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An Unconstrained Optimization Approach to Moment Fitting with Phase Type Distributions

Eliran Sherzer, Yehezkel Resheff, Miklos Telek

TL;DR

This work tackles the problem of fitting a large set of moments to phase-type distributions by turning a constrained PH-parameter optimization into an unconstrained problem via reparameterization. It introduces general, Coxian, and Hyper-Erlang reparameterizations that support gradient-based moment matching for PH sizes up to $n\approx 100$, achieving accurate fits for up to $l=20$ moments with relative errors typically below $0.5\%$. The framework also extends to joint moment–shape fitting and demonstrates a queueing application (PH/PH/1) using QBD to assess how higher moments influence steady-state performance. Practically, the approach enables scalable, flexible PH modeling for complex stochastic systems and provides open-source code to implement these methods.

Abstract

Phase type (PH) distributions are widely used in modeling and simulation due to their generality and analytical properties. In such settings, it is often necessary to construct a PH distribution that aligns with real-world data by matching a set of prescribed moments. Existing approaches provide either exact closed-form solutions or iterative procedures that may yield exact or approximate results. However, these methods are limited to matching a small number of moments using PH distributions with a small number of phases, or are restricted to narrow subclasses within the PH family. We address the problem of approximately fitting a larger set of given moments using potentially large PH distributions. We introduce an optimization methodology that relies on a re-parametrization of the Markovian representation, formulated in a space that enables unconstrained optimization of the moment-matching objective. This reformulation allows us to scale to significantly larger PH distributions and capture higher moments. Results on a large and diverse set of moment targets show that the proposed method is, in the vast majority of cases, capable of fitting as many as 20 moments to PH distributions with as many as 100 phases, with small relative errors on the order of under 0.5% from each target. We further demonstrate an application of the optimization framework where we search for a PH distribution that conforms not only to a given set of moments but also to a given shape. Finally, we illustrate the practical utility of this approach through a queueing application, presenting a case study that examines the influence of the i^{th} moment of the inter-arrival and service time distributions on the steady-state probabilities of the GI/GI/1 queue length.

An Unconstrained Optimization Approach to Moment Fitting with Phase Type Distributions

TL;DR

This work tackles the problem of fitting a large set of moments to phase-type distributions by turning a constrained PH-parameter optimization into an unconstrained problem via reparameterization. It introduces general, Coxian, and Hyper-Erlang reparameterizations that support gradient-based moment matching for PH sizes up to , achieving accurate fits for up to moments with relative errors typically below . The framework also extends to joint moment–shape fitting and demonstrates a queueing application (PH/PH/1) using QBD to assess how higher moments influence steady-state performance. Practically, the approach enables scalable, flexible PH modeling for complex stochastic systems and provides open-source code to implement these methods.

Abstract

Phase type (PH) distributions are widely used in modeling and simulation due to their generality and analytical properties. In such settings, it is often necessary to construct a PH distribution that aligns with real-world data by matching a set of prescribed moments. Existing approaches provide either exact closed-form solutions or iterative procedures that may yield exact or approximate results. However, these methods are limited to matching a small number of moments using PH distributions with a small number of phases, or are restricted to narrow subclasses within the PH family. We address the problem of approximately fitting a larger set of given moments using potentially large PH distributions. We introduce an optimization methodology that relies on a re-parametrization of the Markovian representation, formulated in a space that enables unconstrained optimization of the moment-matching objective. This reformulation allows us to scale to significantly larger PH distributions and capture higher moments. Results on a large and diverse set of moment targets show that the proposed method is, in the vast majority of cases, capable of fitting as many as 20 moments to PH distributions with as many as 100 phases, with small relative errors on the order of under 0.5% from each target. We further demonstrate an application of the optimization framework where we search for a PH distribution that conforms not only to a given set of moments but also to a given shape. Finally, we illustrate the practical utility of this approach through a queueing application, presenting a case study that examines the influence of the i^{th} moment of the inter-arrival and service time distributions on the steady-state probabilities of the GI/GI/1 queue length.

Paper Structure

This paper contains 23 sections, 5 theorems, 31 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature
  3. PH distributions
  4. An optimization approach to moment matching
  5. The general PH re-parametrization
  6. The Coxian re-parametrization
  7. The Hyper-Erlang re-parametrization
  8. Beyond moment matching
  9. Fitting framework
  10. Experiments
  11. Test set
  12. Sampling procedure
  13. Testset coverage
  14. Performance Evaluation
  15. Results
  16. ...and 8 more sections

Key Result

Proposition 1

The image of $\; \mathbb{R}^n \times (\mathbb{R} \setminus \{0\})^n \times \mathbb{R}^{n^2}$ under the following differentiable transformation $\pi$ is the set of valid PH distributions of size $n$ in the Markovian representation (See Eq. eq:parmetrization-1), with no zero elements. In other words, where $\text{diag}(\bm{\gamma}^2)$ is a diagonal matrix whose $i$-th diagonal element is $\gamma_i^

Figures (11)

  • Figure 1: Coxian distribution, it is referred to as CF3 structure in CUMANI
  • Figure 2: Hyper-Erlang distribution with $d_1=4$, $d_2=1$, …, $d_k=3$.
  • Figure 3: SCV, Skewness, and Kurtosis scatter.
  • Figure 4: Success rate $[\%]$ with fitting threshold $\eta=1\%$ (a fit attempt is deemed a success if all moments are within $1\%$ of their respective targets). X-axis labels describe the number of fitted moments $l$ for each bar and the maximum allowed PH size $n$ (short for $n$).
  • Figure 5: Success rate $[\%]$ with fitting threshold $\eta=0.5\%$. See caption of Figure \ref{['fig:three_images_again']} for details.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof