document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=726443>Question 726443</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.\r<BR>\n" );
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document.write( "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\r<BR>\n" );
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document.write( "Parameters<BR>\n" );
document.write( "Total number of channels in the network is c<BR>\n" );
document.write( "Sender: Sends on c_n <= c channels<BR>\n" );
document.write( "	Listens on c_n <= c channels<BR>\n" );
document.write( "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)<BR>\n" );
document.write( "p_m=1-&#8719;_(i=0)^(c_m)(1-min{c_n/(c-i),1})   >= 1-(1-c_n/c)^(c_m )           (1)<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "Question: The probability that a message is still incomplete after i (re)transmissions is then<BR>\n" );
document.write( "P[Y&#8805;i+1]=p[Y>i]=1-(1-(1-p_m )^i )^l                                      (2)<BR>\n" );
document.write( "Here is attempt<BR>\n" );
document.write( "The probability that a packet is received<BR>\n" );
document.write( " p_m<BR>\n" );
document.write( "The probability that a packet is not received<BR>\n" );
document.write( "1-p_m <BR>\n" );
document.write( "The probability that he sent it i times<BR>\n" );
document.write( "(1-p_m )^i<BR>\n" );
document.write( "The probability that he did not send it i times <BR>\n" );
document.write( "1-(1-p_m )^i<BR>\n" );
document.write( "The probability that he did not send l fragments <BR>\n" );
document.write( "(1-(1-p_m )^i )^l<BR>\n" );
document.write( "The probability that he did send l fragments in i times but that they might still be in transit <BR>\n" );
document.write( "1-(1-(1-p_m )^i )^l<BR>\n" );
document.write( "Question: Hence the expected number of packets that have to be transmitted in order to successfully transfer a message is N_*=N(p_m)<BR>\n" );
document.write( "N(p_m )&#8788;&#8721;_(i=0)^&#8734; (P[Y=i]il)<BR>\n" );
document.write( "=&#8721;_(i=0)^&#8734; ((P[Y&#8805;i]-P[Y>i])il)                                          (3)<BR>\n" );
document.write( "=&#8721;_(i=0)^&#8734;((1-(1-(1-p_m )^i )^l )l)                                     (4)<BR>\n" );
document.write( "What do you call the property P[Y=i]=(P[Y>=i]-P[Y>i])?<BR>\n" );
document.write( "What has happened to (3) to get (4). I know they have used (2)?\r<BR>\n" );
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document.write( "Thanks in advance.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #444595 by </B><A HREF=/tutors/aboutme.mpl?userid=lynnlo><B>lynnlo(4176)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lynnlo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=444595\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THIS IS TOO MUCH WORK,FOR YOU TO WANT SOMEONE TO DO \"ALL\"OF IT FOR YOU </SPAN>\n" );
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