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Queueing Delay Minimization in Overloaded Networks

Xinyu Wu, Dan Wu, Eytan Modiano

TL;DR

It is remarked that the explicit min-delay policy design in multi-stage networks facilitates co-optimization with other metrics, such as minimizing total bandwidth, balancing link utilization and node buffer usage, which demonstrates the wider utility of the main results in data center network optimization in practice.

Abstract

We develop link rate control policies to minimize the queueing delay of packets in overloaded networks. We show that increasing link rates does not guarantee delay reduction during overload. We consider a fluid queueing model that facilitates explicit characterization of the queueing delay of packets, and establish explicit conditions on link rates that can minimize the average and maximum queueing delay in both single-hop and multi-stage (switching) networks. These min-delay conditions require maintaining an identical ratio between the ingress and egress rates of different nodes at the same layer of the network. We term the policies that follow these conditions rate-proportional policies. We further generalize the rate-proportional policies to queue-proportional policies, which minimize the queueing delay asymptotically based on the time-varying queue length while remaining agnostic of packet arrival rates. We validate that the proposed policies lead to minimum queueing delay under various network topologies and settings, compared with benchmarks including the backpressure policy that maximizes network throughput and the max-link-rate policy that fully utilizes bandwidth. We further remark that the explicit min-delay policy design in multi-stage networks facilitates co-optimization with other metrics, such as minimizing total bandwidth, balancing link utilization and node buffer usage. This demonstrates the wider utility of our main results in data center network optimization in practice.

Queueing Delay Minimization in Overloaded Networks

TL;DR

It is remarked that the explicit min-delay policy design in multi-stage networks facilitates co-optimization with other metrics, such as minimizing total bandwidth, balancing link utilization and node buffer usage, which demonstrates the wider utility of the main results in data center network optimization in practice.

Abstract

We develop link rate control policies to minimize the queueing delay of packets in overloaded networks. We show that increasing link rates does not guarantee delay reduction during overload. We consider a fluid queueing model that facilitates explicit characterization of the queueing delay of packets, and establish explicit conditions on link rates that can minimize the average and maximum queueing delay in both single-hop and multi-stage (switching) networks. These min-delay conditions require maintaining an identical ratio between the ingress and egress rates of different nodes at the same layer of the network. We term the policies that follow these conditions rate-proportional policies. We further generalize the rate-proportional policies to queue-proportional policies, which minimize the queueing delay asymptotically based on the time-varying queue length while remaining agnostic of packet arrival rates. We validate that the proposed policies lead to minimum queueing delay under various network topologies and settings, compared with benchmarks including the backpressure policy that maximizes network throughput and the max-link-rate policy that fully utilizes bandwidth. We further remark that the explicit min-delay policy design in multi-stage networks facilitates co-optimization with other metrics, such as minimizing total bandwidth, balancing link utilization and node buffer usage. This demonstrates the wider utility of our main results in data center network optimization in practice.
Paper Structure (32 sections, 9 theorems, 62 equations, 17 figures, 2 tables)

This paper contains 32 sections, 9 theorems, 62 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Models, Definitions and Problem Formulation
  3. Network Models: Topology and Dynamics
  4. Single-hop networks
  5. Multi-stage networks
  6. Network Overload
  7. Queueing Delay Characterization
  8. Problem Formulation
  9. Static Min-Delay Policy Design
  10. $N\times 1$ Networks
  11. General Single-Hop and Multi-Stage Networks
  12. $N_S\times N_D$ single-hop networks
  13. Multi-stage networks
  14. Queue-based Min-delay Policy Design
  15. $N\times 1$ Networks
  16. ...and 17 more sections

Key Result

Theorem 1

Given an $N\times 1$ single-hop network with unlimited link capacity. For $\forall T>0$, the set of $\mathbf{g}=\{g_i\}_{i=1}^N$ that minimizes $\bar{D}_{\text{avg}}$ and $\bar{D}_{\text{max}}$ of the packets that arrive within $[t_0,t_0+T]$ where $\mathbf{q}(t_0)=\boldsymbol{0}$ is under which $\bar{D}_{\text{avg}}=\bar{D}_{\text{max}}=\frac{T}{2\mu}\max\{\sum_{i=1}^N \lambda_i-\mu, 0\}$.

Figures (17)

  • Figure 1: An example of a $2\times 1$ single-hop network: Under maxweight policy, link $(s_1,d)$ is always activated while $(s_2,d)$ is blocked since the queue in $s_1$ grows with rate $\lambda_1-c_1=4$ while the queue in $s_2$ grows with rate at most $\lambda_2=3$.
  • Figure 2: (a) A single-hop network structure; (b) A switched network with ingress and egress ports; (c) A server farm with load balancers as ingress and servers as egress.
  • Figure 3: An example of a 4-layer multi-stage network
  • Figure 4: An example of queueing delay characterization of a packet passing two nodes
  • Figure 5: An example of an $N\times 1$ single-hop network
  • ...and 12 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 8 more