Packet Transmission Problem, PTP, And Job Scheduling Problem

Packet Transmission Problem Ptp And Job Scheduling Problems Jsp Ar

Packet Transmission Problem (PTP) and Job Scheduling Problems (JSP) are known NP-complete problems. It is also known that any NP-complete problem can be transformed into another NP-complete problem within polynomial time. Question: How would you transform the PTP to JSP? Following are the descriptions of PTP and JSP.

Packet Transmission Problem (PTP)

There is a data packet that needs to be transmitted to all n sites on the network. Data packet cannot be duplicated by any sites, and each site should receive the data packet only once. The time it takes to transmit this data packet from site i to site j is T(i,j). PTP is to choose the routing so that this data packet can be transmitted to all sites with the least amount of time.

Job Scheduling Problems (JSP)

We have a set of n jobs with the amount of time they need to complete, t1, t2, …, tn, the deadline they need to be completed by, d1, d2, …, dn, and a penalty incurred if the job is not completed by the deadline, p1, p2, …, pn. JSP attempts to order this set of n jobs to incur the smallest penalty.

Paper For Above instruction

Transforming the Packet Transmission Problem (PTP) into the Job Scheduling Problem (JSP) involves establishing an equivalence between the processes of network data dissemination and job scheduling with deadlines and penalties. Both PTP and JSP are NP-complete, implying that solving one efficiently would inherently solve the other, provided the transformation preserves problem constraints and objectives. The key challenge in this transformation lies in mapping the routing and transmission timing considerations of PTP onto the scheduling with deadlines and penalties structure of JSP.

At the core of this transformation is conceptualizing each network site within the PTP as a 'job' in the JSP framework. Specifically, in the PTP, the goal is to find an optimized routing sequence that transmits a data packet to all sites, minimizing overall transmission time. This routing sequence can be viewed as an ordered sequence of 'jobs' that must be completed within certain timelines, akin to scheduling jobs with specific deadlines and penalties for delays, as in JSP.

To initiate the transformation, consider the set of sites in PTP as corresponding to the set of jobs in JSP. Each site i in PTP translates to job i in JSP with processing time t_i, which is analogous to the transmission time required for the packet to reach site i, starting from an initial source node. The transmission time T(i,j) between sites i and j in PTP corresponds to the transition or switch-over times in scheduling. We can define the processing time t_i for each job i as the minimal total transmission time required to reach site i in the routing sequence, beginning from an initial source node treated as a virtual start point.

The deadlines d_i in JSP can be associated with the maximum tolerable transmission times for each site in PTP, which ensures that the packet reaches the site within its acceptable waiting period. The penalty p_i then models the cost incurred if a site does not receive the packet within the deadline, mirroring network failure penalties or Quality of Service (QoS) violations.

To map this fully, we set the following correspondences:

  • Sites in PTP correspond to jobs in JSP.
  • Transmission times T(i,j) correspond to switch-over times or setup costs between jobs, reflecting routing transitions.
  • The total route that minimizes transmission time in PTP corresponds to the job schedule that minimizes total penalty in JSP.
  • Transmission deadlines d_i correspond to job deadlines, determining the latest permissible transmission time for each site.
  • Penalties p_i represent the costs associated with not meeting the deadlines, similar to job penalties.

The transformation involves constructing a JSP instance where the processing times, deadlines, and penalties are derived from the network's transmission times and desired delivery timetables. Then, solving this JSP yields an ordering that, when mapped back, indicates the routing sequence for PTP that minimizes overall transmission time, fulfilling the objective of the original PTP.

Therefore, the process of transforming PTP into JSP encompasses translating network nodes and edge times into scheduling jobs with processing times, deadlines, and penalties, establishing an equivalence between routing optimization and job scheduling under deadlines with penalties. This polynomial-time reduction demonstrates the NP-completeness connection between the two problems and facilitates leveraging existing scheduling algorithms to address network transmission challenges.

References

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