. The “discrepancy” between 4 and e is a large deviation effect. Related results are obtained when np_{n} grows exactly logarithmically and when F is not exponential, but has a tail which decays (at least) exponentially fast. © 1994 John Wiley L Sons, Inc.

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

Pages (from-to) | 361-376 |

Number of pages | 16 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 47 |

Issue number | 3 |

DOIs | |

State | Published - 1994 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*47*(3), 361-376. https://doi.org/10.1002/cpa.3160470307

**Speed of parallel processing for random task graphs.** / Isopi, Marco; Newman, Charles.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 47, no. 3, pp. 361-376. https://doi.org/10.1002/cpa.3160470307

}

TY - JOUR

T1 - Speed of parallel processing for random task graphs

AU - Isopi, Marco

AU - Newman, Charles

PY - 1994

Y1 - 1994

N2 - The random graph model of parallel computation introduced by Gelenbe et al. depends on three parameters: n, the number of tasks (vertices); F, the common distribution of T1,…, Tn, the task processing times, and p = pn, the probability for a given i < j that task i must be completed before task j is started. The total processing time is Rn, the maximum sum of Ti's along directed paths of the graph. We study the large n behavior of Rn when npn grows sublinearly but superlogarithmically, the regime where the longest directed path contains about enpn tasks. For an exponential (mean one) F, we prove that Rn is about 4npn. The “discrepancy” between 4 and e is a large deviation effect. Related results are obtained when npn grows exactly logarithmically and when F is not exponential, but has a tail which decays (at least) exponentially fast. © 1994 John Wiley L Sons, Inc.

AB - The random graph model of parallel computation introduced by Gelenbe et al. depends on three parameters: n, the number of tasks (vertices); F, the common distribution of T1,…, Tn, the task processing times, and p = pn, the probability for a given i < j that task i must be completed before task j is started. The total processing time is Rn, the maximum sum of Ti's along directed paths of the graph. We study the large n behavior of Rn when npn grows sublinearly but superlogarithmically, the regime where the longest directed path contains about enpn tasks. For an exponential (mean one) F, we prove that Rn is about 4npn. The “discrepancy” between 4 and e is a large deviation effect. Related results are obtained when npn grows exactly logarithmically and when F is not exponential, but has a tail which decays (at least) exponentially fast. © 1994 John Wiley L Sons, Inc.

UR - http://www.scopus.com/inward/record.url?scp=84990706843&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990706843&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160470307

DO - 10.1002/cpa.3160470307

M3 - Article

VL - 47

SP - 361

EP - 376

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -