Contact Duration-Aware Routing in Delay Tolerant Networks Tuan Le, Mario Gerla Dept. of Computer Science, UCLA Los Angeles, USA {tuanle, 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. While much work has been done in the design of forwarding algorithms, little work has focused on studying forwarding under the presence of short contact durations. In this paper, we study a single-copy contact duration-aware (CDA) routing strategy. We address two key issues: (1) to which next hop relay node should messages be forwarded and (2) in which order should messages be forwarded. To reduce the transmission cost, we select relay nodes from both current and past contacts based on the one-hop and twohop delivery probability, respectively. We derive the delivery probability from the distribution of contact duration time and inter-contact time. For the message scheduling, messages with the highest delivery probability are prioritized to be transmitted first. Extensive simulation results based on the Cabspotting trace show that our scheme can achieve up to 13% higher delivery rate, 12% lower delay, and 23% lower transmission cost compared to other routing strategies. Keywords—Delay Tolerant Networks; Routing; Relay Selection; Contact Duration; Inter-Contact Time

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], [3], peoplenet [4], ocean sensor networks [5], [6], military networks [7], [8], and vehicular ad-hoc networks [9], [10]. To handle the sporadic connectivity of mobile nodes in DTNs, the storecarry-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 expedite the data delivery process while satisfying any imposed constraints. Routing for DTNs has been widely studied in the past. However, most of the proposed routing schemes assume long contact durations such that all buffered messages can be transferred within a single contact. Recent studies suggest that contact duration is often short in realistic DTN environments [11], [12]. For example, when hand-held devices communicate via Bluetooth that has a typical wireless range of about 10 m, the contact duration tends to be as short as several seconds if the users are walking. For high speed vehicles that communicate via WiFi (802.11g), which has a longer range

(up to 38 m indoors and 140 m outdoors), the contact duration is still short. In the presence of short contact durations, there are two key issues that must be addressed. First is the relay selection issue. We need to select relay nodes that will contact the message’s destination long enough so that the entire message can be successfully transmitted. Second is the message scheduling issue. Since not all messages can be exchanged between nodes within a single contact, it is important to schedule the transmission of messages in such a way that will maximize the network delivery ratio. In this paper, we develop a theoretical framework for the relay selection strategy and message scheduling policy. At the core of this framework is the one-hop and two-hop delivery probability that are derived using a variety of network information, including the distribution of contact duration time, intercontact time, message’s Time-To-Live (TTL), heterogeneous node mobility, and varied message sizes. The one-hop delivery probability is the delivery probability from the current message carrier to the message’s destination via a node that is currently in contact. The two-hop delivery probability is via a node that was met in the past and can be met again in the future. Our main motivation for considering the two-hop delivery probability is to reduce the transmission cost, which in turn helps conserve the scarce DTN network bandwidth. Intuitively, if a node is likely to meet a better relay node than the currently contacted node, it should hold the message until that meeting occurs. By skipping slightly better intermediate relay nodes, the forwarding path can be shortened. To ensure that the added wait time still satisfies the message delivery deadline, we include the inter-contact time (ICT) as a parameter in our theoretical model. We assume exponentially distributed inter-contact times and contact duration times, which is a popular assumption for VANET mobility traces [13], [14], [15], [16]. Lastly, note that the buffer management issue, which determines which messages to be dropped first when the buffer is full, is also an important issue in DTN routing. In this work, since we are mainly concerned with relay selection and since the single-copy model imposes little storage overhead on mobile nodes, we will ignore the issue of buffer management and leave it as future work. The rest of the paper is organized as follows. Section II reviews the related work. Section III states our network assumptions. Section IV describes the theoretical framework for relay selection and message scheduling. Section V presents the experimental results. Section VI concludes the paper and discusses the future work.

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TABLE I. Notations

II. R ELATED W ORK Many relay selection strategies have been proposed for DTNs. PROPHET [17] 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 [18] and MV routing [19] consider the probability of staying at the same location (co-location) as the metric to find the relay. LeBrun et al. [20] use location information of nodes to forward data closer to the destination. 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 [21] and [22], 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 multihop delivery probability computed over the social contact graph. SimBetTS [23] uses egocentric centrality and social similarity for relay selection. BubbleRap [24] combines the observed hierarchy of centrality and observed community structure with explicit labels to select the best forwarding nodes. Several works that use the ICT and its distribution to optimize for relay selection have also been proposed. The first work that takes into account this information is [25], 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. [26] 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. Le et al. [27] define an Expected Minimum Delay (EMD) metric, which takes into account the expected gain in the meeting probability when multiple routes are available. 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. However, these works do not consider short contact duration and thus their applicability is rather limited in the real world. In this work, we formulate a new relay selection strategy based on the exponential distribution of contact duration time and inter-contact time. Furthermore, we derive the two-hop delivery probability through past contacted nodes, which gives the message carrier more choices of relay candidates and also

𝑇 𝑇 𝐿𝑖

Initial Time To Live of message 𝑖

𝑇𝑖

Elapsed time since the creation of message 𝑖

𝑅𝑖

Remaining lifetime of message 𝑖, (𝑅𝑖 𝑇 𝑇 𝐿𝑖 − 𝑇 𝑖 )

𝑤𝑖

Size of message 𝑖

𝐵

Homogeneous communication bandwidth between two nodes

𝐻𝑖

Contact duration time required for a successful transmission of message 𝑖, (𝐻𝑖 = 𝑤𝑖 /𝐵)

𝑋𝑘𝑎,𝑏

Random variable denoting the 𝑘th inter-contact time between node 𝑎 and 𝑏

𝑌𝑘𝑎,𝑏

Random variable denoting the 𝑘th contact duration time between node 𝑎 and 𝑏

𝜆𝑎,𝑏

Inter-contact rate between node 𝑎 and 𝑏

𝜃𝑎,𝑏

Contact-duration rate between node 𝑎 and 𝑏

=

helps shorten the relay path by skipping slightly better relay nodes. 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. In addition, we assume short contact duration and that the inter-contact time is much longer than the contact duration time. We follow a single-copy model, in which, at any point in time, there is at most one copy of the message in the network. Messages are assumed to have the same size and be unfragmented. A message is successfully transmitted if the contact duration is greater than or equal to the message size divided by the available communication bandwidth. Otherwise, the message needs to be re-transmitted in its entirety in the next contact. Each message is associated with a finite Time-ToLive (TTL) value. After the TTL expires, the message will be discarded by its carrier node. Lastly, regarding the distribution of inter-contact time and contact duration time between nodes, recent studies suggest that VANET mobility traces follow an exponential distribution [13], [14], [15], [16], whereas humancarried mobile devices show a truncated power-law distribution [28], [29], [30], [31]. In this paper, we assume an exponentially distributed inter-contact time and contact duration time with rate 𝜆 and 𝜃, respectively, and that different node pairs have different inter-contact rates and contact duration rates under heterogeneous node mobility. The Cabspotting trace [32] used to evaluate our scheme fits best with this assumption. IV. T HEORETICAL F RAMEWORK In this section, we first derive the one-hop and two-hop delivery probability under the presence of short contact durations. We then show how to estimate the inter-contact rate 𝜆 and contact-duration rate 𝜃 of an exponential distribution. Lastly, we outline the relay selection strategy and message scheduling policy based on the derived delivery probabilities.

A. Direct One-Hop Delivery Probability Table I outlines the notations used throughout this section. As stated earlier in section III, a message 𝑖 is successfully transmitted in the 𝑘th meeting if and only if 𝑌𝑘 ≥ 𝐻𝑖 . Otherwise, message 𝑖 needs to be re-transmitted in its entirety in the next contact. Furthermore, we assume 𝑋𝑘 ≫ 𝑌𝑘 . Consider the current message carrier node 𝑠 and destination node 𝑑 of message 𝑖. The probability that message 𝑖 has not been successfully delivered to its destination until the 𝑛th meeting is: (

𝑃𝑖 (𝑛) = 𝑃 0 <

𝑛 ∑

𝑋𝑘𝑠,𝑑 < 𝑅𝑖 ,

𝑘=1

𝑠,𝑑 (0 < 𝑌1𝑠,𝑑 < 𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑛−1 < 𝐻𝑖 ), 𝑌𝑛𝑠,𝑑 ≥ 𝐻𝑖

)

𝑛 ∑

𝑋𝑘𝑠,𝑑

𝑘=1

𝑋𝑘𝑠,𝑑 +

)

𝑛−1 ∑

𝑌𝑘𝑠,𝑑 < 𝑅𝑖

(3)

𝑘=1

However, since we assume 𝑋𝑘𝑠,𝑑 ≫ 𝑌𝑘𝑠,𝑑 , we can simplify Eq. 3 by dropping the 𝑌𝑘𝑠,𝑑 component. It has been established in literature that the sum of 𝑛 independent and identically (i.i.d) exponential ran∑𝑛 distributed 𝑠,𝑑 𝑋 is gamma distributed with dom variables 𝑆𝑛 = 𝑘=1 𝑘 parameters (𝛼 = 𝑛, 𝛽 = 𝜆𝑠,𝑑 ). Since 𝑛 ≥ 1, the distribution of 𝑆𝑛 is an Erlang distribution [33], which has the following cumulative distribution function (CDF): 𝑛−1 𝑛 ) ( ∑ ∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 −𝜆 𝑅 𝑃 0< 𝑋𝑘𝑠,𝑑 < 𝑅𝑖 = 1 − 𝑒 𝑠,𝑑 𝑖 𝑘! 𝑘=1

(4)

𝑘=0

𝑠,𝑑 b) (0 < 𝑌1𝑠,𝑑 < 𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑛−1 < 𝐻𝑖 ): This event states that the contact duration during the 1st, 2nd, ⋅ ⋅ ⋅ , (𝑛 − 1)th meeting between 𝑠 and destination 𝑑 of message 𝑖 is less than 𝐻𝑖 . Thus, message 𝑖 fails to be transmitted during the first 𝑛 − 1 meetings. 𝑠,𝑑 Since 𝐻𝑖 > 0 and 𝑌1𝑠,𝑑 , ⋅ ⋅ ⋅ , 𝑌𝑛−1 are i.i.d exponential random variables with parameter 𝜃𝑠,𝑑 , then:

) ( 𝑠,𝑑 𝑃 0 < 𝑌1𝑠,𝑑 < (𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑛−1 < 𝐻(𝑖 ) ) 𝑠,𝑑 < 𝐻𝑖 = 𝑃 0 < 𝑌1𝑠,𝑑 < 𝐻𝑖 ⋅ ⋅ ⋅ 𝑃 0 < 𝑌𝑛−1 = (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ) ⋅ ⋅ ⋅ (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ) = (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 )𝑛−1

=𝑒

(6)

−𝜃𝑠,𝑑 𝐻𝑖

Combining the results from Eq. 4, 5, and 6, Eq. 2 can be rewritten as: ( 1−

𝑛−1 ∑ 𝑘=0

Next, we will explain the three components of 𝑃𝑖 (𝑛) and show how to compute each of them. ∑𝑛 𝑠,𝑑 < 𝑅𝑖 : This event ensures that a) 0 < 𝑘=1 𝑋𝑘 message 𝑖 will not expire before the 𝑛th meeting. More precisely, the condition should be: 𝑛 ∑

= 1 − (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 )

𝑃𝑖 (𝑛) =

𝑃𝑖 (𝑛) = 𝑃 0 < < 𝑅𝑖 𝑘=1 ) ( ) ( 𝑠,𝑑 < 𝐻𝑖 ⋅ 𝑃 𝑌𝑛𝑠,𝑑 ≥ 𝐻𝑖 ⋅ 𝑃 0 < 𝑌1𝑠,𝑑 < 𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑛−1 (2)

0<

) ( ) ( 𝑃 𝑌𝑛𝑠,𝑑 ≥ 𝐻𝑖 = 1 − 𝑃 𝑌𝑛𝑠,𝑑 < 𝐻𝑖

(1)

Since 𝑋𝑘 and 𝑌𝑘 are independent, Eq. 1 can be re-written as: (

c) 𝑌𝑛𝑠,𝑑 ≥ 𝐻𝑖 : This event ensures that the duration of the 𝑛th meeting between 𝑠 and destination 𝑑 of message 𝑖 lasts long enough for the entire message to be successfully transmitted. The complimentary cumulative distribution function (CCDF) of 𝑌𝑛𝑠,𝑑 can be computed as:

) (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 −𝜆𝑠,𝑑 𝑅𝑖 𝑒 𝑘! ( )𝑛−1 −𝜃𝑠,𝑑 𝐻𝑖 ⋅ 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ⋅𝑒

(7)

Then, the probability for successfully transmitting message 𝑖 is: 𝑃𝑖 =

∞ ∑

𝑃𝑖 (𝑛)

(8)

𝑛=1

As shown in Appendix A, 𝑃𝑖 can be simplified to: 𝑃𝑖 = 1 − 𝑒−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖

∞ [ ∑ ∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 ] ( )𝑛−1 𝑛−1 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 𝑘! 𝑛=1 𝑘=0 (9)

Appendix B further presents the lower and upper bound for 𝑃𝑖 : ] [ 𝑒−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖 (1−𝑒−𝜃𝑠,𝑑 𝐻𝑖 )𝜆𝑠,𝑑 𝑅𝑖 − 1 ≤ 𝑃𝑖 ≤ 𝑒 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ] (10) [ −𝜃𝑠,𝑑 𝐻𝑖 −𝜃 𝐻 𝑒 (1−𝑒 𝑠,𝑑 𝑖 )𝜆𝑠,𝑑 𝑅𝑖 − 1 𝑒 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖

B. Two-Hop Delivery Probability via Nodes Not in Contact This is the probability that a message 𝑖 is delivered indirectly by the current carrier node 𝑠 to destination 𝑑 via an intermediate node 𝑣 that is not currently in contact. To compute this probability, 𝑠 needs to know two key information. First is the elapsed time for 𝑠 to encounter an intermediate node that is not currently in contact. Typically, this information is only available for nodes that 𝑠 has met in the past based on the inter-contact rate information. It is not practical to derive this information for nodes that 𝑠 has never met. Second is the inter-contact rate between the intermediate node and destination of message 𝑖. This requires global network state exchange, in which nodes exchange and merge the list of node encounters during their contacts. Each entry in the node encounter list has the following format: ⟨𝑛𝑜𝑑𝑒 𝑖, 𝑛𝑜𝑑𝑒 𝑗, inter -contact rate𝜆ij , 𝑡𝑖𝑚𝑒𝑠𝑡𝑎𝑚𝑝⟩

(5)

Timestamp denotes the time at which the inter-contact rate 𝜆𝑖𝑗 between node 𝑖 and node 𝑗 is updated. Timestamp is used to resolve a merge conflict, in which two entries have the same

node IDs but with different inter-contact rates. When a merge conflict occurs, we keep the entry with the latest timestamp. By including an intermediate node into the delivery probability, Eq. 1 needs to be modified to account for the intercontact time and contact-duration time between 𝑠 and 𝑣 and between 𝑣 and 𝑑, respectively. Suppose message 𝑖 is delivered from 𝑠 to 𝑣 during the 𝑛th meeting and from 𝑣 to 𝑑 during the 𝑚th meeting. Then the probability that message 𝑖 does not expire until it is delivered to 𝑑 is: 𝑛 𝑚 ) ( ∑ ∑ 𝑃 0< 𝑋𝑎𝑠,𝑣 + 𝑋𝑏𝑣,𝑑 < 𝑅𝑖 𝑎=1

(11)

𝑏=1

Note that in Eq. 11, we omit 𝑌𝑎𝑠,𝑣 and 𝑌𝑏𝑣,𝑑 since we assume 𝑋 ≫ 𝑌 . Since 𝑋1𝑠,𝑣 , ⋅ ⋅ ⋅ , 𝑋𝑛𝑠,𝑣 are i.i.d exponential 𝑛 ∑ random variables, 𝑋𝑎𝑠,𝑣 is gamma distributed with param𝑎=1

eters (𝛼𝑠,𝑣 = 𝑛, 𝛽𝑠,𝑣 = 𝜆𝑠,𝑣 ). Similarly,

𝑚 ∑

𝑋𝑏𝑣,𝑑 is gamma

𝑏=1 𝑚, 𝛽𝑣,𝑑 =

distributed with parameters (𝛼𝑣,𝑑 = 𝜆𝑣,𝑑 ). Eq. 11 is in turn the cumulative distribution function (CDF) of the sum of two independent gamma variables with different parameters. To solve Eq. 11, we use the following theorem: Theorem 1: Let 𝑋𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁 be independent gamma random variables with parameters (𝜆𝑖 , 𝛽𝑖 ), 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁 and let 𝑍 = 𝑋1 + 𝑋2 + ⋅ ⋅ ⋅ + 𝑋𝑁 . The probability density function (PDF) of 𝑍 can be expressed as: ∑

𝑁 𝑁 ( ∞ ˆ ∏ 𝛽ˆ )𝛼𝑞 ∑ 𝛿𝑘 ⋅ 𝑧 𝑞=1 𝛼𝑞 +𝑘−1 ⋅ 𝑒𝑥𝑝(−𝑧/𝛽) 𝑓𝑍 (𝑧) = ∑𝑁 ( ∑𝑁 ) 𝛼 +𝑘 𝑞 𝛽 ˆ 𝑞=1 𝑞 ⋅Γ 𝑞=1 𝑘=0 𝛽 𝑞=1 𝛼𝑞 + 𝑘

(12)

for 𝑧 > 0, where 𝛽ˆ = min𝑁 𝑞=1 (𝛽𝑞 ), and the coefficients 𝛿𝑘 satisfy the recurrence relations: 𝛿0 = 1

and 𝛿𝑘+1

[ 𝑁 ] 𝑘+1 𝛽ˆ )𝑖 1 ∑ ∑ ( = 𝛼𝑗 1 − 𝛿𝑘+1−𝑖 𝑘 + 1 𝑖=1 𝑗=1 𝛽𝑗

The proof of Theorem 1 is shown in [34]. Eq. 11 can then be computed by doing term-by-term integration of Eq. 12. 𝑛 𝑚 ) ( ∑ ∑ 𝑃 0< 𝑋𝑎𝑠,𝑣 + 𝑋𝑏𝑣,𝑑 < 𝑅𝑖 = 𝑎=1 𝑏=1 ] ∑𝑁 ∫ 𝑅𝑖 [ ∏ 𝑁 ( ∞ ˆ 𝛽ˆ )𝛼𝑞 ∑ 𝛿𝑘 ⋅ 𝑧 𝑞=1 𝛼𝑞 +𝑘−1 ⋅ 𝑒𝑥𝑝(−𝑧/𝛽) ∑𝑁 ( ) 𝑑𝑧 𝛽𝑞 ˆ 𝑞=1 𝛼𝑞 +𝑘 ⋅ Γ ∑𝑁 𝛼𝑞 + 𝑘 0 𝑞=1 𝑘=0 𝛽 𝑞=1    𝐴

(13)

where 𝑁 = 2, 𝛽1 = 𝜆𝑠,𝑣 , 𝛽2 = 𝜆𝑣,𝑑 , 𝛼1 = 𝑛, 𝛼2 = 𝑚. Next, we compute the probability that message 𝑖 fails to be transmitted during the 𝑛 − 1 meetings between 𝑠 and 𝑣 and 𝑚 − 1 meetings between 𝑣 and 𝑑. ( 𝑠,𝑣 𝑃 0 < 𝑌1𝑠,𝑣 < 𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑛−1 < 𝐻𝑖 , ) 𝑣,𝑑 𝑣,𝑑 0 < 𝑌1 < 𝐻𝑖 , ⋅ ⋅ ⋅ , 0 < 𝑌𝑚−1 < 𝐻𝑖 = (1 − 𝑒−𝜃𝑠,𝑣 𝐻𝑖 )𝑛−1 (1 − 𝑒−𝜃𝑣,𝑑 𝐻𝑖 )𝑚−1

(14)

Lastly, the probability that message 𝑖 is successfully transmitted during the 𝑛th meeting between 𝑠 and 𝑣 and during the 𝑚th meeting between 𝑣 and 𝑑 is: ) ( 𝑃 𝑌𝑛𝑠,𝑣 ≥ 𝐻𝑖 , 𝑌𝑚𝑣,𝑑 ≥ 𝐻𝑖 = 𝑒−𝜃𝑠,𝑣 𝐻𝑖 𝑒−𝜃𝑣,𝑑 𝐻𝑖 = 𝑒−(𝜃𝑠,𝑣 +𝜃𝑣,𝑑 )𝐻𝑖

(15)

By taking the union of three events in Eq. 13, 14, and 15, we obtain the probability that message 𝑖 is not successfully delivered from 𝑠 to 𝑑 via 𝑣 until the 𝑚th meeting between 𝑣 and 𝑑. 𝑃𝑖 (𝑛, 𝑚) = 𝐴 ⋅ (1 − 𝑒−𝜃𝑠,𝑣 𝐻𝑖 )𝑛−1 (1 − 𝑒−𝜃𝑣,𝑑 𝐻𝑖 )𝑚−1 𝑒−(𝜃𝑠,𝑣 +𝜃𝑣,𝑑 )𝐻𝑖

(16)

Then, the two-hop delivery probability for successfully transmitting message 𝑖 is: 𝑃𝑖 =

∞ ∞ ∑ ∑

𝑃𝑖 (𝑛, 𝑚)

(17)

𝑛=1 𝑚=1

C. Estimating Exponential Parameters The inter-contact rate 𝜆𝑖 between the current node and an encounter node 𝑖 can be computed using their encounter history as follows: 𝑁 𝜆 𝑖 = ∑𝑁

𝑘=1

𝑇𝑘

(18)

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 [35], [36], [37]. 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 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: 𝜆𝑖 (𝑡𝑘+1 ) =

𝑁 +1 + 𝑇𝑁 +1

𝑁 𝜆𝑖 (𝑡𝑘 )

(19)

where 𝑇𝑁 +1 is the value of the new inter-contact time sample, and 𝑇𝑁 +1 = 𝑡𝑘+1 − 𝑡𝑘 . Similarly, the contact-duration rate 𝜃𝑖 can be computed incrementally using the contact duration time. D. Relay Selection Strategy We use a single-copy model in which, at any point in time, there is at most one copy of the message in the network. Suppose a source node 𝑠 encounters a set of nodes 𝑉 = {𝑣1 , 𝑣2 , ⋅ ⋅ ⋅ , 𝑣𝑛 }. We assume that 𝑣𝑖 ∕= 𝑑. Otherwise, 𝑠 can trivially forward the message to the destination immediately. After exchanging the list of node encounters, 𝑠 and each neighbor 𝑣𝑖 independently compute the one-hop delivery probability to 𝑑 using Eq. 9. Each 𝑣𝑖 then advertises the value of 𝑃𝑣𝑖 to 𝑠. Let 𝑃𝑣 = max(𝑃𝑣1 , 𝑃𝑣2 , ⋅ ⋅ ⋅ 𝑃𝑣𝑘 ). If 𝑃𝑠 >= 𝑃𝑣 , 𝑠 will not forward the message. Otherwise, 𝑠 will

TABLE II. Characteristics of the Cabspotting trace

∙

Fig. 1. Relay selection strategy based on one-hop and two-hop delivery probability.

compute the two-hop delivery probability 𝑃𝑟𝑗 to 𝑑 via each intermediate node 𝑟𝑗 that 𝑠 has met in the past, but is not currently in contact. The computation is done using Eq. 17. Let 𝑃𝑟 = max(𝑃𝑟1 , 𝑃𝑟2 , ⋅ ⋅ ⋅ 𝑃𝑟𝑚 ). If 𝑃𝑣 > 𝑃𝑟 , then 𝑠 will forward the message to the current neighbor node 𝑣 with the highest delivery probability 𝑃𝑣 . Node 𝑣 then follows the same strategy until the message is delivered to 𝑑. Fig. 1 summarizes our relay selection strategy. When there is more than one message in the buffer, 𝑠 will form a list of messages that satisfy the condition similar to 𝑃𝑣1 in Fig. 1. Messages from this list are then transferred in descending order of their one-hop delivery probability. The list is updated when a neighboring node is out of contact with 𝑠 or when a new node is in contact with 𝑠. V. P ERFORMANCE E VALUATION In this section, we conduct simulations using a real-life mobility trace to evaluate the performance of our CDA scheme. The simulation setup, performance metrics, and the evaluation results are presented as follows. A. Simulation Setup We implement the proposed relay selection strategy using the opportunistic network simulator ONE 1.5.1 [38]. To obtain meaningful results, we use the real-life Cabspotting trace [32]. 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 [13], [14], [15], [39]. Table II shows the statistics of Cabspotting. 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 2 MB, 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 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.

∙

∙

∙

Trace

Contacts

Duration (days)

Devices

Cabspotting

111,153

30

536

PROPHET [17] 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 [17]. That is, {𝑃𝑖𝑛𝑖𝑡 , 𝛽, 𝛾} = {0.75, 0.25, 0.98}. BubbleRap [24] is a community-based algorithm that routes data based on rankings calculated from the social centrality. A message is first bubbled up using the global ranking until it reaches a node in the same community as the destination. Then the local ranking is used to bubble up the message until the destination is reached or the message expires. MEED [26] selects the route with the minimum expected delay among individual routes to the destination. The observed contact history is used to estimate the expected waiting time for each potential next hop. CDA (our proposed scheme) selects relay nodes based on one-hop and two-hop delivery probabilities. Unlike the above routing schemes, CDA is aware of the contact duration among nodes.

B. Evaluation Metrics We use the following metrics for evaluation: ∙ 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 cost: the average number of relays for each message to be delivered from the source to destination. C. Comparative Results 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. CDA comes second with 74% delivery rate. It outperforms PROPHET, MEED, and BubbleRap by 10%, 12%, and 13%, respectively. The improvement of CDA is a result of two factors. First, CDA is aware of short contact duration and thus it optimizes the relay selection that meets the message deadline. Second, CDA has a broad view of relay selection as it considers both one-hop and two-hop delivery path. Note that BubbleRap has a slightly worse performance with a delivery rate of around 61%. This is perhaps because BubbleRap is impacted by the weak community structure in the Cabspotting trace. Recall that BubbleRap is a community-based algorithm, in which forwarding decisions are made based on the community structure of the network. Fig. 2b depicts the average delay. Again, Epidemic has the best delivery delay, followed by CDA. CDA successfully

Delivery ratio

0.7

2.5 Epidemic CDA PROPHET MEED BubbleRap

2 Average delay (days)

0.8

0.6 0.5 0.4

6 Epidemic CDA PROPHET MEED BubbleRap

5.5 5 4.5 Average cost

0.9

1.5

1

Epidemic CDA PROPHET MEED BubbleRap

4 3.5 3 2.5

0.3 0.5

2

0.2 0.1 0.5

1.5 1

2

4

6 8 TTL (days)

10

12

14

0 0.5

(a) Delivery ratio

1

2

4

6 8 TTL (days)

10

12

14

(b) Average delay

1 0.5

1

2

4

6 8 TTL (days)

10

12

14

(c) Average cost

Fig. 2. Performance comparison using Cabspotting trace.

delivers a message by 5%, 10%, and 12% less time than MEED, BubbleRap, and PROPHET, respectively. Lastly, average cost is compared in Fig. 2c. Epidemic has the highest cost as it floods the message to every network node. The cost of CDA is lower than BubbleRap, MEED, and PROPHET by 14%, 19%, and 23%, respectively. This shows that by considering two-hop relay nodes (in addition to one-hop candidates), CDA effectively eliminates unnecessary relays. VI. C ONCLUSION AND F UTURE W ORK In this paper, we proposed a new relay selection strategy that takes into account the contact duration constraint of message delivery. We derived the one-hop and two-hop delivery probability based on the exponential distribution of inter-contact time and contact duration time. Through considering relay candidates from both current and past contacts, the current message carrier can make a more informed relay selection decision that helps conserve network bandwidth and shorten the routing path. Experimental results using the Cabspotting trace show that our scheme can achieve up to 13% higher delivery rate, 12% lower delay, and 23% lower transmission cost compared to other relay selection strategies. In future work, we plan to derive an expression for the relay selection metric under the power law [28], LogNormal [41], and hyper-exponential distribution [42] of inter-contact times and contact duration times. We also plan to relax the assumption of infinite storage, and evaluate the routing scheme 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] A. Tovar, T. Friesen, K. Ferens, and B. McLeod, “A dtn wireless sensor network for wildlife habitat monitoring,” in Electrical and Computer Engineering (CCECE), 2010 23rd Canadian Conference on. IEEE, 2010, pp. 1–5.

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A PPENDIX A. 𝑃𝑖 Expression Simplification

𝑃𝑖 =

[( ∞ ∑

1−

𝑛−1 ∑

𝑛=1

=𝑒

𝑘=0

−𝜃𝑠,𝑑 𝐻𝑖

−𝑒

[(

(𝜆𝑠,𝑑 𝑅𝑖 )𝑘 −𝜆𝑠,𝑑 𝑅𝑖 𝑒 𝑘!

)

)𝑛−1 −𝜃𝑠,𝑑 𝐻𝑖 ( ⋅ 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ⋅𝑒

]

) ∞ ∑ ( −𝜃𝑠,𝑑 𝐻𝑖 )𝑛−1 1−𝑒 𝑛=1 ∞ ∑

−𝜆𝑠,𝑑 𝑅𝑖

( (

1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖

𝑛=1

∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 )𝑛−1 𝑛−1 𝑘! 𝑘=0

)𝑛−1

(

)] (20)

Note that 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 is in the form of a geometric series with 𝑟 = 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 . Since ∣𝑟∣ < 1, the series converges, and its sum is: ∞ ∑ ( )𝑛−1 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 =

1 1 = 𝑒𝜃𝑠,𝑑 𝐻𝑖 = 1−𝑟 1 − (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ) (21)

𝑛=1

Plugging Eq. 21 into Eq. 20, we obtain the final form of 𝑃𝑖 : 𝑃𝑖 = 1 − 𝑒−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖

∞ [ ∑ ∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 ] ( )𝑛−1 𝑛−1 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 𝑘! 𝑛=1 𝑘=0 (22)

B. Derivation of the Lower and Upper Bound for 𝑃𝑖 First, we observe that: 𝑛−1 ∑ 𝑘=0

∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 ∑ (𝜆𝑠,𝑑 𝑅𝑖 )𝑘 + = = 𝑒𝜆𝑠,𝑑 𝑅𝑖 (23) 𝑘! 𝑘! 𝑘! ∞

∞

𝑘=𝑛

𝑘=0

Above, we use the following Maclaurin series: ∞ ∑ 𝑥𝑘 = 𝑒𝑥 𝑘!

(24)

𝑘=0

where 𝑥 = 𝜆𝑠,𝑑 𝑅𝑖 . By the Taylor’s theorem, the (𝑛−1)th order Taylor polynomial and its remainder term in the Lagrange form are given by: 𝑛−1 ∑ 𝑘=0

(𝜆𝑠,𝑑 𝑅𝑖 )𝑘 𝑒𝜉 + (𝜆𝑠,𝑑 𝑅𝑖 )𝑛 = 𝑒𝜆𝑠,𝑑 𝑅𝑖 , 𝜉 ∈ [0, 𝜆𝑠,𝑑 𝑅𝑖 ] 𝑘! 𝑛!

(25)

Thus, we have: 𝑛−1 ∑ 𝑘=0

(𝜆𝑠,𝑑 𝑅𝑖 )𝑘 𝑒𝜉 = 𝑒𝜆𝑠,𝑑 𝑅𝑖 − (𝜆𝑠,𝑑 𝑅𝑖 )𝑛 𝑘! 𝑛!

Plugging Eq. 26 into Eq.22, we obtain:

(26)

∞ [ ∑ ( )𝑛−1 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 𝑃𝑖 = 1 − 𝑒 𝑛=1 ( )] 𝑒𝜉 𝑒𝜆𝑠,𝑑 𝑅𝑖 − (𝜆𝑠,𝑑 𝑅𝑖 )𝑛 𝑛! { ∞ ∑( )𝑛−1 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 − = 1 − 𝑒−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖 𝑒𝜆𝑠,𝑑 𝑅𝑖 𝑛=1 ( ) ) ]} [( 𝑛 ∞ ∑ (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 )𝜆𝑠,𝑑 𝑅𝑖 𝑒𝜉 − 1 𝑛! 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 𝑛=0 (27) −𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖

Since the first and second sigma sum have the form of a geometric series and a Maclaurin series (Eq. 24), respectively, Eq. 27 can be further simplified to: [

𝑃𝑖 = 1 − 𝑒

𝑒𝜆𝑠,𝑑 𝑅𝑖 1 − (1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ) )] ( 𝜉 −𝜃 𝐻 𝑒 (1−𝑒 𝑠,𝑑 𝑖 )𝜆𝑠,𝑑 𝑅𝑖 −1 𝑒 −𝜃 𝐻

−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖

− =

𝑒

1 − 𝑒 𝑠,𝑑 𝑖 ] [ −𝜃 𝐻 (1−𝑒 𝑠,𝑑 𝑖 )𝜆𝑠,𝑑 𝑅𝑖 −1 𝑒 𝐻

𝜉−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖

1 − 𝑒−𝜃𝑠,𝑑

𝑖

(28)

Since 𝜉 ∈ [0, 𝜆𝑖 𝑅𝑖 ], then we can obtain the following lower and upper bound for 𝑃𝑖 : ] [ 𝑒−𝜆𝑠,𝑑 𝑅𝑖 −𝜃𝑠,𝑑 𝐻𝑖 (1−𝑒−𝜃𝑠,𝑑 𝐻𝑖 )𝜆𝑠,𝑑 𝑅𝑖 − 1 ≤ 𝑃𝑖 ≤ 𝑒 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖 ] (29) [ −𝜃𝑠,𝑑 𝐻𝑖 −𝜃 𝐻 𝑒 (1−𝑒 𝑠,𝑑 𝑖 )𝜆𝑠,𝑑 𝑅𝑖 − 1 𝑒 1 − 𝑒−𝜃𝑠,𝑑 𝐻𝑖