In Heterogeneous computing is possible to allocate best processor in given set processor for each task i.e in the way if parent and child had same processor, in that mean time if parent input has large it takes more time to execute so child has till parent to finish. In that parent and child had different processor, for the critical path to achieve in best way?
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As per my knowledge, the allocation of processors to tasks in distributed computing environment is an optimization problem. You can use various optimization techniques (like evolutionary algorithms) to solve this problem.
Finding a best resource allocation solution for a DAG is an optimization problem this is hard to obtain in polynomial time due to its NP-complete nature. There exist several works that try to formulate the problem and solve it by a heuristic algorithm. You may have a look at the paper "Mapping Workflow Resource Requests for Bandwidth Efficiency in Data Centers", which tries to minimize the network bandwidth allocated for a DAG. You will need to adapt the objective function to your purpose, i.e., minimize the processing time.
Finding the best resource allocation solution for a DAG is an optimization problem this is hard to obtain in polynomial time due to its NP-complete nature. There exist several works that try to formulate the problem and solve it by a heuristic algorithm. You may have a look at the paper "Mapping Workflow Resource Requests for Bandwidth Efficiency in Data Centers", which tries to minimize the network bandwidth allocated for a DAG. You will need to adapt the objective function to your purpose, i.e., minimize the processing time.
Optimal DAG scheduling on an heterogeneous environment is an NP-complete problem. In practice you may want to consider optimizing an objective function using some heuristics. For instance one may consider the makespan another person may minimize the cost given that resource is charged. Usually communication cost is consider to be zero when two consecutive tasks are scheduled on the same processor/machine since it doesn't require the network transfer.
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