Delay-Based Back-Pressure Scheduling in Multi-Hop Wireless Networks

1Bo Ji, 2Changhee Joo and 1Ness B. Shroff

1Department of ECE, The Ohio State University2Department of EECE, Korea University of Technology and Education

Published in IEEE INFOCOM 2011

Outline

• Introduction• System model• Throughput Optimality of Q-BP• Throughput Optimality of D-BP• Numerical Results• Conclusion

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Introduction

• Link scheduling is a critical resource allocation component in multi-hop wireless networks.

• The Queue-length-based Back-Pressure (Q-BP) scheduler has been shown to be throughput-optimal.

• Last packet problem– consider a queue that holds the last packet of a flow, then

the packet does not see any subsequent packet arrivals, and thus the queue length remains very small and the link may be starved for a long time

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Introduction (cont’d)

• Recent works have studied the performance of delay-based scheduling algorithms instead of queue length as link weight.

• However, the throughput performance of the delay-based scheduling schemes has merely been shown for limited cases of single-hop networks.

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Main Contributions of This Paper

• Show that the proposed Delay-based Back-Pressure (D-BP) scheduling scheme is throughput-optimal in multi-hop networks.– We re-derive throughput optimality of Q-BP itself using

fluid limit techniques.– We devise a new delay metric for D-BP and show that it

achieves the optimal throughput performance in multihop wireless networks.

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System model

• We consider a multi-hop wireless network described by a directed graph 𝒢 = ( , 𝒱 ℰ), where 𝒱 denotes the set of nodes and ℰ denotes the set of links.

• Let S denote the set of flows in the network– The route of flow has an ( )-hop length, where each 𝑠 𝐻 𝑠

-th hop link is denoted by 𝑘 ( , )𝑠 𝑘• Let 𝒫 denote the set of all link-flow-pairs• The set of link-flow-pairs that interfere with ( , 𝑠 𝑘):

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System model (cont’d)

• A schedule is a set of (active or inactive) link-flow-pairs, and can be represented by a vector

• Let ℳ𝒫 denote the set of all feasible schedules in 𝒫, and let Co(ℳ𝒫) denote its convex hull.

• Let 𝐴𝑠(𝑡) denote the number of packet arrivals at the source node of flow 𝑠 at time slot 𝑡.

• We call 𝜆𝑠 the arrival rate of flow 𝑠, and let

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System model (cont’d)

• Let Qs,k(t) denote the number of packets at the queue of ( , 𝑠 𝑘) at the beginning of time slot 𝑡.

– where denotes the actual number of packets transmitted from 𝑄 , 𝑠 𝑘 at time slot 𝑡

• Let ,𝑠 𝑘(t) denote the service of Q ,𝑠 𝑘 at time slot t, which takes either 1 if link-flow-pair ( , )𝑠 𝑘 is active, or 0, otherwise. We have

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System model (cont’d)

• The stability of a network

• We define the throughput region of a scheduling policy as the set of rates, for which the network remains stable under this policy.– The optimal throughput region is defined as the union of

the throughput regions of all possible scheduling policies.

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Lyapunov Stability

• Consider a function V(x):RnR such that– V(x)0 with equality if and only if x=0. – 0 with equality if and only if x =0

• Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov.

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Queue-length-based Back-Pressure (Q-BP)

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Throughput Optimality of Q-BP

• In fluid model

• (Throughput Optimality)Q-BP can support any traffic with arrival rate vector that is strictly inside Λ∗.

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Proof of Throughput Optimality

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qs,k(t) 0 because of Q-BP policy

s s,k because of the definition of s Λ∗

.

0

Delay-based Back-Pressure (D-BP)

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𝑊𝑠,𝑘(𝑡) denotes the sojourn time of the HOL packet of queue 𝑄 ,𝑠 𝑘(𝑡)

Throughput Optimality of D-BP

• (Throughput Optimality)D-BP can support any traffic with arrival rate vector that is strictly inside Λ∗.

• Proof:– Lyapunov function:

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In fluid model:

Numerical Results

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The short flow arrives at the network with a finite amount of packets at time 0, and the number of packets follows Poisson distribution with mean rate 10. The long flows have an infinite amount of data and keep injecting packets at the source nodes following Poisson distribution with mean rate 𝜆 at each time slot.

• The last packet problem of Q-BP

Numerical Results (cont’d)

• The last packet problem of Q-BP (cont’d)

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Several last packets of the short flowget excessively delayed under Q-BP.

Numerical Results (cont’d)

• The throughput of different schedulers

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Flow (11 → 10 → 9) has bursty arrivals with a small mean rate

Numerical Results (cont’d)

• D-BP achieves the same throughput region as Q-BP

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Numerical Results (cont’d)

• The tail of the delay distribution under D-BP vanishes much faster than Q-BP.

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Numerical Results (cont’d)

• D-BP has mush shorter delay and achieves better fairness among the flows.

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Conclusion

• This paper develop a throughput-optimal delay-based back-pressure scheme for multi-hop wireless networks

• As a result, the excessively long delays that could be experienced by certain flows under the queue-length-based scheduling schemes are eliminated without any loss of throughput.

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Comments

• This paper presents a method to prove the throughput optimality of schedule policy using fluid limit and Lyapunov technique.

• D-BP is still not practical (because it require a central scheduler to solve the maximum weight independent set problem)

• This paper shows that delay-based scheduling can also achieve throughput optimality while the delays can be reduced– The control of end-to-end delay is still a challenge.

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