Should one suffer at all?

The standard utilitarian view of happiness seems to be 'pleasure and the absence of pain' (J.S. Mill). Whether the good is understood in terms of pleasure or preference satisfaction, a happy life is one in which there are more good and less suffering in it. But is the happiest life one in which there are no suffering at all? Or does one's life as a whole go better if there are some sufferings in it? If there is such thing as "good suffering" (i.e. suffering that is in itself undesirable but enhances one's overall happiness), it seems that utilitarians should not only promote it but also make sure that everyone will experience it in their life course. In this presentation I would like to further develop this idea and examine possible criticisms.