Best effort congestion controls strive to achieve an equitable distribution of network resources among competing flows. However, fair resource allocation becomes undesirable when a bandwidth/delay sensitive application shares a bottleneck with a greedy background application. Less than Best Effort (LBE) Congestion Control Algorithms (CCA) are specially designed for background applications, which do not have strict bandwidth/delay requirements. LBE CCAs give foreground applications higher priority in resource allocation by only utilizing spare bandwidth. This can greatly improve network utility at times of congestion. We propose FlexiS– a Flexible Sender side LBE CCA. Unlike most conventional LBE CCAs, which use queue size based congestion detectors and linear rate controllers, FlexiS employs a queue trend based congestion detector and a cubic increase multiplicative decrease rate controller. We have compared FlexiS with LEDBAT and LEDBAT++. Extensive emulation and preliminary Internet tests showed that: 1) FlexiS has comparatively low impact on concurrent best effort TCP flows; 2) it scales to a wide range of available bandwidths; 3) FlexiS flows in aggregation can effectively utilize available bandwidth; 4) contending FlexiS flows can, in most cases, equally share available bandwidth; 5) FlexiS adapts to route changes quickly; and 6) it maintains low priority even when AQM algorithms or shallow buffers are deployed.