Im trying to understand this inequality but cant understand why it is always true. Its a part of proof which I understand most of it.

R* is the oprimtal rate of transmitting when there is average load on the network.

M is the cache size of each user.

K is tha amount of user.

1/N is the probabilty of asking each file from server(thre are N files in server).

R(with the line above it) is the optimal rate when the load is average(like R*) but the number of different requests of files is j.

M,N,k are the same

and the probability in the picture is the probability of j different request of users from the server. enter image description here

my question is why this inequality is true?


the idea is when j is small there is small amount of files that will satisfy all users and the rate(load) will be low. when the probability is 1/N so almsost every user will ask diffrent file and then the load(R*) will be high.

but still, why is it always true? enter image description here

link for article Coded Caching with Nonuniform Demands



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