I thought my network was experiencing outages. I collected bandwidth (Mbps) measurements every 30 min.
. . .
What would be some ways to analyze the signal to see the time scale of outages?
An "outage" is a random event. Therefore, looking for regularity in outages might be hinting at something more widely already known.
The answers to the questions in the comments are very much dependent to the objectives. But I will try to make this a bit more constructive.
The CDF presented is over 1400 instances at 30 minute intervals and it therefore gives you the probability of sustained download at a particular speed over 1400*30 minutes or about 30 days (If the horizontal axis depicts the index of the measurement). If you were asking if your line would sustain a download speed over the next 15 minutes, the answer would be "I don't know", because you have not measured anything more frequently than 15 minutes.
On their own, wavelets and discrete fourier analysis cannot tell you much. They can tell you that a particular segment of the signal suddenly requires many more components to describe it and this usually happens at discontinuities but this helps in recognising a "spike" which you can already do here by thresholding (for example).
Rather, if you suspect that there is some regularity in the outages and you want to estimate it, then what you can do is some form of time series decomposition.
You can run autocorrelation for example and this will already give you a handle on periodicity. Or a full blown discrete fourier transform (or even wavelet transform) to determine periodic components more accurately.
By determining the Trend and Seasonality components of your signal, you can then create "filters" to discover areas where it veers off its usual course. (And then of course, having discovered the timestamps of those events, you can analyse (or try to predict) the time that passess between them and see if there is anything useful in there).
But notice here, we always need a time reference. For example, when you assess Trend, a trend over what time scale? This will then determine how often you need to observe the signal.
Hope this helps.