This article addresses the problem related to the reliability of path after transmitting the given amount of data with the service-level agreement cooperation in the computer communication network. The links have associated with service performance factor parameter during the data transmission, and each node is associated with the requested service performance factor. In this article, first we have considered the single objective to minimize the transmission time of the quickest path problem. An algorithm for quickest path problem has been proposed for results, and furthermore, its time complexity has been shown. The problem has been extended with bi-objective optimization of the quickest path problem, which minimizes the transmission time and hybrid logarithmic reliability. An algorithm is proposed for getting the number of efficient solutions for the quickest path problem using label-correcting algorithm. The algorithms are implemented and tested on different standard benchmark network problems provided with the set of Pareto front of the results.

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