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Quality of service (QoS) is an important network service requirement to support various real time applications such as video on demand, remote-meeting systems, etc. While QoS parameters are specified by the applications it appears QoS support at routing level would be more appropriate. Here we consider an on-demand source routing modification to the Dijkstra's algorithm, which supports multi-parameter routing to find an optimal path according to the given optimality condition. The modified algorithm is analyzed with respect to its computational requirements and we prove that this algorithm finds a loop-free optimal path, if it exists.