The conventional wisdom is that for every bluray quality 1080p stream you want to be able to transcode separately, you need 2000 passmark points, plus some extra cpu room for any additional load you plan to place on the device (i.e. local playback). My own experience has mirrored this; the i5 vs. i7 question isn't the real question, as there are modern i5 processors with much higher passmarks than some modern i7's (U-branded chips, etc.). The clock and TDP are more important. I have two i5s and two i7s around the house and my testing has suggested that the passmark method is a good approximation (give or take a stream).
For your use case, you'd want a PC with a passmark above 4000, preferably into the 6's to give you some breathing room: that i3 should be able to handle two streams, it's kind of odd that it's not, but maybe local playback is more demanding in your case (are you using integrated graphics that might push down the CPU clock to stay under TDP?). The i5 you mention has a 7k passmark, so should provide more than enough grunt for two streams, and could probably manage three as a pure server.
My five year old i7 2600K (8k-ish passmark) can handle three streams easily, and sometimes four depending on the content. By contrast my new (as of last year) i5 nuc 4250U has a passmark below four thousand and struggles with transcoding more than one stream. But there are U branded i7's (like the one in the new i7 NUC) with lower performance than your i5.
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Topic: Core i7 or Core i5 for a jrmc server streaming video to >1 gizmo/jremote client? (Read 4073 times)