Markovian Arrival Process Parameter Estimation of Quasi-birth-death Queueing Systems with Utilization Data

2026-07-02

AI Summary

This work proposes a parameter estimation method based on the Expectation–Maximization (EM) algorithm for Markovian Arrival Process (MAP)-driven Quasi-Birth–Death (QBD) queueing systems, tailored to realistic scenarios where only coarse-grained data such as system utilization are available. Within a maximum likelihood framework, the approach infers sufficient statistics—including sojourn times, phase transitions, and service dynamics—underlying the hidden states directly from utilization time series. To the best of our knowledge, this is the first method capable of fully estimating MAP-QBD model parameters using solely utilization data. The study further introduces an innovative use of the Akaike Information Criterion (AIC) to automatically select the number of MAP phases, thereby mitigating overfitting. Experimental results demonstrate that the method accurately recovers both arrival and service parameters, offering a practical performance modeling tool for real-world systems lacking fine-grained event logs.

Abstract

Parameter estimation for queueing systems is commonly performed using inter-arrival times, waiting times, or queue-length observations. However, such detailed observations are often unavailable in practical computer systems, where utilization data, such as CPU utilization, is much easier to collect. Utilization data provides only the fraction of time during which the system is busy within each monitoring interval, while the exact arrivals, services, phase transitions, and system states in unobservable periods remain hidden. This paper proposes an expectation-maximization (EM) algorithm for estimating the parameters of Markovian arrival process (MAP)-driven quasi-birth-death (QBD) queueing systems from utilization data. The proposed method formulates the underlying queueing dynamics as a QBD process and derives the expected sufficient statistics for sojourn times, phase transitions, arrivals, and services over both observable and unobservable intervals. These expectations are then used to iteratively update the MAP and service parameters under the maximum likelihood framework. In addition, Akaike's information criterion is introduced to select the appropriate number of MAP phases and mitigate overfitting. The proposed framework enables MAP-based queueing parameter estimation from incomplete utilization observations and provides a practical modeling approach for systems where detailed event-level measurements are difficult to obtain.

Problem

• Markovian Arrival Process
• Quasi-birth-death
• Parameter Estimation
• Utilization Data
• Queueing Systems

Innovation

• Markovian Arrival Process
• Quasi-birth-death process
• Expectation-Maximization algorithm
• Utilization data
• Parameter estimation