[Paper Review] Multi-step Uniformization with Steady-State Detection in Nonstationary M/M/s Queuing Systems

Maciej Rafał Burak|arXiv (Cornell University)|Oct 3, 2014

Advanced Queuing Theory Analysis22 references4 citations

TL;DR

This paper proposes a multi-step uniformization method with adaptive steady-state detection for nonstationary M/M/s queuing systems, enhancing computational efficiency by dynamically allocating error bounds and leveraging convergence properties to detect steady states early. The approach significantly reduces computation time while maintaining strict error bounds, particularly beneficial for real-time call center scheduling under time-varying demand.

ABSTRACT

A new approach to the steady state detection in the uniformization method of solving continuous time Markov chains is introduced. The method is particularly useful in solving inhomogenous CTMC's in multiple steps, where the desired error bound of the whole solution can be distributed not proportionally to the lengths of the respective intervals, but rather in a way, that maximizes the chances of detecting a steady state. Additionally, the convergence properties of the underlying DTMC are used to further enhance the computational savings due to the steady state detection. The method is applied to the problem of modeling a Call Center using inhomogenous CTMC model of a M(t)/M(t)/s(t) queuing system.

Motivation & Objective

• To address the limitations of stationary queuing models in modeling time-varying call center workloads.
• To develop a computationally efficient method for solving inhomogeneous continuous-time Markov chains (CTMCs) in nonstationary M/M/s systems.
• To enable real-time decision-making in call centers under overload or unexpected demand spikes.
• To improve error control in multi-step uniformization by distributing error bounds based on convergence likelihood rather than interval length.
• To detect steady states early using the convergence properties of the embedded DTMC, reducing unnecessary computation.

Proposed method

• Adapts the uniformization method to inhomogeneous CTMCs by computing time-dependent generator matrices Q(t) for each interval.
• Introduces a multi-step approach where error bounds are allocated not proportionally to interval length, but based on the likelihood of steady-state convergence.
• Employs the embedded discrete-time Markov chain (DTMC) to monitor convergence and trigger early termination when steady state is detected.
• Uses adaptive truncation of Poisson probabilities via a custom class (B14) to maintain precision while minimizing computational overhead.
• Applies error estimation and adaptive step size control using embedded Runge-Kutta methods for numerical stability and accuracy.
• Enables parallelization through on-the-fly coefficient generation, supporting efficient multi-threaded execution on multi-core CPUs.

Experimental results

Research questions

• RQ1 Can steady-state detection in the uniformization method reduce computational cost in nonstationary M/M/s systems without compromising accuracy?
• RQ2 How can error bounds in multi-step uniformization be optimally distributed across time intervals to maximize early detection of steady states?
• RQ3 To what extent does leveraging the convergence behavior of the embedded DTMC improve computational efficiency in transient CTMC solutions?
• RQ4 Can the proposed method support real-time scheduling adjustments in call centers under unexpected demand or system overload?
• RQ5 What is the impact of adaptive truncation of Poisson probabilities on both accuracy and performance in large-scale queuing simulations?

Key findings

• The proposed method achieves significant computational savings by detecting steady states earlier than traditional methods, especially in long intervals with slow convergence.
• Error bounds are maintained strictly across the entire solution by dynamically allocating precision where it is most effective, not just based on interval length.
• The use of the embedded DTMC’s convergence behavior enables reliable early termination, reducing unnecessary computation in stable phases.
• The algorithm scales nearly linearly with the number of CPU cores due to low synchronization needs and efficient memory bandwidth usage, enabling fast execution on multi-core systems.
• The custom Poisson probability class (B14) ensures high-precision computation with minimal memory footprint, supporting efficient weighted sum calculations.
• The method outperforms standard uniformization in terms of computational efficiency while maintaining strict error bounds, making it suitable for real-time applications in call centers.