This paper seeks traffic signal control policies which maximise long-run network throughput when demand is within network capacity and also when demand is beyond network capacity. The paper considers smooth versions of two well-known responsive traffic signal control policies: the P0 policy (see Smith, 1979b, 1980 or Smith et al., 2015) and the maxpressure or MP policy (see Varaiya, 2013a). The paper gives two simple example networks where both of these smooth responsive traffic control policies give rise to decreasing throughput for increasing demands; when these demands are beyond the network capacity. The paper then suggests a simple modification of the two responsive traffic control policies and shows that the two modified policies do maximise the long run throughput of the two signal-controlled networks; both when a steady input demand D is within network capacity and also when a steady demand exceeds network capacity. The modified version of P0 is as follows: for a certain positive number K, forD⩽Kuse responsive policyP0,and forD>Kuse the green-times generated byP0whenD=K. Thus in this modification the green-time proportions generated by responsive P0 are frozen as demand passes through K; and remain fixed for all demands D which exceed K. The paper shows that the above modification when applied to a continuous version of MP also maximises throughput in both networks for all values of the steady demand D. Finally the paper considers a signal controlled network with route choice. It is shown that if routes are chosen by travellers in their own best interests then the continuous version of MP does not always maximise throughput, even if the constant input demand D is within the network capacity, but P0 does always maximise throughput, even when the constant input demand D is beyond the network capacity.@en