Histograms and summaries

Histograms and summaries are more complex metric types. Not only does a single histogram or summary create a multitude of time series, it is also more difficult to use these metric types correctly. This section helps you to pick and configure the appropriate metric type for your use case.

Library support

First of all, check the library support for histograms and summaries.

Some libraries support only one of the two types, or they support summaries only in a limited fashion (lacking quantile calculation).

Count and sum of observations

Histograms and summaries both sample observations, typically request durations or response sizes. They track the number of observations and the sum of the observed values, allowing you to calculate the average of the observed values. Note that the number of observations (showing up in Prometheus as a time series with a _count suffix) is inherently a counter (as described above, it only goes up). The sum of observations (showing up as a time series with a _sum suffix) behaves like a counter, too, as long as there are no negative observations. Obviously, request durations or response sizes are never negative. In principle, however, you can use summaries and histograms to observe negative values (e.g. temperatures in centigrade). In that case, the sum of observations can go down, so you cannot apply rate() to it anymore.

To calculate the average request duration during the last 5 minutes from a histogram or summary called http_request_duration_seconds, use the following expression:

  rate(http_request_duration_seconds_sum[5m])
/
  rate(http_request_duration_seconds_count[5m])

Apdex score

A straight-forward use of histograms (but not summaries) is to count observations falling into particular buckets of observation values.

You might have an SLA to serve 95% of requests within 300ms. In that case, configure a histogram to have a bucket with an upper limit of 0.3 seconds. You can then directly express the relative amount of requests served within 300ms and easily alert if the value drops below 0.95. The following expression calculates it by job for the requests served in the last 5 minutes. The request durations were collected with a histogram called http_request_duration_seconds.

  sum(rate(http_request_duration_seconds_bucket{le="0.3"}[5m])) by (job)
/
  sum(rate(http_request_duration_seconds_count[5m])) by (job)

You can approximate the well-known Apdex score in a similar way. Configure a bucket with the target request duration as the upper bound and another bucket with the tolerated request duration (usually 4 times the target request duration) as the upper bound. Example: The target request duration is 300ms. The tolerable request duration is 1.2s. The following expression yields the Apdex score for each job over the last 5 minutes:

(
  sum(rate(http_request_duration_seconds_bucket{le="0.3"}[5m])) by (job)
+
  sum(rate(http_request_duration_seconds_bucket{le="1.2"}[5m])) by (job)
) / 2 / sum(rate(http_request_duration_seconds_count[5m])) by (job)

Note that we divide the sum of both buckets. The reason is that the histogram buckets are cumulative. The le="0.3" bucket is also contained in the le="1.2" bucket; dividing it by 2 corrects for that.

The calculation does not exactly match the traditional Apdex score, as it includes errors in the satisified and tolerable parts of the calculation.

Quantiles

You can use both summaries and histograms to calculate so-called φ-quantiles, where 0 ≤ φ ≤ 1. The φ-quantile is the observation value that ranks at number φ*N among the N observations. Examples for φ-quantiles: The 0.5-quantile is known as the median. The 0.95-quantile is the 95th percentile.

The essential difference between summaries and histograms is that summaries calculate streaming φ-quantiles on the client side and expose them directly, while histograms expose bucketed observation counts and the calculation of quantiles from the buckets of a histogram happens on the server side using the histogram_quantile() function.

The two approaches have a number of different implications:

Histogram Summary
Required configuration Pick buckets suitable for the expected range of observed values. Pick desired φ-quantiles and sliding window. Other φ-quantiles and sliding windows cannot be calculated later.
Client performance Observations are very cheap as they only need to increment counters. Observations are expensive due to the streaming quantile calculation.
Server performance The server has to calculate quantiles. You can use recording rules should the ad-hoc calculation take too long (e.g. in a large dashboard). Low server-side cost.
Number of time series (in addition to the _sum and _count series) One time series per configured bucket. One time series per configured quantile.
Quantile error (see below for details) Error is limited in the dimension of observed values by the width of the relevant bucket. Error is limited in the dimension of φ by a configurable value.
Specification of φ-quantile and sliding time-window Ad-hoc with Prometheus expressions. Preconfigured by the client.
Aggregation Ad-hoc with Prometheus expressions. In general not aggregatable.

Note the importance of the last item in the table. Let us return to the SLA of serving 95% of requests within 300ms. This time, you do not want to display the percentage of requests served within 300ms, but instead the 95th percentile, i.e. the request duration within which you have served 95% of requests. To do that, you can either configure a summary with a 0.95-quantile and (for example) a 5-minute decay time, or you configure a histogram with a few buckets around the 300ms mark, e.g. {le="0.1"}, {le="0.2"}, {le="0.3"}, and {le="0.45"}. If your service runs replicated with a number of instances, you will collect request durations from every single one of them, and then you want to aggregate everything into an overall 95th percentile. However, aggregating the precomputed quantiles from a summary rarely makes sense. In this particular case, averaging the quantiles yields statistically nonsensical values.

avg(http_request_duration_seconds{quantile="0.95"}) // BAD!

Using histograms, the aggregation is perfectly possible with the histogram_quantile() function.

histogram_quantile(0.95, sum(rate(http_request_duration_seconds_bucket[5m])) by (le)) // GOOD.

Furthermore, should your SLA change and you now want to plot the 90th percentile, or you want to take into account the last 10 minutes instead of the last 5 minutes, you only have to adjust the expression above and you do not need to reconfigure the clients.

Errors of quantile estimation

Quantiles, whether calculated client-side or server-side, are estimated. It is important to understand the errors of that estimation.

Continuing the histogram example from above, imagine your usual request durations are almost all very close to 220ms, or in other words, if you could plot the "true" histogram, you would see a very sharp spike at 220ms. In the Prometheus histogram metric as configured above, almost all observations, and therefore also the 95th percentile, will fall into the bucket labeled {le="0.3"}, i.e. the bucket from 200ms to 300ms. The histogram implementation guarantees that the true 95th percentile is somewhere between 200ms and 300ms. To return a single value (rather than an interval), it applies linear interpolation, which yields 295ms in this case. The calculated quantile gives you the impression that you are close to breaking the SLA, but in reality, the 95th percentile is a tiny bit above 220ms, a quite comfortable distance to your SLA.

Next step in our thought experiment: A change in backend routing adds a fixed amount of 100ms to all request durations. Now the request duration has its sharp spike at 320ms and almost all observations will fall into the bucket from 300ms to 450ms. The 95th percentile is calculated to be 442.5ms, although the correct value is close to 320ms. While you are only a tiny bit outside of your SLA, the calculated 95th quantile looks much worse.

A summary would have had no problem calculating the correct percentile value in both cases, at least if it uses an appropriate algorithm on the client side (like the one used by the Go client). Unfortunately, you cannot use a summary if you need to aggregate the observations from a number of instances.

Luckily, due to your appropriate choice of bucket boundaries, even in this contrived example of very sharp spikes in the distribution of observed values, the histogram was able to identify correctly if you were within or outside of your SLA. Also, the closer the actual value of the quantile is to our SLA (or in other words, the value we are actually most interested in), the more accurate the calculated value becomes.

Let us now modify the experiment once more. In the new setup, the distributions of request durations has a spike at 150ms, but it is not quite as sharp as before and only comprises 90% of the observations. 10% of the observations are evenly spread out in a long tail between 150ms and 450ms. With that distribution, the 95th percentile happens to be exactly at our SLA of 300ms. With the histogram, the calculated value is accurate, as the value of the 95th percentile happens to coincide with one of the bucket boundaries. Even slightly different values would still be accurate as the (contrived) even distribution within the relevant buckets is exactly what the linear interpolation within a bucket assumes.

The error of the quantile reported by a summary gets more interesting now. The error of the quantile in a summary is configured in the dimension of φ. In our case we might have configured 0.95±0.01, i.e. the calculated value will be between the 94th and 96th percentile. The 94th quantile with the distribution described above is 270ms, the 96th quantile is 330ms. The calculated value of the 95th percentile reported by the summary can be anywhere in the interval between 270ms and 330ms, which unfortunately is all the difference between clearly within the SLA vs. clearly outside the SLA.

The bottom line is: If you use a summary, you control the error in the dimension of φ. If you use a histogram, you control the error in the dimension of the observed value (via choosing the appropriate bucket layout). With a broad distribution, small changes in φ result in large deviations in the observed value. With a sharp distribution, a small interval of observed values covers a large interval of φ.

Two rules of thumb:

  1. If you need to aggregate, choose histograms.

  2. Otherwise, choose a histogram if you have an idea of the range and distribution of values that will be observed. Choose a summary if you need an accurate quantile, no matter what the range and distribution of the values is.

What can I do if my client library does not support the metric type I need?

Implement it! Code contributions are welcome. In general, we expect histograms to be more urgently needed than summaries. Histograms are also easier to implement in a client library, so we recommend to implement histograms first, if in doubt.