In Distributed Computing Systems When Coordinator Node Fails the Other Nodes of the System Need to Elect Another Leader or Coordinator to Run the System Smoothly. the Bully Algorithm Is a Classical Approach For Electing a Leader In a Distributed Computing System. the Improvement of the Bully Algorithm and Other Modify Algorithms Requiring Less Time Complexity and Minimum Message Passing Over Network. the Performance Analysis of Bully and Other Modify Election Algorithm With Our New Proposed Algorithm Election Administration Would Be Appropriate to Determine Which Algorithm Performs Better Than the Others. This Paper Represents the Normal Case of Election In Which We Compare and Discuss Our Approach With Original Bully Algorithm and Existing Modified Versions of Bully Algorithm.