A massive MIMO system, represented by a base station with hundreds of antennas, is capable of spatially multiplexing many devices and thus naturally suited to serve dense crowds of wireless devices in emerging applications, such as machine-type communications. Crowd scenarios pose new challenges in the pilot-based acquisition of channel state information and call for pilot access protocols that match the intermittent pattern of device activity. A joint pilot assignment and data transmission protocol based on random access is proposed in this paper for the uplink of a massive MIMO system. The protocol relies on the averaging across multiple transmission slots of the pilot collision events that result from the random access process. We derive new uplink sum rate expressions that take pilot collisions, intermittent device activity, and interference into account. Simplified bounds are obtained and used to optimize the device activation probability and pilot length. A performance analysis indicates how performance scales as a function of the number of antennas and the transmission slot duration.