International Journal of Pure and Applied Mathematics Volume 106 No. 7 2016, 57-67 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v106i7.8 PAijpam.eu WEIGHTED LONG RUN FUZZY PROBABILITIES OF FUZZY SEMI-MARKOV MODEL S. Srikrishna 1 Department of Mathematics SSN College of Engineering Kalavakkam, Chennai, Tamilnadu, INDIA Abstract: In this paper, the classical semi-markov model in discrete time is extended to semi-markov model with weighted fuzzy transitions. The definition and the basic equation for interval transitions of a semi-markov model with weighted fuzzy transitions are provided. Also I have analyzed the long run behavior of fuzzy semi-markov model through long run behavior of fuzzy Markov chain and average time spent in each state by means of weighted fuzzy probabilities. The definitions and results for the fuzzy model are provided by means of the weighted fuzzy probabilities and are modeled as generalized triangular fuzzy numbers. These approaches are demonstrated by considering web navigational model. AMS Subject Classification: 60Jxx, 60J10, 60J05 Key Words: generalized triangular fuzzy number, weighted fuzzy transitions, fuzzy semi- Markov model, long run weighted fuzzy transitions 1. Introduction Fuzzy Probability theory is an extension of probability theory to dealing with mixed probability and non probability uncertainty. It provides a theoretical basis to model uncertainties which is only partly characterized by randomness and defines a probability modeling with uncertainties due to lack of trustwor- Received: February 15, 2016 Published: March 3, 2016 c 2016 Academic Publications, Ltd. url: www.acadpubl.eu
58 S. Srikrishna thiness or precision of the data or a lack of pertinent information. Thus to reflect the available information about the true probability distribution which governs the random experiment, the method of fuzzyfying the probability values of the random variable has been given in Buckly [1], [2]. It is called the fuzzy probability and it is represented as fuzzy number. A large amount of work has been done by various researchers to deal with fuzziness of the real life systems. In those works, the notion of fuzzy probability has been applied and it is treated as a fuzzy number. In this paper, homogeneous fuzzy probabilistic semi-markov model (HFPSMM) is proposed as useful tool for predicting the evolution of web access for the specified time duration. In this paper, we modeled a system as homogeneous fuzzy probabilistic semi- Markov model with the following assumptions: Allowing arbitrarily distributed sojourn time in any state. Having the Markovian hypothesis, but in a more flexible manner. The transitions are fuzzy due to uncertainty circumstances in the flow of control between the states Since the transition between the states of a system cannot be precisely measured, this model is examined under the assumption of fuzzy transition probability having an weighted evidence of acceptance represented as generalized triangular fuzzy number. This model also predicts the steady state behavior of the system. This paper is constructed as follows: Section 2 recalls the basic definition. Section 3 defines the homogeneous fuzzy probabilistic semi-markov model with weighted fuzzy transitions. Section 4 deals with the long run behavior of homogeneous fuzzy probabilistic semi-markov model having a weighted evidence of fuzzy probabilities. Section 5 illustrates the above concept for a real time web application. The conclusion is discussed in section 6. All the results and concepts are based on the arithmetic operations of generalized triangular fuzzy number. 2. Preliminaries In this section some basic definitions are reviewed [6]. Definition 1. (Fuzzy Set)
WEIGHTED LONG RUN FUZZY PROBABILITIES... 59 Let Y be an universal set. Then a fuzzy set à on Y is a set of ordered pairs à = {(y, µã(y)) y Y },µã(y) is called the membership function or grade of membership of y in Y. Definition 2. (Fuzzy Number) A fuzzy number à (fuzzy interval) is a special significant type of fuzzy set which is defined on the real number system R with membership function µã : R [0,1] and it must posses the following properties. 1. à must be convex 2. à must be normal 3. Its membership function must be piecewise continuous. Definition 3. (Generalized Triangular Fuzzy Number) A Generalized triangular fuzzy number  [5] can be defined as  = (a 1,a 2, a 3 ;w) and its membership function is defined as µã(x) = w(x a 1 ) a 2 a 1 if a 1 x a 2 w if x = a 2 w(a 3 x) a 3 a 2 if a 2 x a 3 2.1. Arithmetic Operations of Generalized Triangular Fuzzy Numbers Assume that there are two generalized triangular fuzzy numbers [5]  and ˆB, where  = (a 1,a 2,a 3 ;w 1 ) and ˆB = (b 1,b 2,b 3 ;w 2 ). 2.1.1. Generalized Triangular Fuzzy Number Addition  ˆB = (a 1,a 2,a 3 ;w 1 ) (b 1,b 2,b 3 ;w 2 )=(a 1 +b 1,a 2 +b 2,a 3 +b 3 ;min(w 1,w 2 ) 2.1.2. Generalized Triangular Fuzzy Number Multiplication  ˆB = (a 1,a 2,a 3 ;w 1 ) (b 1,b 2,b 3 ;w 2 )=(a 1 b 1,a 2 b 2,a 3 b 3 ;min(w 1,w 2 ) 3. Homogeneous Fuzzy Probabilistic Semi-Markov Model with Weighted Fuzzy Probabilities Consider a random system with finite state space E = {1,2,,m} and let us consider that in a dynamic evolution of this random system, there exists an
60 S. Srikrishna uncertainty in the probability of occurrences, hence the associated probabilities are taken as fuzzy probabilities represented as triangular fuzzy number. Let N= {1,2,...} be a index set. Consider a stochastic process Z = (Z t ) t N which are considered to evolute in discrete time. Then the evolution in discrete time of the system is given by the following chains: X = (X n )n N i.e., the visited states at the time points. S = (S n )n Ni.e., successive time points. T = (T n )n N i.e., successive sojourn times in the visited states. The relation between process Z and X of the successively visited states is given by Z = X N(t), where N(t) = max{n N,S n t} gives the number of jumps in [1,t] N. We will start with defining the fundamental notion of our model which is nothing but homogeneous fuzzy probabilistic semi-markov model (HFPSMM). For this we consider the crisp Markov renewal chain and semi-markov kernel. Semi-Markov kernel is defined as a matrix valued function q(t) = (q ij (t)) satisfy the following properties. 1. q ij (t) 0,i,j E,t N 2. q ij (t) = 0fort = 0,i,j N 3. t=0 j E q ij(t) 1,i E The couple (X,S) = (X n,s n ) n N is called a Markov renewal chain if it satisfies P[X n+1 = j,s n+1 S n = t X 0,...,X n ] = P[X n+1 = j,s n+1 Sn = t X n ]. If the above equation is independent of n, then (X, S) is said to be homogeneous and the discrete time semi-markov kernel q ij (t) is defined by q ij (t) = P[X n+1 = j,t n+1 = t X n = i] If there is uncertainty in the quantity q ij (t), then we compute the generalized fuzzy probability ˆq ij (t)= ( q ij (t),w q ) with a weighted evidence, where q ij (t) can be computed by fuzzifying into triplet q ij (t) = (a,b,c) on [0,1], i.e., a, b, c [0,1] and these generalized fuzzy probabilities forms a transition generalized fuzzy probability matrix ˆq ij (t)=(ˆq ij (t)). Thus we define the homogeneous fuzzy probabilistic semi-markov kernel as ˆq ij (t)=(ˆq ij (t)), where ˆq ij (t)= ( q ij (t),w q ) represented as generalized triangular fuzzy number, whose triplet values satisfies the following properties. 1. q ij (t) (0,0,0),i,j E,t N
WEIGHTED LONG RUN FUZZY PROBABILITIES... 61 2. q ij (t) = (0,0,0)fort = 0,i,j E 3. t=0 j E q ij (t) (1,1,1),i,j E,t N If (X, S) is a homogeneous Markov renewal chain with kernel ˆq(t) = (ˆq ij (t)), then (X, S) is called as fuzzy probabilistic Markov renewal chain, whose values are generalized fuzzy numbers. Next we consider the fuzzy probabilities for the chain (X n )n N. We denote by ˆP = (ˆp ij ) i,j E, transition matrix with generalized fuzzy probabilities, where ˆp ij = ( p ij,w p ) i,j E which are obtained by fuzzifying p ij into triplet p ij =(a, b, c) on [0, 1] i.e., a, b, c [0, 1]. In order to have the fact that the rows in the transition matrix are discrete probability distributions, we introduce the restriction on the p ij such that there are p ij p ij [1] so that n j=1 p ij = 1. When we consider in the evolution of HFPSMC which follows as: The moment the HFPSMC enters any state i, it selects the next state to visit j according to P, its transition fuzzy probability matrix. If state j is selected, then the time that the system remains in state i before moving to state j is a sojourn time distribution in state i. When investigating the evolution of HFPMRC, we are interested in two types of holding times generalized fuzzy probability distributions: Conditional generalized fuzzy probability distribution depending on the next state to be visited and the unconditional generalized fuzzy probability distribution of the sojourn time, having an weighted evidence whose values are represented as a generalized triangular fuzzy numbers. 3.1. Conditional Generalized Fuzzy Probability Distribution of Sojourn Time For this we consider the crisp conditional probability distribution of sojourn times defined as f ij (t) = P[T n+1 = t X n+1 = j,x n = i],t N If there is uncertainty in the probability values f ij (t), then we compute the generalized fuzzy probability ˆf ij (t) = ( f ij (t),w f ) i,j E, having a weighted evidence represented as generalized triangular fuzzy number where f ij (t) can be computed by fuzzifying into triplet f ij (t)=(d,e,f) on[0, 1] i.e., d, e, f [0, 1] and these entries forms the matrixˆf(t) = (ˆf ij (t)). The fuzzy probabilitis f ij (t) can be computed by considering the α cuts. (i.e) we define f ij (t)[α] = [f ij1 (t)(α), f ij2 (t)(α)] for i j=1,2,...,m where
62 S. Srikrishna { f ij1 (t)(α) = min { f ij2 (t)(α) = max { q ij(t) p ij { q ij(t) p ij q ij (t) q ij (t)[α],p ij p ij [α]} q ij (t) q ij (t)[α],p ij p ij [α]} } } 3.2. Unconditional Generalized Fuzzy Probability Distribution of Sojourn Time For this we consider the crisp unconditional probability distributions of Sojourn times defined as h i (t) = P[T n+1 = t X n = i] = j i q ij (t),t N We now compute the generalized fuzzy probability with weighted evidence ĥ i (t) = ( h i (t),w h ) i,j E, represented as generalized triangular fuzzy number, where h i (t)canbecomputedby-cutsasfollows: h i (t)[α] = [h i1 (t)(α), h i2 (t)(α)] for i j=1,2,...,m where h i1 (t)(α) = min { { i j q ij(t) q ij (t) q ij (t)[α]} } h i2 (t)(α) = max { { i j q ij(t) q ij (t) q ij (t)[α]} } The cumulative unconditional sojourn time distribution denoted by H i (t) = t h i (u),t N u=0 WecomputethegeneralizedfuzzyprobabilitywithweightedevidenceĤi(t) = ( H i (t),w H ) i,j E, where H i (t) is a triplet which is obtained by means of α cuts : Hi (t)[α] = [H i1 (t)(α), H i2 (t)(α)] for i j=1,2,...,s where H i1 (t)(α) = min { { u=0,...,t h i(u) h i (u) h i (u)[α]} } H i2 (t)(α) = max { { u=0,...,t h i(u) h i (u) h i (u)[α]} } From theseα - cuts we can find the generalized fuzzy probabilityĥi(t), the unconditional generalized fuzzy probability distribution of sojourn time which represents the generalized fuzzy probability having weighted evidence that the system which entered i to stay t time units in state i before its next transition with the weighted bound. Using fuzzy arithmetic operations, we can write the following u=t+1 h i (u) = 1 t u=1 h i (u) (1)
WEIGHTED LONG RUN FUZZY PROBABILITIES... 63 Now define by H(t), the m x m matrix which has zeros everywhere apart from the diagonal which has in position i the element as in equation 1. Let us introduce the notion of transition function of the FPSMC related to that of FPMRC. The transition function of FPSMC Z is the matrix valued function ˆφ(t) = (ˆφ ij (t);i,j E,t N) defined by ˆφ ij (t) = (ˆδ ij [1 Ĥi(t)]) ( k E u=0,1,...,t ˆp ik ˆf ik (u) ˆφ kj (t u)) where ˆδ ij is defined by ˆδ ij = ( δ ij,w δ ) and δ ij = { (0,0,0), i j (1,1,1), i=j which represents the generalized fuzzy probability of system moving from i to j at t units of time. Now define hadamard product of two matrices ˆQ(u) = ˆP ˆF(u) to be the matrix with elements q ij (u) = ˆp ij ˆf ij (u). Hence ˆφ(t) = Ĥ(t) ([ˆP ˆF(u)] ˆφ ij (t u)) (2) 4. Weighted Long Run Fuzzy Probability Distribution of Homogeneous Fuzzy Semi-Markov Model Homogeneous fuzzy probabilistic semi-markov model can be analyzed for long run performances in the same manner as discrete time fuzzy Markov chain. To do this, we need to know the weighted long run fuzzy probability of the (Fuzzy Markov Model) embedded fuzzy Markov model and the mean residence time in each state or the average time spent in each state with weighted evidence. 4.1. Long Run Generalized Fuzzy Probability of Fuzzy Markov Model We start with crisp regular finite Markov chain and then we proceed to fuzzy finite regular Markov chain for finding long run generalized fuzzy probability of FPMM. We say that crisp Markov chain is regular if p k > 0, for some k which is p k ij > 0 for all i, j. This means that it is possible to go from any state i to many state j in k steps. A property of regular Markov chain is that powers of P converge or lim n P n =, where the rows of are identical. Let w be the unique left eigen values of P corresponding to eigen value one, so that w i > 0
64 S. Srikrishna and i=1,...,m w i = 1. Thatis wp=wfor 1xmvector w. Each rowin is equal to wand P (n) P (0) = w. After along time, thinkingthat each step beinga time interval i, the probability of being in state i is w i,1 i m,independent of the initial condition p (0). In a regular Markov chain, the process goes on forever jumping from state to state. Now proceed to a fuzzy finite regular Markov chain. All of the p ij in the transition matrix P must be known. Suppose some of them are not known precisely and must be estimated and hence are uncertain. We substitute a generalized fuzzy number ˆp ij = ( p ij,w p ) i,j E, where p ij is a triplet having a weighted bound w p. For this uncertain p ij, producing a fuzzy transition ˆP. If some of the p ij are known, like it is zero, we use these values but still written them as a fuzzy p ij. In practice, from the data, we first obtain probability p ij from which we compute the generalized fuzzy probability with weighted evidence ˆp ij. For this, we compute the triplet by assuming the uncertainty is in some of the p ij values but not in the fact that the rows in the transition matrix. So, we put the following restriction on the p ij values, there are p ij = p ij [1] so that P = (p ij ) is a fuzzy matrix. Pick and fix an in [0, 1]. Define Dom[α] as the set of all p ij p ij [α],0 i,j m such that if we form a fuzzy matrix P = (p ij ) with these p ij all row sums equal to 1. Define v = (p 00,p 11,...,p mm ). Row vector v is just all the p ij in a transition matrix P = (p ij ). Then Dom[α] is all the vectors v, where the p ij are in the α cut of p ij, all i,j E such that P is transition matrix for finite Markov chain. For each v Dom[α], set P = (p ij ) and we get P n =. Let Γ(α) = {w/wp = w,0 < w i < 1,w 0 +... + w m = 1, v Dom[α]}. Γ(α) consists of all vectors w, which are the rows in Γ, for every v Dom[α]. Now the rows in ˆ will be all the same where ˆ has a triplet with a weighted bound, so let ŵ = (ŵ 0,ŵ 1,...,ŵ m ) bearow in ˆ, whereeach ŵ 0 = ( w 0,w w ) represented as a generalized triangularfuzzynumber. Thetriplet w 0 canbeestimated asfollows: let w j [α] = [w j1 (α),w j2 (α)]for0 j m, then w j1 (α) = min{w j /w Γ(α)} and w j2 (α) = max{w j /w Γ(α)}, where w j is the jth component in the vector w. Therefore the long run generalized fuzzy probabilities having weighted evidence are given by ŵ 0 = the generalized fuzzy probability of the system in the initial step. ŵ 1 = the generalized fuzzy probability of the system in the first step.
WEIGHTED LONG RUN FUZZY PROBABILITIES... 65 ŵ 2 = the generalized fuzzy probability of the system in the second step., and etc. 4.2. Long Run Generalized Fuzzy Probability Distributions of Fuzzy Probabilistic Semi-Markov Model Consider the transition generalized fuzzy probability matrix ˆP, we can determine the long run transition weighted fuzzy probability ˆ which is mentioned in the previous section. Then the mean residence time or average time vector  = (Āi,w A ), by estimating the triplet by fuzzyfication which as follows: Let Āi = (a 1,b 1,c 1 ) on [0,1], i.e. (a 1,b 1,c 1 ) [0,1] and these entries forms the matrix Ā = (Āi). In the crisp semi-markov model, the steady state probability distribution is characterized by the formula, φ i = A m iπ i j=1 A iπ i. Using α cuts we can compute the fuzzy probability of φ i denoted by φ i. We define φ i [α] = [φ i1 (α), φ i2 (α)] for i j=1,2,...,m where { φ i1 (α) = min { φ i2 (α) = max m A i π i j=1 A iπ i A i Āi[α],π i π i [α] } m A i π i j=1 A iπ i A i Ā i [α],π i π i [α] } Then from these α cuts, we find the long run generalized fuzzy probability ˆφ = ( φ i,w φ )whichrepresentsthelongrungeneralized fuzzyprobabilityofbeing in state i (i = 1,2,...,m) with the weighted evidence for fuzzy probabilistic semi-markov model. 5. Illustration In the following, we illustrate the above defined concepts for a real time application. Let us consider the web navigation of our website www.ssn.edu.in. The operational units are the web pages of Department Computer Science (CS), Information Technology (IT), Electronics and Communication (EC), which are the set of states and the associated connections are the transitions. Since there exists uncertainties in the probabilistic usage information between the state transition, for each transition we associate fuzzy transition defined as generalized transition fuzzy probability. Hence we have modeled fuzzy probabilistic semi-markov model with state space E = CS, IT, EC and transitions as generalized fuzzy probabilities represented as generalized triangular fuzzy number.
66 S. Srikrishna The corresponding transition generalized fuzzy probabilities are given as follows: ˆP = CS IT EC CS IT EC (0.978/0.98/0.982; 0.9) (0.978/0.98/0.982; 0.9) (0.971/0.973/0.975; 0.9) (0.878/0.88/0.882; 0.9) (0.778/0.78/0.782; 0.9) (0.878/0.88/0.882; 0.9) (0.778/0.78/0.782 0.9) (0.988/0.99/0.992; 0.9) (0.967/0.969/0.971; 0.9) We have used fuzzy Weibull distribution to estimate the sojourn time generalized fuzzy probability distribution using MATLAB. This fuzzy Weibull distribution is defined as a generalized triangular fuzzy number by fuzzifying the crisp Weibull distribution values using-cuts. The parameters of the Weibull distribution can be estimated using maximum likelihood estimation method and the values are evaluated using the software MATLAB. Hence the calculated interval generalized fuzzy transition probabilities is given as follows: ˆφ ij = CS IT EC CS IT EC (0.978/0.98/0.982; 0.9) (0.96/0.98/0.982; 0.9) (0.88/0.9/0.92; 0.9) (0.878/0.88/0.882; 0.9) (0.98/0.982/0.984; 0.9) (0.878/0.88/0.882; 0.9) (0.778/0.78/0.782; 0.9) (0.88/0.89/0.892; 0.9) (0.967/0.969/0.971; 0.9) 5.1. Steady State Analysis for the Web Links In section 4, we have seen that weighted fuzzy probability of the system being in each state for embedded fuzzy Markov model(efmm) as ˆπ i, where each row is represented as ŵ = (ŵ 0,ŵ 1,...,ŵ m ) which can be estimated using α cuts. Using the procedure given in section 4, the calculated long run generalized fuzzy probability of EFMM is ((0.983/0.985/ 0.987; 0.9), (0.98/ 0.982/ 0.984;0.9), (0.962/ 0.964/ 0.966;0.9)) and the long run generalized fuzzy probabilities of FPSMM are obtained as follows:ˆφ CS = (0.983/0.985/0.987;0.9);ˆφ IT = (0.85/0.852/0.854;0.9);ˆφ EC = (0.9/0.92/0.94;0.9) From the long run generalized fuzzy probabilities we see that ˆφ IT =(0.85/ 0.852/0.854;0.9) whose triplet has weighted bound of evidence 0.9, which indicates that the state is less frequently visited with high acceptance evidence, when compared to the other states and the most frequently visited states are CS and EC having a high acceptance evidence. Hence more focus should be given to testing links that leads to these states. 6. Conclusion In this paper, I have defined a homogeneous fuzzy probabilistic semi-markov model by means of generalized fuzzy number and presented a fuzzy probabilistic approach to the dynamic evolution of web application defined by interval generalized fuzzy probabilities. By means of this approach, we not only consider uncertainties in the state transition, but also uncertainties in the sojourn
WEIGHTED LONG RUN FUZZY PROBABILITIES... 67 time distributions. This method starts from the idea of evolution of interval generalized fuzzy probabilities and duration time generalized fuzzy probability distributions and this ides allows the approach which results in long run generalized fuzzy probabilities of the web navigational model. Acknowledgments I thank the management of SSN Institutions for providing necessary facilities and support to carry out this work. References [1] J.J. Buckley, Fuzzy Probabilities, Springer Publication. (2005). [2] J. J. Buckly, K.Reilly, and Zheng, X., Fuzzy probabilities for Web Planning, Soft Computing, Vol. 8, (2004), 464-476. [3] Kwank H. Lee, A First Course on Fuzzy Theory and Applications,Springer International Edition, (2005). [4] Li Z. and Tian J., Testing the suitability of Markov Chains as Web Usage Models, Proc. of the 27th Annual International Computer Software and Applications Conference, IEEE Computer Socity, (2003). [5] Amit Kumar, Pushpinder Singh, Parmpreet Kaur, and Amarpreet Kaur, Equality of Generalized Triangular Fuzzy Numbers,International Journal of Physical and Mathematical Sciences, (1), (2010). [6] A. Kaufmann and MM. Gupta, Fuzzy mathematical models in engineering and management science. Elsevier Science Publishers, Amsterdam, Netherlands, (1988).
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