Service Desk has stopped to pull its email from the configured Mail Channel, then, no tickets are getting created by email.
This issue is occurring because the values from com.atlassian.jira.internal.mail.services.MailCleanerJobRunner, com.atlassian.jira.internal.mail.services.MailProcessorJobRunner or com.atlassian.jira.internal.mail.services.MailPullerJobRunner (JIRA Administration > System > Scheduler Details) are not updated as they should be (It's showing the next_run to a past value)
You can double check in your Scheduler Details configuration (JIRA Administration > System > Scheduler Details) to see if com.atlassian.jira.internal.mail.services.MailCleanerJobRunner, com.atlassian.jira.internal.mail.services.MailProcessorJobRunn or com.atlassian.jira.internal.mail.services.MailPullerJobRunne are set with old values for the next_run.
As these values have to be updated and the next_run should be a value in a brief future, your emails won't be processed by Service Desk if you're being affected by this issue. The following documentation has additional details about Service Desk Scheduled Jobs:
This issue is occuring when Service Desk faces any communication problem with its database.
As you'll have to modify these values directly in the database, a simple restart will force JIRA to update these values.