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[(are)-277(briey)-265(introduced.)-252.9(Afterw)12(ards,)-265(the)-277(model)-265(class)-289.1(and)-277(the)]TJ
T*
[(underlying)-265(structured)-289.1(representation)-277(of)-289.1(the)-301.1(state)-313.2(space)-301.1(and)]TJ
0 -1.1684 TD
[(generator)-156.6(matrix)-168.6(are)-192.7(introduced.)-144.5(Section)-180.7(4)-180.7(presents)-180.7(the)-192.7(com-)]TJ
0 -1.1804 TD
[(positional)-240.9(representation)-240.9(of)-252.9(distrib)24.1(ution)-240.9(v)12(ectors)-252.9(and)-252.9(the)-252.9(ba-)]TJ
0 -1.1684 TD
[(sic)-228.9(operations)-204.8(in)36.1(v)24.1(olv)12(ed)-192.7(in)-228.9(the)-228.9(realization)-204.8(of)-216.8(a)-228.9(transient)-216.8(anal-)]TJ
0 -1.1804 TD
[(ysis)-265(using)-265(this)-265(v)12(ector)-252.9(representation.)-228.9(The)-265(follo)24.1(wing)-240.9(section)]TJ
T*
[(e)12(xtends)-385.4(the)-409.5(approach)-373.4(to)-409.5(an)-409.5(adapti)24.1(v)12(e)-385.4(v)12(ersion.)-385.4(Finally)60.2(,)-397.5(the)]TJ
0 -1.1684 TD
[(quality)-240.9(of)-265(the)-252.9(approximation)-228.9(method)-240.9(is)-277(sho)24.1(wn)-252.9(by)-252.9(means)-252.9(of)]TJ
0 -1.1804 TD
[(some)-325.2(e)12(xamples.)-325.2(In)-337.3(Section)-337.3(7)-337.3(some)-337.3(general)-325.2(remarks)-325.2(about)]TJ
T*
[(the)-240.9(quality)-228.9(of)-240.9(the)-240.9(approximation)-204.8(are)-240.9(gi)24.1(v)12(en.)]TJ
/F1 1 Tf
11.9552 0 0 11.9552 58.59 391.14 Tm
[(2.)-692.6(T)70.3(ransient)-261(Analysis)]TJ
/F2 1 Tf
9.9626 0 0 9.9626 70.47 369.06 Tm
[(W)84.3(e)-469.8(consider)-433.6(a)-457.7(nite)-457.7(CTMC)-469.8(with)-457.7(state)-457.7(space)]TJ
/F5 1 Tf
19.5073 0 TD
-0.0043 Tc
(RS)Tj
/F6 1 Tf
2.2043 0 TD
0 Tc
(=)Tj
/F5 1 Tf
-22.904 -1.1684 TD
({)Tj
/F6 1 Tf
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(0)Tj
/F7 1 Tf
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0.1679 Tc
[(,...)24.1(,n)]TJ
/F5 1 Tf
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0 Tc
()Tj
/F6 1 Tf
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(1)Tj
/F5 1 Tf
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(})Tj
/F2 1 Tf
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[(,)-252.9(generator)-240.9(matrix)]TJ
/F8 1 Tf
7.3838 0 TD
(Q)Tj
/F2 1 Tf
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[(,)-265(initial)-277(distrib)24.1(ution)]TJ
/F8 1 Tf
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/F9 1 Tf
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(0)Tj
/F2 1 Tf
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(,)Tj
-23.4516 -1.1804 TD
[(and)-289.1(re)24.1(w)12(ard)-289.1(v)12(ector)]TJ
/F8 1 Tf
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(r)Tj
/F2 1 Tf
0.4698 0 TD
[(.)-301.1(Let)]TJ
/F8 1 Tf
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(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 167.07 344.22 Tm
(t)Tj
/F2 1 Tf
9.9626 0 0 9.9626 173.67 345.66 Tm
[(be)-301.1(the)-301.1(transient)-301.1(distrib)24.1(ution)-277(at)]TJ
-11.5512 -1.1804 TD
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/F2 1 Tf
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[(,)-240.9(which)-240.9(is)-252.9(computed)-216.8(as)]TJ
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(t)Tj
/F6 1 Tf
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(=)Tj
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/F9 1 Tf
6.9738 0 0 6.9738 138.27 303.42 Tm
(0)Tj
/F7 1 Tf
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(e)Tj
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(Q)Tj
/F10 1 Tf
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(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 160.47 304.86 Tm
(=)Tj
/F12 1 Tf
6.9738 0 0 6.9738 174.27 317.34 Tm
()Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
6.9738 0 0 6.9738 170.91 292.86 Tm
(k)Tj
/F9 1 Tf
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/F8 1 Tf
9.9626 0 0 9.9626 187.11 304.86 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 193.47 303.42 Tm
(0)Tj
/F8 1 Tf
9.9626 0 0 9.9626 197.91 304.86 Tm
(P)Tj
/F10 1 Tf
6.9738 0 0 6.9738 205.71 308.94 Tm
(k)Tj
/F7 1 Tf
9.9626 0 0 9.9626 210.63 304.86 Tm
()Tj
/F6 1 Tf
0.6143 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
-0.0006 Tc
[(t,)-168.5(k)]TJ
/F6 1 Tf
1.9995 0 TD
0 Tc
(\))Tj
/F2 1 Tf
4.276 0 TD
(\(1\))Tj
-22.5362 -3.9508 TD
(where)Tj
/F7 1 Tf
2.6237 0 TD
()Tj
/F6 1 Tf
0.9154 0 TD
0.2822 Tc
[(=m)284.4(a)276.3(x)]TJ
/F10 1 Tf
6.9738 0 0 6.9738 123.15 264.06 Tm
0 Tc
(x)Tj
/F6 1 Tf
9.9626 0 0 9.9626 128.07 265.5 Tm
(\()Tj
/F5 1 Tf
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(|)Tj
/F8 1 Tf
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(Q)Tj
/F6 1 Tf
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(\()Tj
/F7 1 Tf
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-0.0054 Tc
[(x,)-173.3(x)]TJ
/F6 1 Tf
1.5779 0 TD
0 Tc
(\))Tj
/F5 1 Tf
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(|)Tj
/F6 1 Tf
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(\))Tj
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(P)Tj
/F6 1 Tf
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(Q)Tj
/F7 1 Tf
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(I)Tj
/F2 1 Tf
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()Tj
/F6 1 Tf
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(\()Tj
/F7 1 Tf
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-0.0006 Tc
[(t,)-168.5(k)]TJ
/F6 1 Tf
1.9995 0 TD
0.2856 Tc
(\)=)Tj
/F7 1 Tf
-22.2429 -1.1804 TD
0 Tc
(e)Tj
/F12 1 Tf
6.9738 0 0 6.9738 63.27 257.34 Tm
()Tj
/F10 1 Tf
0.8947 0 TD
-0.0028 Tc
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 78.03 253.74 Tm
0 Tc
(\()Tj
/F7 1 Tf
0.3854 0 TD
-0.0013 Tc
(t)Tj
/F6 1 Tf
0.9997 0 TD
0 Tc
(\))Tj
/F10 1 Tf
6.9738 0 0 6.9738 95.67 257.34 Tm
(k)Tj
/F7 1 Tf
9.9626 0 0 9.9626 100.71 253.74 Tm
0.0059 Tc
(/k)Tj
/F6 1 Tf
1.06 0 TD
0 Tc
(!)Tj
/F2 1 Tf
0.53 0 TD
[(is)-252.9(the)-252.9(probability)-216.8(that)-252.9(a)-252.9(Poisson)-252.9(process)-240.9(with)]TJ
-5.8178 -1.1804 TD
(rate)Tj
/F7 1 Tf
1.8001 0 TD
-0.0013 Tc
(t)Tj
/F2 1 Tf
1.325 0 TD
0 Tc
[(will)-325.2(mak)12(e)]TJ
/F7 1 Tf
4.3363 0 TD
(k)Tj
/F2 1 Tf
0.8672 0 TD
[(jumps)-313.2(in)-313.2(the)-313.2(interv)24.1(al)]TJ
/F6 1 Tf
8.7165 0 TD
-0.0034 Tc
(\(0)Tj
/F7 1 Tf
0.8913 0 TD
0 Tc
(,)Tj
/F6 1 Tf
0.4457 0 TD
0.0059 Tc
(1])Tj
/F2 1 Tf
0.7829 0 TD
0 Tc
[(.)-301.1(The)-313.2(latter)]TJ
-19.1651 -1.1684 TD
[(representation)-180.7(of)]TJ
/F8 1 Tf
6.8955 0 TD
(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 134.07 228.9 Tm
(t)Tj
/F2 1 Tf
9.9626 0 0 9.9626 139.83 230.34 Tm
[(as)-228.9(an)-216.8(innite)-216.8(sum)-216.8(is)-228.9(called)-216.8(the)]TJ
/F4 1 Tf
12.0968 0 TD
[(r)12(andom-)]TJ
-20.2512 -1.1804 TD
[(ization)-277(method)]TJ
/F2 1 Tf
6.2331 0 TD
[(\(e.g.,)-277([24)24.1(]\))-289.1(and)-289.1(is)-301.1(the)-289.1(most)-301.1(popular)-265(method)]TJ
-6.2331 -1.1684 TD
[(for)-204.8(transient)-228.9(analysis)-216.8(of)-228.9(CTMCs.)-228.9(After)-216.8(a)-228.9(nite)-228.9(truncation)-204.8(of)]TJ
0 -1.1804 TD
[(the)-301.1(innite)-301.1(sum,)-313.2(upper)-289.1(and)-301.1(lo)24.1(wer)-301.1(bounds)-289.1(for)-289.1(the)-313.2(probabil-)]TJ
T*
(ities)Tj
/F8 1 Tf
2.0043 0 TD
(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 84.87 181.98 Tm
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 88.47 183.42 Tm
(\()Tj
/F7 1 Tf
0.3854 0 TD
(x)Tj
/F6 1 Tf
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(\))Tj
/F2 1 Tf
0.7347 0 TD
[(can)-337.3(be)-325.2(easily)-349.3(computed)-301.1(using)-337.3(standard)-313.2(means.)]TJ
-4.6855 -1.1684 TD
[(The)-228.9(instantaneous)-228.9(re)24.1(w)12(ard)-228.9(at)-252.9(time)]TJ
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(t)Tj
/F2 1 Tf
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[(is)-252.9(then)-240.9(gi)24.1(v)12(e)0(n)-228.9(b)0(y)]TJ
/F7 1 Tf
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(E)Tj
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([)Tj
/F7 1 Tf
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(R)Tj
/F10 1 Tf
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(t)Tj
/F6 1 Tf
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0.2763 Tc
(]=)Tj
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(0)Tj
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()Tj
/F12 1 Tf
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()Tj
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(k)Tj
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/F6 1 Tf
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[(t,)-168.5(k)]TJ
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()Tj
/F8 1 Tf
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(r)Tj
/F7 1 Tf
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(.)Tj
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(\(2\))Tj
3.4328 62.3329 TD
[(Bounds)-228.9(after)-228.9(a)-240.9(nite)-240.9(truncation)-216.8(of)-228.9(the)-240.9(innite)-240.9(sum)-228.9(are)-240.9(com-)]TJ
0 -1.1804 TD
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(0)Tj
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()Tj
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(K)Tj
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/F9 1 Tf
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()Tj
/F6 1 Tf
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[(t,)-168.5(k)]TJ
/F6 1 Tf
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()Tj
/F8 1 Tf
0.9154 -1.4093 TD
(r)Tj
/F5 1 Tf
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()Tj
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(E)Tj
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/F7 1 Tf
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(R)Tj
/F10 1 Tf
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(t)Tj
/F6 1 Tf
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(])Tj
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/F8 1 Tf
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(p)Tj
/F9 1 Tf
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(0)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(K)Tj
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()Tj
/F10 1 Tf
6.9738 0 0 6.9738 342.03 689.1 Tm
(k)Tj
/F9 1 Tf
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(k)Tj
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()Tj
/F6 1 Tf
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[(t,)-168.5(k)]TJ
/F6 1 Tf
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()Tj
/F8 1 Tf
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(r)Tj
/F6 1 Tf
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0.222 Tc
[(+\()225.4(1)]TJ
/F5 1 Tf
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()Tj
/F10 1 Tf
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(K)Tj
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()Tj
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(k)Tj
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/F7 1 Tf
9.9626 0 0 9.9626 467.91 697.98 Tm
0 Tc
()Tj
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(\()Tj
/F7 1 Tf
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[(t,)-168.5(k)]TJ
/F6 1 Tf
1.9995 0 TD
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[(\)\))-168(ma)-8.7(x)]TJ
/F10 1 Tf
6.9738 0 0 6.9738 525.87 696.54 Tm
0 Tc
(x)Tj
/F6 1 Tf
9.9626 0 0 9.9626 530.79 697.98 Tm
(\()Tj
/F8 1 Tf
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(r)Tj
/F6 1 Tf
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(\()Tj
/F7 1 Tf
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(x)Tj
/F6 1 Tf
0.5661 0 TD
-0.0034 Tc
(\)\))Tj
/F2 1 Tf
-0.6866 -1.7827 TD
0 Tc
(\(3\))Tj
-22.5482 -1.1684 TD
[(The)-301.1(distrib)24.1(ution)-301.1(of)-313.2(instantaneous)-289.1(re)24.1(w)12(ard)-301.1(can)-313.2(also)-313.2(be)-313.2(com-)]TJ
0 -1.1804 TD
[(puted)-228.9(as)]TJ
/F7 1 Tf
4.1676 -2.2042 TD
(F)Tj
/F6 1 Tf
0.7829 0 TD
([)Tj
/F7 1 Tf
0.277 0 TD
(R)Tj
/F10 1 Tf
6.9738 0 0 6.9738 376.95 633.42 Tm
(t)Tj
/F5 1 Tf
9.9626 0 0 9.9626 383.19 634.86 Tm
()Tj
/F7 1 Tf
1.06 0 TD
(X)Tj
/F6 1 Tf
0.9154 0 TD
0.2643 Tc
(]=)Tj
/F10 1 Tf
6.9738 0 0 6.9738 418.83 647.34 Tm
0 Tc
(n)Tj
/F12 1 Tf
0.7055 0 TD
()Tj
/F9 1 Tf
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(1)Tj
/F13 1 Tf
9.9626 0 0 9.9626 419.19 644.34 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 419.07 623.22 Tm
(x)Tj
/F9 1 Tf
0.6538 0 TD
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(=0)Tj
/F7 1 Tf
9.9626 0 0 9.9626 435.63 634.86 Tm
0 Tc
()Tj
/F6 1 Tf
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(\()Tj
/F8 1 Tf
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(r)Tj
/F6 1 Tf
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(\()Tj
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(x)Tj
/F6 1 Tf
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(\))Tj
/F5 1 Tf
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()Tj
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(X)Tj
/F6 1 Tf
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(\))Tj
/F8 1 Tf
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(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 495.03 633.42 Tm
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 498.51 634.86 Tm
(\()Tj
/F7 1 Tf
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(x)Tj
/F6 1 Tf
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(\))Tj
/F2 1 Tf
3.4087 0 TD
(\(4\))Tj
-22.5483 -2.7824 TD
(where)Tj
/F7 1 Tf
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()Tj
/F6 1 Tf
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(\()Tj
/F7 1 Tf
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(b)Tj
/F6 1 Tf
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0.3699 Tc
[(\)=)-8.7(1)]TJ
/F2 1 Tf
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0 Tc
(for)Tj
/F7 1 Tf
1.4551 0 TD
(b)Tj
/F6 1 Tf
0.807 0 TD
(=)Tj
/F7 1 Tf
1.1443 0 TD
0.0002 Tc
[(tr)-18.4(u)-5.5(e)]TJ
/F2 1 Tf
2.1802 0 TD
0 Tc
(and)Tj
/F6 1 Tf
1.7331 0 TD
(0)Tj
/F2 1 Tf
0.807 0 TD
[(otherwise.)-289.1(Bounds)-277(on)]TJ
-14.8698 -1.1684 TD
[(the)-240.9(distrib)24.1(ution)-228.9(can)-240.9(be)-240.9(computed)-216.8(from)-228.9(the)-252.9(bounds)-216.8(on)]TJ
/F8 1 Tf
21.2488 0 TD
(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 536.79 594.0601 Tm
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T*
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[(,)-385.4(the)-373.4(identity)-373.4(matrix)-361.3(of)-373.4(order)]TJ
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[(.)-289.1(W)84.3(e)-265(as-)]TJ
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[(are)-337.3(scaled)-337.3(such)-337.3(that)]TJ
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[(.)-361.3(Similarly)60.2(,)-349.3(the)-361.3(matrices)-361.4(due)-361.3(to)-361.3(synchro-)]TJ
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[(di)-5.4(ag)]TJ
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[(,)-385.4(a)-373.4(diagonal)-361.3(matrix)-373.4(with)-373.4(the)]TJ
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[(in)-373.4(the)-385.4(main)-361.3(diagonal)-361.3(and)]TJ
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[(is)-385.4(the)-373.4(ba-)]TJ
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[(sic)-252.9(rate)-240.9(of)-240.9(transition)]TJ
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[(rather)-204.8(than)]TJ
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[(.I)385.4(f)]TJ
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T*
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[(kno)24.1(wn,)-349.3(v)12(ector)]TJ
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T*
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[(respecti)24.1(v)12(ely)60.2(.)-277(W)36.1(ith)-313.2(the)-313.2(maximum)-301.1(rates,)-313.2(transition)-301.1(probabil-)]TJ
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25.969 63.1399 TD
[(le)24.1(v)12(el.)-228.9(F)12(o)0(r)-240.9(local)-240.9(transitions)-240.9(we)-252.9(ha)24.1(v)12(e)]TJ
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[(which)-228.9(describes)-240.9(transition)-228.9(probabilities)-228.9(of)-240.9(component)]TJ
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[(As)-265(for)-240.9(matrix)]TJ
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[(in)-252.9(\(8-10\),)-228.9(we)-252.9(deri)24.1(v)12(e)-228.9(a)-252.9(Kroneck)12(er)-228.9(repre-)]TJ
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T*
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(i)Tj
/F9 1 Tf
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(=1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 181.95 687.54 Tm
0 Tc
(I)Tj
/F10 1 Tf
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(n)Tj
/F15 1 Tf
4.9813 0 0 4.9813 191.19 685.14 Tm
(i)Tj
/F5 1 Tf
9.9626 0 0 9.9626 197.07 687.54 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 211.95 699.9 Tm
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 207.03 697.02 Tm
(\015)Tj
/F10 1 Tf
6.9738 0 0 6.9738 208.11 675.78 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
(=1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 223.71 687.54 Tm
0 Tc
(D)Tj
/F9 1 Tf
6.9738 0 0 6.9738 232.47 692.7001 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
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(\))Tj
/F10 1 Tf
-0.8431 -1.1013 TD
(t)Tj
/F6 1 Tf
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([)Tj
/F8 1 Tf
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()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
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(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
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(\))Tj
/F7 1 Tf
0.3854 0 TD
(,)Tj
/F8 1 Tf
0.4698 0 TD
()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6263 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
0.3493 0 TD
-0.0034 Tc
(\)])Tj
/F13 1 Tf
0.6624 1.7104 TD
0 Tc
()Tj
-16.9714 -3.4087 TD
()Tj
ET
q
79.56 0 0 -1.2 121.938 671.808 cm
/Im1 Do
Q
BT
9.9626 0 0 9.9626 201.5101 670.62 Tm
-0.0043 Tc
()Tj
ET
q
79.56 0 0 -1.2 210.378 671.808 cm
/Im1 Do
Q
BT
9.9626 0 0 9.9626 289.9502 670.62 Tm
0 Tc
()Tj
/F2 1 Tf
-14.2854 -1.1684 TD
[(pseudo-transition)-204.8(:=)]TJ
/F8 1 Tf
7.9908 0 TD
(P)Tj
/F12 1 Tf
6.9738 0 0 6.9738 235.47 663.3 Tm
()Tj
/F10 1 Tf
0 -0.9636 TD
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 242.19 658.98 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F7 1 Tf
0.6143 0 TD
(,)Tj
/F8 1 Tf
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()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6143 0 TD
(])Tj
/F2 1 Tf
1.6622 -1.1322 TD
(\(18\))Tj
-22.0424 -1.1804 TD
[(The)-228.9(rst)-240.9(Kroneck)12(er)-216.8(product)-216.8(describes)-228.9(the)-240.9(occurrence)-204.8(of)-240.9(the)]TJ
0 -1.1684 TD
[(synchronized)-216.8(transition,)-252.9(whereas)-252.9(the)-252.9(second)-252.9(and)-252.9(third)-252.9(Kro-)]TJ
0 -1.1804 TD
[(neck)12(er)-168.6(products)-168.6(describe)-180.7(a)-192.7(pseudo-transition.)-156.6(Ob)12(viously)60.2(,)]TJ
/F8 1 Tf
22.4355 0 TD
(P)Tj
/F10 1 Tf
6.9738 0 0 6.9738 291.27 611.1 Tm
(t)Tj
/F2 1 Tf
9.9626 0 0 9.9626 58.59 600.78 Tm
[(is)-240.9(a)-252.9(stochastic)-228.9(matrix.)-228.9(T)84.3(o)-240.9(distinguish)-228.9(pseudo)-228.9(from)-228.9(real)-240.9(tran-)]TJ
0 -1.1684 TD
[(sitions,)-277(we)-289.1(dene)]TJ
/F8 1 Tf
7.3152 0 TD
(P)Tj
/F9 1 Tf
6.9738 0 0 6.9738 139.47 593.34 Tm
(+)Tj
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(t)Tj
/F2 1 Tf
9.9626 0 0 9.9626 149.07 589.14 Tm
[(as)-289.1(the)-277(matrix)-277(resulting)-277(from)-277(the)-277(rst)]TJ
-9.0819 -1.1804 TD
[(part)-228.9(of)-240.9(the)-240.9(sum)-240.9(in)-240.9(\(18\))-228.9(and)]TJ
/F8 1 Tf
10.8272 0 TD
(P)Tj
/F12 1 Tf
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()Tj
/F10 1 Tf
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/F2 1 Tf
9.9626 0 0 9.9626 184.23 577.38 Tm
[(as)-240.9(the)-240.9(matrix)-240.9(resulting)-228.9(from)]TJ
-12.6111 -1.1804 TD
[(the)-240.9(second)-228.9(part)-240.9(of)-240.9(the)-240.9(sum)-240.9(and)]TJ
/F8 1 Tf
12.3943 0 TD
(P)Tj
/F10 1 Tf
6.9738 0 0 6.9738 190.59 564.1801 Tm
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 196.83 565.62 Tm
(=)Tj
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(+)Tj
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0 -0.9636 TD
(t)Tj
/F6 1 Tf
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(P)Tj
/F12 1 Tf
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()Tj
/F10 1 Tf
0 -0.9636 TD
(t)Tj
/F2 1 Tf
9.9626 0 0 9.9626 248.55 565.62 Tm
(.)Tj
-17.8748 -1.1684 TD
[(W)84.3(e)-313.2(no)24.1(w)-277(consider)-277(the)-301.1(multiplication)-265(of)]TJ
/F8 1 Tf
15.779 0 TD
(P)Tj
/F10 1 Tf
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(l)Tj
/F15 1 Tf
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(i)Tj
/F2 1 Tf
9.9626 0 0 9.9626 245.43 553.98 Tm
(and)Tj
/F8 1 Tf
1.721 0 TD
(P)Tj
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(t)Tj
/F2 1 Tf
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(with)Tj
-21.9339 -1.1804 TD
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0 -1.1684 TD
[(representation)-216.8(of)-228.9(v)12(ectors.)-240.9(Thus,)-228.9(assume)-240.9(that)]TJ
/F8 1 Tf
8.0702 -2.2283 TD
(p)Tj
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([)Tj
/F8 1 Tf
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()Tj
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(x)Tj
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(\015)Tj
/F10 1 Tf
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(i)Tj
/F9 1 Tf
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0 Tc
(p)Tj
/F9 1 Tf
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(\()Tj
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([)Tj
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-15.3574 -3.5653 TD
(then)Tj
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()Tj
/F10 1 Tf
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(J)Tj
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/F9 1 Tf
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(\()Tj
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(i)Tj
/F9 1 Tf
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(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 115.95 472.86 Tm
([)Tj
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()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
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(])Tj
/F13 1 Tf
0.277 1.4213 TD
()Tj
/F8 1 Tf
0.9034 -1.4213 TD
(P)Tj
/F10 1 Tf
6.9738 0 0 6.9738 144.39 471.42 Tm
(l)Tj
/F15 1 Tf
4.9813 0 0 4.9813 146.91 470.46 Tm
(i)Tj
/F6 1 Tf
9.9626 0 0 9.9626 150.51 472.86 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F7 1 Tf
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(,)Tj
/F8 1 Tf
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()Tj
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(y)Tj
/F6 1 Tf
0.6263 0 TD
0.2763 Tc
(]=)Tj
/F13 1 Tf
-9.6722 -1.1563 TD
0 Tc
()Tj
0 -0.9034 TD
()Tj
0 -0.2891 TD
()Tj
0 -0.3011 TD
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
0 -1.7947 TD
()Tj
0 -0.3011 TD
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
1.3852 7.4438 TD
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 95.79 453.18 Tm
(i)Tj
/F12 1 Tf
0.3957 0 TD
()Tj
/F9 1 Tf
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(1)Tj
/F13 1 Tf
9.9626 0 0 9.9626 96.75 451.14 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 95.43 434.94 Tm
(j)Tj
/F9 1 Tf
0.5162 0 TD
0.0005 Tc
(=1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 110.91 443.7 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 117.27 447.3 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(j)Tj
/F9 1 Tf
0.5162 0 TD
(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 127.71 443.7 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6023 0 TD
(])Tj
/F13 1 Tf
0.277 1.7104 TD
()Tj
0.9636 -0.9636 TD
()Tj
/F8 1 Tf
1.2768 -0.7468 TD
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 167.91 447.3 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 177.51 443.7 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6023 0 TD
(])Tj
/F8 1 Tf
0.277 0 TD
(P)Tj
/F9 1 Tf
6.9738 0 0 6.9738 196.83 448.86 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F10 1 Tf
-0.8431 -1.1701 TD
(l)Tj
/F6 1 Tf
9.9626 0 0 9.9626 206.43 443.7 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6023 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
0.3493 0 TD
(\))Tj
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/F8 1 Tf
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()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6263 0 TD
(\()Tj
/F7 1 Tf
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(i)Tj
/F6 1 Tf
0.3493 0 TD
-0.0034 Tc
(\)])Tj
/F13 1 Tf
0.831 0.7468 TD
0 Tc
()Tj
-4.7939 -2.0356 TD
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 221.55 423.3 Tm
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 218.55 421.26 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 212.79 405.06 Tm
(j)Tj
/F9 1 Tf
0.5162 0 TD
(=)Tj
/F10 1 Tf
0.8775 0 TD
(i)Tj
/F9 1 Tf
0.3958 0 TD
0.0005 Tc
(+1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 237.15 413.82 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 243.51 417.42 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(j)Tj
/F9 1 Tf
0.5162 0 TD
(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 253.95 413.82 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6023 0 TD
(])Tj
/F13 1 Tf
0.277 1.7104 TD
()Tj
/F2 1 Tf
-4.541 -3.7821 TD
(if)Tj
/F8 1 Tf
0.888 0 TD
()Tj
-0.0241 0 TD
(y)Tj
/F5 1 Tf
0.8913 0 TD
0.2849 Tc
(N)Tj
/F6 1 Tf
1.9152 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3975 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6143 0 TD
(\))Tj
/F7 1 Tf
0.3854 0 TD
(,)Tj
/F8 1 Tf
-18.3774 -1.1804 TD
(0)Tj
/F2 1 Tf
14.5263 0 TD
(otherwise.)Tj
4.6132 -1.1563 TD
(\(19\))Tj
-22.0424 -1.1804 TD
[(The)-373.4(computation)-349.3(follo)24.1(ws)-361.4(from)-373.4(the)-373.4(properties)-361.4(of)-373.4(the)-385.4(Kro-)]TJ
0 -1.1684 TD
[(neck)12(er)-265(product)-252.9([24)12(])-289.1(and)-265(implicitly)-277(sho)24.1(ws)-289.1(that)-277(the)-289.1(multipli-)]TJ
0 -1.1804 TD
[(cation)-265(of)-277(a)-289.1(Kroneck)12(er)-265(product)-252.9(of)-277(v)12(ectors)-265(with)-289.1(a)-289.1(Kroneck)12(er)]TJ
T*
[(sum)-277(of)]TJ
/F7 1 Tf
3.0541 0 TD
(J)Tj
/F2 1 Tf
0.9395 0 TD
[(matrices)-289.1(yields)-277(as)-289.1(a)-289.1(result)-289.1(a)-289.1(sum)-277(of)]TJ
/F7 1 Tf
14.5086 0 TD
(J)Tj
/F2 1 Tf
0.9395 0 TD
[(Kroneck)12(er)]TJ
-19.4417 -1.1684 TD
[(products)-240.9(of)-252.9(v)12(ectors.)-252.9(The)-265(situation)-252.9(is)-277(slightly)-265(more)-252.9(comple)12(x)]TJ
0 -1.1804 TD
[(for)-228.9(Kroneck)12(er)-228.9(products,)-216.8(because)-240.9(pseudo-transitions)-216.8(ha)24.1(v)12(e)-240.9(to)]TJ
T*
[(be)-192.7(considered)-168.6(e)12(xplicitly)60.2(.)-168.6(I)0(f)-192.7(a)-204.8(pseudo-transition)-156.6(occurs)-180.7(in)-192.7(one)]TJ
0 -1.1684 TD
[(component)-313.2(that)-361.4(is)-361.4(in)36.1(v)24.1(olv)12(ed)-325.2(in)-349.3(a)-361.3(synchronization,)-313.2(then)-349.3(the)]TJ
0 -1.1804 TD
[(synchronized)-265(transition)-289.1(is)-301.1(block)12(ed)-289.1(for)-289.1(all)-313.2(others.)-289.1(Thus,)-289.1(the)]TJ
T*
[(v)12(ector)-192.7(after)-192.7(the)-216.8(synchronized)-168.6(transition)-204.8(occurred)-180.7(is)-216.8(gi)24.1(v)12(e)0(n)-192.7(b)0(y)]TJ
/F13 1 Tf
0.4938 -0.5059 TD
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 74.79 243.06 Tm
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 71.79 241.02 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 70.83 224.82 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
(=1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 85.47 233.58 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 91.83 237.18 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 101.31 233.58 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6143 0 TD
(])Tj
/F13 1 Tf
0.277 1.4093 TD
()Tj
/F8 1 Tf
0.9034 -1.4093 TD
(P)Tj
/F10 1 Tf
6.9738 0 0 6.9738 129.75 232.14 Tm
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 133.23 233.58 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F7 1 Tf
0.6023 0 TD
(,)Tj
/F8 1 Tf
0.4698 0 TD
()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6263 0 TD
0.2763 Tc
(]=)Tj
/F13 1 Tf
-8.9495 -1.1081 TD
0 Tc
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 74.79 217.98 Tm
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 71.79 216.06 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 70.83 199.74 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
(=1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 85.47 208.5 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 91.83 212.22 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F6 1 Tf
9.9626 0 0 9.9626 101.31 208.5 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6143 0 TD
(])Tj
/F13 1 Tf
0.277 1.4093 TD
()Tj
/F8 1 Tf
0.9034 -1.4093 TD
(E)Tj
/F9 1 Tf
6.9738 0 0 6.9738 129.51 213.78 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F10 1 Tf
-0.8431 -1.1013 TD
(t)Tj
/F6 1 Tf
9.9626 0 0 9.9626 138.99 208.5 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
-0.012 0 TD
(x)Tj
/F6 1 Tf
0.6023 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
0.3493 0 TD
(\))Tj
/F7 1 Tf
0.3854 0 TD
(,)Tj
/F8 1 Tf
0.4698 0 TD
()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6263 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
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-0.0021 Tc
(\)]+)Tj
/F7 1 Tf
-11.3826 -2.5054 TD
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()Tj
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()Tj
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()Tj
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(\))Tj
/F13 1 Tf
0.554 1.4093 TD
()Tj
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(J)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(i)Tj
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/F8 1 Tf
9.9626 0 0 9.9626 125.19 183.54 Tm
0 Tc
(p)Tj
/F9 1 Tf
6.9738 0 0 6.9738 131.55 187.14 Tm
(\()Tj
/F10 1 Tf
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/F6 1 Tf
9.9626 0 0 9.9626 141.03 183.54 Tm
([)Tj
/F8 1 Tf
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()Tj
-0.012 0 TD
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/F6 1 Tf
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(J)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(i)Tj
/F9 1 Tf
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/F8 1 Tf
9.9626 0 0 9.9626 186.75 183.54 Tm
0 Tc
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/F9 1 Tf
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(\()Tj
/F10 1 Tf
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/F9 1 Tf
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/F6 1 Tf
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([)Tj
/F8 1 Tf
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()Tj
-0.012 0 TD
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/F6 1 Tf
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(])Tj
/F8 1 Tf
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(D)Tj
/F9 1 Tf
6.9738 0 0 6.9738 222.99 188.7 Tm
(\()Tj
/F10 1 Tf
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/F9 1 Tf
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(\))Tj
/F10 1 Tf
-0.8431 -1.0841 TD
(t)Tj
/F6 1 Tf
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([)Tj
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()Tj
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/F6 1 Tf
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(\()Tj
/F7 1 Tf
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(i)Tj
/F6 1 Tf
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(\))Tj
/F7 1 Tf
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(,)Tj
/F8 1 Tf
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()Tj
-0.0241 0 TD
(y)Tj
/F6 1 Tf
0.6263 0 TD
(\()Tj
/F7 1 Tf
0.3854 0 TD
(i)Tj
/F6 1 Tf
0.3493 0 TD
-0.0034 Tc
(\)])Tj
/F13 1 Tf
0.6624 1.4093 TD
-0.0014 Tc
()Tj
/F2 1 Tf
0.1084 -3.1678 TD
0 Tc
(\(20\))Tj
-22.0424 -1.1804 TD
[(The)-240.9(resulting)-228.9(v)12(ector)-228.9(can)-252.9(be)-240.9(represented)-228.9(by)-240.9(three)-240.9(Kroneck)12(er)]TJ
0 -1.1684 TD
[(products.)-240.9(Unfortunately)60.2(,)-240.9(the)-265(last)-289.1(Kroneck)12(er)-252.9(product)-252.9(is)-277(sub-)]TJ
0 -1.1804 TD
[(tracted)-325.2(from)-337.3(the)-337.3(rst)-349.3(tw)12(o)-349.3(such)-337.3(that)-337.3(we)-349.3(ha)24.1(v)12(e)-325.2(to)-349.3(w)12(ork)-337.3(with)]TJ
25.969 63.1399 TD
[(ne)12(gati)24.1(v)12(e)-216.8(and)-240.9(positi)24.1(v)12(e)-228.9(v)12(ectors,)-228.9(e)24.1(v)12(en)-240.9(if)-240.9(all)-252.9(elements)-240.9(of)-240.9(the)-240.9(re-)]TJ
0 -1.1804 TD
[(sulting)-240.9(v)12.1(ector)-228.9(are)-240.9(non-ne)12(gati)24.1(v)12(e)-192.7([8)12(].)]TJ
1.1925 -1.1684 TD
[(Since)-481.8(\(20\))-469.8(contains)-469.8(addition)-469.8(and)-469.8(subtraction)-457.7(of)-481.8(Kro-)]TJ
-1.1925 -1.1804 TD
[(neck)12(er)-325.2(products)-313.2(of)-337.3(v)12(ectors,)-313.2(we)-349.3(ha)24.1(v)12(e)-325.2(to)-337.3(represent)-325.2(both)-325.2(op-)]TJ
T*
[(erations.)-349.3(W)84.3(e)-361.4(start)-361.4(with)-361.4(the)-361.4(addition)-337.3(operation,)-337.3(which)-349.3(rep-)]TJ
0 -1.1684 TD
[(resents)-337.3(a)-349.3(sum)-337.3(of)]TJ
/F7 1 Tf
7.0941 0 TD
(K)Tj
/F2 1 Tf
1.2647 0 TD
[(Kroneck)12(er)-325.2(products)-313.2(of)-349.3(v)12(ectors)-325.2(with)-349.3(a)]TJ
-8.3589 -1.1804 TD
[(single)-361.3(Kroneck)12(er)-337.3(product.)-337.3(W)84.3(e)-385.4(assume)-361.4(that)-361.3(all)-373.4(v)12(ectors)-349.3(are)]TJ
0 -1.1684 TD
[(non-ne)12(gati)24.1(v)12.1(e)-228.9(and)-265(ha)24.1(v)12(e)-265(norm)]TJ
/F6 1 Tf
11.6035 0 TD
(1)Tj
/F2 1 Tf
0.5059 0 TD
[(.)-265(Furthermore)]TJ
/F7 1 Tf
5.7669 0 TD
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 503.19 676.5 Tm
(k)Tj
/F2 1 Tf
9.9626 0 0 9.9626 510.87 677.9401 Tm
[(is)-289.1(the)-265(non-)]TJ
-19.4286 -1.1804 TD
[(ne)12(gati)24.1(v)12(e)-216.8(weight)-240.9(of)-228.9(the)]TJ
/F7 1 Tf
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(k)Tj
/F2 1 Tf
0.5541 0 TD
[(-th)-228.9(v)12(ector)60.2(.)]TJ
/F10 1 Tf
6.9738 0 0 6.9738 360.51 657.54 Tm
(K)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(k)Tj
/F9 1 Tf
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/F7 1 Tf
9.9626 0 0 9.9626 372.99 648.0601 Tm
0 Tc
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 379.47 646.62 Tm
(k)Tj
1.5142 1.5659 TD
(J)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
6.9738 0 0 6.9738 386.07 639.3 Tm
(i)Tj
/F9 1 Tf
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(\()Tj
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/F5 1 Tf
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0 Tc
()Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(K)Tj
/F13 1 Tf
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()Tj
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(k)Tj
/F9 1 Tf
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0 Tc
()Tj
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(k)Tj
/F13 1 Tf
9.9626 0 0 9.9626 391.71 631.62 Tm
()Tj
0.9034 0.6022 TD
()Tj
0 -1.7947 TD
()Tj
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(J)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(i)Tj
/F9 1 Tf
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0 Tc
()Tj
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()Tj
/F10 1 Tf
6.9738 0 0 6.9738 437.79 627.0601 Tm
(k)Tj
/F13 1 Tf
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()Tj
/F10 1 Tf
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(k)Tj
/F9 1 Tf
0.6194 0 TD
0.0005 Tc
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/F10 1 Tf
2.8564 1.9272 TD
0 Tc
()Tj
/F15 1 Tf
4.9813 0 0 4.9813 462.27 620.9401 Tm
(k)Tj
ET
q
23.4 0 0 -0.48 450.018 620.328 cm
/Im1 Do
Q
BT
4.9813 0 0 4.9813 453.27 615.0601 Tm
(K)Tj
/F14 1 Tf
6.9738 0 0 6.9738 452.07 613.74 Tm
(P)Tj
/F15 1 Tf
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(k)Tj
/F17 1 Tf
0.7709 0 TD
0.0081 Tc
(=1)Tj
/F10 1 Tf
6.9738 0 0 6.9738 463.83 608.46 Tm
0 Tc
()Tj
/F15 1 Tf
4.9813 0 0 4.9813 469.11 607.38 Tm
(k)Tj
/F8 1 Tf
9.9626 0 0 9.9626 474.63 617.58 Tm
(a)Tj
/F9 1 Tf
6.9738 0 0 6.9738 480.15 621.1801 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
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(\))Tj
/F10 1 Tf
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0.0048 Tc
(,k)Tj
/F13 1 Tf
9.9626 0 0 9.9626 496.47 637.62 Tm
0 Tc
()Tj
0 -1.7947 TD
()Tj
0.8793 1.7947 TD
()Tj
0 -1.7947 TD
()Tj
/F2 1 Tf
3.1799 0.9997 TD
(\(21\))Tj
-22.0424 -4.1917 TD
[(The)-289.1(resulting)-289.1(v)12(ector)-277(can)-289.1(afterw)12(ards)-289.1(be)-289.1(renormalized)-265(by)-289.1(an)]TJ
0 -1.1804 TD
[(appropriate)-265(choice)-289.1(of)-289.1(the)-289.1(weight)-289.1(such)-289.1(that)-301.1(all)-301.1(v)12(ectors)-289.1(ha)24.1(v)12(e)]TJ
T*
(norm)Tj
/F6 1 Tf
2.4603 0 TD
(1)Tj
/F2 1 Tf
0.5059 0 TD
[(.)-361.4(The)-361.4(representation)-337.3(of)-361.3(the)-361.4(dif)24.1(ference)-337.3(of)-361.4(tw)12(o)-373.4(Kro-)]TJ
-2.9662 -1.1684 TD
[(neck)12(er)-361.4(products)-361.3(of)-373.4(v)12(ectors)-373.4(is)-385.4(dened)-361.3(under)-361.4(some)-373.4(restric-)]TJ
0 -1.1804 TD
[(tions.)-361.4(Again,)-349.3(we)-361.4(assume)-373.4(that)-361.4(all)-373.4(v)12(ectors)-349.3(are)-361.3(non-ne)12(gati)24.1(v)12(e)]TJ
0 -1.1684 TD
[(and)-228.9(ha)24.1(v)12(e)-240.9(norm)]TJ
/F6 1 Tf
6.1055 0 TD
(1)Tj
/F2 1 Tf
0.5059 0 TD
[(.)-240.9(Then)-228.9(we)-252.9(obtain)]TJ
/F7 1 Tf
-4.8287 -1.6261 TD
()Tj
/F9 1 Tf
6.9738 0 0 6.9738 341.55 511.74 Tm
(1)Tj
/F10 1 Tf
1.4454 1.5658 TD
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 348.63 520.62 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 347.67 504.42 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
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/F8 1 Tf
9.9626 0 0 9.9626 362.19 513.1801 Tm
0 Tc
(a)Tj
/F9 1 Tf
6.9738 0 0 6.9738 367.71 516.78 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
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(\))Tj
/F5 1 Tf
9.9626 0 0 9.9626 379.59 513.1801 Tm
()Tj
/F7 1 Tf
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()Tj
/F9 1 Tf
6.9738 0 0 6.9738 396.03 511.74 Tm
(2)Tj
/F10 1 Tf
1.4454 1.5658 TD
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 403.11 520.62 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 402.15 504.42 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
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/F8 1 Tf
9.9626 0 0 9.9626 416.67 513.1801 Tm
0 Tc
(b)Tj
/F9 1 Tf
6.9738 0 0 6.9738 423.03 516.78 Tm
(\()Tj
/F10 1 Tf
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(i)Tj
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(\))Tj
/F5 1 Tf
9.9626 0 0 9.9626 435.39 513.1801 Tm
()Tj
/F9 1 Tf
6.9738 0 0 6.9738 346.59 492.18 Tm
(1)Tj
ET
q
24.6 0 0 -0.48 336.258 490.968 cm
/Im1 Do
Q
BT
/F10 1 Tf
6.9738 0 0 6.9738 336.27 484.74 Tm
()Tj
/F17 1 Tf
4.9813 0 0 4.9813 341.55 483.78 Tm
(1)Tj
/F12 1 Tf
6.9738 0 0 6.9738 345.39 484.74 Tm
()Tj
/F10 1 Tf
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()Tj
/F17 1 Tf
4.9813 0 0 4.9813 356.91 483.78 Tm
(2)Tj
/F13 1 Tf
9.9626 0 0 9.9626 363.63 495.66 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 374.79 493.26 Tm
(J)Tj
0 -1.1529 TD
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/F13 1 Tf
9.9626 0 0 9.9626 389.79 496.26 Tm
0 Tc
()Tj
/F7 1 Tf
0.4577 -0.807 TD
()Tj
/F9 1 Tf
6.9738 0 0 6.9738 400.83 486.78 Tm
(1)Tj
/F8 1 Tf
9.9626 0 0 9.9626 405.27 488.22 Tm
(a)Tj
/F9 1 Tf
6.9738 0 0 6.9738 410.79 491.82 Tm
(\()Tj
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(\))Tj
/F5 1 Tf
9.9626 0 0 9.9626 422.67 488.22 Tm
()Tj
/F7 1 Tf
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/F9 1 Tf
6.9738 0 0 6.9738 439.11 486.78 Tm
(2)Tj
/F8 1 Tf
9.9626 0 0 9.9626 443.55 488.22 Tm
(b)Tj
/F9 1 Tf
6.9738 0 0 6.9738 449.91 491.82 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F13 1 Tf
9.9626 0 0 9.9626 459.51 496.26 Tm
()Tj
/F6 1 Tf
0.7347 -0.807 TD
(=)Tj
/F7 1 Tf
1.0479 0 TD
()Tj
/F9 1 Tf
6.9738 0 0 6.9738 483.75 486.78 Tm
(3)Tj
/F10 1 Tf
1.4454 1.5658 TD
(J)Tj
/F13 1 Tf
9.9626 0 0 9.9626 490.83 495.66 Tm
()Tj
/F10 1 Tf
6.9738 0 0 6.9738 489.99 479.46 Tm
(i)Tj
/F9 1 Tf
0.3957 0 TD
0.0005 Tc
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/F8 1 Tf
9.9626 0 0 9.9626 504.51 488.22 Tm
0 Tc
(c)Tj
/F9 1 Tf
6.9738 0 0 6.9738 509.55 491.82 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3958 0 TD
(\))Tj
/F2 1 Tf
9.9626 0 0 9.9626 536.91 500.94 Tm
(\(22\))Tj
-22.0424 -3.6015 TD
[(The)-192.7(operation)-180.7(is)-216.8(dened)-192.7(only)-192.7(for)]TJ
/F7 1 Tf
13.0674 0 TD
()Tj
/F9 1 Tf
6.9738 0 0 6.9738 454.83 463.62 Tm
(1)Tj
/F7 1 Tf
9.9626 0 0 9.9626 462.03 465.06 Tm
0.2822 Tc
(>)Tj
/F9 1 Tf
6.9738 0 0 6.9738 479.07 463.62 Tm
0 Tc
(2)Tj
/F2 1 Tf
9.9626 0 0 9.9626 485.67 465.06 Tm
[(and)-192.7(non-ne)12(gati)24.1(v)12(e)]TJ
-16.8991 -1.1684 TD
[(v)24.1(alues)-277(in)-289.1(the)-289.1(resulting)-277(v)12(ectors.)-265(Thus,)]TJ
/F8 1 Tf
15.0811 0 TD
(c)Tj
/F9 1 Tf
6.9738 0 0 6.9738 473.31 457.02 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3958 0 TD
(\))Tj
/F5 1 Tf
9.9626 0 0 9.9626 486.51 453.42 Tm
()Tj
/F8 1 Tf
1.1322 0 TD
(0)Tj
/F2 1 Tf
0.8672 0 TD
[(has)-289.1(to)-289.1(hold.)]TJ
-18.9829 -1.1804 TD
[(If)-240.9(the)-240.9(requirement)-216.8(holds,)-228.9(v)12.1(ector)]TJ
/F8 1 Tf
12.7803 0 TD
(c)Tj
/F9 1 Tf
6.9738 0 0 6.9738 450.51 445.26 Tm
(\()Tj
/F10 1 Tf
0.4474 0 TD
(i)Tj
/F9 1 Tf
0.3957 0 TD
(\))Tj
/F2 1 Tf
9.9626 0 0 9.9626 462.51 441.66 Tm
[(can)-240.9(be)-240.9(renormalized)-216.8(to)]TJ
-14.5744 -1.1684 TD
[(get)-204.8(a)-216.8(representation)-180.7(using)-204.8(v)12(ectors)-204.8(with)-204.8(unit)-216.8(norm.)-192.7(As)-216.8(sho)24.1(wn)]TJ
0 -1.1804 TD
[(in)-289.1([8)12.1(],)-289.1(these)-277(conditions)-277(are)-277(observ)12.1(ed)-265(if)-289.1(the)-289.1(operation)-265(is)-289.1(ap-)]TJ
T*
[(plied)-240.9(to)-240.9(\(20\).)]TJ
1.1925 -1.1684 TD
[(W)36.1(ith)-337.3(the)-325.2(operations)-301.1(\(21\))-313.2(and)-313.2(\(22\),)-313.2(a)-325.2(sum)-325.2(of)-325.2(Kroneck)12(er)]TJ
-1.1925 -1.1804 TD
[(products)-313.2(of)-337.3(v)12(ectors)-337.3(can)-337.3(be)-337.3(represented)-313.2(with)-349.3(a)-349.3(single)-337.3(Kro-)]TJ
T*
[(neck)12(er)-204.8(product.)-192.7(Ob)12(viously)60.2(,)-192.7(this)-228.9(representation)-192.7(is)-228.9(an)-216.8(approx-)]TJ
0 -1.1684 TD
[(imation,)-252.9(b)24.1(u)0(t)-265(a)0(s)-265(sho)24.1(wn)-252.9(in)-265([8])-265(for)-252.9(stationary)-252.9(analysis,)-265(the)-253(ap-)]TJ
0 -1.1804 TD
[(proximation)-265(error)-289.1(is)-301.1(often)-289.1(small.)-301.1(An)-301.1(iteration)-289.1(step)-301.1(to)-301.1(com-)]TJ
T*
(pute)Tj
/F8 1 Tf
1.9147 0 TD
(p)Tj
/F10 1 Tf
6.9738 0 0 6.9738 342.87 339.9 Tm
(k)Tj
/F6 1 Tf
9.9626 0 0 9.9626 347.79 336.18 Tm
([)Tj
/F8 1 Tf
0.2891 0 TD
()Tj
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T*
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T*
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T*
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T*
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T*
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T*
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[(,)-301.1(s)0(o)-313.2(that)-301.1(more)-289.1(than)]TJ
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[(state)-445.7(analysis.)-433.6(The)-445.7(second)-421.6(and)-433.6(third)-433.6(column)-433.6(contain)-421.6(the)]TJ
T*
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T*
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T*
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[(the)-265(same)-265(number)-240.9(of)-252.9(iterations)-265(must)-265(be)-265(performed,)-228.9(b)24.1(u)0(t)-252.9(each)]TJ
T*
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[(4-6)-252.9(and)-252.9(7-9)-265(of)-252.9(the)-265(table.)-265(The)-265(quality)-252.9(of)-265(the)-265(results)-265(is)-277(e)12(xcel-)]TJ
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[(impro)12(v)12(e)-216.8(the)-240.9(result.)]TJ
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10.9091 0 0 10.9091 317.31 368.7 Tm
[(6.2.)-704(An)-242(A)99(v)11(ailability)-275(Model)]TJ
/F2 1 Tf
9.9626 0 0 9.9626 329.19 353.46 Tm
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T*
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[(able.)-228.9(W)84.3(e)-240.9(assume)-228.9(that)-228.9(each)-228.9(f)12(ailed)-228.9(part)-228.9(is)-240.9(repaired)-216.8(by)-228.9(a)-228.9(single)]TJ
T*
[(repairman)-301.1(and)-325.2(that)-337.3(repairs)-325.2(are)-337.3(prioritized)-313.2(with)-325.2(preempti)24.1(v)12(e)]TJ
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S
EMC
BT
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0 g
[(Figure)-277(1.)-277(Kanban)-277(e)12(xample)-277(model.)]TJ
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q
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(1)Tj
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1.7706 0 TD
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Q
q
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Q
q
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Q
BT
/F5 1 Tf
9.9626 0 0 9.9626 131.43 496.14 Tm
( )Tj
ET
q
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/Im1 Do
Q
BT
/F2 1 Tf
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[(1.505)-1806.7(1.045)-1565.8(-30.6%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(5.204e-2)-1168.4(1.461e-2)-1156.3(-71.9%)]TJ
ET
q
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Q
BT
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[(11.5)-2059.7(0.49)]TJ
ET
q
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Q
q
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Q
q
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Q
BT
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[(Hea)24.1(vy)-240.9(load)]TJ
ET
q
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Q
q
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Q
q
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Q
BT
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(1)Tj
ET
q
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Q
BT
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[(0.3331)-1300.9(0.3327)-1553.8(-0.1%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(4.266e-5)-1168.4(4.301e-1)-1577.9(0.8%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(4.17)-2059.7(0.03)]TJ
ET
q
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Q
q
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/Im1 Do
Q
q
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/Im1 Do
Q
BT
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(10)Tj
ET
q
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/Im1 Do
Q
BT
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[(1.512)-1806.8(1.438)-1806.8(-4.9%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(3.835e-2)-1168.4(2.821e-2)-1156.3(-26.4%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(23.1)-2059.7(0.09)]TJ
ET
q
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/Im1 Do
Q
q
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/Im1 Do
Q
q
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/Im1 Do
Q
BT
9.9626 0 0 9.9626 131.43 447.66 Tm
(50)Tj
ET
q
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/Im1 Do
Q
BT
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[(3.029)-1806.8(3.040)-1698.3(+0.4%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(2.089e-1)-1168.4(2.043e-1)-1409.3(-2.2%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(94.1)-2059.7(0.31)]TJ
ET
q
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Q
q
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/Im1 Do
Q
q
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/Im1 Do
Q
BT
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(100)Tj
ET
q
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/Im1 Do
Q
BT
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[(3.172)-1806.7(3.190)-1698.3(+0.6%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(2.281e-1)-1168.4(2.263e-1)-1409.3(-0.8%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(175.8)-1806.7(0.59)]TJ
ET
q
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/Im1 Do
Q
q
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/Im1 Do
Q
q
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/Im1 Do
Q
BT
9.9626 0 0 9.9626 129.03 423.42 Tm
(500)Tj
ET
q
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/Im1 Do
Q
BT
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[(3.179)-1806.7(3.196)-1698.3(+0.5%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(2.290e-1)-1168.4(2.273e-1)-1409.3(-0.7%)]TJ
ET
q
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/Im1 Do
Q
BT
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[(813.0)-1806.7(2.76)]TJ
ET
q
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/Im1 Do
Q
q
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Q
BT
/F19 1 Tf
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[(T)84.3(a)0(b)12(l)0(e)-289.1(3)0(.)-277(Results)-277(of)-277(the)-265(MSMQ)-277(system.)]TJ
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Q
q
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/Im1 Do
Q
/EmbeddedDocument /MC2 BDC
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0 J 0 j 0.355 w 10 M []0 d
1 i
230.406 339.66 m
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h
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h
144.367 343.92 m
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s
q
119.91 262.98 113.389 117.144 re
178.597 340.386 m
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171.408 343.224 l
177.084 341.805 l
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W* n
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W n
143.832 341.805 m
177.888 341.805 l
S
Q
171.408 340.386 m
177.084 341.805 l
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177.888 350.319 m
180.016 352.447 l
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220.458 352.447 l
224.715 348.19 l
S
q
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134.703 340.338 m
136.831 342.23 l
132.101 347.623 l
134.797 342.372 l
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W* n
119.91 262.98 113.389 117.144 re
W n
2 j
135.318 365.218 m
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134.75 364.745 l
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133.804 363.941 l
133.189 363.374 l
132.527 362.759 l
131.865 362.049 l
131.25 361.292 l
130.635 360.441 l
130.115 359.542 l
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133.189 344.359 l
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134.324 342.94 l
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S
Q
129.973 345.731 m
134.797 342.372 l
132.101 347.623 l
S
q
119.91 262.98 113.389 117.144 re
144.399 366.732 m
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W* n
119.91 262.98 113.389 117.144 re
W n
2 j
143.832 341.805 m
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144.115 341.994 l
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147.285 360.914 l
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S
Q
149.129 361.34 m
144.352 364.651 l
147.001 359.448 l
S
q
119.91 262.98 113.389 117.144 re
W n
BT
/F2 1 Tf
8.5141 0 0 8.5141 137.4461 373.7325 Tm
/Cs6 cs 0 0 0 scn
(off)Tj
ET
Q
BT
/F2 1 Tf
8.5141 0 0 8.5141 137.4461 329.0335 Tm
/Cs6 cs 0 0 0 scn
(on)Tj
5.25 3.5 TD
(capacity K)Tj
ET
230.406 299.22 m
230.406 296.087 227.863 293.544 224.73 293.544 c
221.597 293.544 219.054 296.087 219.054 299.22 c
219.054 302.353 221.597 304.896 224.73 304.896 c
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h
177.888 305.62 m
209.816 305.62 l
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177.888 294.977 l
209.816 299.234 m
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144.367 275.82 m
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s
q
119.91 262.98 113.389 117.144 re
136.831 300.984 m
134.703 302.876 l
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134.797 300.795 l
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W* n
119.91 262.98 113.389 117.144 re
W n
2 j
135.318 277.949 m
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135.034 278.138 l
134.75 278.422 l
134.324 278.753 l
133.804 279.226 l
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131.865 281.118 l
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129.027 290.058 l
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129.642 292.754 l
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133.804 299.565 l
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S
Q
132.101 295.592 m
134.797 300.795 l
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S
q
119.91 262.98 113.389 117.144 re
142.271 278.375 m
144.399 276.483 l
149.129 281.875 l
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W* n
119.91 262.98 113.389 117.144 re
W n
2 j
143.832 301.363 m
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144.115 301.174 l
144.399 300.89 l
144.825 300.559 l
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145.96 299.518 l
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149.886 287.882 l
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S
Q
147.001 283.767 m
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144.367 303.48 m
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136.953 298.703 134.813 300.843 134.813 303.48 c
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s
q
119.91 262.98 113.389 117.144 re
178.597 299.944 m
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171.408 302.782 l
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W* n
119.91 262.98 113.389 117.144 re
W n
143.832 301.363 m
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S
Q
171.408 299.944 m
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175.759 288.592 m
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194.916 286.463 l
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199.173 286.463 l
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222.587 290.72 l
S
BT
8.5141 0 0 8.5141 135.3176 309.8768 Tm
(on)Tj
ET
q
119.91 262.98 113.389 117.144 re
W n
BT
8.5141 0 0 8.5141 135.3176 263.0494 Tm
(off)Tj
ET
Q
BT
8.5141 0 0 8.5141 180.0165 273.692 Tm
(capacity K)Tj
ET
q
119.91 262.98 113.389 117.144 re
177.604 299.849 m
179.307 302.12 l
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W* n
119.91 262.98 113.389 117.144 re
W n
169.374 341.805 m
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S
Q
171.834 304.106 m
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S
q
119.91 262.98 113.389 117.144 re
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W* n
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W n
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S
Q
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175.05 337.264 l
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S
BT
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()Tj
0 -2 TD
()Tj
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()Tj
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()Tj
/F2 1 Tf
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(1)Tj
ET
q
119.91 262.98 113.389 117.144 re
W n
BT
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(1)Tj
ET
Q
BT
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(2)Tj
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(2)Tj
7.2 -2.7 TD
(2)Tj
ET
q
119.91 262.98 113.389 117.144 re
W n
BT
7.0951 0 0 7.0951 228.9725 305.6198 Tm
(2)Tj
ET
Q
BT
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(2)Tj
ET
EMC
BT
/F19 1 Tf
9.9626 0 0 9.9626 89.79 248.58 Tm
0 g
[(Figure)-277(2.)-277(Overo)12(w)-265(queueing)-265(system.)]TJ
ET
q
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/Im1 Do
Q
BT
/F2 1 Tf
9.9626 0 0 9.9626 58.59 220.74 Tm
[(suf)24.1(cient,)-313.2(since)-325.2(the)-325.2(potential)-313.2(state)-337.3(space)-325.2(equals)-325.2(the)-325.2(reach-)]TJ
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T*
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T*
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([0)Tj
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(,t)Tj
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(])Tj
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[(,)-301.1(and)-289.1(for)-289.1(dif)24.1(ferent)-277(v)24.1(alues)-289.1(of)]TJ
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[(=1)336.5(0)]TJ
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(or)Tj
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[([24])-562.1(W)93.7(.)-428.3(J.)-441.7(Ste)26.8(w)13.4(art.)]TJ
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2.2216 -1.2045 TD
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