In case of in-lecture quizzes, is it unreasonable to fail students who are late or absent? I am teaching a large undergraduate class this semester. One day a week (same day every week) we have a quiz. The syllabus doesn't specify at what moment of the class period the quiz will be given. It does say though that students who missed the quiz aren't allowed to make it up. So, I've had the following situation since the semester started. First and second week I gave the quiz at the end of the class. During week 3 I already had a few students coming more than 20 minutes late, but early enough not to miss the 15 min quiz, which I gave again at the end. So, this week (week 4) I gave the quiz 10 minutes after the start of the class. The actual time of the quiz wasn't announced in advance. And yes, I've had many students coming late and some came when the quiz was already over. These students couldn't have 15 minutes like others if any time at all. Now these students are claiming that I needed to tell them in advance at what time the quiz was supposed to be or emphasize that it can be at any time, or otherwise they follow the pattern. 

All students who came in late were the ones who were late the previous week except one student.

I repeat that it's a big lecture hall, and I don't want to take notes when a specific student actually started his or her quiz and add 15 minutes to that time. Also, after quiz I want all of them to concentrate on the class material. And I don't want to give it at the end all the time as some students will come in late, and that disrupts the flow of a lecture, distracts other students and generally erodes class morale.

I drop quite a lot of lowest quiz grades at the end of the semester to meet those cases when students do have to be absent. So, a good student's grade won't be affected if he or she will miss a couple of quizzes.

I am currently facing lots of negative emails from students who were late and didn't get the grade they would get if they had the same time as everybody else. Do you think it's me being unreasonable? Their main argument is that it should be clearly stated when a quiz is given or emphasized that it can be given at any time, which I find unnecessary.
Do you think it's me being unreasonable?

Sadly, yes. I see two problems here:

You said in a comment that the policy of &quot;the quiz can happen at any time&quot; was not articulated at all. You gave the quiz near the end of the class a couple of times, and the students naturally assumed that that's when you'll always have it. Now you are trying to argue legalistically that since you didn't say when the quiz will occur, you were reserving the right to hold it at any time. However, in my opinion if you are going to argue based on legalisms, the burden is on you to dot all your i's and cross all your t's and make sure that you gave your students such amazingly clear information that they couldn't even imagine arguing with you.
The thing to keep in mind here is that you are an educator, not a prosecutor trying to entrap a criminal in court. The students deserve to have clearly articulated rules so that they can focus their time and energy on the material; in particular, if there's a quiz, they deserve to know the precise time when it will be held, or at the very least to have an explicit announcement that the quiz can be held at variable times. Overall, your approach to dealing with the students who came late as I'm understanding it from your question seems to me to place you in an adversarial, combative position relative to your students, which is not where you want to be as a teacher - it can only serve to distract from your educational objectives and provides for a poor learning environment for your students.

A second problem is that I'm getting a strong feeling from reading your question that you're trying to use the quiz as a crowd control mechanism, which feels wrong to me (and partially as a result has gotten you into the current messy situation). You've devised an elaborate strategy that consists of holding a quiz at a randomly selected time during the class as a solution to the problem of making students come to class, and come on time. The problem of students coming in late may be a very real concern, but random quizzes are simply an inappropriate and ineffective way to address it (and one that can potentially be perceived quite negatively by the students).
A quiz is a form of assessment, and is a legitimate device to use for that purpose, but let's keep things in their right place: deal with the problem of disruptive late-coming students in the appropriate way, and have your assessment in the way that makes sense from an educational standpoint, whether it be quizzes, exams, homework, or even mandatory class attendance if that's important to you and your institution's policies allow it.