Information Dissemination using Epidemic Routing with Delayed Feedback

Yezekael Hayel and Hamidou TembineLIA-CERI, University of Avignon

339, chemin des Meinajaries, Agroparc BP 1228, 84911 Avignon Cedex 9, France.E-mail: {yezekael.hayel,hamidou.tembine} @univ-avignon.fr

Abstract a packet who kill the information in infected node. The aimof the source is to minimize the time to infect N nodes with

Information dissemination is one of the most important less exchanged messages.aspects of routing mechanisms for network management. In this paper we are interested in the phenomenon of delayWe study in this paper the problem of disseminating an in- in the feedback signal and its implication on performancesformation by a source trough all the nodes of a network. and stability of the system.We consider an epidemic routing protocol in an Ad hoc net- The paper is structured as follows. We first provide re-work with a delayed feedback to the source. Indeed, the lated works on dissemination of information in mobile Adsource controls the dissemination of the information in the hoc networks. Then we study the stability and the velocitynetwork by sending an anti-packet, like a vaccine. We show of the delayed logistic equation which model the evolutionthat there exists a sufficient condition for stability of the in- of the information inside the network. After that, we inves-formation dissemination in the network depending on the tigate numerically the impact of the choice of some param-spreading of information in the network and the delay of eters in the delayed logistic equation on the stability of thethe feedback to the source. system in Section 4.

2 Information dissemination in Ad hoc net-Keywords: Information dissemination, delay logistic works

equation, ad hoc networks.Information dissemination or flooding, forms an impor-

1 Introduction tant aspects of routing mechanisms for network manage-ment, service discovery or collaborative sensing like in cog-

We consider a system where a source wants to dissemi- nitive radios networks [18]. A challenging problem for in-nate an information into a network using an epidemic rout- formation dissemination in an ad hoc network is to maxi-ing protocol [3, 4, 10]. The source is aware of the number mize the availability of information throughout the networkof mobiles having the information but this feedback signal while incurring minimal communications and device over-has a delay. Indeed, at the instant t, the source is aware heads. The simplest mechanism for information dissemi-of the number of mobiles having the information at time nation is flooding like in route discovery phase proposedt - T. We denote by x(t) the number of mobile having the in ad hoc networks routing protocols such as AODV andinformation at time t. This notion of dissemination can be DSR. An optimized flooding mechanism is proposed in [15]related to spread disease and a mobile who have the infor- considering resource restrictions in mobiles like power andmation will be called an infected node. Mobiles which have computation capabilities.the information disseminates it to other nodes. When aninfected node meets a non-infected one, the information is 2.1 Gossipingtransmitted.The information dissemination is controlled by the source Datta et al. studied in [14] an epidemic algorithm for se-in order not to infect too many mobiles. The feedback sig- lective information dissemination in wireless mobile ad-hocnal can be provide by sensors usefully distributed inside the networks with autonomous decisions called autonomousnetwork. We denote by N the number of mobiles to be in- gossiping. Autonomous gossiping is a new genre of epi-fected at stationarity. Hence, when the number of infected demic algorithm resembling more closely actual epidemicnodes is higher than N, the source generates like a vaccine, spreading, and it brings the wired or wireless and ad-hoc

1-4244-1455-5/07/$25.OO c2007 IEEE 1

networks are for communication in the whole network and function b e C( [-T, 0], R+ \ {0}) such that x(t) = (t)epidemic algorithms together, specifically in the context of for all t C [-T, 0]. The function X is the initial conditionMANET. This epidemic algorithm spreads data items (i) function. Delay Differential Equation (DDE) are often usedselectively based on vulnerability of other nodes, instead to model evolutionary systems with time dependence andof treating all nodes homogeneously and flooding the net- delay. It is well-known that the delay induces a stabilitywork, (ii) exploiting mobility with flexible casting: the area around the equilibrium of such system. Assumingsame mechanism can be used for either of broadcasting, that the system is under control, the stability area may bemulticasting, geocasting or combination of multicast and chosen efficiently. In this article, we prove that a controlgeocast ,(iii) Self-organization making autonomous gossip- on the stability area induces an effect on the velocity ofing selectivity and self-organizing economic and ecological the system, the speed of convergence. This notion can bemechanisms of rewards and punishments and competition important for dynamic systems. It can be very interestingamong the data items in order for data items to survive and to control the stability of the equilibrium but also the speedpropagate in the network help enforcing selectivity. See also of convergence of the system to the equilibrium.[9] for a related work on optimal broadcasting, local leaderelection problem [12, 13] and gossiping in communication Wright [2] proved that ifnetworks based on self-selection routing protocol. 3

0 < KT < - (2)2.2 Epidemic Routing2 the equilibrium point N attracts all solutions and N is a lo-

Epidemic routing protocol has been proposed for rout- cally asymptotically stable equilibrium.This condition depends not only on the delay T, but also oning packets from a source to a destination using the sparse

density and the highly mobility of nodes in a network. Each the growth parameter K. Then, a system manager is ableto control the stability region, by choosing an K satisfyingnode who carries the informaitonis called infected ande condition (2), but also control the velocity of the dynamic

given by the DDE. The aim here is to find a compromiseAfter, different recovery schemes have been proposed: between this two objectives as it is interesting to control* timers are used to destroy packets in infected nodes, the system evolving like a DDE in order to have small con-

vergence time. The velocity of convergence of the system* the destination generates an anti-packet in order to de- dynamic can be an important measure of performance of the

stroy the information in infected nodes. system as well as the stability.

In [11] authors propose to adapt the forwarding scheme of 3.1 Stabilityeach nodes depending on its environment. Performancesmeasure of such algorithms can be the mean delay to reach We study the choice of the parameter K and the delay Tdestination, the power consumption. In [17], Zhang et al. appearing in (1) that impact the velocity and the stability ofpropose ordinary differential equation model based on lo- the dissemination of the information inside the population.gistic equation to study Epidemic Routing. The parameter K can thus be used to accelerate the rate of

convergence. The stability condition of (1) is given simply3 Logistic equation with delay by

KT < 2The dissemination of the information inside a population I

can e decriedb a elayd Dfferntil Eqatin (DE). In order for the stability to any delay, the product of thecan be described by a Delayed Differential Equahton (DDE). adaptation speed parameter K and delay T(should be 0(1).The main question with a delayed control system is how to Tu h agrtedlyi,tesoe esol h aobtain large dissemination of information while maintain-ing stability. rameter K.

The classical delayed logistic equation [1] is the follow- The linearization of (1) around the stationary point N ising given by

/t

%(t) --Kz(t-T), (3)

x(t) Kx(t) I ( (1) where z(t) = x(t)-N. It is known ([8, pp.3366],[7, pp. 188]or from the theorem of Hartman and Grobman) that the

with N is the quantity of resources, T is the delay and K steady state x* is asymptotically stable for (1) if the triv-is the growth parameter which represents the spreading ial solution of the linearized version (3) is asymptoticallyof information in the network. Assuming that there is a stable.

l'K = 0.25 is astationary pitfor these prmts,for which altheK = 0.47 .. .66

lo'k K = 1.44 1... population have the information.K = 2.5

lo' -- / 1/ 11'' '1'''i116'1'1116111 The resulting trajectories of the population having theinformation, as a function of time, is given in figure 1. For

u r K 0.25, we have stability but the convergence speed isslow. The other extreme is illustrated for K 2.5 which isseen to be unstable: it oscillates rapidly and the amplitude

lo >., +is seen to be greater than N.3 _- ._..........._ Impact of delay We now keep K constant and evaluate

the stability varying the delay between 0.25 and 2.5 timeunits in figure 2. WhenT= 0.25 the system is stable but

of K on velocity and stability the rate of convergence to the stationary point N is not fast.Velocity is not slow. ForT= 2.5 the system is unstable, the

2500 solution oscillates around N = 500.In all these figures the initial condition is x(t)

2000 (, <,<., <_1/N,t (-T,O).0.05 X. '6 Validation of stability conditions. In both figure 1 and0.47- 12. figure 2 we observe cases of stable and of non-stable situa-

tions. All turn out to confirm the stability condition that we1000 _

obtained in previous section.500L i v vNumber of nodes having the information In both fig-

ure 1 and figure 2 we observe that in the oscillating cases,tO 102|0 'f1 0 410 a10> the amplitude can be greater than N. This phenomena can-

not arrive in the system without delay when the initial con-dition is between 0 and N. The solution of the nondelayed

Figure 2. Effect of T on velocity and stability logistic equation (T= 0 in (1)) is given by

3.2 Velocity I(t) N [ 1 + Xo eKtIwhere 10 =x(0).

The rate of convergence to the stationary pointN is equalto R(A*)/T where A* satisfy 1R(A*) > 1 (A) for A where 5 Conclusions and perspectivesA and A* root of the characteristic equation of (3)

x + Ke-TX = O. In this paper, we consider an information disseminationproblem in an Ad hoc network using an epidemic routing

Johari and Tan give in [6] a characterization of the rate of protocol. We consider that the source controls the infor-convergence of (1), for K in the region of stability (0, 7 ). mation dissemination by sending packet for expanding theThey proved that the maximum rate of convergence for the information and anti-packet for destroying the information.system (1) is 1/T when K = 1 . The rate of convergence The source is aware of the number of nodes having the in-

eT

is monotonic increasing and convex from 0 to I and the formation by a delayed feedback signal. We show that thisconvergence is nonoscillatory, for K C (0, 1)1 The rate system can be modeled by a delayed differential equationof convergence is monotonic decreasing for K C ($e 7 ). and we provide a necessary condition for stability of mo-See also [5] for a similar result on the rate of convergence. biles having the information in the network.

An important perspective is to study the impact of the mo-bility on the spreading factor K and also to consider the

4 Numerical illustrations delay not as a constant but as a random variable on the inter-val [0, Tmaxl or a function on time. Then the delay logistic

Impact of the growth parameter K Our first numeri- equation with variable delays [7, 8] iscal experiment studies the convergence of the logistical dy- - (t-(t ))namics for the case of the unit delay as a function of K: ±z(t) =Kx(t) (we check the speed of convergence and the stability of the/dynamics as a function of the gain parameter K. We took with 0 < T(t) < t for all t > 0. Another perspective is toT =1, N =500 and let K vary between 0.25 and 2.5. N consider (1) with a small noise: stochastic logistic equation.

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