unicast barrage relay networks: outage analysis and

1

Unicast Barrage Relay Networks:Outage Analysis and Optimization

S. Talarico 1, M. C. Valenti 1, and T. R. Halford 2

1 West Virginia University, Morgantown, WV.2 TrellisWare Technologies, Inc., San Diego, CA.

Oct. 7th, 2014

____________________________________________________

Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method

5. Network Optimization

6. Conclusion

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Barrage Relays Networks (BRNs)

I BRN is a cooperative MANET designed for use at the tactical edge.

I BRNs use time division multiple access and cooperative communications.

I Packets propagate outward from the source via a decode-and-forward ap-proach as follows:

TDMA Frame

Time Slot

…

time

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1

1

1

1

TDMA Frame

Time Slot

…

time

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1

1

1

1

2

2

2

2

22

2

2

TDMA Frame

Time Slot

…

time

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1

1

1

1

2

2

2

2

22

2

2

3

3

3

3

3

TDMA Frame

Time Slot

…

time

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Controlled Barrage Regions (CBRs)

. . . . . .

silent zone silent zone

active zone active zone active zone

S DR1 R2 S DR1 R2 S DR1 R2

I CBR is the union of multiple cooperative paths within some subregion orzone of the overall network.

I CBRs can be established by specifying a set of buffer nodes (S and D)around a set of N cooperating interior nodes (R1,...,RN ), enabling spatialreuse.

I The nodes operate in a half-duplex mode: during a particular radio framea CBR can be active or silent.

I Since each node transmits each packet at most once, the maximum numberof transmissions per frame is F = N + 1.

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Objective

Optim

ization

Outa

ge a

na

lysis The link outage probabilities are computed in closed form.

The dynamics of how a Barrage relay network evolves over

time is modeled as a Markov process.

An iterative method is used in order to account for co-channel

interference (CCI).

Maximize the transport capacity by finding the optimal:- code rate;

- number of relays;

- placement of the relays;

- size of the network.

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Network Model

I The BRN comprises:

I K CBRs;I M mobiles radios or nodes X = {X1, ..., XM}.

I The instantaneous power of Xi at receiver Xj during time slot t is

ρ(t)i,j = Pig

(t)i,j f(di,j) (1)

where

I Pi is the transmit power;I g

(t)i,j is the power gain due to Rayleigh fading;

I di,j = ||Xi −Xj || is the distance between Xi and Xj ;I f(·) is a path-loss function:

f (d) =

(d

d0

)−αI α is the path loss exponent;I d ≥ d0;

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SINR

The signal-to-interference-and-noise ratio (SINR) for the node Xj during timeslot t, when the signals are diversity combined at the receiver, is (cf., [1]):

γ(t)j =

|X (t)j |

∑k∈G(t)j

g(t)k,jd

−αk,j

Γ−1 +∑i 6∈G(t)j

I(t)i g

(t)i,j d−αi,j

(2)

whereI X (t)

j ⊂ X is the set of barraging nodes that transmit identical packets to

Xj during the tth time slot;

I G(t)j is the set of the indexes of the nodes in X (t)j ;

I Γ is the signal-to-noise ratio (SNR);

I I(t)i is a Bernoulli variable with probability P [I

(t)i = 1] = p

(t)i ;

I p(t)i is the activity probability for the node Xi during time slot t.

[1] V. A. Aalo, and C. Chayawan, “Outage probability of cellular radio systems using maximal ratio combining in

Rayleigh fading channel with multiple interferers”, IEEE Electronics Letters, vol. 36, pp. 1314 - 1315, Jul. 2000.

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Conditional Outage Probability

I An outage occurs when the SINR is below a threshold β.

I β depends on the choice of modulation and coding.

I Conditioning on the network topology X and the set of barraging nodesX (t)j , the outage probability of mobile Xj during slot t is

ε(t)j = P

[γ(t)j ≤ β

∣∣∣X ,X (t)j

]; (3)

I The outage probability depends on the particular network realization, whichhas dynamics over timescales that are much slower than the fading;

I The conditional outage probability is found in closed form [2].

[2] S. Talarico, M. C. Valenti, and D. Torrieri, “Analysis of multi-cell downlink cooperation with a constrained

spatial model”, Proc. IEEE Global Telecommun. Conf (GLOBECOM), Atlanta, GA, Dec. 2013.

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Node and CBR States

I At the boundary between any two time slots, each node can be in one of

the three following node states:

I Node state 0: The node has not yet successfully decoded the packet.I Node state 1: It has just decoded the packet received during previous

slot, and it will transmit on next slot.I Node state 2: It has decoded the packet in an earlier slot and it will

no longer transmit or receive that packet.

I The CBR state is the concatenation of the states of the individual nodeswithin the CBR.

I The CBR state is represented by a vector of the form [S,R1, ..., RN , D]containing the states of the source, relays, and destination.

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Markov States

I The dynamics of how the CBR states evolve over a radio frame can bedescribed as an absorbing Markov process.

I The state space of the process is composed of % Markov states si.

I An absorbing Markov process is characterized by:

I transient states;I absorbing states: CBR outage state and CBR success state.

1. [1000]

7. [2020]

5. [2120]

7. [2200]

6. [2210]

7. [2220]

Slot 1 Slot 2 Slot 3

7. [2000]

8. [2001]

2. [2010]

3. [2100]

4. [2110]

8. [2011]

8. [2101]

8. [2111]

8. [2021]

8. [2121]

8. [2201]

8. [2211]

8. [2221]

p1,2

p1,3

p1,4

p1,7

p1,8

p2,5

p2,7

p2,8

p3,6

p3,7

p3,8

p4,7

p4,8

p5,7

p5,8

p6,7

p6,8

S D

R2

R1

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Transition ProbabilityI The transition probability pi,j is the probability that the process moves

from Markov state si to state sj .

I The transition probability is evaluated as follows:

I If sj is a transient Markov state, it is the product of the individualtransmission probabilities;

I If sj is an absorbing Markov state, it is equal to the sum of theprobabilities of transitioning into the constituent CBR states.

1. [1000]

7. [2020]

5. [2120]

7. [2200]

6. [2210]

7. [2220]


7. [2000]

8. [2001]

2. [2010]

3. [2100]

4. [2110]

8. [2011]

8. [2101]

8. [2111]

8. [2021]

8. [2121]

8. [2201]

8. [2211]

8. [2221]

p1,2

p1,3

p1,4

p1,7

p1,8

p2,5

p2,7

p2,8

p3,6

p3,7

p3,8

p4,7

p4,8

p5,7

p5,8

p6,7

p6,8

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Transition ProbabilityI The transition probability pi,j is the probability that the process moves

from Markov state si to state sj .

I The transition probability is evaluated as follows:

I If sj is a transient Markov state, it is the product of the individualtransmission probabilities;

I If sj is an absorbing Markov state, it is equal to the sum of theprobabilities of transitioning into the constituent CBR states.

1. [1000]

7. [2020]

5. [2120]

7. [2200]

6. [2210]

7. [2220]


7. [2000]

8. [2001]

2. [2010]

3. [2100]

4. [2110]

8. [2011]

8. [2101]

8. [2111]

8. [2021]

8. [2121]

8. [2201]

8. [2211]

8. [2221]

p1,2

p1,3

p1,4

p1,7

p1,8

p2,5

p2,7

p2,8

p3,6

p3,7

p3,8

p4,7

p4,8

p5,7

p5,8

p6,7

p6,8

For instance:

1. [1000]

Slot 1

2. [2010]p1,2

p1,2 = ε(1)R1

(1− ε(1)R2

)ε(1)D

where ε(t)j is found using (3).

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CBR Outage and Success Probabilities

I The state transition matrix P , whose (i, j)th entry is {pi,j}, is

P =

[Q R0 I

](4)

where

I Q is the τ × τ transient matrix;I R is the τ × r absorbing matrix;I I is an r × r identity matrix.

I The absorbing probability bi,j is the probability that the process will beabsorbed in the absorbing state sj if it starts in the transient state si.

I The absorbing probability bi,j is the (i, j)th entry of B, which is:

B = (I −Q)−1 R. (5)

I If the absorbing states are indexed so that the first absorbing state corre-

sponds to a CBR outage, while the second corresponds to a CBR success:I The CBR outage probability is εCBR = b1,1;I The CBR success probability is ε̂CBR = b1,2.

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Inter-CBR InterferenceI Inter-CBR interference causes a linkage between the transition probabilities

in a given CBR and the transmission probabilities of adjacent CBRs.

I In order to compute the transition matrix Pk, an iterative method is used:

Begin by neglecting CCI:( t )

ip [0] 0, i, t= ∀

Compute the

transition matrix Pk(t)[n-1]

Compute the

activity probabilities

Compute the

transition matrix Pk(t)[n]

(t) (t)

k k F P [n]-P [n-1] ?<ζ

Yes

STOP

No

Update the

transition matrix

Evaluate the

transmission probabilities

Evaluate the

transmission probabilities

I Pk[n] is the transition matrix atthe nth iteration for the kth CBR;

I {p(t)i [n]} is the set of activity prob-abilities at the nth iteration;

I || · ||F is the Frobenius norm;

I ξ is a tolerance.

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Example

1 2 3 4 5 6

10−3

10−2

10−1

100

SNR = 0 dB

SNR = 10 dB

Number iterations

CB

R o

utag

e pr

obab

ility

α = 3α = 3.5α = 4

Figure: CBR outage probability for the kth CBRas function of the number of iterations used. Setof curves at the top: SNR = 0 dB. Set of curvesat the bottom: SNR = 10 dB.

Settings:

I Line network;

I Two relays (N = 2);

I CCI from two closest active zones;

I Infinite cascade of CBRs;

I All CBRs have identical topology.

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Network Optimization

I The transport capacity (TC) of a typical CBR is

Υ = dε̂CBR

2 (N + 1)R. (6)

where

I d is the distance between the CBR’s source and destination;I ε̂CBR is the CBR success probability;I N is the number of relays within the CBR;I R = log2 (1 + β) is the code rate (in bit per channel used [bpcu]).

I Let Xk be a vector containing the position of the N relays in the kth CBR.

I The goal of the network optimization is to determine the set Θ = (Xk, R,N, d)that maximizes the TC.

I The optimization can be solved through a convex optimization over (R,N, d)combined by a stochastic optimization over Xk.

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Convex Optimization Surface

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R

ϒ opt(R

| d,

Xk)

SNR = 5 dB

SNR = 0 dB

W/o co−channel interferenceW/ co−channel interference

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

d

ϒ opt(d

| X

k )

SNR = 5 dB

SNR = 0 dB


0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N

ϒ opt(N

| d,

Xk)

SNR = 5 dB

SNR = 0 dB


Settings:

I Line network;

I Unitary CBR;

I CCI from two closest active zones;

I Infinite cascade of CBRs;

I All CBRs have identical topology.

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Optimization Results

SNR CCI R N d Optimal nodes’ location Υopt

0 7 4.452 0 0.3

S D

0.490

3 4.421 5 1.2

R1 R2 R3 R4 R5S D

0.402

5 7 4.611 0 0.4

S D

0.683

3 4.452 5 1.7

R1 R2 R3 R4 R5S D

0.559

10 7 5.028 0 0.5

S D

0.951

3 4.547 5 2.3

R1 R2 R3 R4 R5S D

0.777

Table: Results of the optimization for a line network.

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Outline

1. Introduction

2. Network Model

3. Markov Process

4. Iterative Method


6. Conclusion

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Conclusions

I This paper presents a new analysis and optimization for unicast in a BRN.

I A BRN is analyzed by describing the behavior of each constituent CBR asa Markov process: path loss, Rayleigh fading, and inter-CBR interferenceare taken into account.

I The analysis is used to optimize a BRN by finding

I the optimal number of relays that need to be used in the CBR,I the optimal placement of the relays,I the optimal size of the CBR,I the optimal code rate at which the nodes transmit,

which maximize TC.

I The analysis and the model are general enough that can be extended todifferent types of cooperative ad hoc networks, for which multiple sourcetransmissions are diversity combined at the receiver.

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Thank You

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unicast barrage relay networks: outage analysis and

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