APPLICATIONS OF MATLAB IN SCIENCE AND ENGINEERING - PART 9 potx

53 229 0
APPLICATIONS OF MATLAB IN SCIENCE AND ENGINEERING - PART 9 potx

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

Thông tin tài liệu

ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 5 3.1.1 Generation of the IERG The first of four steps of ISPN analysis is the IERG generation (interval extended reachability graph). From the IERG the set of markings M = T∪ Vis divided into set of tangible markings T and vanishing V. Through the elimination of vanishing markings discussed below, using methods of interval analysis, we obtain the infinitesimal generator matrix [Q] of ICTMC underlying an ISPN model. From a given ISPN, an interval extended reachability graph (IERG) is generated containing markings as nodes and interval stochastic information attached to arcs so as to relate markings to each other. The ISPN reachability graph is a directed graph RG (ISPN)=(V, E), where V = RS(ISPN) and E = { m, t, m   | m, m  ∈ RS(ISPN) and m t → m   are the set of nodes and edges, respectively. If an ISPN model is bounded, the RG (ISPN) is finite and it can be constructed, for example, based on Algorithm 5.1: Computation of the Reachability Graph p. 61 from (Girault & Valk, 2003). The RG (ISPN) is constructed, in this work, using the Algorithm 1 below. The activity defined in Step 2.1 ensures that no marking is visited more than once. Each visited marking is labeled (Step 2.1), and Step 2.2.3 ensures that only unique added markings to V are those that were not previously added. When the marking is visited, only those edges that represents the firing of an enabled transition are added to the set E (Step 2.2.4). =================================================================== Algorithm 1 ( ** IERG generation ** ) Input - A ISPN model. Output - A directed graph RG (ISPN)=(V, E) of a limited network system. 1. Initialize RG (ISPN)= ( { m 0 } , ∅ ) ; m 0 is unlabelled. 2. while there are an unlabeled node m in V do 2.1 Select an unlabeled node m ∈ V label it 2.2 for each enabled transition t in mdo 2.2.1 Calculate m  such that m t → m  ; 2.2.2 if there are m  ∈ V such that m  σ → m  and m” ≤ m’ then the algorithm fails and ends; (no limitation condition was detected). 2.2.3 if there is no m  ∈ V such that m  = m  then V := V ∪ { m  } ;(m  é um nó não etiquetado). 2.2.4 E : = E ∪ { m; t; m  } 3. The algorithm is successful and RG(ISPN) is the interval extended reachability graph. =================================================================== 3.1.2 Elimination of vanishing markings The second of four steps of ISPN analysis is the elimination of vanishing markings, which is the step for generating the ICTMC from a given ISPN. Once the IERG has been generated, it is transformed into an ICTMC by the use of matrix algorithms Bolch et al. (2006). The markings set M = V∪T in the reachability set of an ISPN is partitioned into two sets, the vanishing markings V and the tangible markings T . Let: [P] V =[P] VV | [P] VT (3) 413 ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 6 Will-be-set-by-IN-TECH denote an interval matrix, where • [P] VV - denotes the interval transition probabilities between vanishing markings, • [P] VT - denotes the interval transition probabilities from vanishing markings to the tangible markings. Furthermore, let [U] T =[U] TV | [U] TT (4) denote an interval matrix, where • [U] TV - represents interval transition rates from tangible to vanishing markings; • [U] TT - represents interval transition rates between tangible markings. Now, we obtain the interval rate matrix [U]. This matrix has dimensions | T | × | T | , where T denotes the set of tangible markings. [U]=[U] TT +[U] TV (1 − [P] VV ) −1 [P] VT (5) The interval matrix of the infinitesimal generator is [Q]=[q ] ij , where its entries are given by: [q] ij = ⎧ ⎪ ⎨ ⎪ ⎩ [u] ij if i = j − ∑ k ∈T k = i [u] ik if i = j (6) where T denotes the set of tangible markings. 3.1.3 Steady-state probability vector evaluation Now we describe the third of four steps of ISPN analysis. The steady-state solution of the ICTMC model underlying the ISPN is obtained by solving the interval linear system of equations with as many equations as the number of tangible markings.  [ π ] · [ Q ] = 0 ∑ M∈T [ π ]( M ) = 1 (7) [ π ] is the interval vector for the equilibrium pmf (probability mass function) over the reachable tangible markings, and we write [ π ] ( M) for the interval steady-state probability of a given tangible marking M. Once the interval generator matrix [Q] of the ICTMC associated with a ISPN model has been derived, the steady state probability is calculated so that other respective metrics might be subsequently computed. ISPN models deal with system uncertainties by considering intervals for representing time as well as weights assigned to transition models. The proposed model and the respective methods, adapted to take interval arithmetic into account, allow the influence of simultaneous parameters and variabilities on the computation of metrics to be considered, thereby providing rigorously bounded metric ranges. It is also important to stress that even when only taking into account thin intervals, one may make use of the proposed model, since rounding and truncation errors are naturally dealt with in interval arithmetic, so that the metrics results obtained are certain to belong to the intervals computed. 414 Applications of MATLAB in Science and Engineering ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 7 3.1.4 Interval performance indices The computation of performance indices (metrics) of interest is the fourth and final step in the analysis ISPN. In the case of ISPN steady state analysis, where interval p.m.f. has already been obtained, indices are calculated by interval function evaluation. Interval performance indices are interval functions extended on classical indices (Marsan, Bobbio, Conte & Cumani, 1984). 4. Examples of ISPN models The purpose of this section is to present clearly all steps of ISPN analysis. Two examples are used. One is very simple and can be followed up and have calculations performed without using a computer. The second case, however, you must use a software with an interval arithmetic library as a tool to carry out by all his calculations. Example 1 has only two tangible markings and two vanishing markings. Example 2 has sixteen tangible markings and twelve vanishing markings. The performance evaluations are carried out in MATLAB with the INTLAB toolbox (MATLAB toolbox INTLAB framework). The ISPN model analysis considering only degenerated intervals (points) leads to the classic model GSPN, with verified computations (self-validating). 4.1 Example 1: ISPN analysis of a single machine The model depicted in Figure 1 represents a failure prone machine and finite capacity buffer (Desrochers & Al-Jaar, 1994). Table 1 presents (degenerated) interval rates of timed transition firing per unit time, where [ν] represents the production rate interval, [λ] represents the failure rate interval, and [μ] represents the repair rate interval. Here we have a model equivalent to the GSPN model, because there are only degenerate interval parameters. Fig. 1. The Single Machine module. Transition Value ([t] −1 ) Symbol [t 2 ] [10, 10] [ν] [t 4 ] [3, 3] [μ] [t 5 ] [5, 5] [λ] Table 1. Transition Firing Rates (degenerated intervals) for the Single Machine One-Buffer Transfer Line. As a result of the first step of ISPN analysis we obtain the reachability set (Table 2), and the reachability graph (Figure 2). 415 ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 8 Will-be-set-by-IN-TECH State Marking ( m 1 , m 2 , m 3 , m 4 ) 1 M 0 =(1, 0, 0, 0 ) 2 M 1 =(0, 1, 0, 0 ) 3 M 2 =(0, 0, 1, 0 ) 4 M 3 =(0, 0, 0, 1 ) Table 2. Reachability set and distribution markings from ISPN of Figure 1. Fig. 2. Reachability graph and interval embedded Markov chain Finally, we obtain the matrices [P] VV , [P] VT , [U] TV and [U] TT : [P] VV =  [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0]  [P] VT =  [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0]  [U] TT =  [ 0, 0] [ 5, 5] [ 3, 3] [ 0, 0]  [U] TV =  [ 0, 0] [ 10, 10] [ 0, 0] [ 0, 0]  . Afterwards, carry out the elimination of vanishing markings (Equation 5) to obtain the matrix of rate intervals [U]. The matrix of rate intervals represents an IREMC (Interval Reduced Embedded Markov Chain on Figure 3): [U]=  [ 10, 10] [ 5, 5] [ 3, 3] [ 0, 0]  . M 3 M 1 [t 2 ] [t 5 ] [t 4 ] 0 1 0 0 0 0 0 1 Fig. 3. Interval Reduced Embedded Markov Chain Finally, using Equation 6, we find the infinitesimal generator interval matrix: [Q]=  [ -5, -5] [ 5, 5] [ 3, 3] [ -3, -3]  . The third step of ISPN analysis solves the system of interval linear equations described by Equation (7). The interval linear equations solution is carried out by the verifylss function of the MATLAB toolbox INTLAB. Substituting the last equation of system ( [ π] 1 , [π] 2 ) · [ Q]=0 by the normalization condition [π] 1 +[π] 2 = 1, the linear system ( [ π] 1 , [π] 2 ) · [ A]=[b] is obtained. The solution of this system directly provides the steady state probabilities of tangible states. Considering [A]=  −35 11  and [b]=  0 1  416 Applications of MATLAB in Science and Engineering ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 9 the M-file MATLAB toolbox INTLAB case1v.m, used for calculating verified probabilities and machine production rate, is given bellow: 1. % INPUT: A coeffifiente matrix 2. % b right hand side vector 3. % OUTPUT: x interval probabilities vector solution 4. % P machine production rate 5. format long 6. intvalinit(’displayinfsup’) 7. A=intval([-3,5;1,1]) 8. b=[0;1] 9. x=verifylss(A,b) 10.P=10 * x(1) Executing case1v.m yields: >> case1v ===> Default display of intervals by infimum/supremum (e.g. [ 3.14 , 3.15 ]) intval A = [ -3.00000000000000, -3.00000000000000] [ 5.00000000000000, 5.00000000000000] [ 1.00000000000000, 1.00000000000000] [ 1.00000000000000, 1.00000000000000] b= 0 1 intval x = [ 0.62499999999998, 0.62500000000001] [ 0.37499999999999, 0.37500000000001] intval P = [ 6.24999999999998, 6.25000000000001] >> The verified interval bounds of each state probabilities on tangible states are: [π] ( 1 ) =[0.62499999999998, 6.25000000000001] and [π] ( 2 ) =[0.37499999999999, 0.37500000000001]. Finally we can make the fourth and final step of analysis ISPN, computation of metrics. The machine production rate is [P]=[6.24999999999998, 6.25000000000001] (calculated with the formula [P]=[π] ( 1 ) · [t 2 ]). This results exhibit the enclosure of exact value obtained by GSPN analysis. The ISPN analysis results give us verified results, ensuring that the exact value is certain to belong to the intervals computed. One can, for example, to compare this result with interval P = 6.25 exact value in this simple case. Introducing parameters with input uncertainties Now we calculate a solution in which the parameters are not known exactly, but it is known that they are within certain intervals. Lets consider that rates are [μ]=3 ± 0.01 =[2.99, 3.01] and [λ]=5 ± 0.01 =[4.99, 5.01] intervals. 417 ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 10 Will-be-set-by-IN-TECH As a result from the first step of analysis (by-product of the reachability set), we obtain the matrices [P] VV , [P] VT , [U] TV e [U] TT : [P] VV =  [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0]  [P] VT =  [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0]  [U] TT =  [ 0, 0] [ 4.99, 5.01] [ 2.99, 3.01] [ 0, 0]  [U] TV =  [ 0, 0] [ 10, 10] [ 0, 0] [ 0, 0]  . Afterwards, carry out the elimination of vanishing markings (Equation 5), to obtain the matrix of rate intervals [U]: [U]=  [ 10, 10] [ 4.99, 5.01] [ 2.99, 3.01] [ 0, 0]  . Finally, using Equation 6, we find the infinitesimal generator interval matrix: [Q]=  [ -5.01, -4.99] [ 4.99, 5.01] [ 2.99, 3.01] [ -3.01, -2.99]  . Considering [A]=  −35 11  and [b]=  0 1  the M-file MATLAB toolbox INTLAB case1i.m, used for calculating verified probabilities and machine production rate, is given bellow: 1. % INPUT: A coeffifiente matrix 2. % b right hand side vector 3. % OUTPUT: x interval probabilities vector solution 4. % P machine production rate 5. format long 6. intvalinit(’displayinfsup’) 7. A=infsup([-3.01,4.99;1,1],[-2.99,5.01;1,1]) 8. b=[0;1] 9. x=verifylss(A,b) 10.P=10 * x(1) Executing case1i.m yields: >> case1i ===> Default display of intervals by infimum/supremum (e.g. [ 3.14 , 3.15 ]) intval A = [ -3.01000000000000, -2.99000000000000] [ 4.99000000000000, 5.01000000000000] [ 1.00000000000000, 1.00000000000000] [ 1.00000000000000, 1.00000000000000] b= 0 1 intval x = [ 0.62374656249999, 0.62625343750001] [ 0.37374656249998, 0.37625343750001] intval P = [ 6.23746562499999, 6.26253437500001] >> The interval bounds of each state probabilities on tangible states are: [π] ( 1 ) =[0.62374656249999, 0.62625343750001] 418 Applications of MATLAB in Science and Engineering ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 11 and [π] ( 2 ) =[0.37374656249998, 0.37625343750001]. Finally we can make the computation of machine production rate: [P]=[6.23746562499999, 6.26253437500001] (calculated with the formula [P]=[π] ( 1 ) · [t 2 ]). This result represents the variabilities when the rates are in [μ]=[2.99, 3.01] and [λ]=[4.99, 5.01] intervals. 4.2 Example 2: ISPN analysis of Two-Machine One-Buffer Transfer Line Model Consider the Two-Machine One-Buffer Transfer Line Model in Figure 4 (Desrochers & Al-Jaar, 1994). Table 3 presents (degenerated) interval rates of timed transition firing per unit time, where [ν i ] represents the production rate intervals, [λ i ] represents the failure rate intervals, and [μ i ] represents the repair rate intervals. Here we have a model equivalent to the GSPN model, because there are only degenerate interval parameters. Fig. 4. Two-Machine One-Buffer Transfer Line Model (k = 3) Transition Value ([t] −1 ) Symbol [t 2 ] [1, 1] [ ν 1 ] [t 3 ] [3, 3] [λ 1 ] [t 4 ] [5, 5] [μ 1 ] [t 6 ] [2, 2] [ν 2 ] [t 7 ] [4, 4] [λ 2 ] [t 8 ] [6, 6] [μ 2 ] Table 3. Interval transition firing rates for the Two-Machine One-Buffer Transfer Line model. As a result of the first step of ISPN analysis we obtain the reachability set (Table 4) and the reachability graph (Table 5). Markings enabling the transitions t 1 and t 5 are vanishing, because enabled transitions are immediate (state changes that take negligible amounts of time to occur). Can be identified twelve vanishing markings M 0 , M 2 , M 4 , M 5 , M 7 , M 12 , M 13 , M 17 , M 19 , M 22 , M 24 , M 26 (firing of immediate transitions t 1 and t 5 ) and other markings are tangibles. 419 ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 12 Will-be-set-by-IN-TECH State Marking 1 State Marking 1 1 M 0 =[1, 0, 0, 0, 1, 0, 0, 3] 15 M 14 =[0, 0, 1, 1, 0, 1, 0, 1] 2 M 1 =[0, 1, 0, 0, 1, 0, 0, 2] 16 M 15 =[0, 1, 0, 1, 0, 0, 1, 1] 3 M 2 =[1, 0, 0, 1, 1, 0, 0, 2] 17 M 16 =[0, 1, 0, 2, 0, 1, 0, 0] 4 M 3 =[0, 0, 1, 0, 1, 0, 0, 2] 18 M 17 =[0, 0, 1, 1, 1, 0, 0, 1] 5 M 4 =[0, 1, 0, 1, 1, 0, 0, 1] 19 M 18 =[0, 0, 1, 1, 0, 0, 1, 1] 6 M 5 =[1, 0, 0, 0, 0, 1, 0, 3] 20 M 19 =[1, 0, 0, 2, 0, 0, 1, 1] 7 M 6 =[0, 1, 0, 0, 0, 1, 0, 2] 21 M 20 =[1, 0, 0, 3, 0, 1, 0, 0] 8 M 7 =[1, 0, 0, 1, 0, 1, 0, 2] 22 M 21 =[0, 0, 1, 2, 0, 1, 0, 0] 9 M 8 =[0, 0, 1, 0, 0, 1, 0, 2] 23 M 22 =[0, 1, 0, 2, 1, 0, 0, 0] 10 M 9 =[0, 1, 0, 0, 0, 0, 1, 2] 24 M 23 =[0, 1, 0, 2, 0, 0, 1, 0] 11 M 10 =[0, 1, 0, 1, 0, 1, 0, 1] 25 M 24 =[1, 0, 0, 3, 1, 0, 0, 0] 12 M 11 =[0, 0, 1, 0, 0, 0, 1, 2] 26 M 25 =[1, 0, 0, 3, 0, 0, 1, 0] 13 M 12 =[1, 0, 0, 1, 0, 0, 1, 2] 27 M 26 =[0, 0, 1, 2, 1, 0, 0, 0] 14 M 13 =[1, 0, 0, 2, 0, 1, 0, 1] 28 M 27 =[0, 0, 1, 2, 0, 0, 1, 0] 1- Marking = [ m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , m 7 , m 8 ] Table 4. Reachability set and distribution markings from ISPN of Figure 4. Marking | Firing of transition  New marking M 0 | t 1  M 1 M 1 | T 2  M 2 M 1 | T 3  M 3 M 2 | t 1  M 4 M 2 | t 5  M 5 M 3 | T 4  M 1 M 4 | t 5  M 6 M 5 | t 1  M 6 M 6 | T 2  M 7 M 6 | T 3  M 8 M 6 | T 6  M 1 M 6 | T 7  M 9 M 7 | t 1  M 10 M 8 | T 4  M 6 M 8 | T 6  M 3 M 8 | T 7  M 11 M 9 | T 2  M 12 M 9 | T 3  M 11 M 9 | T 8  M 6 M 10 | T 2  M 13 M 10 | T 3  M 14 M 10 | T 6  M 4 M 10 | T 7  M 15 M 11 | T 4  M 9 M 11 | T 8  M 8 M 12 | t 1  M 15 M 13 | t 1  M 16 M 14 | T 4  M 10 M 14 | T 6  M 17 M 14 | T 7  M 18 M 15 | T 2  M 19 M 15 | T 3  M 18 M 15 | T 8  M 10 M 16 | T 2  M 20 M 16 | T 3  M 21 M 16 | T 6  M 22 M 16 | T 7  M 23 M 17 | t 5  M 8 M 18 | T 4  M 15 M 18 | T 8  M 14 M 19 | t 1  M 23 M 20 | T 6  M 24 M 20 | T 7  M 25 M 21 | T 4  M 16 M 21 | T 6  M 26 M 21 | T 7  M 27 M 22 | t 5  M 10 M 23 | T 2  M 25 M 23 | T 3  M 27 M 23 | T 8  M 16 M 24 | t 5  M 13 M 25 | T 8  M 20 M 26 | t 5  M 14 M 27 | T 4  M 23 M 27 | T 8  M 21 Table 5. Literal description of reachability graph from ISPN of Figure 4. Finally, we obtain the matrices [P] VV , [P] VT , [U] TV and [U] TT : [P] VV = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [0,5, 0,5] [0,5, 0,5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 420 Applications of MATLAB in Science and Engineering ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 13 [P] VT = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [U] TT = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [ 0, 0] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 1, 1] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 421 ISPN: Modeling Stochastic with Input Uncertainties Using an Interval-Based Approach 14 Will-be-set-by-IN-TECH [U] TV = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Afterwards, carry out the elimination of vanishing markings (Equation 5), to obtain the matrix of rate intervals [U] representing the IREMC: [U]= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [ 0, 0] [ 3, 3] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 2, 2] [ 0, 0] [ 5, 5] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 1, 1] [ 0, 0] [ 0, 0] [ 3, 3] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 4, 4] [ 0, 0] [ 0, 0] [ 1, 1] [ 3, 3] [ 0, 0] [ 0, 0] [ 0, 0] [ 0, 0] [ 6, 6] [ 5, 5] [ 0, 0] [ 0, 0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 422 Applications of MATLAB in Science and Engineering [...]... AT(3,1)= [infsup(0. 099 E1,0.101E1),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)]; • AT(3,1)= [infsup(0. 199 E0, 0.201E0),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)]; 5 ISPN MATLAB toolbox INTLAB prototype tool ISPN M-file MATLAB toolbox INTLAB is a prototype for the modeling and evaluation of ISPNs in which exponential... AT % input arc multiplicity of timed transitions, ( - ) minus means input ISPN: Modeling with Input Uncertainties Using an Interval-Based Approach Using an Interval-Based Approach ISPN: Modeling Stochastic Stochastic with Input Uncertainties 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100... 0.0 296 8055852738 ⎜ ⎜ [ 0.0 197 61723201 79 ⎜ ⎜ [ 0.0252725 696 98 29 ⎜ ⎜ [ 0.01 396 9287 495 35 ⎜ ⎜ [ 0.0208645808 692 0 ⎜ ⎝ [ 0.0203258 099 4373 [ 0.01077008114445 ⎛ 0.301623410 591 73] 0.201 292 41213851] 0.100015 790 83720] 0.050 795 91445867] 0.05701 697 985274] 0.05122652318523] 0.03402132703572] 0.0272 898 621 597 5] 0.03607316886027] 0.0 296 80558527 39] 0.0 197 6172320180] 0.0252725 696 9830] 0.01 396 9287 495 36] 0.0208645808 692 2]... 0.36822223532140] = 0. 298 25050636187± 0.0 699 7172 895 954 [0.023460 199 8 293 9, 0.1015200 095 8611] = 0.062 490 10470775± 0.0 390 299 0487836 Table 7 The average machine utilization results obtained with ISPN MATLAB toolbox INTLAB Prototype Tool to Two-Machine One-Buffer Transfer Line Model for three μ1 rate intervals Introducing parameters with input uncertainties: In sequel, the variations in the rates of exponential... 0.100015 790 83720] [ 0.050 795 91445866, 0.050 795 91445867] [ 0.05701 697 985273, 0.05701 697 985274] [ 0.05122652318522, 0.05122652318523] [ 0.03402132703571, 0.03402132703572] [ 0.0272 898 621 597 4, 0.0272 898 621 597 5] [ 0.03607316886026, 0.03607316886027] [ 0.0 296 8055852738, 0.0 296 80558527 39] [ 0.0 197 61723201 79, 0.0 197 6172320180] [ 0.0252725 696 98 29, 0.0252725 696 9830] [ 0.01 396 9287 495 35, 0.01 396 9287 495 36] [ 0.0208645808 692 0,... ) · [A] = [b] is obtained The i =1 solution of this system directly provides the steady state probabilities of tangible states: 424 16 Applications of MATLAB in Science and Engineering Will-be-set-by -IN- TECH [ 0.301623410 591 72 ⎜ [ 0.201 292 41213850 ⎜ ⎜ [ 0.100015 790 837 19 ⎜ ⎜ [ 0.050 795 91445866 ⎜ ⎜ [ 0.05701 697 985273 ⎜ ⎜ [ 0.05122652318522 ⎜ ⎜ [ 0.03402132703571 ⎜ ⎜ [ 0.0272 898 621 597 4 [π ]t = ⎜ ⎜ [ 0.03607316886026... displayed detailing of ISPN analysis as in previous examples Table 7 shows the average machine utilization, UM1 and UM2 for three [μ1 ] rate intervals All exponential rate variabilities have ±1 as errors in the 3rd significant digits: ISPN.m Line 59 modification for each experiment: • AT(3,1)= [infsup(0. 099 E2,0.101E2),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)];... Two-Machine One-Buffer Transfer Line Model for three MR (Machining Rate) = μ1 (degenerated interval) Results obtained with ISPN MATLAB toolbox INTLAB Prototype Tool ISPN: Modeling with Input Uncertainties Using an Interval-Based Approach Using an Interval-Based Approach ISPN: Modeling Stochastic Stochastic with Input Uncertainties Interval rate [ μ1 ] [0. 099 E2, 0.101E2] = 0, 100E2 ± 0, 001E2 [0. 099 E1,... is still being used but ISPN.n will allow you to write your own features and to tailor ISPNs to your own needs 426 18 Applications of MATLAB in Science and Engineering Will-be-set-by -IN- TECH The ISPN.m used for calculating verified probabilities and the machine utilization rate from ISPN model of Figure 4, is given bellow: Uncomment specified lines to display: • Line 191 : Reachability set and distribution... 0.101E1] = 0, 100E1 ± 0, 001E1 [0, 199 E0, 0, 201E0] = 0, 200E0 ± 0, 001E0 425 17 Machine utilization [U M1 ] [0.11 399 6 799 21745, 0.12 493 721523401] = 0.1 194 6700722573± 0.00547020800828 [0. 496 116314 597 60, 0. 696 8857108 498 6] = 0. 596 50101272373± 0.100384 698 12613 [0.544 490 376588 09, 0.70531171756 699 ] = 0.62 490 104707754± 0.0804106704 894 5 [U M2 ] [0.5 796 067 098 2656, 0.61506336243075] = 0. 597 33503612865± 0.01772832630210 . 1.00000000000000] b= 0 1 intval x = [ 0.62 499 999 999 998 , 0.62500000000001] [ 0.37 499 999 999 999 , 0.37500000000001] intval P = [ 6.2 499 999 999 999 8, 6.25000000000001] >> The verified interval bounds of each state. are: [π] ( 1 ) =[0.62 499 999 999 998 , 6.25000000000001] and [π] ( 2 ) =[0.37 499 999 999 999 , 0.37500000000001]. Finally we can make the fourth and final step of analysis ISPN, computation of metrics. The machine production. [infsup(0. 099 E2,0.101E2),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)]; • AT(3,1)= [infsup(0. 099 E1,0.101E1),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)]; • AT(3,1)= [infsup(0. 199 E0, 0.201E0),infsup(2 .99 ,3.01),infsup(4 .99 ,5.01),infsup(1 .99 ,2.01), infsup(3 .99 ,4.01),infsup(5 .99 ,6.01)]; 5.

Ngày đăng: 09/08/2014, 16:21

Từ khóa liên quan

Tài liệu cùng người dùng

  • Đang cập nhật ...

Tài liệu liên quan