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I am trying to prove Theorem 6.2 on page 127 of the book Real-Time Systems by Jane W. S. Liu:

http://www.cse.hcmut.edu.vn/~thai/books/2000%20_%20Liu-%20Real%20Time%20Systems.pdf

It is based on Early Deadline First(EDF) scheduling.

It says on the book that the proof is similar to the proof for Theorem 6.1 on page 124-126. However, I am still stuck.

Here is what I have so far:

enter image description here

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  • \$\begingroup\$ This is not the right forum for this question. I would try stackoverflow.com or math.stackexchange.com \$\endgroup\$ – mhaselup Nov 3 '20 at 5:27
  • \$\begingroup\$ from a first look I noticed something that may be wrong in the sums indices. A sum index is bound not free. So how come you are using k =/= l while k is an index in a sum and l is an index in an independent sum? \$\endgroup\$ – Paul Ghobril Nov 3 '20 at 5:49
  • \$\begingroup\$ It is just showing the tasks in the three terms in 1st eq. are mutually exclusive \$\endgroup\$ – Matt Nov 3 '20 at 6:54
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    \$\begingroup\$ @mhaselup RTOS questions are on-topic here. \$\endgroup\$ – Lundin Nov 3 '20 at 7:18
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Yhe text you quote says it is only a sufficient condition not a necessary condition. If Δ>1 then that implies that t < tΔ + any positive number. Which is exactly what you've proved.

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  • \$\begingroup\$ No, it means that if its density(Δ) <= 1 then tasks can be feasibly scheduled, so density <= 1 is a sufficient condition for feasible scheduling. What I am proving is the contrapositive: if tasks CANNOT be feasibly scheduled(deadline(s) missed) then density > 1. \$\endgroup\$ – Matt Nov 4 '20 at 0:52
  • \$\begingroup\$ The second last sentence may be misleading. I changed it to "So I prove up to..." \$\endgroup\$ – Matt Nov 4 '20 at 1:27

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