[Paper Review] Minimal Evacuation Times and Stability

Leonidas Georgiadis, Georgios S. Paschos+2 more|arXiv (Cornell University)|Nov 20, 2012

Wireless Communication Security Techniques8 references4 citations

TL;DR

This paper establishes a fundamental connection between minimal evacuation time and system stability in time-slotted queueing systems with general arrival processes and admissible policies. It proves that the stability region equals the information-theoretic capacity region for the N-user broadcast erasure channel with feedback, using evacuation time growth rates to characterize stability without restrictive path assumptions.

ABSTRACT

We consider a system where packets (jobs) arrive for processing using one of the policies in a given class. We study the connection between the minimal evacuation times and the stability region of the system under the given class of policies. The result is used to establish the equality of information theoretic capacity region and system stability region for the multiuser broadcast erasure channel with feedback.

Motivation & Objective

• To characterize the stability region of a general time-slotted queueing system using the asymptotic growth rate of minimal evacuation time.
• To establish a connection between queueing-theoretic stability and information-theoretic capacity in multiuser communication systems.
• To demonstrate that the stability region equals the information capacity region for the N-user broadcast erasure channel with feedback.
• To develop a methodology applicable to diverse systems, including wireless networks and control systems, without requiring sample-path assumptions on policies.
• To validate the approach by constructing a policy that achieves evacuation time growth rate matching the inverse of the capacity region.

Proposed method

• Define the minimal evacuation time as the shortest time to clear an initial batch of packets under any admissible policy.
• Use the asymptotic growth rate of this minimal evacuation time to characterize the stability region of the system class.
• Construct a hybrid evacuation policy (π_l) that uses a block-based strategy followed by a one-by-one retransmission if decoding fails.
• Analyze the expected evacuation time of π_l for large batch sizes, showing it approaches 1 per unit of rate when error probability is small.
• Leverage feedback to detect decoding errors and trigger retransmissions, enabling efficient signaling of completion or continuation.
• Prove that the stability region (S_B) equals the capacity region (C) by showing inclusion in both directions using policy constructions and asymptotic analysis.

Experimental results

Research questions

• RQ1 How does the asymptotic growth rate of minimal evacuation time relate to the stability region of a general queueing system?
• RQ2 Can the stability region of a multiuser broadcast erasure channel with feedback be characterized using evacuation time analysis?
• RQ3 Is the information-theoretic capacity region of the N-user broadcast erasure channel with feedback equal to its queueing stability region?
• RQ4 What policy constructions enable the evacuation time to match the theoretical lower bound implied by capacity?
• RQ5 Under what conditions can evacuation time analysis be reliably used to infer stability, and when might it fail?

Key findings

• The stability region of the system is completely characterized by the asymptotic growth rate of the minimal evacuation time over all admissible policies.
• For the N-user broadcast erasure channel with feedback, the information-theoretic capacity region equals the queueing stability region.
• The constructed policy π_l achieves an expected evacuation time per unit rate that approaches 1 as the batch size grows, proving the upper bound on the evacuation time growth rate.
• The proof shows that the stability region S_B is equal to the capacity region C, with equality confirmed via inclusion in both directions: S_D ⊆ R ⊆ S_B.
• The use of feedback allows the transmitter to detect decoding errors and signal continuation or completion, enabling efficient evacuation with minimal overhead.
• Logarithmic feedback overhead for epoch state signaling does not affect the asymptotic stability characterization due to its sublinear growth.