  We investigate the constraints that can be placed on the cosmic string
tension by using the current Pulsar Timing Array limits on the stochastic
gravitational wave background (SGWB). We have developed a code to compute the
spectrum of gravitational waves (GWs) based on the widely accepted one-scale
model. In its simplest form the one-scale model allows one to vary: (i) the
string tension, G\mu/c^2; (ii) the size of cosmic string loops relative to the
horizon at birth, \alpha; (iii) the spectral index of the emission spectrum, q;
(iv) the cut-off in the emission spectrum, n_*; and (v) the intercommutation
probability, p. The amplitude and slope of the spectrum in the nHz frequency
range is very sensitive to these unknown parameters. We have also investigated
the impact of more complicated scenarios with multiple initial loop sizes, in
particular the 2-\alpha models proposed in the literature and a log-normal
distribution for \alpha. We have computed the constraint on G\mu/c^2 due to the
limit on a SGWB imposed by data from the European Pulsar Timing Array. Taking
into account all the possible uncertainties in the parameters we find a
conservative upper limit of G\mu/c^2<5.3x 10^{-7} which typically occurs when
the loop production scale is close to the gravitational backreaction scale,
\alpha\approx\Gamma G\mu/c^2. Stronger limits are possible for specific values
of the parameters which typically correspond to the extremal cases \alpha\ll
\Gamma G\mu/c^2 and \alpha\gg \Gamma G\mu/c^2. This limit is less stringent
than the previously published limits which are based on cusp emission, an
approach which does not necessarily model all the possible uncertainties. We
discuss the prospects for lowering this limit by two orders of magnitude, or
even a detection of the SGWB, in the very near future in the context of the
Large European Array for Pulsars and the Square Kilometre Array.
