research article a statistical analysis based probabilistic routing...

Research ArticleA Statistical Analysis Based Probabilistic Routing forResource-Constrained Delay Tolerant Networks

Jixing Xu, Jianbo Li, Shan Jiang, Chenqu Dai, and Lei You

Information Engineering College of Qingdao University, Ningxia Road 308, Qingdao, Shandong 266071, China

Correspondence should be addressed to Jianbo Li; [email protected]

Received 31 May 2014; Revised 23 August 2014; Accepted 10 September 2014; Published 15 October 2014

Academic Editor: Jianliang Xu

Copyright © 2014 Jixing Xu et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The nonexistence of end-to-end path between the sender and the receiver poses great challenges to the successful messagetransmission in delay tolerant networks. Probabilistic routing provides an efficient scheme to route messages, but most existingprobabilistic routing protocols do not consider whether a message has enough time-to-live to reach its destination. In this paper,we propose an improved probabilistic routing algorithm that fully takes into account message’s time-to-live when predicting thedelivery probability. Based on statistical analysis, we compute and update the expected intermeeting times between nodes. And thenthe probability for a message to be delivered within its time-to-live is computed based on the assumed exponential distribution.Wefurther propose an optimal message schedule policy, by modeling the buffer management problem as 0-1 knapsack, of which themaximum delivery probability sum can be achieved by resorting to the back track technique. Extensive simulations are conductedand the results show that the proposed algorithm can greatly enhance routing performance in terms ofmessage delivery probability,overhead ratio, and average hop count.

1. Introduction

As a new emerging store and forward networking archi-tecture, delay tolerant networks (DTNs) have been widelystudied and applied. In recent years, DTNs have achievedgreat successes in some challenging networks deployed inextreme environment, such as interplanetary Internet, habitatmonitoring networks, underwater sensor networks [1, 2],vehicular ad hoc networks [3], pocket switched networks [4,5], andmobile social networks [6, 7]. However, different fromthe traditional Internet, DTNs are characterized by frequenttopology partitions [8], sparse node density, limited networkresources (e.g., storage, bandwidth, etc.), extremely high end-to-end latency, asymmetric data rate, high bit error rate,heterogeneous interconnection, and so forth. So in DTNs,there may never be a complete end-to-end path between thesender and the receiver. Consequently, the successfulmessagetransmission in DTNs faces great challenges.

In order to cope with the intermittent connectivityproblem, DTN architecture [9, 10] introduces a bundlelayer between the application layer and the transportlayer to implement store-carry-and-forward routing strategy.

Furthermore, with the help of the bundle layer, DTNarchitecture is able to shield heterogeneous networks andcommunicate across the multiple regions that have differenttypes of network architectures and protocols. So based onthe bundle layer, DTNs can relay messages hop by hopuntil encountering the destination nodes. But due to theextremely limited network resource and bandwidth, currentnode should make a clever next hop routing selection tocontrol the number of message copies.

So far, a large number of probabilistic routing strate-gies have been proposed to optimize the next hop routingselection in the absence of global topology knowledge.Most of them (e.g., Prophet [11]) make attempts to predictthe encounter probabilities between nodes and then makerouting decisions based on the computed probability val-ues. There is no denying that these routing strategies canimprove message delivery ratio in opportunistic routing. Wealso should notice that the time-to-live (TTL) of messageis gradually depleted as time progresses. Most traditionalprobabilistic routing protocols always try to replicatemessageto the node with a higher delivery probability, withouttaking into consideration message’s TTL. However, although

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2014, Article ID 623193, 16 pageshttp://dx.doi.org/10.1155/2014/623193

2 International Journal of Distributed Sensor Networks

a selected intermediate node has a higher probability toencounter destination node, the message delivery will stillfail if the message’s TTL is exhausted before meeting thedestination node. In this case, considering the limited bufferresource, the message copy should not be delivered to theintermediate node. From the above case, we can see that ahigher encounter probability can only indicate that the twonodes are more likely to encounter each other. But the twonodes may still need a period of time to encounter eachother. If message’s TTL is exhausted during this period oftime, the message delivery will still fail. From this pointof view, the node is not a good choice although it hasa higher probability to encounter the destination. Conse-quently, an efficient probabilistic routing algorithm that fullytakes into account message’s TTL is expected to be employedin DTNs.

In this paper, from a new perspective, we propose astatistical analysis based probabilistic routing (SAPR) algo-rithm, which predicts the probability that a message canbe successfully delivered to its destination based on itsremaining time-to-live. Firstly, for each pair of nodes, weuse statistical analysis methods to compute the mathematicalexpectation of the intermeeting times (IMTs) between them.Secondly, in the case of fully taking into account message’sremaining time-to-live, we predict the possibility that amessage can be successfully delivered. Finally, we makeprobabilistic routing selections according to the computedprobabilities. In other words, we replicate message to theintermediate node that can make message get a higherdelivery probability. In addition, we also introduce buffermanagement policy in the proposed routing algorithm. Themessage management is modeled as a 0-1 knapsack prob-lem when node’s buffer overflows. By solving the knapsackproblem, we can make sure that each node always keepsthe messages that can maximize the delivery probabilitysum.

The rest of this paper is organized as follows. In Section 2,wemake the routing assumptions and explain the mathemat-ical notations used in this paper. Section 3 gives the detaileddescriptions of the proposed algorithm. The performanceevaluations and comparisons are presented in Section 4.Section 5 discusses some related works. Finally in Section 6,we summarize this paper.

2. Assumptions and Preliminary

In order to analyze and implement the proposed probabilisticrouting algorithm, we make the following assumptions.

(i) The intermeeting time (IMT) between nodes is expo-nentially distributed or has at least an exponential tail.

(ii) Nodes move independently and their mobility isheterogeneous. In other words, different node pairshave different exponential distribution parameter 𝜆.

(iii) The network resource (e.g., storage, bandwidth,energy, etc.) is limited.

Regarding the first assumption, it has been shown thatmany simple synthetic mobility models (e.g., RandomWalk,Random Waypoint, and Random Direction [12, 13]) havesuch a property. Furthermore, it is a known result in thetheory of random walks on graphs that hitting times onsubsets of vertices usually have an exponential tail [14]. And[15] has derived the fact that the expected intermeeting timein RandomWalk model also follows an exponential distribu-tion. Besides, the Exponential Correlated Random Mobilitymodel can also be used to support the first assumption. Sothe assumption that most nodes exhibit random mobilityis reasonable in the opportunistic networks. However, inaddition to exponential distribution, some recent researcheshave suggested that intermeeting time also follows power lawdistribution in some human mobility traces. But recently,by using a diverse set of measured human mobility traces,Karagiannis et al. [16] have argued that the intermeeting timestill exhibits an exponential tail. They find as an invariantproperty that there is a characteristic time, order of half aday, beyond which intermeeting time follows an exponentialdistribution. Within the characteristic time, intermeetingtime follows a power law distribution. This is to say, in manyhuman traces, although intermeeting time follows a powerlaw distribution within a period of time, it still exhibits anexponential tail. Taking the Content datasets of Cambridgehaggle for example, Cambridge spends about twomonths (fargreater than half a day) to collect the trace data. In this case,according to the above conclusion ofThomas, the time periodin which intermeeting time follows exponential distributionis much longer than the time period following power lawdistribution. Thus the entire cumulative distribution can beapproximately seen as a certain exponential distribution. Inaddition, in theMIT trace using Bluetooth devices, up to 60%of intermeeting times observed are above one day (greaterthan half a day), and these large intermeeting times can alsobe found in the traces collected by UCSD and Dartmouth.So in some sense, the distributions of intermeeting times inthese traces can also be seen as exponential. In this paper,our routing is not specifically designed for human mobilitymodel, but in some extent of generality. So we tend toassume the exponential distribution taking into account theabove factors. And the results in our simulations also showthe reasonableness of the assumption. Regarding the secondassumption, it is clear that nodes follow different movingtrajectories and different node pairs usually have differentencounter rates in the real world. Some nodes may encountereach other frequently, but other nodes may never meet eachother.

The mathematical notations used in this paper are listedand explained in Notations section.

3. Statistical Analysis BasedProbabilistic Routing (SAPR)

Before presenting our SAPR algorithm, we first introducesome analysis works and routing models based on the aboveassumptions.

International Journal of Distributed Sensor Networks 3

3.1. Estimating Exponential Distribution Parameter. Here weassume that intermeeting time 𝑋 follows the exponentialdistribution with parameter 𝜆, that is,

𝑋 ∼ Exponential (𝜆) . (1)

With the second assumption, we know that different nodepairs have different parameter 𝜆. For the sake of simplicity, wetemporarily use 𝜆 to uniformly represent these exponentialdistribution parameters in this section. Then we have theprobability density function 𝑓exp(𝑥):

𝑓exp(𝑥) = {

𝜆𝑒−𝜆𝑥

𝑥 > 0

0 𝑥 ≤ 0.(2)

In order to find the functional relationship between 𝐸(𝑋)and 𝜆, we compute the mathematical expectation of theintermeeting times based on the exponential distribution:

𝐸 (𝑋) = ∫

∞

−∞

𝑥𝑓exp(𝑥) 𝑑𝑥 =

1

𝜆. (3)

Now, we can get (4) to estimate the parameter 𝜆 for each nodepair:

𝜆𝑎,𝑏

=1

𝐸 (𝑋𝑎,𝑏

). (4)

3.2. Computing IMT’s Mathematical Expectation. In order tocompute the value of 𝜆

𝑎,𝑏

, we still need to get the mathemat-ical expectation of 𝑋

𝑎,𝑏

. In this section, for each node pair𝑎 and 𝑏, we use the statistical analysis methods to compute𝐸(𝑋𝑎,𝑏

). For this purpose, each node needs to maintain twomatrixes, 𝑀imt and 𝑀count, where 𝑥

𝑘

(𝑁𝑗

) denotes the 𝑘thsample of the intermeeting times between current node andnode 𝑁

𝑗

and correspondingly 𝑐𝑘

(𝑁𝑗

) denotes the count thatthe 𝑘th sample appears. As shown in (5), for each of theother 𝑁 − 1 nodes, 𝑀imt records the current 𝑛 samples byrandom sampling. Then based on these samples, we can usestatistical analysis methods to compute 𝐸(𝑋

𝑎,𝑏

) for nodes𝑎 and 𝑏:

𝑀imt=

𝑁1

...𝑁𝑗

...𝑁|𝑁−1|

1 ⋅ ⋅ ⋅ 𝑘 ⋅ ⋅ ⋅ 𝑛

[[[[[[[

[

𝑥1

(𝑁1

) ⋅ ⋅ ⋅ 𝑥𝑘

(𝑁1

) ⋅ ⋅ ⋅ 𝑥𝑛

(𝑁1

)

... d... d

...𝑥1

(𝑁𝑗

) ⋅ ⋅ ⋅ 𝑥𝑘

(𝑁𝑗

) ⋅ ⋅ ⋅ 𝑥𝑛

(𝑁𝑗

)

... d... d

...𝑥1

(𝑁|𝑁−1|

) ⋅ ⋅ ⋅ 𝑥𝑘

(𝑁|𝑁−1|

) ⋅ ⋅ ⋅ 𝑥𝑛

(𝑁|𝑁−1|

)

]]]]]]]

]

(5)

𝑀count

=

[[[[[[[

[

𝑐1

(𝑁1

) ⋅ ⋅ ⋅ 𝑐𝑘

(𝑁1

) ⋅ ⋅ ⋅ 𝑐𝑛

(𝑁1

)

... d... d

...𝑐1

(𝑁𝑗

) ⋅ ⋅ ⋅ 𝑐𝑘

(𝑁𝑗

) ⋅ ⋅ ⋅ 𝑐𝑛

(𝑁𝑗

)

... d... d

...𝑐1

(𝑁|𝑁−1|

) ⋅ ⋅ ⋅ 𝑐𝑘

(𝑁|𝑁−1|

) ⋅ ⋅ ⋅ 𝑐𝑛

(𝑁|𝑁−1|

)

]]]]]]]

]

. (6)

Assuming current node is node 𝑎, then node 𝑎 can use (7) tocompute the final value of 𝐸(𝑋

𝑎,𝑏

):

𝑃 (𝑥𝑘

(𝑏)) =𝐶𝑘

(𝑏)

∑𝑛

𝑖=1

𝐶𝑖

(𝑏),

𝐸 (𝑋𝑎,𝑏

) =

𝑛

∑

𝑘=1

𝑥𝑘

(𝑏) × 𝑃 (𝑥𝑘

(𝑏))

=∑𝑛

𝑘=1

𝑥𝑘

(𝑏) 𝐶𝑘

(𝑏)

∑𝑛

𝑖=1

𝐶𝑖

(𝑏).

(7)

Note that we use random sampling in this paper. For anode pair, it is easy to compute the interval between twoencounters. By repeating this operation in a random way,we can finally get the sample data. The value of 𝑛 (i.e.,the size of sample data) is not an invariant variable, whichcan be flexibly set to an appropriate value according to

the specific scenario. For the scenario with limited resourcesand computing capacity, we can appropriately reduce thevalue of 𝑛. For the scenario with sufficient resources andbetter computing capacity, we can increase the value so asto make the algorithm more accurate. In this paper, we setthe value of 𝑛 to the number of nodes in the networks.When a node encounters a new node that recently joinedthe networks, it needs to add the node to the above matrixes.Due to the lack of enough sample data about the new node,we first use continuous sampling to quickly get 𝑛 sampledata. After that, we continue to collect data by using randomsampling. Note that we have to use the sample data we havecollected to calculate the mathematical expectation beforegetting enough sample data. In order to accurately estimatethe current expectation of intermeeting times, we always usethe latest sample data in this paper. That is to say, we areconstantly replacing the oldest sample data with the latest oneafter having got 𝑛 sample data.


In order to more accurately compute the 𝐸(𝑋) for a nodepair, we also build an update process to update𝐸(𝑋)when thetwo nodes have not encountered each other for a long periodof time.

Theorem1. For the intermeeting times between any two nodes,assuming their expectation is 𝐸(𝑋) and their variance is𝐷(𝑋),then one can predict the probability that |𝑋 − 𝐸(𝑋)| < 𝑋when setting 𝑋 to a meaningful value; that is,

𝑃 {|𝑋 − 𝐸 (𝑋)| < 𝑋} ≥ 1 −𝐷 (𝑋)

𝑋2

. (8)

Proof. Firstly, we assume the probability density function of𝑋 is 𝑓(𝑥). Then we can get

𝑃 {|𝑋 − 𝐸 (𝑋)| ≥ 𝑋} = ∫|𝑥−𝐸(𝑋)|≥𝑋

𝑓 (𝑥) 𝑑𝑥

≤ ∫|𝑥−𝐸(𝑋)|≥𝑋

|𝑥 − 𝐸 (𝑋)|2

𝑋2

𝑓 (𝑥) 𝑑𝑥

≤1

𝑋2

∫

+∞

−∞

(𝑥 − 𝐸 (𝑋))2

𝑓 (𝑥) 𝑑𝑥.

(9)

For the variable𝑋, we have

𝐷 (𝑋) = ∫

+∞

−∞

(𝑥 − 𝐸 (𝑋))2

𝑓 (𝑥) 𝑑𝑥. (10)

Then with (9), we get

𝑃 {|𝑋 − 𝐸 (𝑋)| ≥ 𝑋} ≤𝐷 (𝑋)

𝑋2

. (11)

Finally, we can get

𝑃 {|𝑋 − 𝐸 (𝑋)| < 𝑋} = 1 − 𝑃 {|𝑋 − 𝐸 (𝑋)| ≥ 𝑋}

≥ 1 −𝐷 (𝑋)

𝑋2

.

(12)

To some extent, Theorem 1 shows the central tendency ofthe intermeeting times. That is to say, we can get the intervalthat most intermeeting times are clustered together in.

Corollary 2. For the intermeeting times between any twonodes, at least 𝜌 of them are clustered together in the interval(𝐸(𝑋) − √𝐷(𝑋)/(1 − 𝜌), 𝐸(𝑋) + √𝐷(𝑋)/(1 − 𝜌)); that is,

𝑃{𝐸 (𝑋) − √𝐷 (𝑋)

1 − 𝜌< 𝑋 < 𝐸 (𝑋) + √

𝐷 (𝑋)

1 − 𝜌} ≥ 𝜌. (13)

Proof. If we set

𝜌 = 1 −𝐷 (𝑋)

𝑋2

, (14)

then we can get the meaningful value of 𝑋:

𝑋 = √𝐷 (𝑋)

1 − 𝜌. (15)

With (8), we can finally get (13). Then setting 𝜌 to an appro-priate value, we can get the interval that most intermeetingtimes appear in.

If two nodes have not encountered each other for a longperiod of time, they should update 𝐸(𝑋) so as to moreaccurately estimate the parameter 𝜆. Now the issue is howand when to update 𝐸(𝑋). According to Corollary 2, if weset 𝜌 to a value close to 1.0, we can find the interval thatmost intermeeting times are distributed in. For simplicity, let𝑥 denote the time that has elapsed since the last encounter.If 𝑥 is in the interval (𝐸(𝑋) − √𝐷(𝑋)/(1 − 𝜌), 𝐸(𝑋) +√𝐷(𝑋)/(1 − 𝜌)), then we consider it a common case and itis not necessary to update 𝐸(𝑋) in this case. And if 𝑥 is inthe interval (0, 𝐸(𝑋)−√𝐷(𝑋)/(1 − 𝜌)], we still do not update𝐸(𝑋). This is because this case will not increase the value of𝐸(𝑋). But if𝑥 is in the range [𝐸(𝑋)+√𝐷(𝑋)/(1 − 𝜌), +∞), weconsider it an abnormal case and it will increase the value of𝐸(𝑋).Therefore in this case, we need to update𝐸(𝑋). For thispurpose, we define the update cycle as√𝐷(𝑋)/(1 − 𝜌); that is,𝐸(𝑋) is updated once every √𝐷(𝑋)/(1 − 𝜌) time units when𝑥 > 𝐸(𝑋). With (11), we have

𝑃{|𝑋 − 𝐸 (𝑋)| ≥ 𝑚√𝐷 (𝑋)

1 − 𝜌} ≤

𝐷 (𝑋)

(𝑚√𝐷 (𝑋) / (1 − 𝜌))

2

≤1 − 𝜌

𝑚2.

(16)

Then, from a statistical point of view, we further assume

𝑃{𝑋 − 𝐸 (𝑋) ≥ 𝑚√𝐷 (𝑋)

1 − 𝜌}

= 𝑃{𝑋 − 𝐸 (𝑋) ≤ −𝑚√𝐷 (𝑋)

1 − 𝜌} .

(17)

Now, we can have

𝑃{𝑋 − 𝐸 (𝑋) ≥ 𝑚√𝐷 (𝑋)

1 − 𝜌} ≤

1

2×1 − 𝜌

𝑚2. (18)

Here, we use its upper bound; that is,

𝑃upper

{𝑋 − 𝐸 (𝑋) ≥ 𝑚√𝐷 (𝑋)

1 − 𝜌} =

1 − 𝜌

2𝑚2. (19)

Finally we use (20) to update 𝐸(𝑋) in the 𝑚th update cyclewhen 𝑥 > 𝐸(𝑋):

𝑚 = round down ( 𝑥 − 𝐸 (𝑋)

√𝐷 (𝑋) / (1 − 𝜌)

)

=

[[[

[

𝑥 − 𝐸 (𝑋)

√𝐷 (𝑋) / (1 − 𝜌)

]]]

]

,


Triggering condition:when node𝑁

𝑎

encounters node𝑁𝑏

Na Executes:(1) For each𝑚

𝑖

in𝑁𝑎

do(2) 𝑅

𝑖

← the remaining TTL of𝑚𝑖

(3) dest← 𝑚𝑖

.𝑑𝑒𝑠𝑡𝑖𝑛𝑎𝑡𝑖𝑜𝑛

(4) update 𝐸(𝑋𝑁

𝑎,dest)

(5) 𝑃𝑚𝑖𝑁

𝑎,dest ← compute delivery probability using (22)

(6) update 𝐸(𝑋𝑁

𝑏,dest)

(7) 𝑃𝑚𝑖𝑁

𝑏,dest ← compute delivery probability using (22)

(8) If 𝑃𝑚𝑖𝑁

𝑏,dest > 𝑃

𝑚

𝑖

𝑁

𝑎,dest then

(9) send𝑚𝑖

to𝑁𝑏

(10) End if(11) End for

Algorithm 1: Next hop routing selection on node𝑁𝑎

.

𝐸 (𝑋) = 𝐸 (𝑋) + 𝑚√𝐷 (𝑋)

1 − 𝜌

× 𝑃upper

{𝑋 − 𝐸 (𝑋) ≥ 𝑚√𝐷 (𝑋)

1 − 𝜌}

= 𝐸 (𝑋) + 𝑚√𝐷 (𝑋)

1 − 𝜌×1 − 𝜌

2𝑚2

= 𝐸 (𝑋) +

√𝐷 (𝑋) (1 − 𝜌)

2𝑚.

(20)

Now, we only need to get the variance𝐷(𝑋) for updating the𝐸(𝑋).With the abovematrixes𝑀imt and𝑀count, we can easilycompute the𝐷(𝑋) for current node 𝑎 and node 𝑏 by using

𝐷(𝑋𝑎,𝑏

) =

𝑛

∑

𝑘=1

(𝑥𝑘

(𝑏) − 𝐸 (𝑋))2

𝑃 (𝑥𝑘

(𝑏))

=∑𝑛

𝑘=1

(𝑥𝑘

(𝑏) − 𝐸 (𝑋))2

𝑐𝑘

(𝑏)

∑𝑛

𝑖=1

𝐶𝑖

(𝑏).

(21)

3.3. Predicting Message’s Delivery Probability. After comput-ing and updating 𝐸(𝑋), we can estimate the parameter 𝜆for each node pair. Then we can predict message’s deliveryprobability according to the message’s remaining TTL.

Theorem 3. Assuming the remaining time-to-live of 𝑚𝑖

is𝑅𝑖

, the mathematical expectation of the intermeeting timesbetween current node and the destination is𝐸(𝑋), the time thathas elapsed since the last encounter with destination node is 𝜏,and then the delivery probability of𝑚

𝑖

on current node is 𝑃𝑚

𝑖

:

𝑃𝑚

𝑖

= 1 − 𝑒−(𝜏+𝑅

𝑖)/𝐸(𝑋)

. (22)

Proof. The probability that a message can be successfullydelivered by a node is equal to the probability that the next

intermeeting time between the node and the destination isnot greater than the sum of message’s remaining TTL and thetime that has elapsed; that is,

𝑃𝑚

𝑖

= 𝑃 {𝑋 ≤ 𝜏 + 𝑅𝑖

}

= ∫

𝜏+𝑅

𝑖

−∞

𝑓exp(𝑥) 𝑑𝑥

= ∫

0

−∞

0 𝑑𝑥 + ∫

𝜏+𝑅

𝑖

0

𝜆𝑒−𝜆𝑥

𝑑𝑥

= −𝑒−𝜆𝑥

𝜏+𝑅

𝑖

0

= −𝑒−𝜆(𝜏+𝑅

𝑖)

+ 1

= 1 − 𝑒−(𝜏+𝑅

𝑖)/𝐸(𝑋)

.

(23)

3.4. Next Hop Selection Strategy. Now we focus on the nexthop selection strategy. When communication opportunityarises, the message should be delivered to the relay node witha higher delivery probability.The detailed process is shown inAlgorithm 1.

3.5. BufferManagement Policy. In DTNs, the buffer resourcesof nodes are usually limited. So when node’s buffer overflows,the resource allocation problem arises. In this case, currentnode needs to determine whether to receive the incomingmessage and which message to drop. To this end, we firstneed to define the optimal objective and then make theoptimal decisions based on the objective. In this paper, ourobjective is to use the limited buffer to maximize the sum ofthe delivery probabilities of messages that can be stored bycurrent node. We formalize the optimal buffer managementas a 0-1 knapsack problem and further solve it by using theback track technique.

Theorem 4. Optimal buffer management is a 0-1 knapsackproblem.


Input:𝑚𝑠𝑔𝐿𝑖𝑠t = [𝑚

1

, 𝑚2

, . . . , 𝑚𝑛

]

(1) sort𝑚𝑠𝑔𝐿𝑖𝑠t in a descending order according to (25)(2) for each𝑚

𝑖

in𝑚𝑠𝑔𝐿𝑖st do(3) 𝑠 [𝑖] ← 𝑚

𝑖

.𝑠𝑖𝑧𝑒

(4) 𝑝[𝑖] ← the delivery probability of 𝑚𝑖

(5) end for(6) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 ← 0, 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃 ← 0, 𝑏𝑒𝑠𝑡𝑃 ← 0(7) BS←the buffer size of current node(8) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑎𝑡ℎ [1, 2, . . . , 𝑛] ← [1, 1, . . . , 1](9) 𝑏𝑒𝑠𝑡𝑃𝑎𝑡ℎ [1, 2 . . . 𝑛] ← [1, 1, . . . , 1](10) call Back Track (1)(11) for each element in 𝑏𝑒𝑠𝑡𝑃𝑎𝑡ℎ do(12) if 𝑏𝑒𝑠𝑡𝑃𝑎𝑡ℎ [𝑖] = 0 then(13) delete or reject𝑚

𝑖

(14) end if(15) end for

Algorithm 2: Buffer management strategy.

Proof. If we view the buffer size of a node, the messages’delivery probabilities, and the sizes of messages as themaximum weight of a knapsack, the values of goods, andthe weights of goods, then the objective of selecting andstoring the messages that can maximize the sum of deliveryprobabilities can be viewed as filling the knapsack with thegoods that can get the maximum value. Consequently, buffermanagement can be modeled as a 0-1 knapsack problem,and then we can further use the technique of solving theknapsack problem to solve the optimal buffer managementproblem.

Definition 5. Theformalization of the optimal buffermanage-ment is as follows, where 𝐵

𝑖

is used to mark whether to storethe message𝑚

𝑖

:

Max𝑛

∑

𝑖=1

𝑃𝑚

𝑖

𝐵𝑖

s.t.𝑛

∑

𝑖=1

size (𝑚𝑖

) 𝐵𝑖

≤ buffer size

𝐵𝑖

= {0, 1} , 1 ≤ 𝑖 ≤ 𝑛.

(24)

The above formalization can make sure that each nodealways keeps the messages that can maximize the deliveryprobability sum. Now the issue is how to solve the optimalproblem.

The common way to solve knapsack problem is dynamicprogramming algorithm. But considering the huge cost ofdynamic programming in this problem, we use the back tracktechnique to solve the knapsack problem in this paper. Thedetailed process is shown in Algorithm 2, which also needsto call Algorithms 3 and 4. Algorithm 4 is to compute theupper bound of the optimal value of right subtree in thesearch process, which is called by Algorithm 3 to determinewhether to continue searching right subtree. That is to say,we cut off the subtree if its upper bound is less than the

Table 1: Simulation settings in RandomWalk.

Parameter Default value (range)Area size 1000m × 1000mNumber of nodes 126Message size 500K–1MTransmission speed 250KBpsTransmission radius 100mMoving speed 0.5–1.5m/sMessage interval 40 s (10–50 s)Node buffer size 50M (5–50M)Time-to-live (TTL) 3 h (2–5 h)Simulation time 24 h𝜌 0.95

current best value. Algorithm 3 is the back track algorithm,which searches the entire solution space tree and recordsthe current optimal solution. In Algorithm 2, line 1 firstsorts messages in a descending order according to the unitvalue computed by (25). Lines 2–9 are to initialize the globalvariables that will be sued in Algorithms 3-4. Note thatthese global variables can also be modified by Algorithms3-4. Finally, lines 11–13 delete or reject those messagesthat are not included in the optimal solution computed byAlgorithm 3. In order to determine which message to dropand which message to receive, the 𝑚𝑠𝑔𝐿𝑖𝑠𝑡 in Algorithm 2should contain both the stored messages and the incomingmessages:

𝑃unit𝑚

𝑖

=𝑃𝑚

𝑖

size (𝑚𝑖

). (25)

4. Simulation

In this section, we use the ONE [17] simulator to con-duct extensive simulations for evaluating the performance


Input:starting index: 𝑖𝑛𝑑𝑒𝑥(1) if 𝑖𝑛𝑑𝑒𝑥 > 𝑛 do(2) 𝑏𝑒𝑠𝑡𝑃 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃(3) 𝑏𝑒𝑠𝑡𝑃𝑎𝑡ℎ ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑎𝑡ℎ(4) return(5) end if(6) if 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 + 𝑠 [𝑖𝑛𝑑𝑒𝑥] ≤ BS do(7) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 + 𝑠[𝑖𝑛𝑑𝑒𝑥](8) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃 + 𝑝[𝑖𝑛𝑑𝑒𝑥](9) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑎𝑡ℎ[𝑖𝑛𝑑𝑒𝑥] ← 1(10) call Back Track (𝑖𝑛𝑑𝑒𝑥 + 1)(11) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆 − 𝑠[𝑖𝑛𝑑𝑒𝑥](12) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃 − 𝑝[index](13) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑎𝑡h[𝑖𝑛𝑑𝑒𝑥] ← 0(14) else(15) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑎𝑡ℎ[𝑖𝑛𝑑𝑒𝑥] ← 0(16) end if(17) if Bound (𝑖𝑛𝑑𝑒𝑥 + 1) > 𝑏𝑒𝑠𝑡𝑃 then(18) call Back Track (𝑖𝑛𝑑𝑒𝑥 + 1)(19) end if

Algorithm 3: Back track process.

Input:starting index: 𝑖𝑛𝑑𝑒𝑥Output:the upper bound of 𝑖𝑛𝑑𝑒𝑥: 𝑏𝑜𝑢𝑛𝑑(1) 𝑙𝑒𝑓𝑡𝑆 ← BS − 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑆(2) 𝑏𝑜𝑢𝑛𝑑 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑃(3) while 𝑖𝑛𝑑𝑒𝑥 ≤ 𝑛 and 𝑠[𝑖𝑛𝑑𝑒𝑥] ≤ 𝑙𝑒𝑓𝑡𝑆 do(4) 𝑙𝑒𝑓𝑡𝑆 ← 𝑙𝑒𝑓𝑡𝑆 − 𝑠[𝑖𝑛𝑑𝑒𝑥](5) 𝑏𝑜𝑢𝑛d← 𝑏𝑜𝑢𝑛𝑑 + 𝑝[𝑖𝑛𝑑𝑒𝑥](6) 𝑖𝑛𝑑𝑒𝑥 ← 𝑖𝑛𝑑𝑒𝑥 + 1(7) end while(8) if 𝑖𝑛𝑑𝑒𝑥 ≤ 𝑛 do(9) 𝑏𝑜𝑢𝑛𝑑 ← 𝑏𝑜𝑢𝑛𝑑 + 𝑝[𝑖𝑛𝑑𝑒𝑥] × 𝑐𝑙𝑒𝑓𝑡𝑆/𝑠 [𝑖𝑛𝑑𝑒x](10) end if(11) return bound

Algorithm 4: Bound process.

of SAPR under various settings. The simulation settings,evaluation metrics, and results are described as follows.

4.1. Simulation Settings. To well evaluate the routing per-formance of SAPR based on our assumptions, we firstconduct simulations based on the synthetic traces generatedby Random Walk model. This is because Random Walkis a typical movement model, in which the intermeetingtimes between nodes follow exponential distributions. So itis very helpful to evaluate our proposed routing algorithm.The detailed simulation settings are shown in Table 1. Weintroduce Epidemic, Prophet, and Source Spray and Waitinto the simulations and comparisons. The reason is thatEpidemic is typical multicopy routing based on flooding,

which can be used to verify the performance improvementsof SAPR. Prophet is typical probabilistic routing based onthe encounter probabilities between nodes, which differsfrom SAPR and can be used to evaluate SAPR from theperspective of probabilistic routing. Source Spray and Waitis typical opportunistic routing that strictly limits the numberofmessage copies, which can be used to evaluate the overheadratio of SAPR. In addition, we also introduce the drop-frontbuffer management policy when implementing the abovethree algorithms in order to evaluate our proposed buffermanagement policy.

Taking into consideration the shortcomings of RandomWalk, we also conduct simulations based on the synthetictraces generated by Helsinki City model. Helsinki City modelis a more realistic mobility scenario, which is based on


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real map data and adds realism, so it is very helpful toevaluate routing protocols. Moreover, we can modify themodel parameters as needed, so that it can reproduce variousempirical mobility properties, which is beneficial to therouting performance evaluations. This is also why we use theHelsinki City model instead of the real traces.

We use 126 nodes in the Helsinki City whose area is4500 × 3400m2. These nodes are divided into 6 groups.Group 1 and Group 3 are pedestrian groups (each groupcontains 40 nodes); Group 2 consists of 40 car nodes.Group 4, Group 5, and Group 6 are tram groups andthey, respectively, consist of two nodes. Pedestrians movewith speeds of 0.5–1.5m/s, cars move with speeds of 2.7–13.9m/s, and trams move with speeds of 7–10m/s. Two

types of devices are introduced in the simulations. One isBluetooth device with transmission speed of 250KBps andtransmission range of 20m. The other is High Speed devicewith transmission speed of 10MBps and transmission rangeof 1000m. Group 1, Group 2, and Group 3, respectively, have4–20MB buffers and they are based on the Shortest PathMapBasedmovementmodel. Group 4, Group 5, andGroup 6 have50MB buffers, respectively, and they are based on the RouteMap Based movement model. Group 1, Group 2, Group 3,Group 5, and Group 6 use Bluetooth devices; Group 4 usesboth Bluetooth devices and High Speed devices. The othersimulation settings are shown in Table 2. In the simulations,we also introduce Epidemic and Prophet for the same reason.Besides taking into consideration themobility of nodes in this


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Table 2: Simulation settings in Helsinki City.

Parameter Default value (range)Message size 500K–1MMessage interval 40 s (30–90 s)Buffer (Groups 1–3) 20M (4–20M)Time-to-live (TTL) 3 h (2–5 h)Simulation time 24 h𝜌 0.95

scenario, we add First Contact to the simulations to evaluatethe overhead ratio of SAPR. These routing algorithms alsoimplement the drop-front buffer management policy.

4.2. Evaluation Metrics. In this paper, the simulations aregrouped into the three categories: varying buffer size, vary-ing message’s time-to-live, and varying message generationinterval. Under the same guideline, we evaluate all routingalgorithms based on the following metrics.

(1) Delivery ratio: this metric is to measure the deliverycapability of each algorithm.

(2) Overhead ratio: it reflects the efficiency of messagetransmission and it is desirable to achieve a lowoverhead ratio.

(3) Average latency: the lower average latency meansbetter routing performance.

(4) Average hop count: it is another routing goal to reducetransmission cost, such as bandwidth and energy.


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(5) Dropped messages: it is desirable to achieve a fewernumber of dropped messages so as to improve theutilization efficiency of storage.

4.3. Simulation Results

4.3.1. Performance Evaluations in Random Walk Model.Figure 1 shows the different routing performance by varyingbuffer size from 5MB to 50MB. Figure 2 shows the differentsimulation results by varying message’s TTL from 2 hoursto 5 hours. Figure 3 shows the performance comparisons byvarying message’s generation interval from 10 s to 50 s.

Regarding Figure 1, we can see that SAPR gets the highestdelivery ratio, the lowest overhead ratio, and the fewest aver-age hop count compared to Epidemic and Prophet. This can

show the accuracy and efficiency of the routing selections ofSAPR. Besides, it can also verify the improvements of SAPRon predicting message’s delivery probability. By limiting thenumber of message copies, Source S and W gets a slightlyhigher delivery ratio than that of SAPR when buffer isinsufficient (i.e., less than 10MB). However, our SAPR getsthe highest message delivery ratio when buffer size is greaterthan 10MB. For the same reason, Source S andWalso gets thelowest overhead ratio and the fewest hop count. But SAPR’sperformance on network overhead and average hop count isvery close to Source S and W.

From Figure 2, we can find that SAPR still outperformsEpidemic and Prophet in terms of overhead ratio and averagehop count. And SAPR still gets the highest delivery ratiowhen message’s TTL is greater than 3 hours. This shows thatSAPR is adapted to the scenario with a longer message TTL.


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Figure 3 shows similar simulation results to Figure 1.Compared to Epidemic and Prophet, SAPR achieves advan-tages in message delivery ratio, overhead ratio, and averagehop count. Moreover, the overhead ratio and average hopcount of SAPR are also close to those of S and W, but SAPRgets a higher message delivery ratio.

Finally from Figures 1–3, we can see that SAPR gets avery low overhead ratio and greatly controls average hopcount. In addition, SAPR also achieves a satisfying messagedelivery ratio. Unfortunately, SAPR does not get advantagesin message’s delivery latency in this scenario.

4.3.2. Performance Evaluations in Helsinki City Model.Figure 4 shows the different routing performance by varying

buffer size from 4MB to 20MB. Figure 5 shows the differentsimulation results by varying message’s TTL from 2 hoursto 5 hours. Figure 6 shows the performance comparisons byvarying message’s generation interval from 30 s to 90 s.

Regarding Figure 4, we can see that SAPR can achieve thehighest message delivery ratio, the lowest overhead ratio, theshortest average latency, and the fewest average hop count. Itcan show once again the accuracy and efficiency of SAPR’sselections.

In Figure 5, SAPR can outperform the other three routingprotocols in terms of delivery latency and average hop count.When message’s TTL is greater than 3 hours, SAPR achievesthe highest message delivery ratio. Moreover, SAPR still getsadvantages in network overhead ratio compared to Epidemicand Prophet.


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Figure 6 shows that SAPR still achieves some advantagesin message delivery ratio, network overhead ratio, deliverylatency, and average hop count compared to the other threealgorithms.

Finally, from Figures 4–6, we can draw the conclusionthat SAPR can enhance routing performance in terms ofmessage delivery ratio, network overhead ratio, average deliv-ery latency, and average hop count compared to Epidemic,Prophet, and First Contact.

4.3.3. Performance Evaluations of Dropped Messages. Figures7(a)–7(c) compare the performance of dropped messagesin Random Walk model by changing buffer size, TTL, andmessage interval. Figures 7(d)–7(f) show the evaluationresults of dropped messages under different settings.

From Figure 7 we can see that the number of droppedmessages of Epidemic is the largest. This is because Epidemicuses flooding strategy to distribute message to every encoun-tered node. In this case, it will spread a large number ofmessage copies to the whole network. When storage resourceis not sufficient, these message copies will be frequentlydropped by nodes. In this case, it is hard to spreadmessage tofarther regions, which is not conducive to a better distributionof messages. On the contrary, the other algorithms all controlmessage redundancy by different schemes, thus droppingfewermessages. In this case, Epidemic is more sensitive to theamount of network resources compared to other algorithms,and its better performance greatly relied on more cacheresources. This can explain the reason why the performanceof Epidemic is not better than that of other algorithms.


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The number of dropped messages of Prophet is fewerthan that of Epidemic; thus Prophet can get better perfor-mance. Compared to Epidemic and Prophet, SAPR dropsfewer messages, and its number of dropped messages isslightly higher than that of Source Spray and Wait. This canindicate that SAPR can greatly control message redundancy,thus improving the utilization efficiency of storage resource.From this point of view, it can explain the better routingperformance of SAPR.

5. Related Works

Prophet is a typical probabilistic routing protocol whichmakes routing selections based on the encounter probabili-ties between nodes. For example, current node will deliver

message to an intermediate node if the node is more likely tomeet the final destination. In addition to using the transitivityproperty to update the encounter probability, Prophet alsoproposes an aging function for the outdated information astime progresses.

For message dropping problem, many traditional policies(e.g., drop-tail, drop-front, random drop, etc.) have beenproposed, which can play a role in opportunistic routing. In[18], Zhang et al. analyze the buffer constrained Epidemicrouting and make the conclusion that drop-front can outper-form drop-tail in DTN context. In [19], a node first deletesthe message that has the largest number of copies in orderto mitigate the impact on routing performance. Based on aspecific community detection algorithm, [20] proposes anefficient buffer management policy for social delay tolerant


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networks, which utilizes social relation and centrality to avoiddropping meaningful messages.

In [21], Li et al. propose an optimal routing strategy byexploiting the heterogeneous features of nodes to enhancethe routing performance. It takes into consideration nodes’heterogeneous contact rates and delivery costs when selectingintermediate nodes to minimize the delivery cost. For mobilesensor networks, [22] provides a reliable routing scheme withan enhanced delaying technique, which estimates connec-tivity based on the ratio of past and present connections.When the connectivity is unreliable, nodes will delaymessagetransmission.

With a home-aware model, CAOR [23] turns mobilesocial networks into a network that only includes commu-nity homes. Then, in the network of community homes,it computes the minimum expected delivery delay by areverse Dijkstra algorithm. In [24], by introducing a metricto accurately detect the quality of friendship, each nodedefines its friendship community as the set of nodes havingclose friendship with itself either directly or indirectly. Thentemporally differentiated friendships are used to make theforwarding decisions of messages.

6. Conclusion

In this paper, we try to improve the probabilistic routingperformance by taking into account the message’s remainingTTL so as to avoid the shortcomings of routing messagesdirectly based on the encounter probabilities between nodes.Our motivation is that the higher encounter probability canonly indicate that the two nodes can meet each other fre-quently. But they may still need a period of time to encountereach other again. However, the message transmission willstill fail if the message’s TTL is exhausted during this periodof time. In this case, an effective scheme that fully takesinto account the message’s remaining TTL when computingmessage’s delivery probability can get a better performance inprobabilistic routing. To this end, by using statistical analysismethods, we propose an efficient scheme to compute andupdate the expectation of the intermeeting times betweennodes. And then, based on exponential distribution, wepredict the probability that a message can be successfullydelivered before its TTL is exhausted.

In addition, we also improve buffer management policyby modeling message dropping problem as a 0-1 knapsackproblem. Then, solving the problem by the back tracktechnique, each node always keeps the messages that canmaximize the delivery probability sum. Extensive simulationsare conducted based on Random Walk model and HelsinkiCity model. The results show that the proposed SAPR cangreatly enhance the routing performance in DTN context.

Notations

𝑋 : Intermeeting time between nodes𝜆: Exponential distribution parameter𝐸(𝑋): Mathematical expectation of𝑋

𝑥𝑘

(𝑁𝑗

): The 𝑘th sample of the intermeeting timebetween current node and𝑁

𝑗

𝑐𝑘

(𝑁𝑗

): The count that the 𝑘th sample appears𝐷(𝑋): Mathematical variance of𝑋𝑚𝑖

: Message𝑚𝑖

𝑅𝑖

: Remaining time-to-live of𝑚𝑖

𝜏: The time that has elapsed since the lastencounter with destination node

𝑃𝑚

𝑖

: The predicted delivery probability of𝑚𝑖

size (𝑚𝑖

): The size of𝑚𝑖

𝐵𝑖

: Drop message𝑚𝑖

if 𝐵𝑖

= 0

𝜌: The probability threshold we set.

Conflict of Interests

All authors do not have any possible conflict of interests.

Acknowledgments

This research is supported in part by Natural Science Foun-dation of Shandong Province under Grant no. ZR2013FQ022,Science and Technology Plan Project for Colleges and Uni-versities of Shandong Province under Grant no. J14LN85,and Foundation Research Project of Qingdao Science andTechnology Plan under Grant no. 12-1-4-2-(14)-jch.

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