Formal Approximations of the Transient Distributions of the M/G/1 Workload Process

TL;DR

We address computing transient distributions for Lévy-driven queues on $\mathbb{R}$, notably the M/G/1 workload process, by approximating with a finite-state Markov chain. The discretization uses a common step $Δ$, truncates the state space to $[0,M]$, and lifts the discrete distribution to a piecewise-constant density on intervals of length $Δ$, with densities written as $\tilde{μ}_k$. We derive explicit transition matrices $P$ via conditional one-jump CDFs and provide formal Wasserstein-distance error bounds that quantify the distance between the true transient and the approximation, including errors from initialization, aggregation, and truncation. Numerical experiments show fast computation times (seconds to minutes) and improved accuracy over Laplace-transform inversion methods, enabling assurances about workload and congestion during transient heavy-load periods.

Abstract

This paper calculates transient distributions of a special class of Markov processes with continuous state space and in continuous time, up to an explicit error bound. We approximate specific queues on R with one-sided Lévy input, such as the M/G/1 workload process, with a finite-state Markov chain. The transient distribution of the original process is approximated by a distribution with a density which is piecewise constant on the state space. Easy-to-calculate error bounds for the difference between the approximated and actual transient distributions are provided in the Wasserstein distance. Our method is fast: to achieve a practically useful error bound, it usually requires only a few seconds or at most minutes of computation time.

Figures (5)

• Figure 1: Bounding the Wasserstein distance
• Figure 2: Transient densities of the M/G/1 workload process started with $Q_0 = 1$ at different times $t$. The service speed is $1$, the job arrival rate is $\lambda = \frac{1}{4}$, job sizes are distributed uniformly over the interval $[1, 5]$. Discretization parameter $\Delta = \frac{1}{500}$, truncation parameter $M = 50$. On the right: the corresponding Wasserstein error bounds.
• Figure 3: Transient densities of the M/G/1 workload process started with $Q_0 = 1$ at time $t=1$. The parameters are the same as in \ref{['fig:mg1_dirac1_transdens']}. Two different discretization parameters as well as the inverse Laplace transform approach are shown.
• Figure 4: On the left: transient densities of the M/G/1 workload process at different times $t$, started with $Q_0 = 0$. The constant service speed is $1$, the job arrival rate is $\lambda = \frac{2}{5}$, and job sizes have an Erlang distribution with scale $6$ and rate $2$ (i.e. expectation 3). The discretization parameter is $\Delta = \frac{1}{100}$, and the truncation parameter is $M = 20$. On the right: the corresponding Wasserstein error bounds.
• Figure 5: Transient densities of a Lévy-driven queue started with $Q_0 = 5$ at different times $t$. The constant positive speed is $1$, the downward jump arrival rate is $\lambda = \frac{1}{3}$, and the jump sizes have a Pareto distribution with minimal value $1$ and shape parameter $\alpha = 1.5$. The discretization parameter is $\Delta = \frac{1}{100}$, the truncation parameter is $M = 55$.