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{PARA 0 "" 0 "" {TEXT -1 2 "Q1" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 2 "a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "G:=piecewise(u<12,0,1):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "lambda:=7.2/60:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "t:=52:   p:=int(1-G,u=0..t)/t:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "Lx:=lambda*t*p:  Ly:=lambda*t*(1-p):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "add(Lx^k/k!*exp(-Lx),k=0..3)
*(1-add(Ly^k/k!*exp(-Ly),k=0..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
$\"+gi'=$\\!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "b)" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Linf:=lambda*12;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LinfG$\"++++S9!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "c)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "exp(-Linf)*lambda*60;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+i)zeq\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "
Q2" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "re
start:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the prote
cted names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L:=[13,20,14,21,25,13]: M:=[23,42,2
2,13,19,20]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "A:=matrix(
7,7,0):  for i to 6 do A[i,i+1]:=L[i]: A[i+1,i]:=M[i] end do: for i fr
om 2 to 6 do A[i,i]:=-A[i,i-1]-A[i,i+1] end do:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "A[1,1]:=-L[1]: A[7,7]:=-M[6]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "s:=linsolve(transpose(A),[0$7]): s:
=evalm(s/add(s[i],i=1..7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-
%'vectorG6#7)#\"&)Hn\"'@S>#\"&Q!QF+#\"&SV&\"'j?e#\"&!eMF0#\"&?'=F+#\"&
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1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "-s[4]*A[4,4]*1.;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'>.Yb#!\"*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 35 "c)    First, make State 1 absorbing" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for i to 7 do A[2,
i]:=0 end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "exponentia
l(A/10.,12/60.)[4,6];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Pj$*o^!#6
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q3" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "restart: i:=6: L:=1/31: M:=1
/23:  a:=0.12:  t:=52.:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "a)   U
se M/M/infinity model" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "p:=exp(-M*t):  PGF:=exp(a*(1-p)/M*(z-1)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(mtaylor(PGF,z,5),z=1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`Rb[*)!#5" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 37 "b)  LGWI model, with 0 initial value!" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E:=exp((M-L)*t):  r:
=M*(1-E)/(L-M*E): p:=(L-M)*E/(L-M*E):PGF:=(p/(1-(1-p)*z))^(a/L):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(mtaylor(PGF,z,5),z=1);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lWG_a!#5" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 54 "c) Find the corresponding distribution function fi
rst." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "t:='t':E:=exp((M-L)
*t): r:=M*(1-E)/(L-M*E): F:=r^i:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "mean:=int(t*diff(F,t),t=0..6000.);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%meanG$\"+>\\-e7!\"(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "sqrt(int((t-mean)^2*diff(F,t),t=0..7000.));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+=Lcl\"*!\")" }}}{EXCHG {PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q4" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "restart:   L:=k->0.6
*sqrt(k):   M:=k->3.*k/(1+k):   d:=n->evalf(product(M(i)/L(i),i=1..n))
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "a)  The usual approach:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "S:=add(d(n),n=0..300):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "1-add(d(n),n=0..15)/S;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g;&os)!#5" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 64 "b) Same approach, making State 10 (instead of State 0)
 absorbing" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "d:=n->evalf(product(M(i)/L(i),i=11..n)):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "S:=add(d(n),n=10..300):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "add(d(n),n=10..15)/S;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+hdeM6!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "c)
  Make State 20 (in addition to State 0) absorbing, switch to CTMC des
cription" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
107 "A:=matrix(19,21,0): for i to 19 do A[i,i]:=M(i): A[i,i+2]:=evalf(
L(i)): A[i,i+1]:=-M(i)-evalf(L(i)) end do:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 82 "with(linalg): linsolve(submatrix(A,1..19,2..20),sub
matrix(-A,1..19,[1,21]))[16,1];" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warnin
g, the protected names norm and trace have been redefined and unprotec
ted\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M(*3ad!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "Q5" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 47 "restart: lam:=3.8:   t:=(52-27)/60.:  L:=lam
*t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "P:=exp(-lam*t*(1-z*e
xp(4.3*(z-1)))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "a) Basic Pois
son process - probability of 3 or more arrivals in that interval:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "1-(1+L+
L^2/2)*exp(-L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8]VB@!#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "b) Cluster process" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "1-eval(mtaylor(P,z,1
4),z=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+t=Zq@!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 155 "c) Each cluster size has Poisson distrib
ution with mean of 4.3, plus 1; the sum is Poisson with mean 8.6, plus
 2 (the Poisson part must be at least 3 then)" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L:=2*4.3:  1-(1+L+L^2/2)*exp
(-L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@NC9**!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "Q6" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 67 "restart: A:=matrix(4,4,[-1,-1,-1,2,3,4,1,-1,
0,-2,2,-2,-4,-1,-1,5]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "findin
g eigenvalues and constituent matrices" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "with(linalg): w:=evalf(eigenvalues(
A));" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm
 and trace have been redefined and unprotected\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"wG6&$\"\"$\"\"!$\"\"#F(F&F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 160 "solve(\{c1+c2=1,w[1]*c1+d1+w[2]*c2+d2=A,w[1]^
2*c1+2*w[1]*d1+w[2]^2*c2+2*w[2]*d2=A^2,w[1]^3*c1+3*w[1]^2*d1+w[2]^3*c2
+3*w[2]^2*d2=A^3\},\n\{c1,c2,d1,d2\}):  assign(%);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 9 "a)  ln(A)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 50 "B:=evalm(c1*ln(w[1])+c2*ln(w[2])+d1/w[1]+d2/w
[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7&7&$!*3i=S
'!\"*$!*5\\`%fF,$!*A+++&F,$\"*'\\)zQ(F,7&$\"+y^Y09F,$\"+Us9$p\"F,$\"*A
+++&F,$!*M^Y0%F,7&$!*WNEW\"F,$!*o-$4\")F,$\"*)>ZJpF,$!*Cnmm'F,7&$!+S&)
zQ<F,F-F/$\"+G3TP=F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "c) check" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expone
ntial(B):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "b) sin(A)" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "B:=evalm(c1*sin(
w[1])+d1*cos(w[1])+c2*sin(w[2])+d2*cos(w[2]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7&7&$\"+)e$eJF!\"*$\"*!ei6k!#5$\"*Po9
;%F,$!+3*p\"e<F,7&$!+\"4r/g\"F,$\")[P+xF,$!*Po9;%F,$\"*;u<o(F,7&$!*],j
V%F,$\"+L[NO:F,$\"*AuH4*F,$\"+#)\\)*z>F,7&$\"+#ej/f#F,F-F0$!+-*\\qh\"F
," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "c) check (against the purely
 imaginary part of):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "exponential(I*A*1.):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "Q7" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "restart:  L:=14.2:  M:=60/18.:  c:=5:  R:=L/M:   G:=a
dd(R^k/k!,k=0..c-1)+R^c/c!/(1-R/c):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "a) We have simple formula" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 4 "R/c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"++++?&)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "b) Average queue \+
length (formula) divided by arrival rate:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 77 "Q:=R^(c+1)/c/c!/G/(1-R/c)^2/L:   Q*60;   (Q*60-16)*
60;  # minutes and seconds" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e7@
g\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")#[vE\"!\"(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 119 "c) We have to use the approch of Q4 Part
 b), this time computing the expected time till (not probability of) a
bsorption" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "lambda:=k->L:
   mu:=k->M*k:    d:=n->evalf(product(lambda(i)/mu(i+1),i=5..n)):   su
m(d(n),n=4..infinity)/mu(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$f
tG`\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q8" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "a)  Any function of:" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "t+int(z*exp(z),z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"tG\"\"\"*&%\"zGF%-%$expG6#F'F%F%F
(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "b) Same as in a), multi
plied by" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "exp(int(-3,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&\"
\"$\"\"\"%\"zGF)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "c)  Base
d on:  g[(z-1)*exp(z)]*exp(-3z) = (z-1)^3, we take  g(x) = x^3 " }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "(t+(z-1
)*exp(z))^3*exp(-3*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&%\"tG\"
\"\"*&,&%\"zGF'F'!\"\"F'-%$expG6#F*F'F'\"\"$F'-F-6#,$*&F/F'F*F'F+F'" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q9" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart:  L:=26.3:  M:=60/(12+37/60
.):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "a)  We use the approach Q7
 Part c)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
126 "lambda:=k->L:   mu:=k->M*k:    d:=n->evalf(product(lambda(i)/mu(i
+1),i=6..n)):   w[5]:=0: w[6]:=sum(d(n),n=5..infinity)/mu(6):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "for i from 6 to 7 do w[i+1]
:=(1+mu(i)/lambda(i))*w[i]-mu(i)/lambda(i)*w[i-1]-1/lambda(i) end do: \+
w[8];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+emJAC!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "b) We have to switch to CTMC description
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "wit
h(linalg): A:=matrix(12,13,0): for i to 11 do A[i+1,i]:=mu(i+1):  A[i+
1,i+2]:=lambda(i+1): A[i+1,i+1]:=-mu(i+1)-lambda(i+1) end do:" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "A[1,1]:=-lambda(1):  A[1,2]:=lambda(1): linsolve(subm
atrix(A,1..12,1..12),[-1$12])[9];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+5b!\\+\"!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "c) We have a s
imple formula for this one (reciprocal of frequency of visits)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1/(lamb
da(8)+mu(9))/((L/M)^8/8!*exp(-L/M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+f^+#o\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q10" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "a) Get the distribution funct
ion first" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "L:=4/3*Pi*0.098*x^3:   F:=1-exp(-L)*(1+L+L^2/2+L^3/6):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "m:=evalf(int(diff(F,x)*x,x=0
..12.));  sqrt(evalf(int(diff(F,x)*(x-m)^2,x=0..12.)));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"mG$\"+!*zsw?!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+W3i`N!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "b)
 Volume of the cylinder is 50*Pi" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "L:=evalf(0.098*Pi*10*5); sqrt(L);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG$\"++/QR:!\")" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+^Q\\BR!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
79 "c) Prob. of having negative z is 2/5 - then, use Binomial distribu
tion (n = 18)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "add(binomial(18,i)*.4^i*.6^(19-i),i=0..7);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+'3X1Q$!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "Q11" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "restart: A:=evalm(matrix(5,5,[0,2,2,3,1,4,0,2,0,2,3,4
,0,6,1,4,0,3,0,5,2,2,1,5,0])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 49 "for i to 5 do A[i,i]:=-add(A[i,j],j=1..5) end do:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 34 "a) State 1 and 4 are absorbing now" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "with(li
nalg): -linsolve(submatrix(A,[2,3,5],[2,3,5]),submatrix(A,[2,3,5],[1,4
]))[1,2];" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names
 norm and trace have been redefined and unprotected\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6##\"$T\"\"$#[" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
29 "b)  only State 5 is absorbing" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "w:=linsolve(submatrix(A,1..4,1..4),[-1$4]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "tau:=linsolve(submatrix(A,1.
.4,1..4),-2*w):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "w[2];   \+
 sqrt(tau[2]-w[2]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$\\\"\"$'H
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"$)))!\"\"\"'&o&=#\"\"\"\"\"#
F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "c) State 2 is absorbing, b
ut we have to take one transition first, and then use formula of total
 probability" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "aux:=convert(submatrix(A,[2],[3,4,5]),vector):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a:=linsolve(submatrix(A,3..5,3..5),
submatrix(A,3..5,1..2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"-evalm(aux&*a)[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%W<\"%$4\"" 
}}}}{MARK "115 0 0" 108 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }