Analysis of Markovian Arrivals and Service with Applications to Intermittent Overload

TL;DR

A characterization of mean queue length in the MAMS system is derived, with explicit bounds for all arrival and service chains at all loads, using a somewhat novel framework based around the concepts of relative arrivals and relative completions.

Abstract

In many important real-world queueing settings, arrival and service rates fluctuate over time. We consider the MAMS system, where the arrival and service rates each vary according to an arbitrary finite-state Markov chain, allowing intermittent overload to be modeled. This model has been extensively studied, and we derive results matching those found in the literature via a somewhat novel framework. We derive a characterization of mean queue length in the MAMS system, with explicit bounds for all arrival and service chains at all loads, using our new framework. Our bounds are tight in heavy traffic. We prove even stronger bounds for the important special case of two-level arrivals with intermittent overload. Our framework is based around the concepts of relative arrivals and relative completions, which have previously been used in studying the MAMS system, under different names. These quantities allow us to tractably capture the transient correlational effect of the arrival and service processes on the mean queue length.

Figures (5)

• Figure 1: A time-varying arrival process with intermittent overload. The system spends $5/6$ of its time on average with a low arrival rate of $\lambda_L=0.2$, and $1/6$ of its time with a high arrival rate of $\lambda_H=2$. The average arrival rate is $\lambda = 0.5$, which is below the service rate of $\mu=1$.
• Figure 2: The arrival chain of the two-level arrival process.
• Figure 3: Expected arrivals by time $t$, starting in arrival states $H$ and $L$, and in the steady-state $Y$. Relative arrivals $\Delta(H)$ and $\Delta(L)$ are the limiting differences between these functions. Setting: $\lambda_H = 2, \lambda_L=0.2, \alpha_H = 0.5, \alpha_L=0.1, \mu=1, \lambda = 0.5$.
• Figure 4: Simulation and bounds for mean queue length $\mathbbm{E}[Q]$ in two-level system with intermittent overload, under varying switching rates $\alpha_H, \alpha_L$. Setting: $\lambda_H = 2, \lambda_L=0.2, \alpha_H = 5 \alpha_L, \mu=1, \rho=0.5$. Overall switching rate $\alpha := (\alpha_H + \alpha_L)/2$. Bounds given in \ref{['cor:two-level-e-q-bounds']}: The upper bound which is tight for small $\alpha$ is \ref{['eq:two-level-e-q-upper-slow']}, for large $\alpha$ is \ref{['eq:two-level-e-q-upper-fast']}, and the lower bound is \ref{['eq:two-level-e-q-lower']}. Simulated $10^8$ arrivals.
• Figure 5: Simulation and bounds for mean queue length $\mathbbm{E}[Q]$ in two-level system with intermittent overload, under varying load $\rho$, for three different values of $\alpha_L$. Setting: $\lambda_H = 4 \rho, \lambda_L=0.4 \rho, \alpha_H = 5\alpha_L, \mu=1$. Bounds given in \ref{['cor:two-level-e-q-bounds']}: Upper bound is minimum of \ref{['eq:two-level-e-q-upper-fast']} and \ref{['eq:two-level-e-q-upper-slow']}. Simulated $10^8$ arrivals.

Theorems & Definitions (46)

• Lemma 3.1
• Definition 3.1
• Definition 3.2
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.1
• proof
• Corollary 3.1