At early stages the dynamics of cosmic string networks is expected to be influenced by an excessive production of small loops at the scales of initial conditions l _ { min } .
To understand the late time behavior we propose a very simple analytical model of strings with a non-scaling population of loops .
The complicated non-linear dynamics is described by only a single parameter N \sim 2 / ( 1 - C ( l _ { min } ) ) where C ( l ) is a correlation function of the string tangent vectors .
The model predicts an appearance of two new length scales : the coherence length \xi \sim t / N ^ { 2 } and the cross-correlation length \chi \sim t / N .
At the onset of evolution N \sim 10 and at late times N is expected to grow logarithmically due to cosmological stretching and emission of small loops .
The very late time evolution might be modified further when the gravitational back-reaction scale grows larger than l _ { min } .