C

Consiglio Nazionale delle Ricerche

Optimal energy vs. delay trade off in opportunistic networks

E. Biondi, C. Boldrini, A. Passarella, M. Conti

IIT TR-13/2015

Technical Report

Ottobre 2015

Iit

Istituto di Informatica e Telematica

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 1

Optimal energy vs. delay trade off inopportunistic networks

Elisabetta Biondi, Chiara Boldrini, Andrea Passarella, and Marco Conti

Abstract—Opportunistic communications have been recently proposed as a key strategy for offloading traffic from 3G/4G cellularnetworks, which is particularly beneficial in case of crowded areas where many users are interested in similar contents. To conserveenergy, duty cycling schemes are typically applied, and therefore contacts between nodes may become intermittent and sporadic alsoin dense networks. It is thus of paramount importance to accurately tune the duty cycling policy in order to meet energy requirementswithout compromising the quality of communications. In this paper, building upon a model of duty cycling in opportunistic networks thatwe have validated in a previous work, we study how to optimise the value of the duty cycle in order to provide probabilistic guaranteeson the delay experienced by messages. More specifically, for a broad range of end-to-end delay distributions, and for the most relevantcases of inter-contact times distributions found in real traces, we provide closed-form approximated solutions for deriving the optimalduty cycle such that the probability that the delay is smaller than a target value z is greater than or equal to a configurable probability p.

Index Terms—Computer Society, IEEEtran, journal, LATEX, paper, template.

F

1 INTRODUCTION

Opportunistic networks have been conceived at the in-tersection between Mobile Ad hoc NETworks (MANET) andthe Delay Tolerant (DTN) paradigm. In the conventionalmodel, they exploit the movements of the nodes of the net-work (people with their smart, handheld devices like tabletsand smartphones) in order to deliver messages to their desti-nations according to the store-carry-and-forward paradigm:nodes hold messages while they move and forward them toother nodes that are in radio contact, until messages reachtheir final destination. Opportunistic communications wereinitially seen as a standalone solution for those scenariosin which the nodes of the network were sparse and theinfrastructure unavailable (disaster/emergency scenarios,developing countries, etc.). Recently, however, they havebecome one of the key strategies for mobile data offloading[?], whose main goal is to offload the traffic from cellularnetworks to other types of networks (e.g., WiFi infrastruc-tured or MANET) in a synergic way, in order to address theoverloading of the 3G/4G infrastructure.

In case of crowded environments (and thus dense net-works) overloading may be even more critical, and oppor-tunistic networking techniques can be usefully applied, asfollows. Due to the typical Zipf-like shape of content inter-est, it is likely that large fractions of users in the crowd areinterested in few, very popular contents (e.g., those mostlyrelated to the area where the crowd gathers). Multicast canbe a solution to reduce the traffic load only when contentrequests can be synchronised. When requests are generateddynamically by users, exploiting communications betweenusers’ devices is a more flexible solution, as content can be

• E.Biondi, C.Boldrini, A.Passarella, and M.Conti are with the Institute forInformatics and Telematics of the Italian National Research Council, ViaG.Moruzzi 1, 56124 Pisa, Italy.E-mail: firstname.lastname@iit.cnr.it

This work was partly funded by the EC under the EINS (FP7- FIRE 288021),MOTO (FP7 317959), and EIT ICT Labs MOSES (Business Plan 2015)projects.

sent through the cellular network only to a few of them,exploiting opportunistic communications for the rest. TheDevice-to-Device technology addresses this goal to someextent, and is currently proposed in latest LTE releases. Inthis paper we focus on offloading through ad-hoc WiFi orBluetooth technologies, as this approach permits to exploitadditional portions of the spectrum (and, therefore, addi-tional bandwidth) with respect to that allocated to cellularnetworks. A possible roadblock in this scenario is the factthat direct communications consume significant energy. Toaddress this, nodes are typically operated in duty cyclingmode, by letting their WiFi (or Bluetooth) interfaces ONonly for a fraction of time. The joint effect of duty cyclingand mobility is that, even if the network is dense, theresulting patterns in terms of communication opportunitiesis similar to that of conventional opportunistic networks, asdevices are able to directly communicate with each otheronly when they come in one-hop radio range and bothinterfaces are ON.

The net effect of implementing a duty cycling scheme isthus the fact that some contacts between nodes are missedbecause the nodes are in power saving mode. Hence, de-tected intercontact times, defined as the time between twoconsecutive contact events during which a communicationcan take place for a pair of nodes, are longer than intercon-tact times determined only by mobility, when a duty-cyclingpolicy is in place. This heavily affects the delay experiencedby messages, since the main contribution to message delayis in fact due to the intercontact times. In our previouswork [?], we have focused on exponentially distributedintercontact times and we have studied how these aremodified by duty cycling, obtaining that intercontact timesremain exponentially distributed but their rate is scaled bythe inverse of the duty cycle (see Proposition 1, Section 3).Building upon this result, we have then investigated howthe first moments of the end to end delay vary with the dutycycle for a number of opportunistic forwarding schemes. In

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 2

addition, we have found that energy saving and end-to-enddelay both scale linearly with the duty cycling, althoughthe linear coefficient of scaling for the delay depends on theoriginal distribution of the inter-contact times.

Our work in [1] assumed that the value of the duty cyclewas given and studied its effects on important performancemetrics such as the delay, the network lifetime, and the num-ber of messages successfully delivered to their destination.More in general, the duty cycling can be seen as a parameterthat can be configured, typically, based on some targetperformance metrics. To this aim, the main contribution ofthis paper is a mathematical model that allows us to tunethe duty cycle in order to meet a given target performance,expressed as a probabilistic guarantee (denoted as p) on thedelay experienced by messages. Considering probabilistic,instead of hard, guarantees, allows us to cover a very broadrange of application scenarios also beyond best-effort cases– all but those requiring real-time streaming. Specifically, westudy the case of exponential, hyper-exponential and hypo-exponential delays (please recall that any distribution fallsinto one of these three cases, at least approximately [2]),for original inter-contact times following an exponential ora Pareto distribution, which are two of the most relevantcases according to the analysis of real traces (see, e.g.,[3]). In all these cases we derive the optimal duty cyclegiven a fixed maximum delay z and a given probabilisticguarantee on that maximum delay. For the simple case ofexponential delays we are able to provide an exact solution.For the other cases, we derive an approximated solution andthe conditions under which this approximation introducesa small fixed error ε (which is always below 0.14) onthe target probability p. Specifically, in the worst case, theapproximated duty cycle introduces an error on the targetprobability p of about 0.1 (hyper-exponential case) and 0.14(hypo-exponential case), while in the other cases the error iswell below these thresholds.

The paper is organised as follows. In Section 2 weoverview the literature on duty cycle optimisation for op-portunistic networks. After having introduced the networkand duty cycle model that we consider in this work andrecalled some preliminary results (Section 3), we providean analytical model for the end-to-end delay in presenceof duty cycling (Section 4). Then, in Section 5, we derivethe optimal duty cycles for the case of exponential, hyper-exponential, and hypo-exponential delays. Finally, Section 6concludes the paper.

2 RELATED WORK

There are not many contributions in the DTN literaturestudying the optimisation of the duty cycling policy. In [4],using a fixed duty cycle scheme, Wang et al. study the rela-tionship between the probability of missing a contact andthe associated energy consumption (considered inverselyproportional to the contact probing interval). Building uponthese results, [4] provides some heuristic algorithms toachieve an optimal contact probing. Differently from thiswork, in this paper we mathematically define the optimi-sation problem and we provide an analytical, closed form,result.

In [5], Gao and Li focus on the design of an adaptiveduty cycle that minimises wakeups during intercontacttimes (which are useless, from a contact probing stand-point). Differently from [5], we have chosen to optimisethe duty cycle directly, based on the performance goal thatwe want to achieve. While it is true that an optimisationbased on intercontact times impacts directly on the delayperformance, it is not straightforward how to control theone based on the other. With our model, instead, we candirectly go from the requirements in term of probability ofstaying below a fixed delay threshold to the correspondingduty cycle value. In addition, differently from [5], we focuson a fixed duty cycle, similar to [6] [7] [4]. It is still an openresearch point which duty cycling strategy is to be preferred.However, preliminary results in [4] show that, under someassumptions, fixed duty cycle is the optimal strategy.

Another contribution focused on duty cycle optimisationis [8], in which Altman and Azad study the optimisation ofnode activation in DTN relying on a fluid approximationof the system dynamics. However, the problem analysedis different from the one studied in this paper, since in[8] nodes, once activated, remains active. In addition, thismodel is based on the assumption of i.i.d. intercontact times,while it has been shown that realistic intercontact times areintrinsically heterogeneous. For this reason, here we focuson heterogenous (but still independent) intercontact times.

3 PRELIMINARIES

The duty cycle is for us any energy saving protocol thatkeep the device alternating between ON and OFF statesin order to reduce the activity of the device and thusthe battery consumption. In the ON state the device andall its interfaces are active, and thus it is able to detectcontacts with other devices and to share the messageswith them. Otherwise in the OFF state all the connectivityinterfaces are switched off and the communication cannotbe achieved. This simple abstract scheme is able to capturethe most popular wireless technology (please see [9] for thedetails). In the literature are developed both deterministicand stochastic duty cycle policies, but in this work we willfocus only on the deterministic one. However, as seen in[10], the deterministic case is able to capture the averagebehaviour of the stochastic case too. The main parameter ofa deterministic duty cycle scheme is denoted with ∆ definedas the ratio between the ON state and the total period of theduty cycle (that is the sum of the ON and the OFF state).It is well known that in opportunistic networks the delayexperienced by the messages is mostly due to the time thateach node has to wait before meeting another node, becausethe message reaches its final destination thought multipleexchanges between pair of nodes. This time is measuredin the literature by the intercontact times. As the effect ofthe duty cycle is to reduce the number of contacts betweennodes, the intercontact times obviously are changed by thethe duty cycle. In [10] we have studied the effect of theduty cycle on the intercontact times, supposing to knowthe distribution of the intercontact times when no energy-saving protocols are applied in the network. In particular,when the intercontact times without duty cycle policies are

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 3

exponential, the intercontact times are still exponential withrate scaled by the factor ∆. The exact result is the following:

Proposition 1. Considering a tagged pair of nodes i andj with exponential intercontact time of rate λij , thedetected intercontact time, i.e. the effective intercontacttime when a duty cycling policy is in place, features ap-proximately an exponential distribution with rate ∆λij ,as long as λijT � 1, where T is the duty cycling period.

When the intercontact times are distributed as a Paretothen the tail of the distribution is still Pareto with thesame exponent and furthermore we can derive the first twomoments of the detected intercontact times. Summing up,we have the following:

Proposition 2. Considering a tagged pair of nodes i and jwith Pareto intercontact time S of rate λij and exponentαij and supposing that λijT

αij−1 � 1, where T is the dutycycling period, the detected intercontact time decays as aPareto random variable with exponent αij and the first,the second and the coefficient of variation of the detectedintercontact times S are given by:

E[S] =1

∆E[S] (1)

E[S2] =

(2

∆− 1

)E[S]2 +

1

∆(E[S2] + E[S]2) (2)

cv2S

= ∆ cv2S + (1−∆) (3)

We have yet seen in [1] that the condition λijT � 1 holds forthe majority of contact traces available in the literature. Forthe Pareto case. The results cited above are obtained withoutconsidering the contact durations between nodes. The sameanalysis can be made using the results in [10] for contactduration non negligible, but as seen in the same work, themodel with negligible contacts fits well the majority of thetraces available in the literature.

4 THE EFFECT ON THE DELAY

In this section we exploit the results on the detected inter-contact times summarized in section 3 in order to computethe first two moments of the delay for a set of representativeforwarding strategies designed for opportunistic networks.The first step in this direction is to use the first two momentsof the detected pairwise intercontact times for approximat-ing the distribution of the intercontact time itself. Please notethat while the analysis in the previous section focused on atagged pair of nodes, in this section we study the wholenetwork. So, when the intercontact times are exponentiallydistributed, denoting with their rates as λij for node pairi, j, we have shown that, when assumption λijT � 1holds, the detected intercontact times follow approximatelyan exponential distribution. When the intercontact times arePareto distributed, denoting with their rates as λij and withtheir exponent as αijfor node pair i, j, we have shown that,when assumption λijT

αij−1 � 1 holds, the moments of thedetected intercontact times change as in Proposition 2. Usingthis approximation, in the following we solve the analyticalmodel proposed in [11] for both real intercontact times anddetected intercontact times. The goal of this evaluation is to

study how the first two moments of the delay are affectedby energy saving techniques.

The forwarding model that we exploit represents theforwarding process in terms of a Continuous Time MarkovChain (CTMC) [12]. The chain has as many states as thenodes of the network and transitions between states dependboth on the meeting process between nodes (i.e., theirintercontact times) and on the forwarding protocol in use.Denoting the delay of messages from a generic node i toa tagged node d as Did, and using standard Markov chaintheory, it is possibile to derive the first two moments of Did

as in [11]. We report this result below for the convenienceof the reader. Please note that this model (as most analyticalmodels in the literature) assumes buffers and bandwidthlarge enough for accommodating all messages.Lemma 1 (Delay’s first and second moment). The first and

second moment of the delayDid for a message generatedby node i and addressed to node d can be obtained fromthe minimal non-negative solutions, if they exists, to thefollowing systems, respectively:

∀i 6= d : E[Did]=1∑

j∈Ri λij+

∑j∈Rs−{d}

λij∑z∈Ri λiz

E[Dj ]

∀i 6= d :

E[(Did)2]=

2

[∑j∈Ri λij ]

2+

∑j∈Rs−{d}

λij∑z∈Ri λiz

E[D2jd] +

+2∑

j∈Rs−{d}

λij∑z∈Ri λiz

1∑j∈Ri λij

E[Djd]

where we denote with Ri the set of possible relaystowards destination d when the message is on node i(this set depends on the forwarding strategy in use).

In [11], we have defined a set of abstract policies ableto capture significant aspects of popular state-of-the-art for-warding strategies. In the following we will focus on thesepolicies. Under the Direct Transmission (DT) forwardingscheme, the source of the message is only allowed to handit over to the destination itself, if ever encountered. With theAlways Forward (AF) policy, the message is handed overby the source, and the following relays to the first nodesencountered. Both DT and AF are social-oblivious (alsoknown as context-oblivious or randomized) policies, i.e.,they do not exploit information on node social relationshipsand contact behavior. In [11] two social-aware policies werealso defined. In social-aware policies, each intermediate for-warder hands over the message to nodes that have a higherprobability of bringing the message closer to the destination,according to some predefined forwarding metrics. The firstof these policies is Direct Acquaintance (DA), in which theforwarding metric is the contact rate with the destination( 1E[Sid] ): a better forwarder is one with a higher contact

rate with respect to the node currently holding the message.The second policy is Social Forwarding (SF), for which theforwarding metric is β 1

E[Si,d] + (1 − β)∑j∈Pi wij

1E[Sjd] ,

where wij =λij∑

j∈Piλij

. With respect to the DA policy, whichonly captures direct meetings with the destination, SF is also

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 4

able to detect indirect meetings, allowing nodes to selectrelays that not only meet the destination frequently but alsomeet nodes that meet the destination frequently.

In the rest of the paper we are going to indicate withD∆ the random variable which represent the delay of themessage when the duty cycle ∆ is active and with D therandom variable which represent the delay of the messagewhen the duty cycle is not active (i.e. D = D1). We nowexplore the effect of the duty cycle on the moments of thedelay for the exponential intercontact times and for thePareto intecontact times. We will see that the results areextremely different and in particular in the Pareto case, wewill need to go deeper for finding the desired results.

4.1 The exponential case

First of all we analyse the case of nodes intercontact timesexponentially distributed. The fitting analysis presented in[3] has shown that contact rates in the traces follow aGamma distribution. Below, we focus on the distributionparameters for the RollerNet scenario reported in [13], i.e.,shape ξ = 4.43, rate r = 1088. We consider a networkmade up of 25 nodes and we solve the forwarding modeldescribed above in the case of duty cycle equal to 5

15 , 1015

and 1 (no duty cycling). Figures 1 and 2 show the CDF ofthe moments of the delay in this case. As expected, both thefirst and second moment become larger as we reduce theON interval in the duty cycle. In fact, as discussed before,the net effect of duty cycling is to effectively reduce thenumber of usable contacts to only those happening duringan ON period. The shorter the ON period, the fewer theusable contacts every T , the longer the delay.

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time [s]

Delta=1Delta=5/15

Delta=10/15

(a) DT

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time [s]

Delta=1Delta=5/15

Delta=10/15

(b) AF

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time [s]

Delta=1Delta=5/15

Delta=10/15

(c) DA

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time [s]

Delta=1Delta=5/15

Delta=10/15

(d) SF

Fig. 1. CDF of the first moment of the delays for the different forwardingalgorithms

Let us now see what happens to the coefficient of vari-ation c(D∆) of the delay. From Figure 3 it can be seenthat both c(D∆) < 1 and c(D∆) ≥ 1 are possible. Thismeans that the delay can be approximated with a hyper-exponential or a hypo-exponential distribution [2]. It isinteresting to observe from Figure 3 that the coefficient ofvariation does not depend on the duty cycle ∆ (in fact,all curves overlap). This means that, in the case of expo-nential intercontact times, the duty cycle does not affect

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time2 [s2]

Delta=1Delta=5/15

Delta=10/15

(a) DT

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time2 [s2]

Delta=1Delta=5/15

Delta=10/15

(b) AF

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

CD

F

time2 [s2]

Delta=1Delta=5/15

Delta=10/15

(c) DA

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000

Delta=1Delta=5/15

Delta=10/15

(d) SF

Fig. 2. CDF of the second moment of the delays for the different forward-ing algorithms

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3C

oeffi

cien

t of v

aria

tion Delta=0.1

Delta=0.2Delta=0.3Delta=0.4Delta=0.5Delta=0.6Delta=0.7Delta=0.8Delta=0.9Delta=1.0

(a) DT

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Coe

ffici

ent o

f var

iatio

n Delta=0.1Delta=0.2Delta=0.3Delta=0.4Delta=0.5Delta=0.6Delta=0.7Delta=0.8Delta=0.9Delta=1.0

(b) AF

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Coe

ffici

ent o

f var

iatio

n Delta=0.1Delta=0.2Delta=0.3Delta=0.4Delta=0.5Delta=0.6Delta=0.7Delta=0.8Delta=0.9

Delta=1

(c) DA

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Coe

ffici

ent o

f var

iatio

n Delta=0.1Delta=0.2Delta=0.3Delta=0.4Delta=0.5Delta=0.6Delta=0.7Delta=0.8Delta=0.9Delta=1.0

(d) SF

Fig. 3. CDF of the coefficient of variation of the delay with different ∆

the variability of the delay experienced by messages, andthus for every duty cycle ∆, c(D∆) = c(D). As the coeffi-cient of variation is defined as the square root of the ratiobetween the standard deviation and the mean value, i.e.c(D∆)2 = σ(D∆)

E[∆] , if we are able to capture the dependenceof the first moment of the delay E[D∆] from the dutycycle ∆, we immediately can derive the dependence of thesecond moment of the delay E[D2

∆] from ∆. We firstly triedwith the inversion proportionality, i.e. with E[D∆] = E[D]

∆ .For testing this equation, in Figure 4 we plotted E[D]

∆E[D∆] .Figure 4 (derived for the same parameters of the RollerNetscenario used above) tells us that the ratio stays around 1independently of the specific duty cycle value ∆. This resultis very interesting, because it shows that under exponentialintercontact times the delay experienced by the message isscaled by the factor ∆. In summary, what we can derive isthe following preposition:

Proposition 3. When the node intercontact times are expo-nentially distributed with rates λij for node pair (i, j),that verify λijT � 1, then:

E1 The expected delay when a duty cycling policy is inplace (denoted as E[D∆]) is approximately equal to the

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 5

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

∆

R∆

(a) DT

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

∆

R∆

(b) AF

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

∆

R∆

(c) DA

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

∆

R∆

(d) SF

Fig. 4. E[D]∆E[D∆]

varying ∆ in the different forwarding algorithms

expected delay E[D] with no duty cycle scaled by afactor 1

∆ , i.e, E[D∆] = E[D]∆ .

E2 The second moment of the delay when a duty cyclingpolicy is in place (denoted as E[D2

∆]) is approximatelyequal to the second moment of the delay E[D2] with noduty cycle scaled by a factor 1

∆2 , i.e, E[D2∆] = E[D2]

∆2 .E3 The dependence of the coefficient of variation c(D∆) of

the delay from ∆ is negligible, i.e. c(D∆) = c(D).

Proof: E1 and E2 are the consequences of the analysismade above. The last easily derive by E1 and E2, using thedefinition of coefficient of variation.

4.2 The Pareto case

In this section we want to study the case of a networkin which the intercontact times between nodes are Paretodistributed, i.e. with PDF given by the function f(x) =

α bα

(b+x)α+1 , for all x > 0. For this part we decided to analysea network of nodes where the pairs intercontact times arePareto distributed with parameters taken from a real trace.As we need finite first and second moments, we selectedfrom the Rollernet trace the distributions of interontacttimes that are Pareto distributed with shape α bigger than2. Then, we considered a network of 11 nodes and withintercontact times between the pairs of scale b equal to the120s, the average scale of the selected intercontact times,and shape α randomly extract from the shapes of theselected intercontact times. Using different duty cycles, asmade in Section 4.1, and then analysing the moments ofthe delay of the messages sent in the network, we discoverthat the relations E1-E3 valid for exponential intercontacttimes, are not working here. In figures 5 and 6 we cansee that the dependences of the moments with duty cycleE[D∆], E[D2

∆] from the respective moments without dutycycle E[D], E[D2] are linear. However, their slope are notgoing like 1

∆ and 1∆2 respectively.

From a deeper analysis, we discover that the two-orderlogarithm of the ratios E[D∆]/E[D] and of E[D2

∆]/E[D2]

(a) DA

(b) DT

Fig. 5. Behaviour of E[D∆] respect to E[D]

(a) DA

(b) DT

Fig. 6. Behaviour of E[D2∆] respect to E[D2]

are linearly dependent from ∆, and precisely for a general

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 6

pair (i, j) of nodes, holds1:

log2

(log

(E[D

(i,j)∆ ]

E[D(i,j)]

))= m

(i,j)1 ∆ + q

(i,j)1 , (4)

log2

(log

(E[(D

(i,j)∆ )2]

E[(D(i,j))2]

))= m

(i,j)2 ∆ + q

(i,j)2 , (5)

for some coefficients m(i,j)1 , q

(i,j)1 ,m

(i,j)2 , q

(i,j)2 ∈ R.

The coefficients of equations (4) and (5) vary for each pair.Thus, using linear regression on the coefficients for all thedifferent pairs we than obtain several functions that givethe first (respectively the second) moment of the delay witha certain duty cycle, knowing the first (respectively thesecond) moment of the delay without duty cycle.

(a) DA

(b) DT

Fig. 7. First moment of the delay with different duty cycles for differentforwarding algorithms

In Figure 7, 8 and 9, can be seen the comparison ofthe empirical data with result of the fitting of the first, thesecond and the coefficient of variation for a tagged pair ofnode with different duty cycles and forwarding policies. Ascan be seen, the function proposed is able to fit well theempirical data. As expected, the moments of the delay arelarger when duty cycle is active. In Figure 9 can be seenthe coefficient of variation of the delay. We can see herethat the effect of the duty cycle is to reduce the coefficientof variation an in particular by applying a duty cycle, anhyper-exponential delay can become hypo-exponential (seefor example Figure 9). We omitted the plots for the otherpairs for lack of space, but the behaviour is the same.Summarizing, we have derived the following properties,which will be used extensively through the paper:Proposition 4. When the node intercontact times are Pareto

distributed, then the node pair (i, j) we have taht:

1. We here used a formulation with the logarithm to base 2 and e, butby the rules of changing base, the same formula is valid for other basesopportunely changing the coefficients m and q.

(a) DA

(b) DT

Fig. 8. Second moment of the delay with different duty cycles for differentforwarding algorithms

(a) DA

(b) DT

Fig. 9. Coefficient of variation of the delay with different duty cycles fordifferent forwarding algorithms

P1 The expected delay when a duty cycling policy is inplace (denoted as E[D

(i,j)∆ ]) is approximately given by:

E[D(i,j)∆ ] = exp(β

(i,j)1 2γ

(i,j)1 ∆) · E[D(i,j)].

P2 The second moment of the delay when a duty cyclingpolicy is in place (denoted as E[(D

(i,j)∆ )2]) is approxi-

mately given by:

E

[(D

(i,j)∆

)2]

= exp(β(i,j)2 2γ

(i,j)2 ∆) · E

[(D(i,j)

)2].

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 7

P3 The dependence of the coefficient of variation c(D(i,j)∆ )

of the delay from ∆ is approximately given by:

c(D(i,j)∆ ) =

√√√√√√√(c(D(i,j)) + 1) exp

(β

(i,j)2 2γ

(i,j)2 ∆

)exp

(2β

(i,j)1 2γ

(i,j)1 ∆

) − 1.

Proof: P1 and P2 are the consequences of the analysismade above, and in particular the parameters are derivedfrom 4 and 5 using γ

(i,j)k = m

i,j)k and β

(i,j)k = 2q

(i,j)k for

k = 1, 2. The last easily derive by P1 and P2, using thedefinition of coefficient of variation.

5 SETTING THE DUTY CYCLE

In this section we are going to solve the problem of optimiz-ing the duty cycle for each pair of nodes, so all the variablesare referred to the node pair considered. For simplicity ofnotation, we here omit the superscript (i, j) of the variableswhich indicates the pair considered, and thus for examplethe delay of the message sent between the pair (i, j), will bedenoted with D instead of D(i,j). In the following section,when we will analyse the whole network we will come backto the complete notation.

In this section we discuss how to derive the optimalduty cycle ∆opt such that the delay of a tagged messageremains, with a certain probability p, under a target fixedthreshold z or, in mathematical notation, ∆opt = min{∆ :P{D∆ < z} ≥ p}. Since the delay increases with ∆, thelatter is equivalent to finding the solution to the following2:

∆opt = {∆ : P{D∆ < z} = p}. (6)

In the following we will denote the CDF of D∆ as F∆(x).In order to find the solution to Equation 6, the distributionof the delay D∆ should be known. Although it is in generalunfeasible to obtain an exact closed form for the distributionof D∆ (except for some trivial cases, such as when thesource node can only deliver the message to the destinationdirectly), it is often possible to compute its moments, eitherexactly or approximately, under different distributions forintercontact times, as shown, e.g., in [14] [15]. When the firsttwo moments of the delay can be derived, it is possible toapproximate its distribution with either a hypo-exponentialor hyper-exponential random variable, using the momentmatching approximation technique [2]. So, exploiting prop-erties derived in Section 4, according to the coefficient ofvariation c∆ as

√E[D2]E[D]2 − 1, when c∆ is greater than one,

D∆ can be approximated using a 2-stages hyper-exponentialdistribution with the same moments of D∆, as stated in thefollowing Lemma.Lemma 2 (Hyper-exponential approximation). The two mo-

ments matching approximation of D∆ with coefficientof variation c ≥ 1 is a 2-stages hyper-exponential dis-tribution with parameters (λ1, p1), (λ2, p2) given by thefollowing: p1 = 1

2

(1 +

√c2∆−1

c2∆+1

)λ1 = 2p1

E[D∆]

{p2 = 1− p1

λ2 = 2p2

E[D∆](7)

2. In the rest of the paper, for convenience of notation, we will dropsubscript opt since all ∆ we derive are the optimal ones.

Vice versa, when the coefficient of variation of the delayis smaller than 1 (but greater than 1√

2[16]), D∆ can be

approximated with an hypo-exponential distribution withCDF FX(x) = 1− µ2

µ2−µ1e−µ1x + µ1

µ2−µ1e−µ2x, for all x ≥ 0,

according to the following lemma.Lemma 3 (Hypo-exponential approximation). The two mo-

ments matching approximation of D∆ with a coefficientof variation. In the general case c∆ < 1, the two mo-ments matching approximation of the hypoexponentialis provided in [2]. However, with this approximation, µ1

and µ2 are such that they continuously oscillate as c∆increases. Hence, the techniques described in this papercannot be exploited in this case. c∆ ∈ ( 1√

2, 1) is an hypo-

exponential distribution with rates µ1, µ2 given by thefollowing: µ1 = 2

E[D∆] ·1

1+√

1+2(c2∆−1)

µ2 = 2E[D∆] ·

1

1−√

1+2(c2∆−1)

(8)

In the rest of the section, we will analyse the optimi-sation problem in Equation 6 assuming that D∆ featuresan exponential (Section 4.1), hyper-exponential (Section 5.2)or hypo-exponential distribution (Section 5.3). Please notethat all three cases are possible starting from exponential orPareto intercontact times.

5.1 The exponential delayIf it was well defined, the simplest case would be when thedelay features a coefficient of variation c equal to one. Inthis hypothesis, in fact, the distribution of the delay wouldbe exponential and thus the solution would come out fromthe inversion of (6), that in this case is the following:

1− e−λ∆z = p, (9)

where the parameter is λ∆ = E[D∆]−1. Unfortunately, inthe case of Pareto intercontact times, the coefficient of vari-ation very with ∆ and this means that if ∆∗ is the solutionof (9), the coefficient of variation c(D∆∗) can be (and ingeneral is) different from 0. For this reason, in this casethe exponential distribution (and thus the solution of (9))represents only a possible approximation of the distributionof the delay (respectively the solution of the problem). Forthe exponential intercontact times instead, the problem iswell posed as the coefficient of variation c of the delay isindependent from ∆ and the equation (9) gives the exactsolution when c = 1.

In both cases of exponential and Pareto intercontacttimes however, we are able to invert equation (6). Then,it is straightforward to derive Theorem 1 for exponentialintercontact times and Theorem 2 for Pareto intercontacttimes.Theorem 1 (Exponential intercontact times). If the inter-

contact times are exponentially distributed, one approx-imation of the optimal duty cycle using an exponentialdistribution for the delay D∆ is given by the following:

∆ = − log(1− p)λz

, (10)

where we indicate with λ the parameter of the exponen-tial distribution obtained with ∆ = 1, i.e., λ = E[D]−1.

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 8

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Del

ta

z [s]

p=0.8p=0.4

Fig. 10. ∆ optimum for the exponential delay, varying the target delaythreshold z.

Proof: We know that λ∆ = 1E[D∆] , hence, since

E[D∆] ∼ E[D]∆ (Property E1), we have that λ∆ = λ∆. Thus,

we can rewrite Equation (6) as 1 − e−λ∆z = p, from which∆ can be easily obtained.Theorem 2 (Pareto intercontact times). If the intercontact

times are Pareto distributed, the optimal duty cyclewhen D∆ features an exponential distribution is givenby the following:

∆ =1

γlog2

log(− zλ

log(1−p)

)β

, (11)

where,as before, we indicate with λ the parameter of theexponential distribution obtained with ∆ = 1, i.e., λ =E[D]−1 and with the parameters β and γ those of P1obtained with the linear regression.

Proof: The proof works exactly as the proof of Theo-rem 1, but using properties P1 instead of E1 .

For the sake of example, in Figure 10 we plot ∆ obtainedfrom Theorem 1 setting p = 0.8. E[D] is set to 154s, whichis the average expected delay obtained in Section 5.1 forthe policy DA. Figure 10 shows that, as expected, when thetarget delay threshold is too small, it is impossible to achieveit with a probabilistic guarantee p, regardless of the value ofthe duty cycle. Instead, starting from z = − log (1−p)

λ , ∆ isinversely proportional to z.

Varying the parameters z and p, Equation 6 describes asurface in R3, and more precisely the surface K given by:

K = {(z, p,∆) ∈ R× [0, 1]× [0, 1] : P{D∆ < z} = p} .(12)

Given a certain duty cycle ∆ ∈ (0, 1], we can thus describeK as the union of its level sets K∆ or, in other terms, K =⋃

∆∈(0,1]K∆ where:

K∆ = {(z, p) ∈ R× [0, 1] : P{D∆ < z} = p} . (13)

K∆ is thus the set of pairs (z, p) that can be obtained witha given duty cycling ∆. It can be useful to plot K∆ fordifferent ∆ in order to study whether it is possible to slightlycompromise on the target performance in order to achievea lower duty cycle. Assuming that we want z = 250s, inthe exponential case (Figure 11) we can achieve it with aprobability 0.8 with ∆ = 1 or with 0.68 with ∆ = 0.7, thussaving battery lifetime. Similarly, if we want to guarantee atarget probability p = 0.8, with ∆ = 1 we obtain approxi-mately z = 250s. If we are more flexible in terms of z, wecan choose level set K0.7 which gives z = 350s. This kindof analysis can be performed also for the hyper-exponentialand hypo-exponential delays, with similar results.

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

p

z [s]

Delta=0.1Delta=0.3Delta=0.5Delta=0.7Delta=1.0

Fig. 11. Level set K∆ for different values of ∆ for the exponential delay,varying the target delay threshold z.

5.2 The hyper-exponential delay

When the coefficient of variation of the delay is greaterthan one, the delay can be approximated with an hyper-exponential distribution as stated in Lemma 2. This meansthat Equation 6 becomes 1− p1e

−λ1z − p2e−λ2z = p, where

parameters (λ1, p1), (λ2, p2) are given by Equation 7. InSections 5.2.1 and 5.2.2 we are going to analyse the casesof exponential and Pareto intercontact times respectively.

5.2.1 The exponential intercontact times

From Equation (7), λ1 and λ2 depend on ∆ (while p1 andp2 do not), thus, denoting with λ0

1 and λ02 the rates when

∆ = 1 and exploiting property P2, we can write Equation 6as follows:

1− p1e−λ0

1∆z − p2e−λ0

2∆z = p. (14)The exact solution ∆ to this equation cannot be found ana-lytically because Equation 14 cannot be inverted. However,in Theorem 3 below, we show how to obtain an approx-imated solution ∆a that introduces a small error at mostequal to ε.

Theorem 3 (Exponential intercontact times). Let us λ0

denote E[D]−1 and λ01, λ

02 the rates of the hyper-

exponential delay (Equation 7) for ∆ = 1. When delayD∆ has coefficient of variation greater than one, givena threshold z of the delay and a target probability p,for every fixed ε ≥ min{ε1, ε2} (whose definition isprovided in the proof below), the duty cycle defined by:

∆a =

1z

[− 1−p−p2

λ02p2

+

+ 1λ0

1W

(p2

1

p22eλ0

1(1−p−p2)

λ02p2

)]if ε1 < ε2

− log 1−pλ0z if ε1 ≥ ε2,

(15)where W is the Lambert function3, verifies that|F∆a

(z) − p| ≤ ε and so it is a good approximation ofthe solution to Equation 14.

Proof: We will provide below an intuitive sketch ofthe proof whose detailed version can be found in [17]. Theidea for finding an approximate solution to Equation 14 isto identify an approximation F (z) that is close to F∆(z)under some conditions. So, we build a function F for whichit is possible to solve Equation 14 and for which min{ε1, ε2}

3. The Lambert function is defined as W (x)eW (x) = x, for all x ≥− 1e

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 9

is the error introduced (we will clarify this point below).Specifically, we have identified the following function:

F (z) =

{1− p1e

−λ01∆z − p2(1− λ0

2∆z) if ε1 < ε2

1− e−λ0∆z if ε1 ≥ ε2

(16)Let us denote with F1(z) and F2(z) the two parts of F (z)in the above equation. In F1(z), we have approximated thethird term on the left hand side of Equation 14 using theTaylor expansion, after noting that this term contributes toF∆(z) less and less as the coefficient of variation c increases.Vice versa, the pure exponential behaviour (F2(z)) domi-nates when c is close to 1. Both F1(z) and F2(z) can besolved to find ∆, from which Equation 15 follows.

The quality of these two approximations depends on thedesired tolerance to the error that we inevitably introducewhen we approximate F∆(z). If we tolerate a large error,either approximation can be chosen. Instead, if we wantto achieve the smallest error, depending on the coefficientof variation of D∆ we might have to prefer the one or theother. In the following we briefly discuss how to identify theminimum error introduced by F1(z) and F2(z), which wedenote with ε1 and ε2 respectively. Let us start with F1(z).We want to find the region for which |F∆a

(z) − p| ≤ ε or,equivalently, |F∆a

(z) − F(∆a)1 (z)| ≤ ε, where we denote

with superscript (∆a) the fact that the CDF is computedusing the approximated solution for ∆. Solving the aboveinequality, we find that it holds for all p < pmax, wherepmax is a function of c and ε (due to lack of space, we donot report its formula here, please refer to [17] for details).Specifically, pmax monotonically increases with ε. So, if wewant to derive the minimum error for which inequality|F∆a

(z) − p| ≤ ε holds for all p, we have to solve equationpmax(c, ε) = 1. We obtain the following:

ε1 =

(a− 1)

(−(a− 1)W

((a+1)2e

a+1a−1

(a−1)2

)+ a+ 1

)2

4(a+ 1)2, (17)

where again W (x) denotes the Lambert function and a isdefined as

√c2−1c2+1 .

Let us now consider F2(z). We are able to prove thatfunction |F∆a

(z) − p| has a maximum in p∗. We derive p∗

by finding the p in which the derivative of |F∆a(z) − p|

becomes zero. Then, ε2 can be computed as ε2 = |F∆a(z)−

p∗|, obtaining the following:

ε2 =1

2(a+ 1)−2/a

(a√

2− a2 + 1)1/a

·

·(

(a− 1)(a+ 1)2

a√

2− a2 + 1− a√

2− a2 + 1

a+ 1+ 2

), (18)

where again a =√

c2−1c2+1 . Thus, for both ε1 and ε2 we have

derived a closed-form expression that tells us that the errorthat we make with our approximation is fixed for a given c.

In Figure 12, we show how ε1 and ε2 vary with respectto the coefficient of variation c. As expected, for small c(recall that we are in the hyper-exponential case, so c > 1by definition) the exponential assumption F2 allows us toachieve smaller errors. The opposite is true for large c. The

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 1.5 2 2.5 3 3.5 4 4.5 5

ep

silo

n

c

epsilon1epsilon2

epsilon minimum

Fig. 12. Error introduced by F1(z) and F2(z), varying c.

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Delta

z [s]

Approximated (c=1.3)Exact (c=1.3)

Approximated (c=3.0)Exact (c=3.0)

Fig. 13. ∆ optimum (approximated vs exact) for the hyper-exponentialdelay, varying the target delay threshold z, when the target probability isp = 0.8 and c ∈ {1.3, 3}.

worst case is reached for c ∼ 1.5, when the minimum erroris around 0.1, which is still low. In Figure 13 we plot howthe optimal duty cycle varies with z, setting the target prob-ability to p = 0.8, for two values of coefficient of variation(c = 1.3 and c = 3). In both cases the approximation isgood (the exact value is computed with standard numericaltechniques to solve Equation 14). Specifically, when c = 1.3the minimum error that can be achieved is 0.06 and isprovided by F2(z), hence confirming the predominance ofthe exponential behaviour for c close to 1. Vice versa, whenc = 3 the minimum error is 0.026 and is provided by F1(z).It is also interesting to notice that smaller duty cycles can beachieved when c increases, i.e., when the variability of thedelay is higher.

5.2.2 The Pareto intercontact timesWhen the intercontact times are Pareto distributed, theequation that has to be inverted becomes much more com-plicated than the one analysed in Section 5.2.1. In fact, asc∆ depends from ∆, all the parameters of the distributionp1, p2, λ1, λ2 depend from ∆. Inverting this equation is notpossible; however in the following theorem we describe anapproximated solution and we provide an upper bound ofthe error ε.Theorem 4 (Pareto intercontact times).

Let us indicate, as before, with λ the parameter of theexponential distribution obtained with ∆ = 1, i.e., λ =E[D]−1 and with β and γ the parameters of P1 obtainedwith the linear regression. Consider then pa1 , p

a2 , λ

a1 , λ

a2

the parameters obtained from the following equations4: pa1 = 12

(1 +

√c2−1c2+1

)λa1 =

2pa1E[D∆]

{pa2 = 1− pa1λa2 =

2pa2E[D∆]

(19)

When delay D∆ has coefficient of variation greater thanone, given a threshold z of the delay and a target

4. We will explain in the proof how they are obtained.

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 10

probability p, for every fixed ε ≥ min{ε1, ε2} (whosedefinition is provided in the proof below), the duty cycledefined by:

∆a =

1γ log(2) log

[1β log

(2pa1 p

a2 z

1−p−pa2+pa2 W

(− p

a1pa2e−(z−1+

1−ppa2

)))]

if ε1 < ε2

1γ log2

(log(− zλ

log(1−p) )β

)if ε1 ≥ ε2,

(20)where W is the Lambert function, verifies that |F∆a

(z)−p| ≤ ε and so it is a good approximation of the solutionto Equation 14.

Proof: We illustrate a brief sketch of the proof. Let Fbe the CDF of the hyper exponential distribution. As previ-ously said, one thing that makes this problem more difficultthan the analogous for exponential intercontact times, is thedependence of the coefficient of variation c(D∆), and thusof the parameters p1 and p2 defined in equation (7), from ∆.The strategy that we used, consists in finding two differentapproximations of F , as in Theorem 3 and then choosing theones which leads to the lower error. The first approximationis given by the function F1 that can be obtained passing bytwo steps:

1) We approximate the parameters p1 and p2 with twoother parameters pa1 and pa2 that are independent form∆. Then, using those parameters and the correspondentλa1 and λa2 , we consider the hyper exponential distribu-tion with CDF f .

2) We approximate the function f with Taylor for findingthe function F1.

Let now start with 1. By P3, we have that c(D∆) =√κ(∆) (c2 + 1)− 1, where:

k(∆) = exp(β22γ2∆ − β12γ1∆+1

). (21)

As the parameters β1, β2, γ1, γ1 are obtained by analysingexperimental data, we are not able to fully characterize thefunction κ(∆). However, in our all experimental data, wesaw that κ(∆) ≤ 1 and that is close to 1. We can observethat the first fact means that c(D∆) < c, i.e. the duty cyclereduces the coefficient of variation. On the other side, we usethe closeness to 1, for approximating c(D∆) ' c. Thus theapproximated parameters becomes those of equation (19).The function f is then given by:

f(z) = 1− pa1e−λa1z − pa2e−λ

a2z. (22)

For the approximation in 2, we apply Taylor to the functionf as in the proof of Theorem 3. Thus F1 is:

F1(z) = 1− pa1e−λa1z − pa2(1− λa2z). (23)

As second approximation for the function F , we choose theexponential distribution i.e. the following CDF:

F2(z) = 1− e−z/E[D∆]. (24)

Using F1 and F2 instead of F in equation (14), the identitybecomes invertible and the solutions are the two pieces ofequation (20), respectively ∆1

a and ∆2a. Obviously we choose

the one or the other depending on which from F1 or F2 bestapproximate the CDF F . We define ε1 and ε2 as the distanceof solution from the target probability threshold p chosen. Inother words, for i = 1, 2,

εi(z, p) = |P (∆ia)− p|, (25)

that depend only from the thresholds and the others param-eters of the problem. This concludes the proof.

5.3 The hypo-exponential caseWhen the coefficient of variation c of the delayD∆ is smallerthan one, following Lemma 3, it is possible to approximatethe delay with a hypo-exponential distribution. In partic-ular, using property P2, if we denote with µ0

1 and µ02 the

parameters obtained when ∆ = 1 in Equation 8, we canrewrite Equation 6 making explicit the dependence on ∆:

1− µ02

µ02 − µ0

1

e−µ01∆z +

µ01

µ02 − µ0

1

e−µ02∆z = p. (26)

As in the hyper-exponential case, this equation can not bedirectly inverted for finding ∆, but it is possible to derivean approximate solution for which a small fixed (for a givenc) error is introduced.Theorem 5. Let µ0

1 and µ02 be the parameters given by Equa-

tion 8 with ∆ = 1. When the delay D∆ has coefficient ofvariation smaller than one, the duty cycle defined by:

∆a =

{− 1µ0

1zlog[(1− p) · µ

02−µ

01

µ02

]if ε1 < ε2

− log 1−pλ0z if ε1 ≥ ε2,

(27)

verifies that |F∆a(z)− p| ≤ ε (with ε ≥ min{ε1, ε2}, seethe proof in [17]), and so it is a good approximation ofthe solution to Equation 26.

Due to lack of space and since the rationale follows thatof the proof for Theorem 3, we omit the proof of the abovetheorem, which can however be found in [17].

In Figure 14 we plot ε1 and ε2 varying c. When c is closeto one, both approximations are very good. For values ofc roughly in the interval (0.83, 0.97), F1(z) provides betterresults, while, for low values of c, F2(z) is to be preferred. InFigures 15(a) and 15(b) we show how the optimal duty cyclevaries with z, setting the target probability to p = 0.8, fortwo values of coefficient of variation (c = 0.75 and c = 0.9,respectively). In both cases the approximation and the exactvalue are very close. In Figure 15(a) the minimum error thatcan be achieved is 0.13 and is provided by F2(z), while inFigure 15(b) the minimum error is 0.05 and is provided byF1(z).

6 CONCLUSION

In this work we have studied how to optimise the dutycycle in order to guarantee, with probability p, that thedelay of messages remains below a threshold z, assumingthat intercontact times are either exponentially or Paretodistributed. We have provided an exact solution for the casein which the delay follows an exponential distribution, andapproximated solutions for the cases in which the coefficientof variation of the delay is greater than or smaller than 1.We have also demonstrated that the approximation of ∆

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 11

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.75 0.8 0.85 0.9 0.95 1

ep

silo

n

c

epsilon1epsilon2

epsilon minimum

Fig. 14. Error introduced by F1(z) and F2(z), varying c.

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Delta

z [s]

Delta approximated Delta exact

(a) c = 0.75

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Delta

z [s]

Delta approximated Delta exact

(b) c = 0.9

Fig. 15. ∆ optimum (approximated vs exact) for the hypo-exponentialdelay, varying the target delay threshold z, with target probability p = 0.8and c ∈ {0.75, 0.9}.

introduces an error ε whose formula we have provided andthat is small and fixed for a given coefficient of variation cof the delay, in the case of exponential intercontact times.

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