Reference Card

1 Probability and Related Topics:

Notation

\(O_i\) denotes sample point \(i\).
\(\mathcal{S}\) denotes the sample space.
\(P\{O_{i}\}\) denotes the probability of \(O_i\).
\(f(x)\) denotes the probability that the random variable \(X\) takes on the value \(x\).
\(E(X)\) denotes the expected value of the random variable \(X\).
\(P(A)\) denotes the probability of event \(A\).
\(P(A|B)\) denotes the conditional probability of event \(A\) given event \(B\).
\(\lambda\) denotes the arrival rate.
\(\mu\) denotes the service rate.

Important Facts/Results

\( P(A \mbox{ or } B \mbox{ or both}) = P(A) + P(B) - P(A \mbox{ and } B) \)

\( P(A \mbox{ and } B) = P(A|B) \cdot P(B) \)

Assuming certain assumptions are satisfied, the expected size of a queueing system is given by \(\lambda / (\mu - \lambda)\).

Assuming certain assumptions are satisfied, the expected time in a queueing system is given by \(1 / (\mu - \lambda)\).

2 File Sharing and Related Topics:

Notation

\(N\) denotes the number of machines that want the file
\(F\) denotes the size of the file (in bits)
\(u_S\) denotes the source's upload rate
\(u_i\) denotes the upload rate of machine \(i\)
\(d_i\) denotes the download rate of machine \(i\)
\(d_{\mbox{min}} = \min\{d_1, d_2, \ldots, d_N\}\)
\(D\) denotes the total time

Important Facts/Results

\( D_{\text{Clint-Server}} = \max \left\{ \frac{NF}{u_S}, \frac{F}{d_{\mbox{min}}} \right\} \)

\( D_{\text{Peer-to-Peer}} \approx \max \left\{ \frac{F}{u_S}, \frac{F}{d_{\mbox{min}}}, \frac{NF}{u_\mbox{total}} \right\} \)

3 Some HTML Elements:

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