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**1 - 2**of**2**### Makespan Minimization of Task Graphs with Random Task Running Times

- In Interconnection Networks and Mapping and Scheduling Parallel Computations, D. F. Hsu et al. (Eds.), AMS, DIMACS series
, 1994

"... . The problem of scheduling a set of tasks on two parallel and identical processors is considered. The executions of tasks are constrained by precedence relations. The running times of the tasks are independent random variables with a common exponential distribution. The goal of scheduling is to min ..."

Abstract
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. The problem of scheduling a set of tasks on two parallel and identical processors is considered. The executions of tasks are constrained by precedence relations. The running times of the tasks are independent random variables with a common exponential distribution. The goal of scheduling is to minimize the makespan, i.e. the maximum task completion time. A simple optimal preemptive policy is proven to stochastically minimize the makespan when the precedence graph belongs to a class of forest-cut graphs. 1. Introduction Parallel programs are usually represented by task graphs which are directed acyclic graphs where vertices represent tasks and arcs represent precedence relations between tasks. The executions of these tasks have to satisfy these precedence constraints in such a way that a task can start execution only when all its predecessor tasks have completed execution. For any given task graph, the scheduling problem consists in assigning tasks to a set of processors in such a wa...

### Scheduling with Limited Processor Availability

"... In scheduling theory the basic model assumes that all machines are continuously available for processing throughout the planning horizon. This assumption might be justified in some cases but it does not apply if certain maintenance requirements, breakdowns or other constraints that cause the machine ..."

Abstract
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In scheduling theory the basic model assumes that all machines are continuously available for processing throughout the planning horizon. This assumption might be justified in some cases but it does not apply if certain maintenance requirements, breakdowns or other constraints that cause the machines not to be available for processing have to be considered. In this chapter we discuss results related to deterministic scheduling problems where machines are not continuously available for processing. Examples of such constraints can be found in many areas. Limited availabilities of machines may result from pre-schedules which exist mainly because most of the real world resources planning problems are dynamic. A natural approach to cope with a dynamic environment is to trigger a new planning horizon when the changes in the data justify it. However, due to many necessities, as process preparation for instance, it is mandatory to take results of earlier plans as fixed which obviously limits availability of resources for any subsequent plan. Consider e.g. ERP (Enterprise Resource Planning) production planning systems