similar to in social networks, trust evaluation and management .... âM k=1 bik = 1,âi. Ï = (Ï1,Ï2, ..., ÏN ) is the trust distribution over states where Ïi = P[Y0 = si] ...
2011 IEEE International Conference on Privacy, Security, Risk, and Trust, and IEEE International Conference on Social Computing
A Probabilistic-Based Trust Evaluation Model using Hidden Markov Models and Bonus Malus Systems Kevin Xuhua Ouyang, Binod Vaidya, Dimitrios Makrakis Broadband Wireless and Internetworking Research Laboratory School of Electrical Engineering and Computer Science University of Ottawa, Ottawa, Canada K1N 6N5 Email: {xouyang, bvaidya, dimitris}@site.uottawa.ca Abstract—In the paper, the uncertainty of trust is transformed into a probability vector denoting the probability distribution over possible trust states that are hidden from observation but determined by an entity’s expected performance. We suggest the use of Hidden Markov Models (HMMs) for estimating the unknown probability distributions in peer-to-peer interactions. HMMs allow us to explicitly consider an entity’s unobserved trustworthiness that influences it’s occurrences of behavioral patterns. The proposed hidden Markov processes are associated with a specified Bonus-Malus System (BMS) that is interpreted as a Markov chain with constant transition matrix and is used to simplify the structure of model and to reduce the computational complexity of parameter estimations in HMMs. The maximum likelihood estimators of the unknown HMM parameters are obtained using EM algorithm. An application of the model in the scenario of detection of probabilistic packet-drop attack has been investigated. The simulations demonstrate that the approach is capable of accurately estimating the (hidden) trust states probability distribution as well as the expected performance for the entities that have different observed behavioral patterns.
In the paper, we propose a quantitative trust evaluation model to identify the trustworthiness of (one-hop) entities using Hidden Markov Models (HMMs) and Bonus-Malus Systems (BMSs). By constructing the HMM-BMS model, the trust evaluation will be assessed by performing matrix manipulations that describe the behaviors of the entities in terms of random variables and their trust probability distributions. Our main contributions are as follows: 1) a probabilitybased model is presented to express the uncertainty of trust, in which the trust is represented as a probability vector that gives probability distribution of the possible trust states; 2) we suggest the use of HMMs to model unobserved trustworthiness of the entities and their observed behaviors, dealing with the problem of maximum likelihood parameter estimation; 3) we introduce a Bonus-Malus System to the hidden Markov process in order to fasten the convergence speed in the estimation of the parameters. The primary goal of the trust model is to assess uncertainty of trust for entities in a peer to peer manner and to estimate their expected performance outcomes by solving the HMMs. The novelties of our work lie in that: 1) by considering that the trust evaluation process for securing communication in peer-to-peer environments can be viewed as a measure of relative uncertainty, we attempt to address the probabilistic uncertainty analysis based on historical data using theory of probability and mathematical statistics. Specifically, these observed outcome data with respect to the expected performance of an entity often exhibit variability that can be captured by simply fitting a mathematical model; 2) by using Bonus-Malus Systems in connection with HMM processes, we explore a computationally efficient parameter estimation technique for HMMs while having no effect on the estimation accuracy. The rest of the paper is organized as follows: some related work is reviewed in Section II. Section III presents the trust model based on Hidden Markov Models and Bonus-Malus Systems. In Section IV, an application of the model to detect packet dropping attack is investigated. Performance evaluation and discussion are addressed in the following section. The paper ends with conclusion.
I. I NTRODUCTION Trust is a particularly important component of security in the global computing environments. In the context of routing and reliable forwarding of data, trust is considered as a probability evaluation of an entity that propagates information to the destination entity while remaining integrity of the data. Much similar to in social networks, trust evaluation and management among entities is the foundation in peer-to-peer systems for constructing security authentication mechanisms, such as key management and secure routing. Given the trustworthiness of the entities in a wireless network, the routing path between the source and destination will be determined as the most trusted path rather than the shortest path [1]. The trust evaluation systems in distributed systems are responsible for capturing and updating trust information based on the peer-to-peer interactions. The problem of trust evaluation in wireless context has been examined and formulated via various mathematical methods, such as fuzzy logic model [2][8], graph theory [1], Bayesian network [5] [6] and Markov Process [7]. In principle, the trust value assigned to an entity is quantified into a definite real number in a range, and then mathematical equations or algorithms are used to derive and update the trust value per transaction between peers. Finally, a criteria is defined to determine whether an entity is trustworthy or not. 978-0-7695-4578-3/11 $26.00 © 2011 IEEE DOI
II. R ELATED W ORK Hidden Markov Modeling techniques have been applied to the trust modeling in a wide range of contexts. We survey in 1004
trust is defined as a probability vector with N entries: the probabilities of state 1 (full trust) , state 2, and so on, till state N (full distrust), respectively. For example, if a trustor is observing one of its trustee, which is within its transmission range, the local trust distribution of the trustee on the trustor can be represented N by a probability vector π = (π1 , π2 , · · · , πN ), where i=1 πi = 1 and πi is the probability that the trustee is at trust level i (i = 1, 2, . . . , N ). Initially, the trustor has no definite trust information on its trustees, but it knows general trends. It is reasonable to believe that the hidden characteristics of trust are revealed by the transactional behaviors of the trustees that are observed by the trustors. Hence with transaction evaluations in a performancebased manner, the trustors seek to identify trust for their trustees.
the section some trust models that are based on HMMs. [12] presented a comparison of the hidden Markov trust model and the Beta reputation system. The authors argued that the hidden Markov trust model has more parameters than the Beta model, thus it can be more fine-tuned and adaptable to dynamic environments. However, this also leads to challenges related to the parameter estimation. [9] proposed a trust model for autonomous agents in multi-agent environments based on HMMs and reinforcement learning. Trusting agent in the system rates all other agents after an interaction and uses an HMM per agent to decide and predict whether or not the agent is malicious. The HMM is updated from observations which come in the form of ratings after direct experiences or recommendations requested from other intermediaries. A new trust establishment framework for web services using mixture of local experts using a coarse-grained Hidden Markov model was proposed in [11] to make optimal trust decision for interacting with a provider agent in current/future time. The model is based on the prediction to find the similar patterns in the whole time space over which analysis of the dynamic nature of trustworthiness of the trustee is performed and use theses patterns for deciding the future trust values. [13] introduced a HMM-based trust model to approximate the behavior of the principal using HMMs which maximize the likelihood of the available history of observations. The approximate behavior model is then used to evaluate the estimated predictive probability distribution given any sequence of observations. Contrary to these existing work that involves high computational complexity in parameter estimation for modeling HMM processes, we will extend HMMs to include a Bonus-Malus System, which can simplify the structure of the HMM-based trust model, thereby providing a comparable faster parameter estimation.
B. Hidden Markov Model Since the formulation of hidden trustworthiness is nearly identical to that assigned by an hidden Markov model, we suppose in what follows that the trust state transition process in our model obeys a Markov Chain. Let us consider an entity’s performance stochastic output X = Xt |T1 (T ∈ N) that is visible to the trustor and depends on another stochastic sequence of trust states Y = Yt |T1 that is an hidden Markov chain. Thus, the trust evaluation process for a trustee can be modeled mathematically as a five-tuple (discrete) Hidden Markov Model τ = (S, O, A, B, π) where S = {s1 , s2 , ..., sN } is the finite set of trust states; O = {o1 , o2 , ..., oM } is the finite set of possible observations in each state; A = {aij } is the state transition probability (N × N ) matrix where aij = P [Yt = sj |Yt−1 = si ], 1 ≤ i, j ≤ N and N j=1 aij = 1, ∀i; B = {bik } is the state-dependent emission (N × M ) matrix where bik = P [Xt = ok |Yt = si ], 1 ≤ i ≤ N ; 1 ≤ k ≤ M M and k=1 bik = 1, ∀i. π = (π1 , π2 , ..., πN ) is the trust distribution over states N where πi = P [Y0 = si ], 1 ≤ i ≤ N and i=1 πi = 1. In the paper, the trust model deals with two discrete time stochastic processes {Xt ; Yt } for the trustee peers, where any observed variable Xt is conditioned on the Markov Chain Yt at any t. Each trustee peer has N trust states giving the form of the state vector π, but the trustor cannot observe them directly, that is, they are hidden from it. Our objective is to estimate the trust distribution π and the expected performance of the trustees, given their transaction observations. To model the HMM described, we need a procedure for estimating the state transition matrix A, the emission matrix B and the trust state vector π. We denote the parameters as
III. S YSTEM M ODEL A. Trust In distributed environments, two entities that carry out information interactions come to be known as trustor and trustee. A trust relationship is formed and evaluated over time by a trustor on a trustee, after the latter performs tasks for the former. As shown in Table I, we suppose that there exists N different trust levels (states) a trustor has for its individual trustee. TABLE I T RUST L EVELS Trust Level 1 Full Trust 2 ↓ .. .. . . N −1 ↑ N Full Distrust
A probability-based model is developed for expressing the posterior probability distribution of trust levels. Specifically,
θ = (A, B, π).
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C. Assumptions
where
In order to formulate the mathematical expressions describing the system, the following assumptions have been made: 1. In each of these N states, the entities are behaving consistently in a predictive manner. This defines the probability of bik that is grouped in the matrix B. 2. The Markov chain is time-homogeneous, ergodic, then the transition matrix A is the same after each step. 3. The Markov chain is irreducible and aperiodic, then the state distribution π is unique stationary, which leads to the equation π = πA.
ak
= =
··· ··· ··· ··· ···
aN −1 aN −2 ··· 0 a1
ak ≈
1≤k≤M
(2)
(3)
Tk , T
k = 1, 2, . . . , M.
(4)
where Tk is interpreted as the number of observations that M occur in the particular performance subdivision k, and k=1 Tk = T . As a result, the information of the matrix A can be obtained using statistically-based observation. Following, by using the equality π = πA, we then get the estimator of the trust distribution vector π. Finally, we need to determines the matrix B using optimization methods. In particular, we search for the maximum likelihood estimators of (M − 1)N emission probabilities bik , 1 ≤ i ≤ N ; 1 ≤ k ≤ M − 1, i.e. all but the last ones in each row of the B (the last entries are obtained by matrix M difference, since k=1 bik = 1, ∀i). Therefore, the parameter θ has been normalized as the vector to be estimated with the maximum likelihood method
N N −1 N . .. N
aN + · · · + aM aN −1 + aN + · · · + aM ··· 1 − a1 1 − a1
bik πi ,
We now search for some estimators of the parameter θ. Given a HMM τ , a trustee’s performance stochastic output X = Xt |T1 (T ∈ N) and the stochastic sequence of trust states Y = Yt |T1 . ak in eq. (2) can be statistically approximated by
θ = (b11 , ..., b1M −1 , b21 , ..., bN M −1 ) .
of M ≥ N , we have N × N state transition a2 0 ··· 0 0
E. Parameters Estimation
TABLE II T RUST S TATE T RANSITION RULE
In the case matrix ⎛ a1 ⎜ a1 ⎜ A=⎜ ⎜ ··· ⎝ 0 0
P [Xt = ok |Yt = si ]P [Yt = si ]
By introducing the concept of BMS as an incentive mechanism, the HMM has been accordingly normalized to the expressions eq. (1) and eq. (3) where the matrix A is determined by M parameters ak , k = 1, 2, . . . , M , while B is determined by N ×M parameters bik , 1 ≤ i ≤ N ; 1 ≤ k ≤ M . Summarily, Eq. (1) gives the invisible state transitions in the Markov chain Yt and, Eq. (3) states how the transitions affect their performance outputs Xt .
Table II specifies a hypothetical trust transition model having N trust states based on the BMS.
N −1 N −2 N . .. N
(a1 , . . . , ak , . . . , aM ) = πB
The Bonus-Malus System [4] is widely used in automobile insurance to determine the premium paid by a customer according to his individual claim history. Consider that policyholders are subdivided into a finite number of BMS classes, in which all transfers of the policies within these classes are regulated by defined transition rules. The class assignment to every policyholder in a given year is uniquely determined by both the class where he belonged in the previous year and the number of claims reported during the year. The fundamental principle of BMS is that the higher the claim frequency of a policyholder, the higher the insurance costs that on average are charged to the policyholder. This principle is also valid for estimating trust of peers depending on their transaction performance. An entity ascends the higher trust level after one or more successful transaction(s), or descends the lower trust level after one or more unsuccessful transaction(s).
··· ··· ··· . .. ···
P [Xt = ok , Yt = si ]
si ∈S
M and k=1 ak = 1. It is clear that ak is the probability that the trustor has observed the trustee’s behavior has fallen into performance subdivision k for any time instant t. Furthermore, based on eq. (2) we have
(1)
2 1 3 . .. Min(M + 1, N )
si ∈S
D. Bonus-Malus System
1 1 2 . .. Min(M, N )
P [Xt = ok ] =
si ∈S
This also denotes that the trust information of the entities remain unchanged as the system evolves.
Trust state Performance 1 Performance 2 . .. Performance M
=
Let Φ be the parameter space. Let x = (x1 , ..., xT ) be the vector of the observed data, i.e. the sequence of T outputs of stochastic process Xt |T1 performed by a trustee, and y = (y1 , ..., yT ) be the vector of the unobserved trust states Yt |T1 of the trustee. The likelihood function L(θ; x, y) may be found by multiplying all the
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
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is less than or equal to a sufficiently small arbitrary value. If the algorithm converges at the (k + 1)th iteration, then θ(k+1) , ln L(θ(k+1) ; x) is a stationary point and then (k+1) (k+1) (k+1) (k+1) θ(k+1) = (b11 , ..., b1M −1 , b21 , ..., bN M −1 ) is the maximum likelihood estimator of the unknown parameter θ. Once the matrix B is determined, according to eq. (2) the expected value of Xk for any t that is given by E(Xt ) = μ i πi . (9)
probabilities associated with each trust transition and emission in the performance of the trustee. Formally, we have L(θ; x, y) = P (X1 = x1 , · · · , XT = xT , Y1 = y1 , · · · , YT = yT ) = P (y1 )P (x1 |y1 )P (y2 |y1 )P (x2 |y2 ) · · · P (yT |yT −1 )P (xT |yT ) T
= πy1 by1 x1 ayt−1 yt byt xt
(5)
si ∈S
t=2
where μi = E(bik ), ∀k. Please note that eq. (10) gives an estimate of the expected performance of trustees.
The maximum likelihood estimate of the unknown parameters θ is determined by the marginal likelihood of the observed data L(θ; x) = P (X1 = x1 , · · · , XT = xT |θ) T
= ... πy 1 b y 1 x 1 ayt−1 yt byt xt y1 ∈S
IV. A PPLICATION TO PACKETS - DROPPING ATTACK An application of the HMM-BMS trust model is provided in this section to evaluate entities’ trustworthiness in the multihop Mobile Ad Hoc Networks (MANETs), in that the problem of the packet-dropping attack in the typical routing scenario will be examined. Not forwarding packets or dropping packets for others to prevent communication from being established in the networks is known as packet-dropping attack, which is probably the most difficult to detect among various types of service attacks when considering the dynamic source routing protocols for MANETs [10]. In the context, a quantitative measure of an entity’s trustworthiness is with respect to its expected packet drop rate, i.e. the higher the expected packet drop rate, the lower trustworthiness the entity obtains. We define four (N = 4) trust states that a trustor entity has for its trustee peers shown in Table III.
(6)
t=2
yT ∈S
where ayt−1 yt is the entries in matrix A that is obtained by eq. (4). Thus, the method of maximum likelihood estimates θ by finding a value of θˆ that maximizes the objective function eq. (7) θˆ = arg max L(θ; x) θ∈Φ
subject to eq. (3). In order to solve the optimization problem, we shall perform the EM algorithm [3], which is based on an iterative procedure with two steps at each iteration. The first step, E step, computes the expectation of the log-likelihood evaluated using the current estimate for the parameter θ; the second one, M step, computes parameter maximizing the expected log-likelihood found on the E step. We will explain the iterative scheme of the EM algorithm as follows. We define the following function at the E step: Q(θ; θ ) = Eθ (ln L(θ; x)),
TABLE III T RUST S TATE Trust State 1 Full 2 High 3 Medium 4 Low
(7)
where θ ∈ Φ. It’s proved in [3] that maximizing eq. (7) is to maximize eq. (8) with respect to θ. Let θ(k) be the vector of estimates obtained at the k th iteration: (k)
(k)
(k)
TABLE IV T RUST S TATE T RANSITION RULE
(k)
θ(k) = (b11 , ..., b1M −1 , b21 , ..., bN M −1 ) ,
Trust Level Performance 1: < 1 packet Performance 2: [1,10) packet Performance 3: [10,20) packets Performance 4: ≥ 20 packets
at the (k + 1)th iteration, the E and M steps are defined as below: E step: given θ(k) , compute Q(θ; θ(k) ) = Eθ(k) (ln L(θ; x));
(8)
1 1 2 3 4
2 1 3 4 4
3 2 4 4 4
4 3 4 4 4
Table IV specifies a Bonus-malus system having four performance subdivisions (M = 4). Correspondingly, the transition matrix for the trustees is given by ⎛ ⎞ a1 a2 a3 a4 ⎜ a1 0 a2 a3 + a4 ⎟ ⎟ A=⎜ ⎝ 0 a1 0 1 − a1 ⎠ 0 0 a1 1 − a1
M step: search for that θ(k+1) which maximize eq. (9), i.e. such that Q(θ(k+1) ; θ(k) ) ≥ Q(θ; θ(k) ), for any θ ∈ Φ. Following, the E any M steps are repeated in an alternating way until the sequence of log-likelihood value ln L(θ(k) ; x) converge, i.e. until the difference
where a1 is the probability of no packet dropping, a2 is the probability of less than 10 packet dropping, a3 is the
ln L(θ(k+1) ; x) − ln L(θ(k) ; x)
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the most trusted peer (trustee) to the trustor. The probability distribution over the outcome values of π1 for 21 trustee peers are as shown in Figure 1. It was found that a small increase in the packet-drop rate leads to a dramatic decrease in the value of the probability component π1 till the value approaches zero.
probability of more than 10 but less than 20 packet dropping, a4 is the probability of more than 20 packets dropping. We further consider the number of dropped packets on a trustee peer detected by the trustor peer is a (discrete) random variable with a mean μ. Our goal is to estimate the parameter space θ, and ultimately to identify the hidden probability distribution π over the trust states set S of the Markov Chain Yt and the parameter μ for the individual trustee peers.
1 0.9 0.8
V. P ERFORMANCE E VALUATION
0.7
π1 Probability
To evaluate the performance of the HMM-based trust model, we developed a testbed with MATLAB and performed Monte Carlo simulations. A. Metrics
0.6 0.5 0.4
Specifically, we mainly focus on the following three metrics. (1) Accuracy of trust estimation. The objective of the trust model is to estimate the uncertainties for entities involved. Once the probability spaces of trust states are obtained, it is interesting to find out how accurately our model assess trust values for entities against their actual packet-drop mean μ. (2) Accuracy of estimating the expected performance of entities. Since trust probability distributions (π) of an entity are determined by its actual packet-dropping mean (μ), this can be verified through Eq. 10. (3) Rate of convergence of the model. This is another important metric for evaluating and analyzing the effectiveness of a trust model. In such case, we are concerned about the speed at which a trustor uncovers the hidden trust patterns of its trustees by observing their behaviors in terms of the number of lost packets in forwarding transactions.
0.3 0.2 0.1 0
2
Fig. 1.
4
6
8
10 12 Trustee Entity Peers
14
16
18
20
π1 Probability Distribution over 21 Trustee Peers
Figure 2 depicts the comparison of the estimated expected packet-drop rates against the actual rates (i.e. 0, 0.1, 0.2, . . . , 1, 1.5, 2.5, . . . , 10.5) for the 21 trustee peers. We can see the trust model is able to find the actual packetdrop intensities accurately as the system evolves. 12
B. Simulation Setup The experiments are carried out in accordance with the following structures. An entity (trustor) is observing 21 peer entities’ (trustees) behaviors. They are labeled with i = 1, 2, · · · , 21, in which entity 1 will always forward packets for the trustor when requested to do so; the probability distribution of the number of dropped packets of entities 2-11 follow Poisson distribution with mean of 0.1, 0.2, · · · , 1, respectively; the number of dropped packets of entities 12-21 have (discrete) uniform distribution with the ranges [1,2],[2,3],...[10,11] respectively. It’s obvious that entity 1 is referred as to as the most trustworthy entities in the network. In the experiments, no fragmentation is used. Packet number per transaction is a uniform discrete random variable between 80 and 100.
Expected Packet−drop Rate
10
8
6
4
2
0
C. Simulation Results
Actual Rate Estimated Rate
0
5
Fig. 2.
The first and main focus of the HMM-BMS model is to identify trustworthiness of trustee peers which are determined by their actual individual expected packet drop number per transaction. According to the trust definition in Section III, the entity that has the highest value of trust state 1 (π1 ) is referred to as
10 15 Trustee Entity Peers
20
25
The Estimated vs Actual Expected Packet-drop Rate
Figure 3 shows the results of estimating the actual packetdrop intensity (μ = 1) of entity 11, given the different transaction numbers (T ) ranging from 1 to 80, observed by the trustor. We can see that it is required for a trustor to take about 50 transactions with an trustee to find out its pattern
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Finally, we have applied the model to a case study of packet-drop attacks in the multi-hop mobile ad hoc networks. The experimental results demonstrated that through the observed performance outputs, our model is able to estimate the expected performance of the entities that fully accounts for their trustworthiness and to effectively give the distribution of probabilistic trust.
of trust information for which the packet-drop performance matches. From the control theory point of view, it can be said that the system is expected to take 50 steps to converges to a steady state. 1.8
1.6
R EFERENCES Expected Packet−drop Rate
1.4
[1] G. Theodorakopoulos and J.S. Baras. On trust models and trust evaluation metrics for ad hoc networks. IEEE Journal on Selected Areas in Communications, 24(2), Feb. 2006, pp. 318-328 [2] Junhai Luo, Xue Liu b, Mingyu Fan. A trust model based on fuzzy recommendation for mobile ad-hoc networks. Computer Networks, Volume 53, Issue 14, 18 September 2009, pp. 2396-2407. [3] Dempster, A.P.; Laird, N.M.; Rubin, D.B. Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. 1977, Series B (Methodological) 39 (1): pp. 1-38 [4] Denuit, M. Modern actuarial risk theory: using R. 2nd ed. Berlin, Heidelberg : Springer-Verlag, 2008. [5] S Chinni, J Thomas, G Ghinea, Z Shen. Trust model for certificate revocation in ad hoc networks. Ad Hoc Networks, 2008. Volume 6, Issue 3. pp.441-457 [6] Y. Wang and J. Vassileva. Bayesian network-based trust mode. In 2003 IEEE / WIC International Conference on Web Intelligence, (WI 2003), IEEE Computer Society, 2003, pp. 372-378 [7] Ouyang, K.X.; Vaidya, B.; Makrakis, D.; A trust evaluation model using controlled Markov process for MANET. Wireless and Mobile Computing, Networking and Communications (WiMob), 2010 IEEE 6th International Conference on. 11-13 Oct. 2010, pp. 630-637 [8] S. Schmidt, R. Steele, T.S. Dillon and E. Chang, Fuzzy trust evaluation and credibility development in multi-agent systems, Applied Soft Computing 7 (2) (2007), pp. 492-505 [9] Moe, Marie Elisabeth Gaup; Tavakolifard, Mozhgan; Knapskog, Svein Johan. Learning Trust in Dynamic Multiagent Environments using HMMs. Proceedings of The 13th Nordic Workshop on Secure IT Systems (NordSec 2008). Copenhagen, Denmark, Oct, 2008. [10] X. Zhang et al., ”Malicious Packet Dropping: How It Might Impact the TCP Performance and How We Can Detect It,” Symp. Security Privacy, May 1998, pp. 263C72. [11] Sarangthem Ibotombi Singh, Smriti Kumar Sinha. A New Trust Model using Hidden Markov Model Based Mixture of Experts. International Conference on Computer Information Systems and Industrial Management Applications (CISIM), Oct. 8-10, 2010. pp., 502-507 [12] M. E. G. Moe , B. E. Helvik and S. J. Knapskog. Comparison of the Beta and the Hidden Markov models of trust in dynamic environments. Trust Management III, IFIP Advances in Information and Communication Technology, Springer Boston, vol. 300, 2009, pp.283-297. [13] ElSalamouny, E., Sassone, V., Nielsen, M. HMM-based trust model. In: Proc. 6th International Workshop on Formal Aspects in Security and Trust. Volume 5983 of LNCS., Springer (2009) pp.21-35
1.2
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0.4
0.2
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Fig. 3.
10
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30 40 50 Number of Observed Transactions (T)
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Estimate the actual packet-drop intensity (λ = 1) for entity 11
D. Discussion It is found from the simulations that the rate of convergence (metric 3) depends linearly on the value of M . However, the value of N is predefined and has no effect on the rate. This can be verified by eq. 1, 3 and 4. We have described the scenario of M ≥ N in the previous sections. In fact, the similar results can be obtained for the case of M < N . We have successfully reduced parameter estimation complexity by making use of a Bonus-Malus system in the HMMs process. We should indicate that the way of constructing the Bonus-Malus system (i.e. Table II) may vary somewhat from case from case, which depends on the specific requirements that must be met in order to be eligible for incentives. VI. C ONCLUSION As reviewed in Section II, HMMs have been used to model and approximate the dynamic behaviors of the entities in modern open network systems. In the paper, we examined the problem by introducing a new Bonus-Malus System, which can be interpreted as a Markov chain with constant transition matrix, to simplify the structure of the trust evaluation model. The trust state transition of an entity is modelled as a HMM process and has an definite effect on its observed performance. We also showed that the maximum likelihood estimators of the parameters of the HMMs are suitably obtained using EM algorithm. The trustworthiness of entities, with respect to a given task in a quantitative manner, is not directly visible from others and is represented by a probability vector that gives the probability distribution of the trust states in a finite space set. The advantages of the HMM-BMS model provides a comparable faster parameter estimation than the existing work.
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