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8 The Mathematics of Scheduling. 8.1 The Basic Elements of Scheduling 8.2 Directed Graphs (Digraphs) 8.3 Scheduling with Priority Lists 8.4 The Decreasing-Time Algorithm 8.5 Critical Paths 8.6 The Critical-Path Algorithm 8.7 Scheduling with Independent Tasks.

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8 The Mathematics of Scheduling

8.1 The Basic Elements of Scheduling

8.2 Directed Graphs (Digraphs)

8.3 Scheduling with Priority Lists

8.4 The Decreasing-Time Algorithm

8.5 Critical Paths

8.6 The Critical-Path Algorithm

8.7 Scheduling with Independent Tasks

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The Decreasing-Time Algorithm

Our first attempt to find a good priority list is to formalize what is a seeminglysensible and often used strategy: Do the longer jobs first and leave the shorter jobsfor last. In terms of priority lists, this strategy is implemented by creating a priority list in which the tasks are listed in decreasing order of processing times–longest first, second longest next, and so on. (When there are two or more taskswith equal processing times,we order them randomly.)

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The Decreasing-Time Algorithm

A priority list in which the tasks are listed in decreasing order of processingtimes is called, not surprisingly, a decreasing-time priority list, and the process ofcreating a schedule using a decreasing-time priority list is called the decreasing-time algorithm.

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Example 8.9 Building That Dream Home on Mars: Part 4

The figure shows, once again, the project digraph for the Martian Habitat Unitbuilding project. To use the decreasing-time algorithm we first prioritize the 15tasks in a decreasing-time priority list.

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Example 8.9 Building That Dream Home on Mars: Part 4

Decreasing-time priority list:AD(8), AP(7), IW(7), AW(6), FW(6), AF(5), IF(5), ID(5), IP(4), PL(4), HU(4), PU(3), PD(3), EU(2), IC(1)

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Example 8.9 Building That Dream Home on Mars: Part 4

Using the decreasing-time algorithm withN = 2 processors, we get thisschedule with project finishing time Fin = 42hours.

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Weaknesses of this Algorithm

When looking at the finishing time under the decreasing-time algorithm, onecan’t help but feel disappointed. The sensible idea of prioritizing the longer jobsahead of the shorter jobs turned out to be somewhat of a dud–at least in thisexample! What went wrong? If we work our way backward from the end, we cansee that we made a bad choice at T = 33hours. At this point there were threeready tasks [PD(3), EU(2), and IC(1)], and both processors were available.

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Weaknesses of this Algorithm

Based on the decreasing-time priority list, we chose the two “longer” tasks, PD(3)and EU(2)ahead of the short task, IC(1). This was a bad move! IC(1) isa much more criticaltask than the other two because we can’t start task FW(6) until we finish task IC(1). Had we looked ahead at some of the tasks for whichPD, EU, and IC are precedent tasks, we might have noticed this.

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Weaknesses of this Algorithm

An even more blatant example of the weakness of the decreasing-time algorithm occurs at the very start of this schedule when the algorithm fails to takeinto account that task AF(5) should have a very high priority. Why? AF(5) isone of the two tasks that must be finished before IF(5) can be started, and IF(5) must be finished before IW(7) can be started, which must be finished beforeIP(4) and ID(5) can be started, and so on down the line.