Tips to handle colleagues' response time when SLA deadlines are tight?

[Robert Werner]

Imagine a scenario where you are a JSD Agent and are supporting a number of corporate customers.  Further imagine one morning you arrive at work and suddenly within minutes 5 different Priority 1 requests arrive from different customers.  You quickly respond to them, thus satisfying the Time to First Response clause in your SLAs.  You also ably create JIRA tickets for your engineering colleagues to resolve the issues.  Now you have 5 different clocks counting down for the Time to Corrective Action clause in your SLAs.

Here's my question: With such pressure on you, and very little time to ensure that all 5 issues are resolved in time, what best practices have you followed to successfully ensure that all are resolved in time?

Clearly you don't want to micro-manage anyone, because that'll only make things worse but you also don't want even one colleague to not attend to what is clearly a pressing problem.  Perhaps there are some JSD plugins or external software you've used to assist with the management of such a situation?

Robert