We analyze alternating traffic crossing a narrow one-lane bridge on a two-lane road. Once a car begins to cross the bridge in one direction, arriving cars from the other direction must wait, forming a queue, until all the arrivals in the first direction finish crossing the bridge. Such a situation can often be observed when road-maintenance work is being carried out. Cars are assumed to arrive at the queues according to independent Poisson processes and to cross the bridge in a constant time. In addition, once cars join the queue, each car needs a constant starting delay, before starting to cross the bridge. We model the situation where a signal controls the traffic so that the signal gives a priority to one direction as long as a new car from the same direction arrives in a fixed time. For this model, we get a closed form for the first two moments of the waiting time of cars arriving at the bridge, and then numerically obtain Pareto optimal solutions of holding times to minimize the mean waiting time and its standard deviation.