  Motivated by the classification of non-equilibrium steady states suggested by
R.K.P. Zia and B. Schmittmann in J. Stat. Mech. P07012 (2007), we propose to
measure the violation of the 'detailed balance' criterion by the $p$-norm
($||K||_p$) of the matrix formed by the probability currents. Its asymptotic
analysis, for the totally asymmetric exclusion process, motivates the
definition of a 'distance' from equilibrium $K^*$ obtained for $p=1$. In
addition, we show that the latter quantity and the average activity $A^*$ are
both related to the probability distribution of the entropy production.
Finally, considering the open-ASEP and open-ZRP, we show that the current of
particles gives an exact measure of the violation of 'detailed balance'.
