2017 IEEE 14th International Conference on Mobile Ad Hoc and Sensor Systems

A Relay Selection Strategy Based on Power-Law and Exponentially Distributed Contacts in DTNs Tuan Le, Pengyuan Du, Mario Gerla Dept. of Computer Science, UCLA Los Angeles, USA {tuanle, pengyuandu, gerla}@cs.ucla.edu

Abstract—Delay Tolerant Networks (DTNs) are sparse mobile ad-hoc networks in which there is typically no complete path between the source and destination. Although many routing algorithms for DTNs have been proposed, prior works generally focus on utilizing the delivery probability of network nodes and the social network structure for data forwarding. In this work, we investigate the use of the inter-contact time (ICT) distribution to derive a new metric that selects the next relay node with the least expected minimum delay (EMD) among all possible routes to the destination. We address the case of exponential and powerlaw ICTs, which are the most popular assumptions for ICTs that have emerged in recent literature. Extensive simulation results based on the Cabspotting and Cambridge Haggle traces show that our proposed metric can achieve up to 21% higher delivery rate and 23% lower delay than existing schemes. Keywords—Delay Tolerant Networks; Power Law; Exponential; Inter-Contact Time; Routing; Relay Selection

I. I NTRODUCTION Delay Tolerant Networks (DTNs) [1] are characterized as sparsely connected, highly partitioned, and intermittently connected ad-hoc networks. In these challenging environments, end-to-end communication paths between node pairs are rarely available. There are many practical applications of DTNs, including wildlife tracking sensor networks [2], peoplenet [3], ocean sensor networks [4], military networks [5], and vehicular ad-hoc networks [6]. To handle the sporadic connectivity of mobile nodes in DTNs, the store-carry-and-forward method is used. That is, messages are temporarily stored and carried by a node until an appropriate communication opportunity with the next relay hop arises. A key challenge in DTN routing is to determine a proper relay selection strategy in order to minimize the number of packet replicas in the network, and to expedite the data delivery process. Routing for DTNs has been widely studied in the past. To select the best relay node, a variety of network information, including dynamic network information (e.g., location information, traffic information, and encounter information), and static network knowledge (e.g., social relationships among nodes), is utilized. Recently, several relay selection metrics based on the ICTs have been proposed [7], [8], [9]. These works select relays based on the minimum expected delay (MED) among individual routes to the destination, which is typically computed using Dijkstra’s algorithm. With this metric, the addition of new routes does not contribute to the expected 2155-6814/17 $31.00 © 2017 IEEE DOI 10.1109/MASS.2017.12

gain in the delivery probability of a node. That is, a node with hundreds of routes to the destination is not considered to be better than a node with a single lower delay route to the destination. However, fewer routes also imply less robustness. In resource-constrained DTNs, a route may become unavailable due to a variety of reasons. For example, intermediate nodes (such as handheld devices) may run out of memory or battery. When a route fails, the overall delivery delay will depart significantly from its initial estimation, especially in the case of a single route. Thus, MED does not effectively cope with unforeseeable changes in the node contact topology. Furthermore, prior works either ignore the distribution of ICTs or assume exponentially distributed ICTs, which is not applicable to all mobility traces. Recent studies reveal that VANET mobility traces follow an exponential distribution [10], [11], [12], [13], whereas human-carried mobile devices show a truncated power-law distribution [14], [15], [16], [17]. Although less common, other plausible hypotheses for ICTs include LogNormal [18] and hyper-exponential distribution [19]. In this work, we propose an alternative relay selection metric based on the expected minimum delay (EMD). This metric more accurately estimates the actual delay by accounting for the expected gain in the meeting probability when multiple routes are available. That is, the addition of each new route from the encounter node to the destination increases the likelihood of the node being chosen as a relay node. Furthermore, in addition to the case of exponential ICTs, we derive an EMD form for power-law ICTs, which is conceptually more complex. We provide tight lower and upper bounds for EMD (power-law case) using Hermite-Hadamard and CauchySchwarz inequalities, respectively. Lastly, note that computing EMD requires knowledge of the entire network topology (for example, knowledge of the set of all routes to each destination). Acquiring such global information is expensive in terms of the control overhead. Alternatively, we propose a distributed computation algorithm, where each node computes EMD using only the advertised EMD values from its direct neighbors. This not only eliminates the need for storing and exchanging expensive network topology information (mainly, the edges and edge updates), but can also significantly reduce the time and computational complexity of calculating EMD. The rest of the paper is organized as follows. Section II reviews the related work. Section III states our network 117

assumptions. Section IV describes the design of the relay selection strategy. Section V outlines the estimation of the ICT model parameters. Section VI presents the experimental results. Section VII concludes the paper and discusses the future work. II. R ELATED W ORK Many relay selection strategies have been proposed for DTNs. PROPHET [20] uses the past history of encounter events to predict the probability of future encounters. An encounter node is selected as a next relay node if it has a higher delivery predictability than the current node. CAR [21] and MV routing [22] consider the probability of staying at the same location (co-location) as the metric to find the relay. Le et al. [23] propose a queue length control mechanism to select relay nodes on less congested paths to achieve better network load balancing. Since node mobility patterns are highly volatile and hard to control, attempts at exploiting the stable social network structure for data forwarding have emerged. In [24] and [25], the one-hop delivery probability is defined in terms of the social-tie relationship between two nodes. Messages are forwarded to a node with a higher multi-hop delivery probability computed over the social contact graph. SimBetTS [26] uses egocentric centrality and social similarity for relay selection. BubbleRap [27] combines the observed hierarchy of centrality and observed community structure with explicit labels to select the best forwarding nodes. However, these works do not consider using the ICT and its distribution to optimize for relay selection. The first work that takes into account this information is [7], in which the authors introduce the Minimum Expected Delay (MED) metric. MED computes the expected waiting time between pairs of nodes using the known contact schedule, and uses it to represent the delay cost for edges in the contact graph. The least delay cost routing path for each message is then computed at the source and is fixed during the entire lifetime of the message. A major drawback with MED is that it fails to exploit superior edges which become available after the route has been computed. To overcome this drawback, Jones et al. [8] propose a variant of MED, which they call Minimum Estimated Expected Delay (MEED). Instead of using the known contact schedule, MEED uses the observed contact history to estimate the expected waiting time for each potential next hop. Furthermore, MEED allows message carriers other than the source node to recompute the least delay cost path to the destination of the message each time a contact arrives. This allows nodes to discover better relay nodes at a later time after message creation, thus improving the delay. Liu et al. [9] define an Expected Delay (ED) metric, which estimates the expected time it takes to deliver a message with a given remaining hop count. ED assumes that the ICTs between different node pairs are exponentially distributed and independent of each other. ED is computed considering the joint expected delay of all possible descendant forwarders in the forwarding tree. A message is forwarded/replicated to an encounter node with a smaller expected delay to the destination.

Overall, these works tend to select a route (i.e. the next relay node) with the minimum expected delay among individual routes to the destination. However, this metric does not take into account the aggregation of expected delays from multiple routes available. This aggregation is important to cope with unforeseeable changes in the node contact topology, such as when a route suddenly becomes unavailable. In this work, we propose an alternative relay selection metric called Expected Minimum Delay (EMD). With this metric, a relay node that is on multiple paths to the destination tends to be more preferred over a relay node that has a single route to the destination. We formulate EMD for both cases of exponential and power-law ICT distributions, which are the most common assumptions for ICTs in practice. III. A SSUMPTIONS We assume a DTN network with an infinite forwarding bandwidth and storage at each mobile node. Nodes can transfer messages to each other when they are within communication range. We follow a multi-copy model, in which messages are replicated during a transfer while a copy is retained. We assume a long contact duration so that all buffered messages can be replicated to their next relay hops within a single contact. Furthermore, messages are assumed to have the same size and be unfragmented. Once transmitted, a message will always successfully arrive at the encounter node in its entirety. Each message is also associated with a Time-To-Live (TTL) value. After the TTL expires, the message will be discarded by its source node and intermediate nodes. Lastly, we assume that different node pairs have different inter-contact rates under heterogeneous node mobility. Also, note that the distribution of the ICTs varies with the mobility trace. For the Cabspotting trace [28], ICTs follow an exponential distribution with rate λ [12], [29]. For Cambridge Haggle traces [30], ICTs follow a power-law distribution with shape α and scale xmin [14], [31]. IV. R ELAY S ELECTION S TRATEGY In this section, we first describe the formulation of the EMD metric and a general routing strategy based on EMD. We then compute EMD in the case of exponential and power-law ICT distribution, respectively. Lastly, we outline a distributed computation algorithm for EMD. A. General Framework Suppose that a node s has a message m in its buffer that is intended for a destination node d. At each time slot t, s probes the environment to discover other mobile nodes in the vicinity. Node s then updates EM D(s, d) as follows:   n  EM D(s, d) = E min Is,i + EM D(vi , d) i=1

(1)

where Is,i is a random variable representing the inter-contact time between s and its neighbor vi , ∀i ∈ {1, · · · , n} : EM D(vi , d) ≤ EM Dold (s, d), and EM D(d, d) = 0. Let

118

vˆ = arg minni=1 EM D(vi , d). Then, s replicates message m to vˆ subject to the following constraint: EM D(ˆ v , d) < EM Dnew (s, d)

n

i=1

λˆn =

i=1 n 

=

  Pr Is,i > x =



Pr[X > x] =

j=1

min

i=1

ximin

E[X] =

∞ 0

if x < 0 if x ≥ 0

Pr[X > x] =

e j=1 ⎪ ⎪ n ⎪ ⎪ ⎩ e−λi (x−ci ) i=1

if ci < x < ci+1

(6)

if cn < x < ∞

By the definition of expectation, we obtain a closed-form expression for E[X] as follows;

E[X] =

=

∞ 0 c1 0

i=1

ci

+ = c1 +

n−1  i=1

i

e−λj (x−cj ) dx

j=1

∞ n cn i=1

e−λi (x−ci ) dx

 e−λˆn cn +cˆn ˆ 1  −λˆi ci +cˆi e − e−λi ci+1 +cˆi + λˆi λˆn (7)

if xn min + cn < x < ∞

xi+1 min +ci+1



ximin +ci

i  x − cj −αj +1

j=1

∞

xn min +cn

xjmin

dx

n  x − ci −αi +1 i=1

ximin

dx

(10)

Since it is not possible to obtain a closed-form expression for the second and third term in Eq. 10, we instead aim to find a good estimation for E[X]. Our approach is based on the properties of convex functions. Theorem 1: If f and g are convex, both increasing (or decreasing), and positive functions on an interval, then f · g is convex. The proof of Theorem  1 is shown in Appendix A. Consider function fj (x) =

x−cj xjmin

−αj +1

. Its second derivative is:

fj (x) = αj (αj − 1)

n−1  ci+1

(9)

< xi+1 min + ci+1

+

Pr[X > x]dx 1dx +

n−1 

(5)

if 0 < x < c1 −λj (x−cj )

if ximin + ci < x

Pr[X > x]dx

i=1

1 e−λx

(8)

if x ≥ ximin

if 0 < x < x1min + c1

⎪ ⎪ ⎪ −αi +1 ⎪ n  ⎪ x−ci ⎪ ⎩

= (x1min + c1 ) +

Without loss of generality, we assume that 0 ≤ c1 ≤ c2 ≤ · · · ≤ cn . The CCDF of X can then be expressed as: ⎧ 1 ⎪ ⎪ ⎪ i ⎪ ⎨

x ximin

if 0 < x < ximin

−αi +1

By the definition of expectation, E[X] can be obtained as follows:

The CCDF of an exponential random variable Is,i is:   Pr Is,i > x =

1 

⎧ 1 ⎪ ⎪ ⎪  i  ⎪ x−cj −αj +1 ⎪ ⎪ ⎨ j x

B. Exponential ICTs 

λi c i

i=1

Without loss of generality, we assume that 0 ≤ x1min +c1 ≤ + c2 ≤ · · · ≤ xnmin + cn . The CCDF of X is:

Pr[Is,i > x − ci ]

The next subsections will derive CCDF of X and compute E[X] when Is,i follows the exponential and power-law distribution, respectively.

cˆn =

λi ,

x2min

(4)

i=1

λj c j

j=1 n

The CCDF of a power-law random variable Is,i has the following form:

i=1

Pr[Is,i + ci > x]

j=1 n

i

C. Power-Law ICTs

  n Pr[X > x] = Pr min(Is,i + ci ) > x =

cˆi =

λj ,

i=1

(3)

Assume that random variables Is,i are independent to each other. Then the complimentary cumulative distribution function (CCDF) of X is:

n 

i

λˆi =

(2)

If EM D(vi , d) is computed by the neighbors and advertised to s (see Subsection IV-D), then EM D(vi , d) can be represented as a known quantity ci . Let X be a random variable representing the minimum delay over all possible routes through s’s neighbors. Then, X can be expressed as: X = min(Is,i + ci )

where

(x − cj )−αj −1 (xjmin )−αj +1

(11)

For all x in an interval Ij = [xjmin + cj , +∞), f  (x) ≥ 1 > 0. Thus, fj (x) is convex on Ij . Furthermore, (xjmin )2 fj (x) ≥ 1, ∀x ∈ Ij . Thus, provided α > 1, fj (x) is a positive, decreasing, and convex function on Ij .

i Let Fi (x) = j=1 fj (x), where xjmin + cj ≤ ximin + ci . Consider a subinterval Ii = [ximin + ci , xi+1 min + ci+1 ] ∈ Ij . Since fj (x) is positive, decreasing, and convex on Ij , it is

119

also positive, decreasing, and convex on the subinterval Ii . Thus, by Theorem 1, Fi (x) is convex on Ii . The Hermite-Hadamard inequality states that if f (x) is convex on [a, b], then the following chain of inequalities hold: f

a + b 2

≤

1 b−a



b a

f (x)dx ≤

f (a) + f (b) 2

+∞

n  x − ci −αi +1

ximin

i=1

xn min +cn

2 n+1

(12)



 xi

min

Ai xi+1 min +ci+1



ximin +ci

j=1

C· 

xjmin

n b a

2 fi (x)dx ≤ n + 1 i=1

dx ≤

Bi

n

i=1

x1min + c1 +

i  x − cj −αj +1

j=1

ximin +ci

xjmin

2 i+1  

n 



i

fi (a) + fi (b)



1− 1 (14) n

j=1



a

n

+ ci ) +

fj (xi+1 min

(15)

+ ci+1 )



 = 

ximin +ci

ximin +ci

≤

xjmin 

i

 x − c −αj +1 j

xjmin

 x − c −αj +2 xi+1 min +ci+1 j  = j  i −αj + 2 xmin x +ci

i=1

n−1  ⎪ ⎪ Ci ⎩

b a

f 2 (x)dx ·



b a

+ D (19)

g 2 (x)dx

(20)

i



j=1

(16)



 x − c 2(−αj +1) j

120

 = 

xjmin

ximin +ci

dx

(21)

xi+1 +c 1/2 xjmin  x − cj −2αj +3  min i+1  i −2αj + 3 xjmin xmin +ci  

i=1

xjmin

From Eq. 16, we can see that the upper bound in Eq. 15 holds when αj > 2. Similarly, we can obtain an upper bound for the third term in Eq. 10 as follows:

1/2

Gi

xn min +cn

min

dx ≤

xi+1 min +ci+1

+∞

n  x − ci −αi +1

xi+1 min +ci+1

fj (x)dx =

j=1

ximin +ci

j=1

where

2

i  x − cj −αj +1

n



Ci



f (x)g(x)dx

1− 1



xi+1 min +ci+1

Ai ≤ E[X] ≤ x1min + c1 +

⎧n−1  ⎪ ⎪ Bi ⎨

The new upper bounds for the second and third term in Eq. 10 are presented in Eq. 21 and Eq. 22, respectively.

1

fj (x)dx

i=1

b



ximin +ci

fj (ximin

n−1 

i=1

xi+1 min +ci+1



(18)

It is possible to obtain a tighter upper bound for E[X] using the Cauchy-Schwarz inequality:

dx ≤

xi+1 min +ci+1



i=1

Applying Eq. 14 to the set of functions f1 (x), · · · , fi (x) on the interval Ii gives:



In summary, the second term in Eq. 10 has one lower bound Ai and two upper bounds Bi and Ci . The third term has one upper bound D. Thus, E[X] can be bounded as follows:

i=1

xi+1 min +ci+1

n

−αi +2 ximin  ci − cn − xn min = i −αi + 2 xmin

fi (x)dx

n 

1− 1

xn min +cn

(13)

n

a



+ cn ) + 0



+∞

 x − c −αi +1 i ximin

fi (x)dx = xn min +cn

1

b

fi (xn min D

+∞

Fi (ximin + ci ) + Fi (xi+1 min + ci+1 )  2 



(17)

where

i where C = (xi+1 min + ci+1 ) − (xmin + ci ). An alternative upper bound for the middle term in Eq. 13 can be obtained using Corollary 5.2 in [32], which states that if f1 , f2 , · · · , fn are positive, convex, and continuous functions on [a, b], then the following inequality holds:



n  i=1



i  x − cj −αj +1

n

fi (x)dx



 + ci + xi+1 min + ci+1 ≤  2 

1

+∞

n

i=1 n xmin +cn

Applying Eq. 12 to Fi (x) on the interval Ii gives: C · Fi 

dx ≤

n i=1



ximin 

n

dx ≤

 +∞

 x − c 2(−αi +1) 1/2 i ximin

i=1 n xmin +cn

−2αi +3 ximin  ci − cn − xn min i −2αi + 3 xmin  H

1/2 

(22)

Note that these upper bounds hold when α > 3/2. Eq. 19 can then be rewritten as: x1min + c1 +

n−1 

Ai ≤ E[X] ≤ x1min + c1 +

i=1

n−1 

Gi + H

value of λi (tk ). There is no need to keep track of the entire encounter history. Then, λi (tk+1 ) can be updated at the next encounter event tk+1 with node i as follows:

(23)

Finally, we approximate E[X] by taking the average of its tight lower and upper bounds. n−1 

E[X] ≈ x1min + c1 +

i=1

(Ai + Gi ) + H 2

(24)

D. Distributed Computation of EMD EMD can be computed recursively using Eq. 1. However, this centralized method requires knowledge of the entire network topology, which is expensive in terms of the control overhead (information exchange during encounters) and storage requirement. Furthermore, it is computationally complex and may incur significant processing overhead. Thus, to scale with large network size with resource constraints, we propose a distributed computation approach that is similar in nature to the decentralized Distance-Vector routing algorithm [33] in traditional wired networks. In this approach, each node monitors only the link costs to its direct neighbors (i.e. the ICTs - Is,i in Eq. 3). Each node also maintains a vector of expected minimum delay (VEMD) from itself to each known destination. When nodes meet each other, they exchange their VEMDs. Based on the neighbors’ advertised VEMDs (ci in Eq. 3), each node independently updates its own VEMD by computing Eq. 7 or Eq. 24 (depending on the ICT distribution of the mobility trace) in a single iteration. The major overhead of computing these equations comes from sorting the advertised EMDs from a node’s neighbors to obtain a set {0 ≤ c1 ≤ c2 ≤ · · · ≤ cn }. Thus, the time complexity is on the order of O(n log n) if using a quick sort or a merge sort algorithm. V. E STIMATING PARAMETERS OF THE ICT M ODELS In this section, we show how to estimate the inter-contact rate λ of an exponential distribution and the shape α and scale xmin of a power-law distribution.

N λi =  N k=1

Tk

(25)

where {T1 , T2 , · · · , TN } are the inter-contact time samples. It is reasonable to estimate λi this way since, in reality, the intercontact time distribution is quite stable due to the regularity inherent in human mobility patterns [34], [35], [36]. To reduce the storage overhead, λi can be updated incrementally by maintaining the most recent encounter time tk with node i, the current number of samples N , and the current

N λi (tk )

(26)

where TN +1 is the value of the new inter-contact time sample, and TN +1 = tk+1 − tk . B. Power-Law Model We estimate xmin and α using the Kolmogorov-Smirnov (KS) statistic [37] and the Maximum Likelihood Estimator (MLE), respectively. Each node k independently collects and maintains intercontact time samples x = {x1 , x2 , · · · } for each encounter node i. Fig. 1 presents steps (written in R code [38]) to estimate ximin and αi of the power-law ICT between the current node k and an encounter node i. The input x to function EstimateParams is a vector of empirical observations of inter-contact time samples. Line 7 iterates over the ICT dataset and uses each unique data as xmin . Line 9 truncates the dataset to include only data greater than or equal to the chosen xmin . Line 11 estimates α based on the chosen xmin , using the direct MLE:  α=1+n

n  i=1

ln

xi

−1

xmin

(27)

The derivation detail of Eq. 27 is given in Appendix B. Note that the α value on line 11 is not yet the final α value for our fitted power-law model. Line 12 computes the empirical CCDF, which is a step function S(x), defined as the fraction of the full dataset that are greater than or equal to some value x. If the data is sorted in ascending order x1 ≤ x2 ≤ · · · ≤ xn as on line 4, then the corresponding values for the empirical CCDF, 1 13 computes in order, are S(x) = {1, n−1 n , · · · , n }. Line

−α+1 x the fitted theoretical CCDF: P (x) = xmin . Line 14 computes the KS statistic, which is the maximum distance between the CCDFs of the data and the fitted model:

A. Exponential Model The inter-contact rate λi between the current node and an encounter node i can be computed using their encounter history as follows:

N +1 + TN +1

λi (tk+1 ) =

i=1

D = max |S(x) − P (x)| x≥xmin

(28)

Line 19 estimates the final fitted x ˆmin as the value of xmin from the dataset that minimizes D. Line 20 then truncates the dataset based on x ˆmin . Line 23 finds the corresponding fitted α ˆ using Eq. 27. VI. P ERFORMANCE E VALUATION In this section, we conduct extensive simulations using real-life mobility traces to evaluate the performance of our proposed relay selection strategy. The simulation setup, performance metrics, and the evaluation results are presented as follows.

121

1: EstimateParams ← function(x) { 2: xmins = unique(x) 3: dat = numeric(length(xmins)) 4: sdat = sort(x) 5: # Compute dist between empirical and theoretical CCDF 6: 7: for (i in 1:length(xmins)) { 8: xmin = xmins[i] 9: tdat = sdat[sdat >= xmin] 10: n = length(tdat) 11: alpha = 1 + n * (sum(log(tdat/xmin))) ˆ (-1) 12: sx = (n:1)/n 13: px = (tdat/xmin) ˆ (-alpha+1) 14: dat[i] = max(abs(sx-px)) 15: } 16: # Estimate final value of xmin and α 17: 18: D = min(dat[dat>0], na.rm=TRUE) 19: xmin = xmins[which(dat==D)] 20: sdat = x[x >= xmin] 21: sdat = sort(sdat) 22: n = length(sdat) 23: alpha = 1 + n * (sum(log(sdat/xmin))) ˆ (-1) 24: 25: return(list(“xmin”=xmin, “alpha”=alpha)) 26: }

# Obtain a vector of all unique values of x # Create a vector to store KS statistics # Sort values of x in ascending order

# Choose next xmin candidate # Truncate data below this xmin value # # # #

Estimate alpha using direct MLE Construct a vector of empirical CCDF values Construct a vector of fitted theoretical CCDF values Compute the KS statistic

# # # #

Find the smallest D value Find the corresponding xmin value that minimizes D Truncate data below this xmin value Sort values of x in ascending order

# Estimate alpha based on the fitted xmin

Fig. 1. Estimating parameters xmin and α of a power-law ICT distribution. TABLE I. Characteristics of the Cabspotting trace

A. Simulation Setup We implement the proposed relay selection strategy using the opportunistic network simulator ONE 1.5.1 [39]. To obtain meaningful results, we use the Cabspotting trace [28] and traces from the Cambridge Haggle dataset [30]. Cabspotting contains GPS coordinates of 536 taxis collected over 30 days in the San Francisco Bay Area. The ICTs in this trace have been previously shown to follow an exponential distribution [12], [29]. Table I shows the statistics of Cabspotting. Cambridge Haggle dataset contains a total of five traces of Bluetooth device connections by people carrying mobile devices (iMotes) for a number of days. The traces are Intel, Cambridge, Infocom, Infocom2006, and Content. However, we do not include the Intel trace in the evaluation because it has a very small number of mobile iMotes (only 8 iMotes). These traces are collected by different groups of people in office environments, conference environments, and city environments, respectively. Bluetooth contacts are classified into two groups: (1) internal contacts - iMotes’ sightings of other iMotes, and (2) external contacts - iMotes’ sightings of other types of Bluetooth devices (non-iMotes). Note that these traces contain no record of contact between non-iMotes. Furthermore, the ICTs in these traces follow a power-law distribution [14], [31]. Table II shows the statistics of the four Cambridge Haggle traces. We assume nodes have an infinite buffer capacity. Each node initially has five source messages in its buffer. Each message is of the same size of 10KB, and is intended for a random destination node in the network. Furthermore, we assume that messages have a homogeneous TTL value, which is varied for different simulations. For statistical convergence, the results

Trace

Contacts

Duration (days)

Devices

Cabspotting

111,153

30

536

TABLE II. Characteristics of four Cambridge Haggle traces Trace

Contacts

Duration (d.h:m.s)

iMotes

NoniMotes

Cambridge

6,732

6.1:34.2

12

211

Infocom

28,216

2.22:52.56

41

233

Infocom2006

227,657

3.21:43.39

98

4,519

Content

41,587

23.19:50.18

54

11,418

reported in this section are averaged from 20 simulation runs. We evaluate the performance of the following relay selection strategies: • Epidemic routing [40] is a flooding-based routing algorithm. It is optimal in terms of delivery ratio and delay, but is very inefficient in terms of network resource consumption and the amount of network traffic generated. • PROPHET [20] selects relay nodes with higher delivery predictability to the destination. The delivery predictability is inferred using the past history of encounter events. In our simulations, we use the same parameters as specified by the authors in [20]. That is, {Pinit , β, γ} = {0.75, 0.25, 0.98}. • MEED [8] selects the route with the minimum expected delay among individual routes to the destination. However, it does not take into account the aggregation of expected delays from multiple routes available.

122

0.8

2

Epidemic PROPHET MEED EMD

0.8 0.7

0.5 0.4

1.5

1

0.3 0.5 0.2 0.1 0.5

Epidemic PROPHET MEED EMD

0.8 0.7

0.5 0.4

2

4

6 8 TTL (days)

10

12

(a) Delivery ratio

14

0 0.5

1

2

4

6 8 TTL (days)

10

12

0.3 0.2 0.1 1

1.5

2

2.5

3 3.5 TTL (days)

4

4.5

5

5.5

0 0.5

6

0.8

2

2.5

(b) Infocom

Epidemic PROPHET MEED EMD

0.45 0.4

Epidemic PROPHET MEED EMD

0.35 Delivery ratio

0.6 Delivery ratio

1.5 TTL (days)

0.5

0.9

0.7

EMD (our proposed metric) selects the route with the least expected minimum delay among all possible routes to the destination. Unlike MEED, EMD accounts for the expected gain in the meeting probability when multiple routes are available.

1

(a) Cambridge

(b) Average delay

Fig. 2. Performance comparison using Cabspotting trace.

•

0.4

0.2

0 0.5

14

0.5

0.3

0.1 1

Epidemic PROPHET MEED EMD

0.6

0.6 Delivery ratio

Average delay (days)

Delivery ratio

0.7 0.6

0.9

0.9

2.5 Epidemic PROPHET MEED EMD

Delivery ratio

0.9

0.5 0.4 0.3

0.3 0.25 0.2 0.15

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0.1

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0.05

0 0.5

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1.5

2 TTL (days)

2.5

3

0 4

3.5

6

8

10

12 14 TTL (days)

16

18

20

22

(d) Content

(c) Infocom2006

Fig. 3. Delivery ratio vs message TTL in Cambridge Haggle traces.

B. Evaluation Metrics We use the following metrics for evaluation:

1.5

3

2.5

Epidemic PROPHET MEED EMD

Epidemic PROPHET MEED EMD Average delay (days)

•

Delivery ratio: the proportion of messages that have been delivered out of the total messages created. Average delay: the average interval of time for each message to be delivered from the source to destination.

Average delay (days)

•

2

1.5

1

1

0.5

0.5

C. Comparative Results

0 0.5

123

1.5

2

2.5

3 3.5 TTL (days)

4

4.5

5

5.5

0 0.5

6

1

1.6

2

2.5

14 Epidemic PROPHET MEED EMD

12

Average delay (days)

1.8

1.5 TTL (days)

(b) Infocom

(a) Cambridge 2

Average delay (days)

The results from Cabspotting trace and Cambridge Haggle traces are presented as follows: 1) Cabspotting Trace: Fig. 2a compares the delivery ratio among the schemes. As expected, Epidemic has the highest delivery ratio of around 82%. This is achieved at the expense of very high network resource consumption, and thus is not practical. EMD comes second with 72% delivery rate. It outperforms PROPHET and MEED by 8% and 10%, respectively. Fig. 2b depicts the average delay. Again, Epidemic has the best delivery delay, followed by EMD. EMD successfully delivers a message by 13% and 17% less time than MEED and PROPHET, respectively. 2) Cambridge Haggle Traces: Fig. 3 shows the delivery ratio of the compared schemes under four different human mobility traces. EMD achieves a delivery rate of up to 12% and 21% higher than PROPHET and MEED, respectively. The improvements of EMD are more significant in environments with more regular mobility patterns such as a campus environment (Fig. 3a) and city environment (Fig. 3d), and less significant in environments with relatively random mobility such as conference environments (Fig. 3b and 3c). In terms of the average delay, Fig. 4 shows that EMD successfully delivers messages by up to 20% and 23% less time than MEED and PROPHET, respectively. Similar to the delivery rate results, the improvements of EMD are more profound in Cambridge and Content traces, which feature more regular mobility patterns.

1

1.4 1.2 1 0.8 0.6

Epidemic PROPHET MEED EMD

10 8 6 4

0.4 2 0.2 0 0.5

1

1.5

2 TTL (days)

2.5

(c) Infocom2006

3

3.5

0 4

6

8

10

12 14 TTL (days)

16

18

20

22

(d) Content

Fig. 4. Delivery delay vs message TTL in Cambridge Haggle traces.

VII. C ONCLUSION AND F UTURE W ORK In this paper, we proposed a new relay selection metric based on the expected minimum delay (EMD). EMD more accurately estimates the actual delay to the destination compared to the commonly used minimum expected delay metric. We derived EMD for both cases of exponential and powerlaw ICTs, which are the most popular assumptions for ICTs in the literature. Experimental results using Cabspotting and Cambridge Haggle traces show that EMD can achieve up to 21% higher delivery rate and 23% lower delay than existing schemes. In future work, we plan to derive an expression for EMD under the LogNormal [18] and hyper-exponential distribution

[19] of the ICTs. We also plan to relax the assumptions of infinite forwarding bandwidth and storage, and evaluate EMD in conjunction with buffer management policies. R EFERENCES [1] K. Fall, “A delay-tolerant network architecture for challenged internets,” in Proceedings of the 2003 conference on Applications, technologies, architectures, and protocols for computer communications. ACM, 2003, pp. 27–34. [2] P. Juang, H. Oki, Y. Wang, M. Martonosi, L. S. Peh, and D. Rubenstein, “Energy-efficient computing for wildlife tracking: Design tradeoffs and early experiences with zebranet,” in ACM Sigplan Notices, vol. 37, no. 10. ACM, 2002, pp. 96–107. [3] M. Motani, V. Srinivasan, and P. S. Nuggehalli, “Peoplenet: engineering a wireless virtual social network,” in Proceedings of the 11th annual international conference on Mobile computing and networking. ACM, 2005, pp. 243–257. [4] J. Partan, J. Kurose, and B. N. Levine, “A survey of practical issues in underwater networks,” ACM SIGMOBILE Mobile Computing and Communications Review, vol. 11, no. 4, pp. 23–33, 2007. [5] Z. Lu and J. Fan, “Delay/disruption tolerant network and its application in military communications,” in Computer Design and Applications (ICCDA), 2010 International Conference on, vol. 5. IEEE, 2010, pp. V5–231. [6] J. Ott and D. Kutscher, “A disconnection-tolerant transport for drivethru internet environments,” in INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings IEEE, vol. 3. IEEE, 2005, pp. 1849–1862. [7] S. Jain, K. Fall, and R. Patra, Routing in a delay tolerant network. ACM, 2004, vol. 34, no. 4. [8] E. P. Jones, L. Li, J. K. Schmidtke, and P. A. Ward, “Practical routing in delay-tolerant networks,” Mobile Computing, IEEE Transactions on, vol. 6, no. 8, pp. 943–959, 2007. [9] C. Liu and J. Wu, “On multicopy opportunistic forwarding protocols in nondeterministic delay tolerant networks,” Parallel and Distributed Systems, IEEE Transactions on, vol. 23, no. 6, pp. 1121–1128, 2012. [10] H. Zhu, L. Fu, G. Xue, Y. Zhu, M. Li, and L. M. Ni, “Recognizing exponential inter-contact time in vanets,” in INFOCOM, 2010 Proceedings IEEE. IEEE, 2010, pp. 1–5. [11] K. Lee, Y. Yi, J. Jeong, H. Won, I. Rhee, and S. Chong, “Maxcontribution: On optimal resource allocation in delay tolerant networks,” in INFOCOM, 2010 Proceedings IEEE. IEEE, 2010, pp. 1–9. [12] E. Wang, Y. Yang, and J. Wu, “A knapsack-based buffer management strategy for delay-tolerant networks,” Journal of Parallel and Distributed Computing, vol. 86, pp. 1–15, 2015. [13] T. Le, H. Kalantarian, and M. Gerla, “A joint relay selection and buffer management scheme for delivery rate optimization in dtns,” in World of Wireless, Mobile and Multimedia Networks (WoWMoM), 2016 IEEE 17th International Symposium on A. IEEE, 2016, pp. 1–9. [14] A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, and J. Scott, “Impact of human mobility on opportunistic forwarding algorithms,” Mobile Computing, IEEE Transactions on, vol. 6, no. 6, pp. 606–620, 2007. [15] I. Rhee, M. Shin, S. Hong, K. Lee, S. J. Kim, and S. Chong, “On the levy-walk nature of human mobility,” IEEE/ACM Transactions on Networking (TON), vol. 19, no. 3, pp. 630–643, 2011. [16] T. Karagiannis, J.-Y. Le Boudec, and M. Vojnovi´c, “Power law and exponential decay of intercontact times between mobile devices,” Mobile Computing, IEEE Transactions on, vol. 9, no. 10, pp. 1377–1390, 2010. [17] T. Le, H. Kalantarian, and M. Gerla, “A buffer management strategy based on power-law distributed contacts in delay tolerant networks,” in Computer Communication and Networks (ICCCN), 2016 25th International Conference on. IEEE, 2016, pp. 1–8. [18] P.-U. Tournoux, J. Leguay, F. Benbadis, V. Conan, M. D. De Amorim, and J. Whitbeck, “The accordion phenomenon: Analysis, characterization, and impact on dtn routing,” in INFOCOM 2009, IEEE. IEEE, 2009, pp. 1116–1124. [19] C. Boldrini, M. Conti, and A. Passarella, “Performance modelling of opportunistic forwarding under heterogenous mobility,” Computer Communications, vol. 48, pp. 56–70, 2014. [20] A. Lindgren, A. Doria, and O. Schel´en, “Probabilistic routing in intermittently connected networks,” ACM SIGMOBILE mobile computing and communications review, vol. 7, no. 3, pp. 19–20, 2003.

[21] M. Musolesi and C. Mascolo, “Car: context-aware adaptive routing for delay-tolerant mobile networks,” Mobile Computing, IEEE Transactions on, vol. 8, no. 2, pp. 246–260, 2009. [22] B. Burns, O. Brock, and B. Levine, “Mv routing and capacity building in disruption tolerant networks,” in INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings IEEE, vol. 1, March 2005, pp. 398–408 vol. 1. [23] T. Le and M. Gerla, “A load balanced social-tie routing strategy for dtns based on queue length control,” in Military Communications Conference, MILCOM 2015-2015 IEEE. IEEE, 2015, pp. 383–387. [24] T. Le, H. Kalantarian, and M. Gerla, “A novel social contact graph-based routing strategy for workload and throughput fairness in delay tolerant networks,” Wireless Communications and Mobile Computing, vol. 16, no. 11, pp. 1352–1362, 2016. [25] ——, “A novel social contact graph based routing strategy for delay tolerant networks,” in Wireless Communications and Mobile Computing Conference (IWCMC), 2015 International. IEEE, 2015, pp. 13–18. [26] E. M. Daly and M. Haahr, “Social network analysis for information flow in disconnected delay-tolerant manets,” Mobile Computing, IEEE Transactions on, vol. 8, no. 5, pp. 606–621, 2009. [27] P. Hui, J. Crowcroft, and E. Yoneki, “Bubble rap: Social-based forwarding in delay-tolerant networks,” Mobile Computing, IEEE Transactions on, vol. 10, no. 11, pp. 1576–1589, 2011. [28] M. Piorkowski, N. Sarafijanovic-Djukic, and M. Grossglauser, “CRAWDAD dataset epfl/mobility (v. 2009-02-24),” Downloaded from http://crawdad.org/epfl/mobility/20090224/cab, Feb. 2009, traceset: cab. [29] A. Krifa, C. Barakat, and T. Spyropoulos, “Optimal buffer management policies for delay tolerant networks,” in Sensor, Mesh and Ad Hoc Communications and Networks, 2008. SECON’08. 5th Annual IEEE Communications Society Conference on. IEEE, 2008, pp. 260–268. [30] J. Scott, R. Gass, J. Crowcroft, P. Hui, C. Diot, and A. Chaintreau, “CRAWDAD dataset cambridge/haggle (v. 2006-09-15),” Downloaded from http://crawdad.org/cambridge/haggle/20060915, Sep. 2006. [31] J. Leguay, A. Lindgren, J. Scott, T. Friedman, and J. Crowcroft, “Opportunistic content distribution in an urban setting,” in Proceedings of the 2006 SIGCOMM workshop on Challenged networks. ACM, 2006, pp. 205–212. ´ and T. F. M´ori, “Sharp integral inequalities for products [32] V. O. CSISZAR of convex functions,” JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only], vol. 8, no. 4, pp. Paper–No, 2007. [33] Wikipedia, “Distance vector routing protocol.” [Online]. Available: https://en.wikipedia.org/wiki/Distance-vector routing protocol [34] M. C. Gonzalez, C. A. Hidalgo, and A.-L. Barabasi, “Understanding individual human mobility patterns,” Nature, vol. 453, no. 7196, pp. 779–782, 2008. [35] C. Song, Z. Qu, N. Blumm, and A.-L. Barab´asi, “Limits of predictability in human mobility,” Science, vol. 327, no. 5968, pp. 1018–1021, 2010. [36] S. Sch¨onfelder and K. W. Axhausen, Urban rhythms and travel behaviour: spatial and temporal phenomena of daily travel. Ashgate Publishing, Ltd., 2010. [37] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical recipes: The art of scientific computing (cambridge,” 1992. [38] R. C. Team, “R language definition,” 2000. [39] A. Ker¨anen, J. Ott, and T. K¨arkk¨ainen, “The ONE Simulator for DTN Protocol Evaluation,” in SIMUTools ’09: Proceedings of the 2nd International Conference on Simulation Tools and Techniques. New York, NY, USA: ICST, 2009. [40] A. Vahdat, D. Becker et al., “Epidemic routing for partially connected ad hoc networks,” Technical Report CS-200006, Duke University, Tech. Rep., 2000.

A PPENDIX A. Proof of Theorem 1 By definition, if f and g are positive and convex on an interval I, then the following Jensen’s inequality holds:

124

∀x1 , x2 ∈ I,∀t ∈ [0, 1] : f (tx1 + (1 − t)x2 ) ≤ tf (x1 ) + (1 − t)f (x2 ) g(tx1 + (1 − t)x2 ) ≤ tg(x1 ) + (1 − t)g(x2 )

(29)

Taking the product of two convex functions of the weighted means from Eq. 29, we obtain: f (tx1 + (1 − t)x2 ) · g(tx1 + (1 − t)x2 ) ≤ [tf (x1 ) + (1 − t)f (x2 )] · [tg(x1 ) + (1 − t)g(x2 )] (30) = tf (x1 )g(x1 ) + (1 − t)f (x2 )g(x2 ) + t(1 − t) · (f (x2 ) − f (x1 )) · (g(x1 ) − g(x2 ))

Considering the third term in Eq. 30, we have t(1 − t) > 0 since t ∈ [0, 1]. If f and g are both increasing (or decreasing), then (f (x2 ) − f (x1 )) · (g(x1 ) − g(x2 )) ≤ 0. Thus, the third term is less than or equal to zero. Eq. 30 can then be rewritten as: f (tx1 + (1 − t)x2 ) · g(tx1 + (1 − t)x2 ) ≤ (31) tf (x1 )g(x1 ) + (1 − t)f (x2 )g(x2 )

Eq. 31 shows that the product f · g satisfies Jensen’s inequality. Thus, f · g is convex. B. Derive the MLE Estimator for the Power-Law α Parameter Consider a power-law distribution described by a probability density function p(x): p(x) =

α−1 xmin



xi

−α (32)

xmin

Assume that we already know the lower bound xmin at which power-law behavior holds. Given a dataset containing n observations xi ≥ xmin , the probability that the data were drawn from the model (i.e., the likelihood of the data given the model) is: p(x|α) =

 −α n α−1 xi xmin xmin i=1

(33)

Taking the logarithm of the likelihood function, we obtain: L = ln p(x|α) = ln =

n 

 −α n α−1 xi xmin xmin i=1

ln(α − 1) − ln xmin − α ln

i=1

= n ln(α − 1) − n ln xmin − α

n  i=1

xi



xmin ln

(34)

xi xmin

Then, we differentiate the log likelihood with respect to α and equate to 0: ∂L =0 ∂α n  n xi ⇔ − ln =0 α − 1 i=1 xmin

(35)

Solving for α, we obtain the MLE for the shape parameter:  α=1+n

n  i=1

ln

xi xmin

−1 (36)

125