[Paper Review] Joint queue length distribution of multi-class, single server queues with preemptive priorities

Andrei Sleptchenko, Selen, Jori+2 more|TU/e Research Portal (Eindhoven University of Technology)|Nov 12, 2014

Advanced Queuing Theory AnalysisBusiness, Management and Accounting4 citations

TL;DR

This paper presents an exact recursive method using matrix-analytic techniques to compute the equilibrium joint queue length distribution in an M/M/1 queue with N customer classes and preemptive priorities, leveraging embedded M/G/1-type Markov processes. The method avoids truncation and infinite series, enabling fast, accurate computation of system performance metrics such as spare parts availability in under 5 seconds for 3-class systems.

ABSTRACT

In this paper we analyze an $M/M/1$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an $M/M/1$ queueing system with non-preemptive priorities between customer classes.

Motivation & Objective

• To develop an exact method for computing the joint equilibrium queue length distribution in multi-class, single-server queues with preemptive priority scheduling.
• To avoid numerical approximation errors from truncation or infinite series in existing methods for multi-class priority queues.
• To enable accurate performance evaluation in real-world applications such as spare parts logistics with repairable components.
• To extend the method to non-preemptive priority systems by adapting the embedded Markov process framework.
• To demonstrate computational efficiency and accuracy through numerical experiments on a 3-class spare parts repair system.

Proposed method

• The method uses matrix-analytic techniques to model the system as a multi-dimensional Markov process with state space N₀^N, tracking class-specific queue lengths.
• It identifies embedded Markov processes on levels where no higher-priority customers are present, which are of M/G/1-type, enabling recursive solution via first-passage probabilities.
• Equilibrium probabilities are computed recursively by analyzing excursions from each class-level, using one-step analysis to derive transition probabilities.
• The approach leverages the fact that lower-priority customers observe the system as an M/G/1 queue with 'vacations' corresponding to higher-priority service interruptions.
• For non-preemptive systems, the state space is extended to include the class of the customer currently in service, and recursive computation is adapted accordingly.
• The algorithm is implemented in Java and uses a probability mass truncation threshold ε = 10⁻⁶ to ensure accuracy while maintaining computational efficiency.

Experimental results

Research questions

• RQ1 Can the joint queue length distribution in an M/M/1 queue with N preemptive priority classes be computed exactly without truncation or infinite series?
• RQ2 How can matrix-analytic methods be applied recursively to compute equilibrium probabilities in a multi-dimensional Markov process with priority-based state transitions?
• RQ3 What is the computational performance of the proposed method in realistic spare parts logistics scenarios with multiple repairable parts?
• RQ4 How does the joint distribution enable accurate system availability analysis under different priority assignments and basestock levels?
• RQ5 Can the method be extended to non-preemptive priority systems by modifying the state description and embedded process structure?

Key findings

• The method computes the exact joint queue length distribution for N-class M/M/1 queues with preemptive priorities using recursive matrix-analytic techniques, avoiding numerical approximation.
• For a 3-class spare parts repair system, computation times ranged from 0.02 to 34.35 seconds depending on load and priority assignment, with ε = 10⁻⁶ ensuring high accuracy.
• System availability was accurately computed and maximized under different priority assignments, with values exceeding 0.9994 even at 95% utilization.
• The highest availability (0.9999) was achieved when high-priority parts were repaired first, and low-priority parts had minimal queueing.
• The method demonstrated robustness across varying utilization levels and priority rankings, with mean queue lengths and basestock levels accurately reflected in system performance.
• The extension to non-preemptive systems is feasible by including the service class in the state description and adapting the recursive computation of boundary probabilities.