Determining Limit Size for API Requests

Over the 10 hours, I have collected the number of new issues that have come in. Each hour is as follows:

data = [3, 5, 4, 1, 0,
        0, 0, 1, 4, 7]

For a next time period, I want to estimate the limit size that I should use to get number of new issues that will come in.

Note

I am not interested in the mean of the data, but rather in a quantile of the data in order to determine the upper bound for what might come in.

I will using a Poisson-Gamma model to determine the issue distribution.

The 90th percentile of the predictive distribution will be used to determine the limit.

Modeling the Data

The Gamma distribution is used as the prior and the poisson_gamma function is used to calculate the posterior distribution (just another Gamma distribution).

from conjugate.distributions import Gamma
from conjugate.models import poisson_gamma, poisson_gamma_predictive

prior = Gamma.from_occurrences_in_intervals(occurrences=3, intervals=1)
posterior: Gamma = poisson_gamma(n=len(data), x_total=sum(data), prior=prior)

Determining the Limit Size from the Predictive Distribution

I have scenario where the limit needs to be determined for the next 35 minutes. Despite having measured intervals of 1 hour, the predictive distribution can be used to determine the number of issues that will come in the next 35 minutes by scaling down n by the fraction of the hour.

minutes = 35
window_size = minutes / 60
posterior_predictive = poisson_gamma_predictive(
    distribution=posterior,
    n=window_size,
)

With this predictive distribution, any statistics can be calculated. For our case, we are interested in the 90th percentile so we can use the dist attribute to access the numpy distribution which has the ppf method for the quantiles.

The limit size for the next 35 minutes is 3 issues.

percentile = 0.90
limit = posterior_predictive.dist.ppf(percentile).item()
# 3.0

Visualizing the Predictive Distributions

We will compare the prior and posterior predictive distributions to see where the limit size falls.

Tip

Use the poisson_gamma_predictive function with the prior distribution for the prior predictive distribution.

prior_predictive = poisson_gamma_predictive(
    distribution=prior,
    n=window_size,
)

We can plot the prior and posterior predictive distributions using either plot_pmf or plot_cdf methods and the picked limit.

import matplotlib.pyplot as plt

def plot_picked_limit(limit: float, percentile: float, ax: plt.Axes) -> plt.Axes:
    alpha = 0.25
    kwargs = {
        "color": "black",
    }
    x_kwargs  = {
        "marker": "x",
        "markersize": 10,
        "label": "picked limit",
    }
    line_kwargs = {
        "alpha": alpha,
        "linestyle": "--",
    }
    ax.plot([limit], [percentile], **x_kwargs, **kwargs)
    ax.plot([0, limit], [percentile, percentile], **line_kwargs, **kwargs)
    ax.plot([limit, limit], [0, percentile], **line_kwargs, **kwargs)
    return ax

max_value = limit * 2.0
ax = plt.gca()
prior_predictive.set_max_value(max_value).plot_cdf(ax=ax, label="prior")
posterior_predictive.set_max_value(max_value).plot_cdf(ax=ax, label="posterior")
plot_picked_limit(limit, percentile, ax)
padding = 0.025
ax.set(
    ylim=(0 - padding, 1 + padding),
    xlabel="Number of Issues",
    title=f"The predictive {percentile:.0%} in next {minutes} minutes",
)
ax.legend(title="Predictive Distribution")

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