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Online Distributed Queue Length Estimation

Aditya Bhaskara, Sreenivas Gollapudi, Sungjin Im, Kostas Kollias, Kamesh Munagala

TL;DR

This work studies how to monitor the length of a FIFO queue using decentralized one-way pings from packets that only know their own position in the queue. It introduces two policy families: Poa, where packets ping only on arrival with a height-dependent probability, and Pico, where packets ping continuously with a rate inversely proportional to their height and waiting time; the server reconstructs the queue height from received pings. The authors prove near-optimal error guarantees for Poa under constant-rate and Poisson departures, provide matching lower bounds, and show Poa is insufficient for highly bursty or adversarial arrivals/departures, motivating Pico, which achieves an \\epsilon-reconstruction with a bounded area-based argument and explicit bounds on ping counts. The results extend to networks of queues and illuminate the trade-offs between communication (pings) and estimation accuracy, offering practical guidance for distributed congestion monitoring in traffic and similar systems. Overall, the paper advances decentralized monitoring by characterizing when simple arrival-only pings suffice and when continuous, adaptive pinging is necessary, with concrete quantitative guarantees.

Abstract

Queue length monitoring is a commonly arising problem in numerous applications such as queue management systems, scheduling, and traffic monitoring. Motivated by such applications, we formulate a queue monitoring problem, where there is a FIFO queue with arbitrary arrivals and departures, and a server needs to monitor the length of a queue by using decentralized pings from packets in the queue. Packets can send pings informing the server about the number of packets ahead of them in the queue. Via novel online policies and lower bounds, we tightly characterize the trade-off between the number of pings sent and the accuracy of the server's real time estimates. Our work studies the trade-off under various arrival and departure processes, including constant-rate, Poisson, and adversarial processes.

Online Distributed Queue Length Estimation

TL;DR

This work studies how to monitor the length of a FIFO queue using decentralized one-way pings from packets that only know their own position in the queue. It introduces two policy families: Poa, where packets ping only on arrival with a height-dependent probability, and Pico, where packets ping continuously with a rate inversely proportional to their height and waiting time; the server reconstructs the queue height from received pings. The authors prove near-optimal error guarantees for Poa under constant-rate and Poisson departures, provide matching lower bounds, and show Poa is insufficient for highly bursty or adversarial arrivals/departures, motivating Pico, which achieves an \\epsilon-reconstruction with a bounded area-based argument and explicit bounds on ping counts. The results extend to networks of queues and illuminate the trade-offs between communication (pings) and estimation accuracy, offering practical guidance for distributed congestion monitoring in traffic and similar systems. Overall, the paper advances decentralized monitoring by characterizing when simple arrival-only pings suffice and when continuous, adaptive pinging is necessary, with concrete quantitative guarantees.

Abstract

Queue length monitoring is a commonly arising problem in numerous applications such as queue management systems, scheduling, and traffic monitoring. Motivated by such applications, we formulate a queue monitoring problem, where there is a FIFO queue with arbitrary arrivals and departures, and a server needs to monitor the length of a queue by using decentralized pings from packets in the queue. Packets can send pings informing the server about the number of packets ahead of them in the queue. Via novel online policies and lower bounds, we tightly characterize the trade-off between the number of pings sent and the accuracy of the server's real time estimates. Our work studies the trade-off under various arrival and departure processes, including constant-rate, Poisson, and adversarial processes.

Paper Structure

This paper contains 25 sections, 23 theorems, 42 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Algorithmic Intuition and Technical Contributions
  4. Related Work
  5. The Queue Monitoring Problem
  6. Poa Policy for the Constant-rate Departure Case
  7. Optimality of Poa Policies for Constant-rate Departures
  8. The Poa Policy and its Analysis
  9. Proof of Theorem \ref{['thm:ping']}
  10. Optimality Results and Lower Bounds for Poa Policies
  11. General Arrivals and Departures: The Pico Policy
  12. The Pico Policy
  13. Algorithmic Intuitions and Analysis Overview
  14. Analysis
  15. Bounding Under-estimation Error
  16. ...and 10 more sections

Key Result

Lemma 3.1

For any pinging policy for constant-rate departures, an equivalent policy (in terms of the expected number of pings and the reconstruction error) has packets only sending pings on arrival.

Figures (1)

  • Figure 1: Visualization of Pico. Suppose the packet $i = j(t, y)$ of height $y$ pings at current time $t$. The packet has had height at least $y$ since it arrival and therefore the rectangle $\mathcal{L}_{it}$, colored blue, of height $h$ and width equal to $i$'s waiting time at time $t$ fits under the time varying height profile. The server projects $\mathcal{L}_{it}$ into the future. The projection $\mathcal{R}_{it}$, colored green, is a vertical and horizontal extension of $\mathcal{L}_{it}$. The server's estimation is the union of the extended rectangles.

Theorems & Definitions (38)

  • Lemma 3.1
  • Theorem 1
  • Theorem 2
  • Lemma 3.2
  • proof
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 28 more