[Paper Review] The maximum queue length for heavy tailed service times

Misja Nuyens|ArXiv.org|Aug 5, 2003

Advanced Queuing Theory Analysis10 references3 citations

TL;DR

This paper establishes that in an M/G/1 queue with heavy-tailed, log-convex service time distributions under the Foreground-Background (FB) service discipline, the tail of the maximum queue length over a busy cycle is exponentially bounded by $\rho^n$, where $\rho$ is the system load. This bound implies significantly longer buffer overflow times in unstable systems compared to FIFO, demonstrating FB's superior performance under heavy-tailed traffic.

ABSTRACT

In this paper we study the maximum queue length $M$ (in terms of the number of customers present) in a busy cycle in the M/G/1 queue. Assume that the service times have a logconvex density. For such (heavy-tailed) service-time distributions the Foreground Background service discipline is optimal. This discipline gives service to the customer(s) that have received the least amount of service so far. It is shown that under this discipline $M$ has an exponentially decreasing tail. From the behaviour of $M$ we obtain asymptotics of the maximum queue length $M(t)$ over the interval $(0,t)$ for $t o\infty$. These are applied to calculate the time to overflow of a buffer, both in stable and unstable queues.

Motivation & Objective

• To analyze the maximum queue length in a busy cycle for an M/G/1 queue with heavy-tailed service times.
• To investigate how the service discipline, specifically Foreground-Background (FB), affects the tail behavior of the maximum queue length.
• To derive an exponential upper bound on the tail probability $P(M > n)$ for the maximum queue length $M$ under the FB discipline.
• To compare buffer overflow times between FB and FIFO disciplines in both stable and unstable M/G/1 queues with heavy-tailed service times.
• To demonstrate that the FB discipline significantly delays buffer overflow in unstable systems with heavy-tailed service times.

Proposed method

• Uses a coupling argument between two M/G/1 FB queues with different service time distributions to compare maximum queue length distributions.
• Introduces an auxiliary service discipline, FB*, where the initial customer of a busy period is served only when no others are present, to derive a key stochastic inequality.
• Applies the regenerative structure of the queue length process to relate the maximum queue length over a finite interval to the busy cycle maximum.
• Employs a coupling of service times via inverse transform sampling to ensure identical behavior up to a threshold $p$, enabling stochastic dominance of queue length processes.
• Uses the bound $P(M > n) \leq \rho^n$ to derive asymptotic behavior of $M(t)$, the maximum queue length over $[0,t]$ as $t \to \infty$.
• Applies the result to unstable queues by coupling with a stable queue having a truncated and exponentially tilted version of the original service time distribution.

Experimental results

Research questions

• RQ1 What is the tail behavior of the maximum queue length in an M/G/1 queue with heavy-tailed, log-convex service times under the FB discipline?
• RQ2 How does the FB discipline compare to FIFO in terms of buffer overflow time for heavy-tailed service time distributions?
• RQ3 Can an exponential upper bound on the tail of the maximum queue length be derived that is independent of the specific form of the service time distribution?
• RQ4 How does the time to buffer overflow scale in unstable M/G/1 FB queues with heavy-tailed service times?
• RQ5 What is the impact of the log-convexity assumption on the performance bounds of the FB discipline?

Key findings

• The tail of the maximum queue length $M$ in an M/G/1 FB queue with log-convex service time density satisfies $P(M > n) \leq \rho^n$, where $\rho$ is the system load.
• This bound is independent of the specific form of the service time distribution, applying to both heavy-tailed (e.g., Pareto, Weibull with $\beta < 1$) and light-tailed (e.g., gamma) distributions with log-convex densities.
• For unstable queues with $\rho = \infty$, the time to buffer overflow can be extremely large if the arrival rate is low, as demonstrated by a numerical example with $\lambda = 0.01$ and $t = 10^{40}$ yielding overflow probability less than 0.02.
• The coupling argument shows that $M_F(t) \leq_{st} M_G(t) + K_p(t)$, where $K_p(t)$ is a Poisson process with rate $\lambda(1-p)$, enabling comparison between systems with different service time distributions.
• In the example with a Pareto service density $f(x) = 1/(x+1)^2$, the time to overflow for a buffer of size $d=1000$ is shown to exceed $10^{40}$ with high probability when $\lambda = 0.01$, due to the stability of a truncated and exponentially tilted version of the service time distribution.
• The result confirms that the FB discipline can dramatically extend buffer overflow times in unstable systems with heavy-tailed service times, outperforming FIFO significantly.