In vehicular networks, vehicles periodically exchange messages that include their positions and IDs. On the other hand, adversaries can violate the vehicles' location privacy by obtaining the vehicles' movement history from the messages. In recent studies, mix-zone-based schemes protect the vehicles' location privacy. In the schemes, vehicles utilize pseudonyms instead of their IDs in encrypted messages, and therefore adversaries cannot track them by their IDs. However, since the messages are encrypted by public keys of Road Side Units (RSUs), the vehicles can change their pseudonyms only around the RSUs. In this paper, we propose a Dynamic Mix-zone scheme considering Communication delay for Location Privacy (DMCLP). The DMCLP enables vehicles to change their pseudonyms even if there are no RSUs around them by removing necessity of RSUs from the mix-zone-based schemes. To remove the necessity, the DMCLP adopts both Identity-Based Signature (IBS) and proposed two-stage mix-zones. The IBS enables vehicles to certify the correctness of public keys by themselves. Therefore, it removes the necessity to tell the correctness to trusted authorities through RSUs. Furthermore, since RSUs guarantee connectivity of vehicular networks, influence of communication delay on pseudonym change becomes a problem without RSUs. The proposed two-stage mix-zones mitigate the influence of communication delay by removing vehicles that will cause communication delay from pseudonym change processes. Simulation results show that the proposed scheme improves update frequency of pseudonyms by 50 at most in 1000 seconds compared with the previous mix-zone-based scheme.