Policy-dependent multi-hop distances in linear adhoc networks

Golaleh RahmatollahiInstitute of Communications TechnologyLeibniz Universitat Hannover, Germany

Email: gola@ikt.uni-hannover.de

Giuseppe AbreuCenter for Wireless Communications

University of Oulu, FinlandEmail: giuseppe@ee.oulu.fi

Abstract—In this paper we derive upper bounds on the multi-hop distributions of distances from a source to a destinationN hops away yielding closed-form expressions for the averagenumber of hops required to reach the given destination underdifferent routing policies. Furthermore, we present a simplemethod to estimate the probability distribution P (N |D) of thenumber of hops given a certain Euclidean distance for eachhopping strategy by using the results of the average numberof hops .

The results are useful to quantify the impact of hopping strate-gies on the performance and efficiency of multi-hop networks aswell as to analyze the reliability in terms of delay and accuracyof certain applications such as flooding and localization.

I. INTRODUCTION

It has been shown that hopping policy in terms of closest,furthest and random hop strategy has a strong impact ofthe performance of wireless ad hoc networks regarding forinstance capacity. In [1] the authors obtain better transmissioncapacity results when using closest neighbor strategy thanchoosing furthest hop policy.

However, in [2] the costs and efficacy of relay selectionwas analyzed with the outcome that furthest neighbor relayselection is less sensitive to the number of contending nodescompared to hopping to the closest neighbor. This indicatesthat the relay selection overhead can be reduced by applyingfurthest neighbor selection.

Random hopping strategies are considered in flooding andlocalization applications where the position of sensor nodes areunknown. In [3] the authors introduce an efficient localizationalgorithm for multi-hop ad hoc networks allowing accurateself-localization where packets are forwarded to random nodeswhich act as relay nodes. The examples show that the choiceof the hopping policy is a matter of the considered application.

What is generally true is that the number of hops has animpact on the time required for packets to reach a certaindestination, while the hop length strongly impacts on the relia-bility of the communication. Hence, in order to quantify costs,delay and accuracy in multi-hop wireless ad hoc networksa detailed study of policy-dependent distance distribution isrequired leading to a challenging task.

As the random deployment of nodes in ad hoc networksyields random inter-sensor distances and correlation in therandom distances on a hop-by-hop basis the relation betweenhop distance and Euclidian distance can not be obtained by

using simple geometric calculations leading to a requiredstatistical study. Furthermore, especially for planar networksthe additional problem arises that hops not always occur on astraight line from source to destination leading to intractableexpressions due to the swing problem.

In [4] the authors derive distance distributions for linearnetworks based on gaussian approximation being inaccuratefor few hops. Whereas, the authors of [5] present a recursiveexpression for the distance distribution in planar networkswhich is not tractable for many hops. In this article we obtainsimple and clear expressions for the distance distribution inlinear ad hoc networks and derive closed-form formulae forthe expected number of hops by considering asymptotic lowerbounds to eliminate correlation. Later, the results can be usedfor planar networks to map the linear results and eliminate theswing problem. Using the derived distributions lower boundson the expected number of hops required to reach a givenEuclidean distance under each hopping policy are obtainedin simple closed-forms. The expressions are found to be tightcompared to experimental results obtained by simulations. Theresults indicate that the number of hops linearly depends on thedistance with slope determined by the policy. The evaluation ofthe probability distribution of N given an Euclidean distanceD implies that for a sufficient large distance D our proposedmethod of using the average number of hops delivers accurateresults compared to simulations.

The remainder of this paper is as follows: in section IIwe introduce the system model and its general parametersand derive the single-hop and multi-hop probability densityfunction (pdf) for each routing policy. In section III we deriveanalytically the expected progress per hop and obtain simpleclosed-form expression for the average number of hops. Weuse this results and show how to obtain simply the probabilitydistribution P (N |D) of the number of hops N given a certaindistance D to the destination in section IV. In section V weverify our analytical results by simulations and conclude thepaper in section VI.

II. SINGLE AND MULTI-HOP DISTANCES

We consider a linear multi-hop wireless sensor network witha source node, a destination node and N relay nodes. Letxi with i = 0, 1...N denote the location of the nodes wherex0 = 0 is the source’s location and xN = D the destination’s

978-1-4244-7157-7/10/$26.00 ©2010 IEEE 241

location respectively. In the following, we assume {xi}Ni=0 tobe independently and uniformly distributed random variableson (0, D). Given a fixed transmission range R equal for allnodes, node i−1 and i are connected if their distance is equalto ri = xi − xi−1 ≤ R.

Due to the independent and uniform deployment of thenodes, the occurrence probability of a node can be modeledas a one dimensional point Poisson process (PPP). For ourconsiderations we assume λR to be sufficient large so thatthere exists a path from source to destination indicating thatthere is always a receiving relay in range of a transmittingnode. This assumption is feasible, since our goal is to usethe pdf results of linear networks for dense planar networkswhere the probability that there exists a path from source todestination is sufficient high.

We further assume that the next hop always occurs intothe direction of the destination node. The source node trig-gers the transmission by sending the packet to a relay nodebeing in range. In the following analytical considerations wedistinguish between three different routing strategies in termsof random, closest and furthest hop as the choice of a certainhop policy affects the statistics of the distance distribution.The last hop from xn−1 to xn is not affected by the routingstrategy. Once a node in range of the destination receives thepacket the next hop will certainly reach the destination.

First, we calculate the probability density function (pdf)by deviating the cumulative distribution function (cdf) ofthe single-hop distance. Hence, we are able to calculate theexpected progress per hop and the expected number of hopsgiven the Euclidian distance D.

A. Single-hop statistics1) Random hop: Fig. 1 (a) depicts the random hop strategy

where each transmitting node chooses randomly a relay nodein range. As the nodes are distributed uniformly the cdf F r(ri)and pdf fr(ri) of the random hop strategy can be expressedby:

F r(ri) =riR

(1a)

fr(ri) =1R

0 ≤ ri ≤ R. (1b)

Hence, in case of random hops we obtain uniform andindependent distributions.

Note, that in case of random hops the node density λ doesnot affect the statistics as we assume that there is always arelay node in range. Thus, the transmitting node picks oneof the nodes in range as relay leading to a process which isindependent of λ.

2) Closest hop: Fig. 1 (b) shows the single-hop distancein the case where routing is performed to the closest relayindicating that all nodes have to be considered to reach thedestination. In order to calculate the statistics we use thecharacteristics of the PPP as it describes the probability of theoccurence of k nodes with density λ within a certain range rby [7]:

P (k) =e−λr(λr)k

k!. (2)

Hence, the probability of having at least one node in ri isgiven by 1− e−λri yielding the cdf F c(ri) and its pdf f c(ri)expressed by:

F c(ri) =1− e−λri1− e−λR (3a)

f c(ri) =λe−λri

1− e−λR 0 ≤ ri ≤ R, (3b)

where the denominator∫ R

0f c(ri)dri = 1− e−λR denotes the

restriction to R. Due to the fact that every node has to beconsidered as a relay in the closest hop policy the outcomein terms of the cdf and pdf respectively is independentlydistributed.

3) Furthest hop: The minimum path of a source destinationpair relies on the furthest hop strategy as the most distant nodeacts always as relay node. It is the most investigated case inliterature and its single-hop statistic is described in [6]. Fig. 1(c) indicates that the hop to the furthest node yields a vacantsegment rei including no nodes which has to be consideredin terms of a condition. Hence, the cdf can be calculated likein the case of closest hop times the probability of having nonodes in rei . Its cdf and pdf are expressed by [6]:

F f (ri) =e−λR(eλri − eλrei−1)

1− e−λ(R−rei−1)

(4a)

ff (ri) =λe−λ(R−ri)

1− e−λ(R−rei−1)

rei−1 ≤ ri ≤ R. (4b)

Obviously, we obtain a recursive expression where the nexthop distance is a function of the previous empty segment rei−1.For sufficient large λR it follows that rei−1 << R and allowsus to neglect rei−1. In the following we will consider rei−1 = 0and rei−1 = E(rei ). The latter case relies on the idea that theempty gap can be approximated by its mean value to avoidcorrelations.

B. Multi-hop statistics

The mulihop distance is defined by the sum of the single-hop distances ri and is given by dN =

∑Ni=1 ri. In order to

obtain the statistics of the multi-hop distance we focus on thepdf of the sum of independent random variables which can becalculated by the convolution of the single-hop pdf [7].

fdN (dN ) =∫ dN

0

f(r)f(dN − r)dr. (5)

In the following we will introduce the pdf of the multi-hopdistance dN for the different routing strategies.

1) Random hop: Since we deal with uniformly and in-dependently distributed random variables ri ∈ (0, R) whenhaving random hop strategy the sum of them leads to the so-called Irwin-Hall distribution given by:

frdN (dN ) =

∑Nk=0(−1)k

(N

k

)(dN − k)N−1 sgn(dN − k)

2R(N − 1)!0 ≤ dN ≤ NR.

(6)

242

0x

...

1x 2x 1Nx Nx

D

R

RR

Nr2r1r

(a) Random hop strategy

0x

...

1x 3x 1Nx Nx

D

RR

Nr2r1r

2x 4x

3r 4r

(b) Closest hop strategy

0x

...

1x 2x 1Nx Nx

D

R

R

Nr

er11r

2rer2

(c) Furthest hop strategy

Fig. 1. Illustration of the different routing strategies: (a) random, (b) closestand (c) furthest

The special case for N = 2 leads to the triangular dis-tribution as we convolve two uniformly distributed randomvariables. Due to the Central Limit theorem the distributionconverges to the Normal distribution for increasing N .

2) Closest hop: In the closest hop case the single-hopdistribution relies on the PPP which has an exponential distri-bution. Hence, the pdf of the sum yields an Erlang distributionexpressed by:

f cdN (dN ) =λNd

(N−1)N e−λdN

(1− e−λR)NΓ(N)0 ≤ dN ≤ NR, (7)

where Γ(N) = (N − 1)! denotes the Gamma function.Like the random hop strategy the distribution converges to

the Normal distribution as we deal with i.i.d. random variablesri.

3) Furthest hop: In order to determine the multi-hop statis-tics we focus on the distribution of the vacant segment rei . Due

to the fact that dN = NR − deN , ffdN (dN ) can be expressedby ffdN (dN ) = fde

N(NR− dN ).

In case of rei−1 = 0 the vacant segment distributioncorresponds to the closest hop distribution. Hence, ffdN (dN ) =f cdN (NR− dN ):

ffdN (dN ) =λN (NR− dN )(N−1)e−λ(NR−dN )

(1− e−λR)NΓ(N)0 ≤ dN ≤ NR.

(8)

For the case where we assume rei−1 = E(rei ) the distributionis similar. However, due to the definition range of ri withR − rei−1 ≤ ri ≤ R where re0 = 0 we can express the multi-hop pdf by:

ffdN (dN ) =λN (NR− dN )(N−1)e−λ(NR−dN )

(1− e−λR)(1− e−λ(R−E(rei)))N−1Γ(N)

NE(rei ) ≤ dN ≤ NR.(9)

The denominator of the pdf consist of (1 − e−λR)(1 −e−λ(R−E(rei )))N−1 implying that for i = 1 the range of r1

corresponds to 0 ≤ r1 ≤ R and the range for i > 1 isrei−1 ≤ ri ≤ R. Assuming rei−1 = E(rei ) = 0 equation (9)results in equation (8) again .

III. AVERAGE NUMBER OF HOPS

The expected number of hops can be calculated by theassumption that every single-hop distance ri is identicallydistributed with a certain mean E(ri) which we derive in thefollowing for different hopping strategies. Hence, we deter-mine the mean number of hops given an Euclidian distancegenerally by E(N) = D

E(ri).

A. Random hop

The expected progress per hop Er(r) for random hop policyis given by:

Er(ri) =∫ R

0

rifr(ri)dri =

R

2. (10)

B. Closest hop

The expected progress per hop is calculated analogously asin the random case and given by:

Ec(ri) =∫ R

0

rifc(ri)dri =

1− e−λR(λR− 1)λ(1− e−λR)

. (11)

The characteristic of Ec(ri) is depicted in Fig. 2 (a) andindicates the asymptotic behavior. By increasing λ the meanhop distance is decreased as the number of nodes increasesand with this automatically their distances.

Note, that for the closest hopping policy the simple consid-eration that every node in the network serves as a relay nodeleads to the fact, that the number of hops is always equal tothe number of nodes in the network. Hence, by calculatingthe total number of nodes in the network given by the PPPwe obtain automatically the number of hops.

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C. Furthest hop

We obtain for Ef (ri) when neglecting rei−1 :

Ef (ri) = R− 1− e−λR(λR− 1)λ(1− e−λR)

. (12)

Since ri = R−rei and the distribution of the vacant segmentbecomes equal to the closest hop case if rei−1 = 0 we obtainthe mean value by subtracting R from the expected value ofthe closest hop case.

The expected value Ef (ri) when approximating rei−1 by itsmean leads to a numerically solvable expression.

First we calculate the expected value of E(rei−1) = E(rei )given by:

E(rei ) =1− e−λ(R−rei−1)(1 + λ(R− rei−1))

λ(1− e−λ(R−rei−1))

. (13)

As rei−1 = E(rei ) = R − Ef (ri) we can simplify theequation and obtain after some analytical manipulations:

ln(1− λEf (ri)λR− λEf (ri)− 1

)− λEf (ri) = 0. (14)

Hence, for a certain λR we can calculate Ef (ri) numeri-cally. Fig. 2 (b) shows the characteristics of the expected valueand implies the asymptotic behavior. By increasing λR themean hop progress converges to the maximum transmissionrange. It must have the opposite characteristic compared tothe closest hop case.

IV. PROBABILITY DISTRIBUTION P (N |D)

There exist some work in the literature investigating ofhow to obtain the probability distribution of the number ofhops. One method for instance shown in [4] makes use of themulti-hop probability density function which is assumed to begaussian. They derived the expression of P (N |D) by usingBaye’s Rule. Their analytical expression is given by:

P (N |D)

=(∫DD−R fdn−1(x)dx)

∏N−2j=1 (

∫D−R0

fdj (x)dx)∑Nmaxi=Nmin

((∫DD−R fdi−1(x)dx)

∏i−2j=1(

∫D−R0

fdj (x)dx)),

where fdn−1 denotes the multi-hop pdf of the nth-1 hop.The expression indicates that P (N |D) consists of two parts.First the probability that the nth-1 hop falls into the rangeof the destination and the probability that all nth-2 hops donot normalized by the sum of all. Hence, the product of theintegral over fdj for all nth-2 hop must be computed leading toa complex calculation. Thus, in the following we will presenta simple method to obtain P (N |D) for each hopping strategyby making use of the knowledge that the distribution of thetotal number of nodes in the whole network is given by thePoisson process.

In order to determine the probability distribution of thenumber of hops N given a distance D we first refer to theclosest hop policy where we know that the number of hopsare supposed to be equal to the number of nodes in the network

0 5 10 15 200.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Expected single-hop distance for increasing !R

Ec(r

i)

!R

(a) Closest hop strategy

0 5 10 15 200.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Ef(r

i)

!R

(b) Furthest hop strategy

Fig. 2. Characteristic of the expected single-hop distance Ec(ri) and Ef (ri)vs. λR for closest (a) and furthest (b) hop strategy

as we have to consider each single node when hopping to thedestination. Thus, this leads to a trivial solution as the numberof nodes in the network rely on the Poisson process whichgives us the probability of having k nodes within a certainrange D given a density λ. Hence, the number of nodes k inour network follows this distribution and can be expressed byP (k) = e−λD(λD)k

k! . As we assumed the network to be fullyconnected by taking λR sufficient large we can expect theprobability distribution of the number of hops for the closesthop strategy P c(N |D) to be equal to the Poisson process.However, note that the Poisson process delivers non-zerovalues for all k. In order to be able to use the Poisson processto determine P c(N |D) the minimum number of hops given byNmin = D

R has to be considered. For N < Nmin, P c(N |D)is supposed to be zero and a normalization is required. Finally,

244

the probability distribution P c(N |D) can be given by:

P c(N |D) =

0 if N < Nmin

e−λD(λD)N

N !∑Nmax

i=NminP (i|D)

if N ≥ Nmin .

For the random and furthest case the same distribution musthold though not all nodes act as relays. However, in order todetermine P (N |D) we consider the number of nodes bypassedat each hop which is actually a random variable but can bedescribed by its average NB when assuming large number ofhops . Hence, the total number of nodes NN in the networkwhich also is a random variable and given by the Poissonprocess can be expressed by NN = N + NBN ⇔ N =NN

(1+NB). This indicates that the number of hops N is given by

the total number of nodes with the known distribution scaledby a constant. Hence, we can obtain the probability distributionfor the random and furthest policy by using the distribution ofthe closest hop policy and the scaling factor:

P r,f (N |D) =

{0 if N < Nmin1aP

c(Na |D) if N ≥ Nmin ,

where a = 1(1+NB)

. Note, that NB is different for each policyand can be determine by calculating the mean of the Poissonprocess for the expected progress per hop E(r) = r given insection III according to each strategy. Thus, NB is given by:

NB = E(e−λr(λr)NB

NB !).

As we replace the random variable NB by its average valuethe proposed method should deliver better results for largenumber of hops and Euclidean distance, respectively.

V. EVALUATION AND SIMULATION RESULTS

Fig.3 depicts the analytical evaluation of the mean numberof hops E(N) for the different routing strategies compared toexperimental results obtained by simulations. The experimen-tal results are obtained by performing K = 10000 simulationruns per point. In general, the analytical results are very similarto the experimental results

In Fig. 4 the mean number of hops versus the Euclideandistance is evaluated for the independence assumption rei−1 =0, for the case where rei−1 = E(rei ) and the trivial case wherewe assume that hops always occur to the maximum rangeR, (rei = 0). Note, that for the trivial case where rei = 0the mean number of hops E(N) = D

R which stands for thelowest bound. The upper bound is given by the assumptionthat rei−1 = 0 as we reach the smallest progress per hop forthis case. Finally, the case for rei−1 = E(rei ) leads to a meannumber of hops in between the upper and lower bound.

For the evaluation results of P (N |D) we refer to Fig. 5where the experimental and the analytical results for eachhopping strategy are depicted for a Euclidean distance of D=10 m. The diamond markers are the probability of N for theanalytical case mapped onto the experimental results depicted

0 5 10 15 200

5

10

15

20

25

30

35

40

45

50

Analytical and experimental mean number of hops

E(N

)

D [m]

Closest analyticalClosest experimentalRandom analyticalRandom experimentalFurthest analyticalFurthest experimental

Fig. 3. Comparison of E(N) vs. D, R = 1 m, λ = 1 nodes/m

0 5 10 15 200

5

10

15

20

25

30

35

Comparison of mean number of hops for furthest hop strategy

E(N

)

D [m]

rei!1 = 0

rei!1 = E(re

i )rei = 0

Fig. 4. Comparision of E(N) vs. D for furthest hop strategy, rei−1 = 0,

rei−1 = E(re

i ), rei = 0

as boxes with a mean value and the variance of the simulation.The results indicate that for a sufficient large node density anddistance the proposed method yield accurate results comparedto results obtained by simulations even for the furthest andrandom case where we consider NB by its average value.Hence, we showed a very easy and low complex way tocalculate the probability of N accurately.

VI. CONCLUSION

In this paper we present closed-form expressions for thesingle-hop statistics of the distance distribution in linear wire-less sensor networks for different routing policies in termsof the pdf and expected values for the progress per hopand number of hops respectively. Furthermore, we exploit the

245

20 25 30 35 40 45 50 55 60 65

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Closest hopping strategy: P (N |D) given λ =4nodes/m, D/R=10P

(N|D

)

N

(a) Closest hop strategy

12 15 18 21 24 27

0

0.05

0.1

0.15

0.2

Random hopping strategy: P (N |D) given λ =4nodes/m, D/R=10

P(N

|D)

N

(b) Random hop strategy

11 12 13

0

0.1

0.2

0.3

0.4

0.5

Furthest hopping strategy: P (N |D) given λ =4nodes/m, D/R=10

P(N

|D)

N

(c) Furthest hop strategy

Fig. 5. Evaluation of P (N |D) for each hopping policy for D/R=10, λ=4nodes/m

single-hop information to obtain the pdf for multi-hop distancedistribution by applying convolution. We verify our analyticalresults by simulations and show that the mean number ofhops obtained by our analytics are highly consistent withthe experimental results. Furthermore, we tested our proposedmethod to calculate the probability distribution of the numberof hops given an Euclidean distance and showed that it fitsquite good to the experimental results when assuming a large

distance where the number of hops are sufficient large. Thiscan be explained by the choice of the number of bypassednodes which is actually a random variable but was replacedby its average value.

Using the results for one dimensional networks and mapthem to two dimensional ones is our immediate next step.The obtained results can further be used to quantify e.g. thecommunication costs of multi-hop self-localization algorithmsand better evaluate the total cost of relay selection. Theinformation can also be exploit to enhance the work done in[1] by analyzing the aggregate information efficiency takinginto account the number of hops.

ACKNOWLEDGMENT

This work was performed in the EUWB project and co-funded by the European Commission.

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[3] S. Severi, D. Dardari and G. Abreu, Efficient and accurate localizationin multihop networks, Asilomar Conference on Signals, Systems andComputers, 2009.

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