SOLUTION: I have very basic knowledge on probability. I am reading a paper and came across finding the probability of receiving a message in a computer network. The problem: What is the n

Question 726443: I have very basic knowledge on probability. I am reading a paper and came across finding the probability of receiving a message in a computer network.
The problem: What is the number of packets that have to be transmitted in order to successfully transfer a message in a computer network without an attacker/jammer
Parameters
Total number of channels in the network is c
Sender: Sends on c_n <= c channels
Listens on c_n <= c channels
Question: Show that the probability that a packet is received is given by the following formula (on this one I have no idea what is happening)
p_m=1-∏_(i=0)^(c_m)(1-min{c_n/(c-i),1}) >= 1-(1-c_n/c)^(c_m ) (1)
A packet consists of l fragments. A message is said to have been transmitted if all the l packets have been received by the receiver.
Let Y be the number of times that the sender is required to retransmit (some packets may be lost on the way or jammed by an attacker, so they have to be transmitted a number of times) the sequence of packets in order to successfully transfer the message.
Question: The probability that a message is still incomplete after i (re)transmissions is then
P[Y≥i+1]=p[Y>i]=1-(1-(1-p_m )^i )^l (2)
Here is attempt
The probability that a packet is received
p_m
The probability that a packet is not received
1-p_m
The probability that he sent it i times
(1-p_m )^i
The probability that he did not send it i times
1-(1-p_m )^i
The probability that he did not send l fragments
(1-(1-p_m )^i )^l
The probability that he did send l fragments in i times but that they might still be in transit
1-(1-(1-p_m )^i )^l
Question: Hence the expected number of packets that have to be transmitted in order to successfully transfer a message is N_*=N(p_m)
N(p_m )≔∑_(i=0)^∞ (P[Y=i]il)
=∑_(i=0)^∞ ((P[Y≥i]-P[Y>i])il) (3)
=∑_(i=0)^∞((1-(1-(1-p_m )^i )^l )l) (4)
What do you call the property P[Y=i]=(P[Y>=i]-P[Y>i])?
What has happened to (3) to get (4). I know they have used (2)?
Thanks in advance.

Answer by lynnlo(4176) (Show Source): You can put this solution on YOUR website!
THIS IS TOO MUCH WORK,FOR YOU TO WANT SOMEONE TO DO "ALL"OF IT FOR YOU
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