Exact analysis of transient behavior of finite-capacity MAP-driven queues

Abstract

This paper studies the workload distribution of a finite-capacity queue driven by a spectrally one-sided Markov additive process (MAP). Our main result provides the Laplace-Stieltjes transform of the workload at an exponentially distributed time, thereby uniquely characterizing its transient distribution. The proposed approach combines several decompositions with established fluctuation-theoretic results for spectrally one-sided Lévy processes. For the special case of Markov-modulated compound Poisson input, we additionally derive results for the idle time and the cumulative amount of lost work. We conclude this paper with a series of numerical experiments.

Figures (4)

• Figure 1: The three scenarios in the decomposition in the proof of Lemma \ref{['lemma: decomp chi(x)']}, with $K=4$ and $x=2$: ( a) hitting $0$ first, ( b) hitting $K$ first, and ( c) killing first. The vertical dashed line corresponds to a sample of the killing time $T_\beta$.
• Figure 2: Results for instance 1. Left panels: mean ( a) and variance ( c) of the workload; right panels: probability of an empty system ( b) and a full system ( d). All quantities are shown as functions of time, with initial workload $x=0$, for different initial background states.
• Figure 3: Results for instance 2. Mean ( a) and variance ( b) of the workload, as functions of time, with initial workload $x=3$, for different initial background states. The purple line corresponds to $d=1$ and $\varphi(\alpha)=\alpha + \frac{1}{2}\alpha^2$; the dashed lines are the mean and variance of the corresponding workload in stationarity.
• Figure 4: Results for instance 3. Mean ( a) and variance ( b) of the workload, and probability of an empty system ( c), as functions of time, with initial workload $x=K=4$, for different initial background states.

Theorems & Definitions (38)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Remark 1
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7