Fundamental Lower Bound for Node Buffer Size in Intermittently Connected Wireless Networks Yuanzhong...

Fundamental Lower Bound Fundamental Lower Bound for Node Buffer Size infor Node Buffer Size in

Intermittently Connected Wireless NetworksIntermittently Connected Wireless Networks

Yuanzhong Xu, Xinbing Wang

Shanghai Jiao Tong University, China

Buffer in Intermittently Connected Network Presentation 2

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IntroductionIntroduction MotivationsMotivations ObjectivesObjectives

Model and Model and AssumptionAssumption

Percolation of Active NodesPercolation of Active Nodes

Lower Bound In Supercritical CaseLower Bound In Supercritical Case

Lower Bound In Subcritical CaseLower Bound In Subcritical Case

Discussion on Length of Time SlotDiscussion on Length of Time Slot

ConclusionConclusion


MotivationMotivation

Under certain constraints, wireless networks areUnder certain constraints, wireless networks are

Only Only intermittently connectivityintermittently connectivity::

A complete path from the source to the destination does not A complete path from the source to the destination does not exist all the time. exist all the time.

Example:Example: Wireless sensor networks:

Node sleeping mode for energy saving. ([7, Dousse]) CR networks:

Secondary users wait for availability of wireless channel. ([8, Ren & Zhao])

Mobile networks:Nodes move out of reach. ([9, Kong & Yeh])


MotivationMotivation

Intermittently connectivity requires adequate node buffer, Intermittently connectivity requires adequate node buffer, even with even with infiniteinfinite channel capacity and processing speed, channel capacity and processing speed,

Temporarily store the packets not ready to be sent out.Temporarily store the packets not ready to be sent out.


ObjectivesObjectives

We focus on node buffer occupation in We focus on node buffer occupation in static random wireless static random wireless networksnetworks with with intermittent connectivity intermittent connectivity due to due to node inactivitynode inactivity::

1.1. How does the minimum buffer requirement for each node How does the minimum buffer requirement for each node increase as the network size grows?increase as the network size grows?

2.2. What is the What is the achievable achievable lower bound for node buffer size?lower bound for node buffer size?


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IntroductionIntroduction








Model and Assumption – I/VModel and Assumption – I/V Node LocationsNode Locations: : First consider a Poisson point process on R2 with density λ Locations of nodes are points within the region:

Direct LinksDirect Links:: Each node covers a disk shaped area with radius ½Each node covers a disk shaped area with radius ½ Two nodes have a Two nodes have a direct link direct link if and only if they overlap:if and only if they overlap:


Model and Assumption – II/VModel and Assumption – II/V Assumption on Node DensityAssumption on Node Density: :

There exists an infinite connected cluster in R2(giant cluster)

Nodes in giant cluster: connected nodes

Number of connected nodes is proportional to n

Larger density => larger connected proportion:

,1n

nc


Model and Assumption – III/VModel and Assumption – III/V Node inactivity – Node inactivity – Each node switch between active state Each node switch between active state

and inactive state:and inactive state: Time is slotted, slot length: States of each active nodes are i.i.d. among time slots. The probability to be inactive is for all nodes. States of different nodes are i.i.d.

Model Notation:


Model and Assumption – IV/VModel and Assumption – IV/V Traffic Pattern of Connected Nodes – Traffic Pattern of Connected Nodes – Random UnicastRandom Unicast Permanent source-destination pairs (connected). Each source messages to destination in constant rate . Transmission in multi-hop. BufferingBuffering In each hop, if the transmitter or the receiver is inactive, the

messages should be buffered in the transmitter until both are active.


Model and Assumption – V/VModel and Assumption – V/V Assumption on Capacity and Processing speedAssumption on Capacity and Processing speed Channel capacity is large enough to be viewed as infinity,

compared to the actual utilization.

Node processing speed is also infinite, compared to the state-switching frequency.

Message slotMessage slot Messages generated by u during time slot t whose

destination is v have the same route, and exist in the same node at the end of a time slot. Denoted by .

Size of one message slot:


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Active nodes density: Active nodes density: Threshold for probability of inactivity:Threshold for probability of inactivity:

Supercritical Case: Supercritical Case:

active giant exists in each time active giant exists in each time slot slot

Subcritical CaseSubcritical Case::

no active giant in each time slotno active giant in each time slot


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Supercritical case:Supercritical case:Achievable Lower bound – Achievable Lower bound – Constant Order:Constant Order:

For a connected node in with , at time For a connected node in with , at time slot , slot ,

Lower bound Lower bound at least needs to buffer messages generating by itself when at least needs to buffer messages generating by itself when it is inactive.it is inactive.

achievable ?achievable ?


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – I/VII– I/VII

Buffer occupation in w at time slot tBuffer occupation in w at time slot t


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – II/VII– II/VII

Achievable with Achievable with Optimal Routing Scheme (ORS)Optimal Routing Scheme (ORS)

3 stages of ORS in (for source and destination ) :3 stages of ORS in (for source and destination ) :

I.I. Stage 1 – Source relay & bufferingStage 1 – Source relay & bufferingRelay along a linear pathRelay along a linear pathNodes keep a copy until one node Nodes keep a copy until one node in the path belongs to active giant.in the path belongs to active giant. ( (source buffering path – nodes thatsource buffering path – nodes thathave received the messagehave received the message); );

II.II. Stage 2 – Transmit via active giantStage 2 – Transmit via active giant

III.III. Stage 3 – Destination relayStage 3 – Destination relay


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – III/VII– III/VII



I.I. Stage 1 – Source relay & bufferingStage 1 – Source relay & buffering

II.II. Stage 2 – Transmit via active giantStage 2 – Transmit via active giantTransmit from the source buffering path to the nearest node Transmit from the source buffering path to the nearest node a f to destination. (without latency) a f to destination. (without latency)

III.III. Stage 3 – Destination relayStage 3 – Destination relay


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – IV/VII– IV/VII



I.I. Stage 1 – Source relay & bufferingStage 1 – Source relay & buffering

II.II. Stage 2 – Transmit via active giantStage 2 – Transmit via active giant

III.III. Stage 3 – Destination relayStage 3 – Destination relayAlong the Along the shortest path shortest path from tofrom todestination.destination.(destination buffering path)(destination buffering path)


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – V/VII– V/VII

Why finite buffer in ORS?Why finite buffer in ORS?

Active giant spreads all over the network:Active giant spreads all over the network:

Only considering active giant in one time slot Only considering active giant in one time slot – – Vacant components Vacant components are small.are small.

[17][17]

In Stage I,In Stage I,

buffering path is circulated bybuffering path is circulated bya vacant component.a vacant component.

Small source buffering pathSmall source buffering path

(finite expectation of size)(finite expectation of size)


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – VI/VII– VI/VII


Active giant spreads all over the network:Active giant spreads all over the network:

Only considering active giant in one time slot Only considering active giant in one time slot – – Vacant components Vacant components are small.are small.

[17][17]

In Stage III, In Stage III, destination is circulated by a vacant destination is circulated by a vacant component with size at least the distance component with size at least the distance between between and destination and destination

Small destination buffering pathSmall destination buffering path

(finite expectation of size)(finite expectation of size)


Lower Bound In Supercritical Case Lower Bound In Supercritical Case – VII/VII– VII/VII


Finite-sized source buffering path and Finite-sized source buffering path and destination buffering pathdestination buffering path

Finite Latency;Finite Latency;One node only buffers messages withOne node only buffers messages withnear near sources or destinations.sources or destinations.

Finite buffer occupation in ORS.Finite buffer occupation in ORS.


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Lower Bound In Subcritical Case Lower Bound In Subcritical Case – I/III– I/III

Lower bound:Lower bound: The smallest latency of a message slot for u to v The smallest latency of a message slot for u to v

satisfies:satisfies:

By Little’s Law, average buffer occupation among all By Little’s Law, average buffer occupation among all connected nodes :connected nodes :

– – The lower bound is .The lower bound is .

)()()1()(

)(nLnL

ntS

L


Lower Bound In Subcritical Case Lower Bound In Subcritical Case – II/III– II/III

AchievabilityAchievability Assuming transmission path from to is along the Assuming transmission path from to is along the

straight line connecting them:straight line connecting them:

Hop by hop transmission, path between neighboring flag Hop by hop transmission, path between neighboring flag nodes is the shortest one.nodes is the shortest one.

In this scheme, In this scheme, lower bound is achieved. lower bound is achieved.


Lower Bound In Subcritical Case Lower Bound In Subcritical Case – III/III– III/III

Proof Sketch of AchievabilityProof Sketch of Achievability

Giant component (including both active and inactive nodes)Giant component (including both active and inactive nodes)

If belongs to the path from to ( ), then If belongs to the path from to ( ), then it it cannot be far from the line connecting them cannot be far from the line connecting them ( )( )

Prove that serves for Prove that serves for S-D pairs S-D pairs

lower bound is achieved.lower bound is achieved.


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In both supercritical and subcritical cases, lower bounds for In both supercritical and subcritical cases, lower bounds for buffer occupation scales linearly to the length of time slot.buffer occupation scales linearly to the length of time slot.

With constant probability of node inactivity,With constant probability of node inactivity,

smaller results in smaller buffer requirements. smaller results in smaller buffer requirements.

When , it is equivalent to no inactivity but channel When , it is equivalent to no inactivity but channel

capacity decreases to , similar to TDMA.capacity decreases to , similar to TDMA.2)1( p

C

C


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DiscussionDiscussion on Length of Time Sloton Length of Time Slot




We study the lower bounds for node buffer in intermittently We study the lower bounds for node buffer in intermittently connected network.connected network.

In supercritical case, the achievable lower bound does not In supercritical case, the achievable lower bound does not increase as the network size grows.increase as the network size grows.

In subcritical case, the achievable lower bound isIn subcritical case, the achievable lower bound is ..

In both cases,In both cases, lower bounds for buffer occupation scales lower bounds for buffer occupation scales linearly to the length of time slot.linearly to the length of time slot.

Thank you !Thank you !


ReferenceReference


Intermittently Connected Long PathIntermittently Connected Long Path

A path of n nodesA path of n nodes

With finite channel capacity,With finite channel capacity,assume every node send all messages in its buffer to the assume every node send all messages in its buffer to the next node:next node:

Buffer occupation of nodes on the path has large variance Buffer occupation of nodes on the path has large variance in time domain.in time domain.



With finite channel capacity,With finite channel capacity,assume every node send all messages in its buffer to the assume every node send all messages in its buffer to the next node:next node:



Non-empty ratio: the proportion of time slots during which Non-empty ratio: the proportion of time slots during which buffer is empty.buffer is empty.



Improvement by a simple mechanism –Improvement by a simple mechanism –Restrict the maximum amount of messages sent in one Restrict the maximum amount of messages sent in one time slot of each hop. time slot of each hop.

p = 0.5, in one time slot each hop can transmit at most 30 message slots.

Fundamental Lower Bound for Node Buffer Size in Intermittently Connected Wireless Networks Yuanzhong...

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