How to plot the outage probability?

Questions
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1

How to plot the outage probability (f) of the gamma function in empirical distribution (by simulation) and theoretical distribution in the same figure for comparison purpose

f = 1 - Gamma[ m, (m ((2^(2 R) - 1)/S)^(1/n))/ t] (Gamma[m, (m ((2^(2 R) - 1)/S)^(1/n))/t]/Gamma[m]^2)
Plot[f, {S, 0, 7}]

we assume that R=1, n=3, m=1, t=1

2

This should work:

f=1-Gamma[m,(m ((2^(2 R)-1)/S)^(1/n))/t] (Gamma[m,(m ((2^(2 R)-1)/S)^(1/n))/t]/Gamma[m]^2)
Plot[f/.{m->1,t->1,n->3,R->1},{S,0,7}]

You have to substiture values for m,t,n,R otherwise you can’t plot it.

3

Thanks for your interest, but my enquiry is about the comparison between analytical results with simulation results such as the below graph as an example:

picture.jpg
picture.jpg357×258 48.9 KB

4

What about something like

f[m_, S_] := 
 With[{ R = 1, n = 3, t = 1}, 
  1 - Gamma[
     m, (m ((2^(2 R) - 1)/S)^(1/n))/
      t] (Gamma[m, (m ((2^(2 R) - 1)/S)^(1/n))/t]/Gamma[m]^2)]

data = Table[{S, f[#, S] + RandomReal[{-1, 1}]/100}, {S, 0.01, 
      7, .1}] & /@ {1, 2, 3};

Show[
 Plot[Evaluate[f[#, S] & /@ {1, 2, 3}], {S, 0, 7}],
 ListPlot[data]
 ]

3066plot.png
3066plot.png360×225 11 KB

5

In the probability speak, the function f is known as a SurvivalFunction. We can define the probability distribution, describing the outage characteristics (either a duration, or an onset, you are not being specific) as

In[80]:= f = (FullSimplify[#1, S > 0] & )[
FunctionExpand[With[{R = 1, n = 3, m = 1, t = 1},
1 - Gamma[m, (m*((2^(2*R) - 1)/S)^(1/n))/t]*(Gamma[m, (m*((2^(2*R) - 1)/S)^(1/n))/t]/Gamma[m]^2)]]]

Out[80]= 1 - E^(-((2*3^(1/3))/S^(1/3)))

In[81]:= odist =
ProbabilityDistribution[{"SF", f}, {S, 0, Infinity}];

We can now sample from this distribution using RandomVariate:

sample = RandomVariate[dist, 10^6];

The distribution has heavy-tail:

In[52]:= Probability[x > 1000, x \[Distributed] sample] // N

Out[52]= 0.24759

Here we plot a histogram of the data, displaying the logarithm of the probability density function:

hist = Histogram[sample, {0, 1000, 1}, "LogPDF"];
plot = Plot[Log@PDF[TruncatedDistribution[{0, 1000}, dist], S] // Evaluate, {S, 0.1, 1000}, PlotRange -> All, PlotStyle -> Thick, Exclusions -> None];
Show[hist, plot]

Here is a screen-shot:

Capture.PNG
Capture.PNG757×802 11 KB

6

Hi Oleksandr,

In fact, I am willing to inquire if I can plot the following function with scattered data instead of histogram graph as I want to compare this with the analytical graph below

f = Product[ 1 - (Gamma[m, (m ((2.25)/10^(S/10))^(1/n))/\[CapitalOmega]] Gamma[ m, (m ((1.5)/10^(S/10) )^(1/n))/\[CapitalOmega]])/ Gamma[m]^2, {i, 1, M}] /. Subscript -> Part;
ticks = {10^-4, 10^-3, 10^-2, 10^-1, 1};
LogPlot[f, {S, 0, 40}, PlotRange -> {10^-4, 1}, Ticks -> {Automatic, ticks}, GridLines -> Automatic, Frame -> True]

Assuming that :

m=1, n=1, M=3, \[CapitalOmega] = 2

plot.jpg
plot.jpg342×216 19.7 KB