How to Fit Queueing Models Using Only CPU Utilization Data

Most queueing theory assumes you have detailed arrival and service time logs. In real production systems, you're lucky to get CPU utilization snapshots—the fraction of time the system was busy in each monitoring interval. The exact arrivals, service completions, and internal state transitions stay hidden.

A new paper on arXiv proposes an expectation-maximization (EM) algorithm that fits Markovian arrival process (MAP) parameters for quasi-birth-death (QBD) queueing systems using only that utilization data. No inter-arrival times, no waiting time records, no queue-length traces. Just busy fractions.

Why Utilization Data Is All You Get

Detailed event-level measurements are expensive or impossible to collect in production. CPU utilization, on the other hand, is available from every OS, every hypervisor, every container runtime. The paper treats the unobservable underlying queue dynamics as a QBD process and derives expected sufficient statistics for sojourn times, phase transitions, arrivals, and services over both observable and unobservable intervals. These expectations feed into iterative maximum likelihood updates of the MAP and service parameters.

The EM Algorithm That Connects the Dots

The key insight: even though you never see individual arrivals or services, the utilization signal carries enough information to recover the underlying rate parameters—if you model the hidden process correctly. The derived EM steps update phase-transition probabilities, arrival rates, and service rates simultaneously. This is a concrete, implementable algorithm, not a hand-wavy framework.

Picking the Right Number of Phases with AIC

Overfitting is a real risk when you have so few observables. The paper introduces Akaike's information criterion (AIC) to select the appropriate number of MAP phases, balancing model fit against complexity. That makes the method usable in practice: you can tune the model to your data without guessing the phase count.

This gives operators a practical way to model system behavior from the most basic telemetry available—CPU utilization alone—and opens the door to applying MAP/QBD models in environments that previously couldn't support them.

Source: Markovian Arrival Process Parameter Estimation of Quasi-birth-death Queueing Systems with Utilization Data
Domain: arxiv.org