The present disclosure discloses a connection control method, which includes that: a first network element receives an Up Link (UL) S1 message used for establishing or resuming an S1 interface related to User Equipment (UE) from a second network element; and the first network element sends a Downlink (DL) S1 message to the second network element, the DL S1 message including at least one of first indication information and second indication information. The present disclosure also discloses a connection control method and device, a service processing method and device, a processor and a storage medium.