WEIGHTED LONG RUN FUZZY PROBABILITIES OF FUZZY SEMI-MARKOV MODEL. S. Srikrishna. SSN College of Engineering Kalavakkam, Chennai, Tamilnadu, INDIA
|
|
- Beverly Hicks
- 5 years ago
- Views:
Transcription
1 International Journal of Pure and Applied Mathematics Volume 106 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /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:
2 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)
3 WEIGHTED LONG RUN FUZZY PROBABILITIES 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 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 ) 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 ) 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
4 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
5 WEIGHTED LONG RUN FUZZY PROBABILITIES 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 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
6 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)
7 WEIGHTED LONG RUN FUZZY PROBABILITIES 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 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
8 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 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.
9 WEIGHTED LONG RUN FUZZY PROBABILITIES ŵ 2 = the generalized fuzzy probability of the system in the second step., and etc 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 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.
10 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/ ) (0.988/0.99/0.992; 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
11 WEIGHTED LONG RUN FUZZY PROBABILITIES 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), [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).
12 68
A Study on Homogeneous Fuzzy Semi-Markov Model
Applied Mathematical Sciences, Vol. 3, 2009, no. 50, 2453-2467 A Study on Homogeneous Fuzzy Semi-Markov Model B. Praba Department of Mathematics SSN College of Engineering Kalavakkam, Chennai - 603110,
More informationSTEADY-STATE BEHAVIOR OF AN M/M/1 QUEUE IN RANDOM ENVIRONMENT SUBJECT TO SYSTEM FAILURES AND REPAIRS. S. Sophia 1, B. Praba 2
International Journal of Pure and Applied Mathematics Volume 101 No. 2 2015, 267-279 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i2.11
More informationPAijpam.eu SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEMS USING RANKING FUNCTION
International Journal of Pure and Applied Mathematics Volume 106 No. 8 2016, 149-160 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v106i8.18
More informationKalavakkam, Chennai, , Tamilnadu, INDIA 2,3 School of Advanced Sciences. VIT University Vellore, , Tamilnadu, INDIA
International Journal of Pure and Applied Mathematics Volume 09 No. 4 206, 799-82 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: 0.2732/ijpam.v09i4.4 PAijpam.eu
More informationFuzzy Inventory Control Problem With Weibull Deterioration Rate and Logarithmic Demand Rate
Volume 7 No. 07, 5-44 ISSN: -8080 (printed version); ISSN: 4-95 (on-line version) url: http://www.ijpam.eu ijpam.eu Fuzzy Inventory Control Problem With Weibull Deterioration Rate and Logarithmic Demand
More informationPAijpam.eu ON FUZZY INVENTORY MODEL WITH ALLOWABLE SHORTAGE
International Journal of Pure and Applied Mathematics Volume 99 No. 205, 5-7 ISSN: 3-8080 (printed version; ISSN: 34-3395 (on-line version url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v99i.
More informationIEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2011, Professor Whitt Class Lecture Notes: Tuesday, March 1.
IEOR 46: Introduction to Operations Research: Stochastic Models Spring, Professor Whitt Class Lecture Notes: Tuesday, March. Continuous-Time Markov Chains, Ross Chapter 6 Problems for Discussion and Solutions.
More informationAn M/M/1 Queue in Random Environment with Disasters
An M/M/1 Queue in Random Environment with Disasters Noam Paz 1 and Uri Yechiali 1,2 1 Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978,
More informationParametric and Non Homogeneous semi-markov Process for HIV Control
Parametric and Non Homogeneous semi-markov Process for HIV Control Eve Mathieu 1, Yohann Foucher 1, Pierre Dellamonica 2, and Jean-Pierre Daures 1 1 Clinical Research University Institute. Biostatistics
More informationFuzzy Optimization and Normal Simulation for Solving Fuzzy Web Queuing System Problems
Fuzzy Optimization and Normal Simulation for Solving Fuzzy Web Queuing System Problems Xidong Zheng, Kevin Reilly Dept. of Computer and Information Sciences University of Alabama at Birmingham Birmingham,
More informationPAijpam.eu THE ZERO DIVISOR GRAPH OF A ROUGH SEMIRING
International Journal of Pure and Applied Mathematics Volume 98 No. 5 2015, 33-37 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v98i5.6
More informationSolving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Number Matrices by Singular Value Decomposition
Intern. J. Fuzzy Mathematical Archive Vol. 3, 23, 6-22 ISSN: 232 3242 (P), 232 325 (online) Published on 8 November 23 www.researchmathsci.org International Journal of Solving Fully Fuzzy Linear Systems
More informationType-2 Fuzzy Shortest Path
Intern. J. Fuzzy Mathematical rchive Vol. 2, 2013, 36-42 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 15 ugust 2013 www.researchmathsci.org International Journal of Type-2 Fuzzy Shortest Path V.
More informationSMSTC (2007/08) Probability.
SMSTC (27/8) Probability www.smstc.ac.uk Contents 12 Markov chains in continuous time 12 1 12.1 Markov property and the Kolmogorov equations.................... 12 2 12.1.1 Finite state space.................................
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
More informationData analysis and stochastic modeling
Data analysis and stochastic modeling Lecture 7 An introduction to queueing theory Guillaume Gravier guillaume.gravier@irisa.fr with a lot of help from Paul Jensen s course http://www.me.utexas.edu/ jensen/ormm/instruction/powerpoint/or_models_09/14_queuing.ppt
More informationChapter 2 SOME ANALYTICAL TOOLS USED IN THE THESIS
Chapter 2 SOME ANALYTICAL TOOLS USED IN THE THESIS 63 2.1 Introduction In this chapter we describe the analytical tools used in this thesis. They are Markov Decision Processes(MDP), Markov Renewal process
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationPAijpam.eu OBTAINING A COMPROMISE SOLUTION OF A MULTI OBJECTIVE FIXED CHARGE PROBLEM IN A FUZZY ENVIRONMENT
International Journal of Pure and Applied Mathematics Volume 98 No. 2 2015, 193-210 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v98i2.3
More informationChapter 5. Continuous-Time Markov Chains. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 5. Continuous-Time Markov Chains Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Continuous-Time Markov Chains Consider a continuous-time stochastic process
More informationLIMITING PROBABILITY TRANSITION MATRIX OF A CONDENSED FIBONACCI TREE
International Journal of Applied Mathematics Volume 31 No. 18, 41-49 ISSN: 1311-178 (printed version); ISSN: 1314-86 (on-line version) doi: http://dx.doi.org/1.173/ijam.v31i.6 LIMITING PROBABILITY TRANSITION
More informationApplication of the Fuzzy Weighted Average of Fuzzy Numbers in Decision Making Models
Application of the Fuzzy Weighted Average of Fuzzy Numbers in Decision Making Models Ondřej Pavlačka Department of Mathematical Analysis and Applied Mathematics, Faculty of Science, Palacký University
More informationLecture 11: Introduction to Markov Chains. Copyright G. Caire (Sample Lectures) 321
Lecture 11: Introduction to Markov Chains Copyright G. Caire (Sample Lectures) 321 Discrete-time random processes A sequence of RVs indexed by a variable n 2 {0, 1, 2,...} forms a discretetime random process
More informationA Method for Solving Intuitionistic Fuzzy Transportation Problem using Intuitionistic Fuzzy Russell s Method
nternational Journal of Pure and Applied Mathematics Volume 117 No. 12 2017, 335-342 SSN: 1311-8080 (printed version); SSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special ssue ijpam.eu A
More informationFuzzy Eigenvectors of Real Matrix
Fuzzy Eigenvectors of Real Matrix Zengfeng Tian (Corresponding author Composite Section, Junior College, Zhejiang Wanli University Ningbo 315101, Zhejiang, China Tel: 86-574-8835-7771 E-mail: bbtianbb@126.com
More informationON THE CONSTRUCTION OF HADAMARD MATRICES. P.K. Manjhi 1, Arjun Kumar 2. Vinoba Bhave University Hazaribag, INDIA
International Journal of Pure and Applied Mathematics Volume 120 No. 1 2018, 51-58 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v120i1.4
More informationChapter 16 focused on decision making in the face of uncertainty about one future
9 C H A P T E R Markov Chains Chapter 6 focused on decision making in the face of uncertainty about one future event (learning the true state of nature). However, some decisions need to take into account
More informationMarkov decision processes and interval Markov chains: exploiting the connection
Markov decision processes and interval Markov chains: exploiting the connection Mingmei Teo Supervisors: Prof. Nigel Bean, Dr Joshua Ross University of Adelaide July 10, 2013 Intervals and interval arithmetic
More informationKaraliopoulou Margarita 1. Introduction
ESAIM: Probability and Statistics URL: http://www.emath.fr/ps/ Will be set by the publisher ON THE NUMBER OF WORD OCCURRENCES IN A SEMI-MARKOV SEQUENCE OF LETTERS Karaliopoulou Margarita 1 Abstract. Let
More informationContinuous time Markov chains
Chapter 2 Continuous time Markov chains As before we assume that we have a finite or countable statespace I, but now the Markov chains X {X(t) : t } have a continuous time parameter t [, ). In some cases,
More informationMarkov Chains. Arnoldo Frigessi Bernd Heidergott November 4, 2015
Markov Chains Arnoldo Frigessi Bernd Heidergott November 4, 2015 1 Introduction Markov chains are stochastic models which play an important role in many applications in areas as diverse as biology, finance,
More informationSTOCHASTIC PROCESSES Basic notions
J. Virtamo 38.3143 Queueing Theory / Stochastic processes 1 STOCHASTIC PROCESSES Basic notions Often the systems we consider evolve in time and we are interested in their dynamic behaviour, usually involving
More informationIrreducibility. Irreducible. every state can be reached from every other state For any i,j, exist an m 0, such that. Absorbing state: p jj =1
Irreducibility Irreducible every state can be reached from every other state For any i,j, exist an m 0, such that i,j are communicate, if the above condition is valid Irreducible: all states are communicate
More informationExample: physical systems. If the state space. Example: speech recognition. Context can be. Example: epidemics. Suppose each infected
4. Markov Chains A discrete time process {X n,n = 0,1,2,...} with discrete state space X n {0,1,2,...} is a Markov chain if it has the Markov property: P[X n+1 =j X n =i,x n 1 =i n 1,...,X 0 =i 0 ] = P[X
More informationHigh-dimensional Markov Chain Models for Categorical Data Sequences with Applications Wai-Ki CHING AMACL, Department of Mathematics HKU 19 March 2013
High-dimensional Markov Chain Models for Categorical Data Sequences with Applications Wai-Ki CHING AMACL, Department of Mathematics HKU 19 March 2013 Abstract: Markov chains are popular models for a modelling
More informationLecture 10: Semi-Markov Type Processes
Lecture 1: Semi-Markov Type Processes 1. Semi-Markov processes (SMP) 1.1 Definition of SMP 1.2 Transition probabilities for SMP 1.3 Hitting times and semi-markov renewal equations 2. Processes with semi-markov
More information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side
More informationSTAT STOCHASTIC PROCESSES. Contents
STAT 3911 - STOCHASTIC PROCESSES ANDREW TULLOCH Contents 1. Stochastic Processes 2 2. Classification of states 2 3. Limit theorems for Markov chains 4 4. First step analysis 5 5. Branching processes 5
More informationReliability Analysis of a Fuel Supply System in Automobile Engine
ISBN 978-93-84468-19-4 Proceedings of International Conference on Transportation and Civil Engineering (ICTCE'15) London, March 21-22, 2015, pp. 1-11 Reliability Analysis of a Fuel Supply System in Automobile
More informationA Fuzzy Approach to Priority Queues
International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 2, Number 4 (2012), pp. 479-488 Research India Publications http://www.ripublication.com A Fuzzy Approach to Priority Queues
More information= P{X 0. = i} (1) If the MC has stationary transition probabilities then, = i} = P{X n+1
Properties of Markov Chains and Evaluation of Steady State Transition Matrix P ss V. Krishnan - 3/9/2 Property 1 Let X be a Markov Chain (MC) where X {X n : n, 1, }. The state space is E {i, j, k, }. The
More informationMath 304 Handout: Linear algebra, graphs, and networks.
Math 30 Handout: Linear algebra, graphs, and networks. December, 006. GRAPHS AND ADJACENCY MATRICES. Definition. A graph is a collection of vertices connected by edges. A directed graph is a graph all
More informationMarkov Chains Handout for Stat 110
Markov Chains Handout for Stat 0 Prof. Joe Blitzstein (Harvard Statistics Department) Introduction Markov chains were first introduced in 906 by Andrey Markov, with the goal of showing that the Law of
More informationNetwork Analysis of Fuzzy Bi-serial and Parallel Servers with a Multistage Flow Shop Model
2st International Congress on Modelling and Simulation, Gold Coast, Australia, 29 Nov to 4 Dec 205 wwwmssanzorgau/modsim205 Network Analysis of Fuzzy Bi-serial and Parallel Servers with a Multistage Flow
More informationMATH Mathematics for Agriculture II
MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3
More informationFuzzy Geometric Distribution with Some Properties Dr. Kareema Abed AL-Kadim 1, Abdul Hameed Ashwya 2
Fuzzy Geometric Distribution with Some Properties Dr. Kareema Abed AL-Kadim 1, Abdul Hameed Ashwya 2 1 Department of Mathematics, Babylon University, College of Education for Pure Sciences, Hilla, Iraq,
More informationMarkov Repairable Systems with History-Dependent Up and Down States
Markov Repairable Systems with History-Dependent Up and Down States Lirong Cui School of Management & Economics Beijing Institute of Technology Beijing 0008, P.R. China lirongcui@bit.edu.cn Haijun Li Department
More informationSolution of Fuzzy System of Linear Equations with Polynomial Parametric Form
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 7, Issue 2 (December 2012), pp. 648-657 Applications and Applied Mathematics: An International Journal (AAM) Solution of Fuzzy System
More informationPAijpam.eu NEW H 1 (Ω) CONFORMING FINITE ELEMENTS ON HEXAHEDRA
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 609-617 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v109i3.10
More informationOPTIMAL SOLUTION OF BALANCED AND UNBALANCED FUZZY TRANSPORTATION PROBLEM BY USING OCTAGONAL FUZZY NUMBERS
International Journal of Pure and Applied Mathematics Volume 119 No. 4 2018, 617-625 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v119i4.4
More informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More information2. Transience and Recurrence
Virtual Laboratories > 15. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 2. Transience and Recurrence The study of Markov chains, particularly the limiting behavior, depends critically on the random times
More informationFully fuzzy linear programming with inequality constraints
Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 5, No. 4, 2013 Article ID IJIM-00280, 8 pages Research Article Fully fuzzy linear programming with inequality
More informationSolution of Fuzzy Maximal Flow Network Problem Based on Generalized Trapezoidal Fuzzy Numbers with Rank and Mode
International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 9, Issue 7 (January 2014), PP. 40-49 Solution of Fuzzy Maximal Flow Network Problem
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca
More informationLecture 4a: Continuous-Time Markov Chain Models
Lecture 4a: Continuous-Time Markov Chain Models Continuous-time Markov chains are stochastic processes whose time is continuous, t [0, ), but the random variables are discrete. Prominent examples of continuous-time
More informationRELIABILITY ANALYSIS OF A FUEL SUPPLY SYSTEM IN AN AUTOMOBILE ENGINE
International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 973-9424, Vol. 9 No. III (September, 215), pp. 125-139 RELIABILITY ANALYSIS OF A FUEL SUPPLY SYSTEM IN AN AUTOMOBILE ENGINE R. K. AGNIHOTRI 1,
More informationReinforcement Learning
Reinforcement Learning March May, 2013 Schedule Update Introduction 03/13/2015 (10:15-12:15) Sala conferenze MDPs 03/18/2015 (10:15-12:15) Sala conferenze Solving MDPs 03/20/2015 (10:15-12:15) Aula Alpha
More informationResearch Article A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs
Abstract and Applied Analysis Volume 2012, Article ID 950482, 12 pages doi:101155/2012/950482 Research Article A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs Emrah Akyar Department of
More informationMORE NUMERICAL RADIUS INEQUALITIES FOR OPERATOR MATRICES. Petra University Amman, JORDAN
International Journal of Pure and Applied Mathematics Volume 118 No. 3 018, 737-749 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.173/ijpam.v118i3.0
More informationNotes on Measure Theory and Markov Processes
Notes on Measure Theory and Markov Processes Diego Daruich March 28, 2014 1 Preliminaries 1.1 Motivation The objective of these notes will be to develop tools from measure theory and probability to allow
More informationA Divide-and-Conquer Algorithm for Functions of Triangular Matrices
A Divide-and-Conquer Algorithm for Functions of Triangular Matrices Ç. K. Koç Electrical & Computer Engineering Oregon State University Corvallis, Oregon 97331 Technical Report, June 1996 Abstract We propose
More informationInput Control in Fuzzy Non-Homogeneous Markov Systems
Input Control in Fuzzy Non-Homogeneous Markov Systems Maria Symeonaki 1 and Giogros Stamou 2 1 Panteion University, Department of Social Politics, 136 Syngrou Av., 176 71, Athens, Greece (e-mail: msimeon@panteion.gr,
More informationHomogeneous Backward Semi-Markov Reward Models for Insurance Contracts
Homogeneous Backward Semi-Markov Reward Models for Insurance Contracts Raimondo Manca 1, Dmitrii Silvestrov 2, and Fredrik Stenberg 2 1 Universitá di Roma La Sapienza via del Castro Laurenziano, 9 00161
More informationSummary: A Random Walks View of Spectral Segmentation, by Marina Meila (University of Washington) and Jianbo Shi (Carnegie Mellon University)
Summary: A Random Walks View of Spectral Segmentation, by Marina Meila (University of Washington) and Jianbo Shi (Carnegie Mellon University) The authors explain how the NCut algorithm for graph bisection
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationAn Overview of Methods for Applying Semi-Markov Processes in Biostatistics.
An Overview of Methods for Applying Semi-Markov Processes in Biostatistics. Charles J. Mode Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 Overview of Topics. I.
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Department of Mathematics University of Manitoba Winnipeg Julien Arino@umanitoba.ca 19 May 2012 1 Introduction 2 Stochastic processes 3 The SIS model
More informationTHIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING DIFFERENTIAL EQUATION IN FUZZY ENVIRONMENT. S. Narayanamoorthy 1, T.L. Yookesh 2
International Journal of Pure Applied Mathematics Volume 101 No. 5 2015, 795-802 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu PAijpam.eu THIRD ORDER RUNGE-KUTTA
More informationStochastic inventory system with two types of services
Int. J. Adv. Appl. Math. and Mech. 2() (204) 20-27 ISSN: 2347-2529 Available online at www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Stochastic inventory system
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationLecturer: Olga Galinina
Lecturer: Olga Galinina E-mail: olga.galinina@tut.fi Outline Motivation Modulated models; Continuous Markov models Markov modulated models; Batch Markovian arrival process; Markovian arrival process; Markov
More informationDefinition A finite Markov chain is a memoryless homogeneous discrete stochastic process with a finite number of states.
Chapter 8 Finite Markov Chains A discrete system is characterized by a set V of states and transitions between the states. V is referred to as the state space. We think of the transitions as occurring
More informationFuzzy Order Statistics based on α pessimistic
Journal of Uncertain Systems Vol.10, No.4, pp.282-291, 2016 Online at: www.jus.org.uk Fuzzy Order Statistics based on α pessimistic M. GH. Akbari, H. Alizadeh Noughabi Department of Statistics, University
More informationMATH 564/STAT 555 Applied Stochastic Processes Homework 2, September 18, 2015 Due September 30, 2015
ID NAME SCORE MATH 56/STAT 555 Applied Stochastic Processes Homework 2, September 8, 205 Due September 30, 205 The generating function of a sequence a n n 0 is defined as As : a ns n for all s 0 for which
More informationCDA6530: Performance Models of Computers and Networks. Chapter 3: Review of Practical Stochastic Processes
CDA6530: Performance Models of Computers and Networks Chapter 3: Review of Practical Stochastic Processes Definition Stochastic process X = {X(t), t2 T} is a collection of random variables (rvs); one rv
More informationMarkov Chains (Part 4)
Markov Chains (Part 4) Steady State Probabilities and First Passage Times Markov Chains - 1 Steady-State Probabilities Remember, for the inventory example we had (8) P &.286 =.286.286 %.286 For an irreducible
More informationNote Set 5: Hidden Markov Models
Note Set 5: Hidden Markov Models Probabilistic Learning: Theory and Algorithms, CS 274A, Winter 2016 1 Hidden Markov Models (HMMs) 1.1 Introduction Consider observed data vectors x t that are d-dimensional
More informationUncertainty Runs Rampant in the Universe C. Ebeling circa Markov Chains. A Stochastic Process. Into each life a little uncertainty must fall.
Uncertainty Runs Rampant in the Universe C. Ebeling circa 2000 Markov Chains A Stochastic Process Into each life a little uncertainty must fall. Our Hero - Andrei Andreyevich Markov Born: 14 June 1856
More informationCDA5530: Performance Models of Computers and Networks. Chapter 3: Review of Practical
CDA5530: Performance Models of Computers and Networks Chapter 3: Review of Practical Stochastic Processes Definition Stochastic ti process X = {X(t), t T} is a collection of random variables (rvs); one
More informationSOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX
Iranian Journal of Fuzzy Systems Vol 5, No 3, 2008 pp 15-29 15 SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX M S HASHEMI, M K MIRNIA AND S SHAHMORAD
More informationParametric Models Part III: Hidden Markov Models
Parametric Models Part III: Hidden Markov Models Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Spring 2014 CS 551, Spring 2014 c 2014, Selim Aksoy (Bilkent
More informationBulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 67-82
Bulletin of the Transilvania University of Braşov Vol 10(59), No. 1-2017 Series III: Mathematics, Informatics, Physics, 67-82 IDEALS OF A COMMUTATIVE ROUGH SEMIRING V. M. CHANDRASEKARAN 3, A. MANIMARAN
More informationDefinition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
More informationECE 541 Project Report: Modeling the Game of RISK Using Markov Chains
Contents ECE 541 Project Report: Modeling the Game of RISK Using Markov Chains Stochastic Signals and Systems Rutgers University, Fall 2014 Sijie Xiong, RUID: 151004243 Email: sx37@rutgers.edu 1 The Game
More informationMATH 446/546 Test 2 Fall 2014
MATH 446/546 Test 2 Fall 204 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 546 level. Please read and follow all of these
More informationAPPLYING SIGNED DISTANCE METHOD FOR FUZZY INVENTORY WITHOUT BACKORDER. Huey-Ming Lee 1 and Lily Lin 2 1 Department of Information Management
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 1349-4198 Volume 7, Number 6, June 2011 pp. 3523 3531 APPLYING SIGNED DISTANCE METHOD FOR FUZZY INVENTORY
More informationAdrian I. Ban and Delia A. Tuşe
18 th Int. Conf. on IFSs, Sofia, 10 11 May 2014 Notes on Intuitionistic Fuzzy Sets ISSN 1310 4926 Vol. 20, 2014, No. 2, 43 51 Trapezoidal/triangular intuitionistic fuzzy numbers versus interval-valued
More informationAustralian Journal of Basic and Applied Sciences, 5(9): , 2011 ISSN Fuzzy M -Matrix. S.S. Hashemi
ustralian Journal of Basic and pplied Sciences, 5(9): 2096-204, 20 ISSN 99-878 Fuzzy M -Matrix S.S. Hashemi Young researchers Club, Bonab Branch, Islamic zad University, Bonab, Iran. bstract: The theory
More informationCS412: Lecture #17. Mridul Aanjaneya. March 19, 2015
CS: Lecture #7 Mridul Aanjaneya March 9, 5 Solving linear systems of equations Consider a lower triangular matrix L: l l l L = l 3 l 3 l 33 l n l nn A procedure similar to that for upper triangular systems
More informationBirth and Death Processes. Birth and Death Processes. Linear Growth with Immigration. Limiting Behaviour for Birth and Death Processes
DTU Informatics 247 Stochastic Processes 6, October 27 Today: Limiting behaviour of birth and death processes Birth and death processes with absorbing states Finite state continuous time Markov chains
More informationprinting Three areas: solid calculus, particularly calculus of several
Math 5610 printing 5600 5610 Notes of 8/21/18 Quick Review of some Prerequisites Three areas: solid calculus, particularly calculus of several variables. linear algebra Programming (Coding) The term project
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationarxiv: v2 [physics.data-an] 19 Apr 2013
Noname manuscript No. (will be inserted by the editor) Performability analysis of the second order semi-markov chains: an application to wind energy production Guglielmo D Amico Filippo Petroni Flavio
More informationAn Application of Interval Valued Fuzzy Matrices in Medical Diagnosis
Int. Journal of Math. Analysis, Vol. 5, 2011, no. 36, 1791-1802 An Application of Interval Valued Fuzzy Matrices in Medical Diagnosis A. R. Meenakshi and M. Kaliraja Department of Mathematics, Karpagam
More informationThe Google Markov Chain: convergence speed and eigenvalues
U.U.D.M. Project Report 2012:14 The Google Markov Chain: convergence speed and eigenvalues Fredrik Backåker Examensarbete i matematik, 15 hp Handledare och examinator: Jakob Björnberg Juni 2012 Department
More informationAdaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018
Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Nonlinearity of adaptation! Parameter-adaptive filtering! Test for whiteness of the residual!
More informationAdomian decomposition method for fuzzy differential equations with linear differential operator
ISSN 1746-7659 England UK Journal of Information and Computing Science Vol 11 No 4 2016 pp243-250 Adomian decomposition method for fuzzy differential equations with linear differential operator Suvankar
More information