I've worked in environments that had similar, poor cultures regarding IT helpdesk/servicedesk. I have personal experience improving those cultures and I'll outline below my advice on the subject.
these 20-30 really make the work unenjoyable...How would you deal with those intrusive customers...?
To answer your question: Don't take it personally. You are an IT professional, a paid employee. Your job is not to be a phone-answering-machine, but to coordinate multiple complex issues across a business.
Importantly, as a service role, it's easy to forget you are not the last-stop. Who does IT call when they have issues? As you experience issues in your job, you need to reach out to your teammates who support you (as appropriate): Management, HR, coworkers, etc.
Ultimately, the right person to resolve this is the person responsible for IT operations. If that person is you, be excited! You get to mold your job into something you enjoy—not everyone has this privilege. If you hate the idea of that, you have the wrong job.
The more often they call, the more unwilling I am to pick up their phone or call back once I am finished with my current task.
Is this an emotional response or are there negative consequences (disciplinary action from management) to them calling a lot? If there are no consequences don't be worried about this. This is simply poor culture and can mostly be resolved. There are always hard-to-work-with people. Once you realize this, you can stop emotionally investing in these people and move on.
From what you described, it seems there is a poor helpdesk/servicedesk culture.
The first thing you can do is be consistent. Build a routine that works and stick to it. If a customer has an issue with your routine, realize that you resolved your problem and that the customer is complaining about their problem (their poor culture/expectations of you).
Second, I would identify all the issues you are facing, what attempts you have made to resolve them, and what you suggest the resolution should be (with professional sources). Take this to your management and get their responses (emotional investment) to all the issues.
Finally, with the clear understanding from management, build a plan from their feedback and implement it. Your customers may complain, and management may make changes or change their mind completely. Make the changes to your plan and continue—don't let this discourage you, it's completely normal—but still discouraging.
If management does not invest, or makes changes to the plan that are unsustainable, realize that this problem is un-resolvable. At that point, it becomes about your engagement; are you willing to support management's poor decisions?
If management supports you, make sure to enjoy yourself, you've accomplished something!
On here, there are a lot of good—and bad—answers on what you can do or change. My suggestion is take all three "stakeholders" into account when identifying needed changes. The stakeholders being you, the business, and the customer.
I've seen a lot of IT departments that design things in their (selfish) favor. Making themselves hard to contact, allowances for long turnaround times, etc. These often punish the customer and management allows this because they are clueless and/or helpless.
On the contrary side, I've seen IT departments that have little say in how they operate. Without the ability to improve their department, they are buried in inefficiency.
Here are some suggestions to improve your department:
- Implement a limited-availability window during a slow time period, such as Friday afternoons, to work on issues affecting your own department. This gives you time to improve your department without worrying about break-fix issues.
- Identify common issues and implement complete fixes or automate their resolution. Reducing common issues reduces call volume. I went from ~60 calls a day to ~8 over a period of four years by taking time to implement full solutions instead of workarounds.
- Display your helpdesk procedure at the primary location where your customers interact with you. Such as on the webpage for your ticket system or as a recording on your helpdesk phone, etc. Having your customers know how to contact you, what to expect, and how to appropriately escalate their issues, goes a long way to improving your interactions with the customers—and allows you to clearly show to management when customers aren't following the procedure.