584 4wC �8rar
� Monte Carlo
[� ar 10000 �o�,1By�ot'W (1−W ) ,
HKh7� 1%. I m ZM�� (1−W )m < (0.01)m, R9 m = 3, Sol*�
�= (1−W )3 < (0.01)3 = 10−6. Mf
M�J6�
� 1012 ,|d�dL�
104 ��dSo�& 10−6 ,HK�Ml�B3℄0
[���z!��5K�
�a!ary��
�� R tO (MC3.R), � 10000 �ar�*&�! D = 3d y��
�=y
W3d ≤ 1/3, �P TJ,�5ZOq~ WD < 0.01?
hLh,�5JKy x, ; (W3d)
x < 0.01, m 3x > 100, |
x >
lg 100
lg 3
=
2
0.47712
= 4.1918.
|�5Z,JK6�& 4.1918D ≈ 13d, Oq~o��
�=J�� 0.01.
Myotd�4l��P!�R~ WD < 10
−6, ;�JKy
TD = 3x = 39d.
iS (��5Z�JKy 39d y�o~o��
�5Z,�= WD < 10
−6.
10.4 )1�� 8z�%gux
~Q� (Queueing Theory) �m:rt�_� [3^P�rt�_.
�&4o,�=KT�ÆV8_tÆ�_��h����vi,lr
��
~Q���:r3t,lMk�M�t~QaEy���(℄���P,:
rar
[�6<
aE
[[#� ��<
~Q�,q1�v�
10.4.1 7y�/5�fD1�3
1. 7yf�3
� 10.13 �V�$
u*��A�8|ezU�%7|ae��3V�$�%
�(�T=U�%Æur ae�x�2�B%��T�B$H��E�}[
a$rmtÆCFAl
FJgFp~Q�_�
R�FAl�~Qt�_�Ol_$r[3~Qt�_�hI3N�3
t�$r&��~Q.��0�t��;��3tNe 10.6 ���
$r�Q Q� tÆC
- - -
�O �{
t�_
e 10.6: t�_,Y�
2. 7y�/5�fD1�3
(1) �O3t
�O3t Y�$r�'{$r �yu"�MNy�
�_�P\H,y���Qmy�_,H��B℄ !�_otÆr�,��
'K� G�tÆr���9=,bA�
10.4.2 7y(L(0f&V
1. &V4�
aEar,'�>� � �t{O�� Yo,y����~QaEo,
O�� �y�Yo,��N6dyq�$r&��dyq�$r�f (tÆ
np) .�~�BGar�U�'�>��
(1) y�>���8�_Yodl� ,y��N$r&�m$r�f�
(2) �#>��!#6t�_o$r,�#�
(3) �_�E>���_,�E�N�_ u"`\H��_o$r,�#�k
D6h�tÆNC0�tÆ.�
�!MU�'�>���A>�S<
!�
2. Poisson /Nf(0
6~QtÆ�_o�[eÆh$r,&�y��0�,y�tÆ Poisson 3
t�~��R� Poisson 3t,ar wksh,�
588 4wC �8rar
��=Y}oY�!:r3t 'Ky λ , Poisson 3ty��6��\
+TG�,:r>��*tÆR#y λ ,b#kK�|
fTi(t) =
{
λe−λt, t > 0,
0, t ≤ 0,
i = 2, 3, · · · ,
+�,kK9#y
FTi(t) =
{
1− λe−λt, t > 0,
0, t ≤ 0,
i = 2, 3, · · · .
~���
t = −1
λ
ln(1− FTi(t)).
�� FTi(t) ∼ U(0, 1), ; 1−FTi(t) ∼ U(0, 1), ~��ar Poisson 3t&�
,y�����y
ti = −1
λ
ln ui, i = 1, 2, · · · , (10.31)
�o ui ∼ U(0, 1).
10.4.3 h[�7y(Lf(0
.�i~QaEo�e�,aE M/M/S/∞, |$r&��_,+�&
�y���G��*tÆR#y λ ,b#kK
|�O3ty Poisson 3t��
tÆC,tÆy�iG�℄kK�*tÆR#y µ ,b#kK�V*�_u�
�)�0N
(~Q�
1. S = 1 fJ} (M/M/1/∞)
5�4�
�l>�
t — y�>� NA — 6 t yq&��_,$r�#
tA — $r,&�y� n — 6 t yq!#��_,$r#
tD — $r,�fy� T — �tÆy�
#�>� (t k y
>�)
wt — �8Yo� ,y� wn — �8�_o,$r#
ws — �8_l� &�l� ,��y�
10.4 ~�OLu�Bd$$KP 589
(0�� I
(1) z�L�h t = NA = 0, ao$r&��_,z�y� T0, h tA = T0,
tD =∞(�y�_o�$r). h k = 0.
(2) �8�_�E�h k = k + 1, wt(k) = t, wn(k) = n. N2 tA < T , ;h
ws(k) = min(tA, tD)− t,
BM� (3); p;h
ws(k) =
{
0, tD =∞,
tD − t, tD <∞,
BM� (8).
(3)N2 tA < tD, ;h t = tA, NA = NA+1($r&��# +1), n = n+1(�
_o$r# +1), ao�l$r&��_,y� tA.
(4) N2 n = 1, aotÆC_$r,�fy� tD.
(5) N2 tA ≥ tD, ;h t = tD, n = n− 1(�_o$r# −1).
(6) N2 n = 0(�_o�$r), h tD = ∞; p;aotÆC_$r,�f
y� TD.
(7) � (2).
(8) (�y tA ≥ T , J40�?$r�dnp�_o$r,tÆ). N2 n >
0(�_o`�$r), Fh t = tD, n = n − 1(�_o$r# −1). N2 n > 0, a
otÆC_$r,�fy� TD, BM� (2); p;� (9).
(9) �8Qf (Ls) ��aE0y� (Ws) �$r.�,�= (Pwait).
Ls =
1
t
∑
k
ws(k) · wn(k),
Ws =
1
NA
∑
k
ws(k) · wn(k),
Pwait =
1
t
∑
wn(k)≥1
ws(k),
Y
�8��{ Ls � Ws � Pwait.
R NS(tO_� queue1.R)
590 (q� R#F~�
queue1<-function(lambda, mu, T){
k<-0; wt<-0; wn<-0; ws<-0;
tp<-0; nA<-0; n<-0; t<-0
r<-runif(1); tA<--1/lambda*log(r); tD<-Inf
repeat{
k<-k+1; wt[k]<-t; wn[k]<-n
if (tA < T){
ws[k]<-min(tA, tD)-t
if (tA < tD){
t<-tA; n<-n+1; nA<-nA+1
r<-runif(1); tA<-t-1/lambda*log(r)
if (n==1){
r<-runif(1); tD<-t-1/mu*log(r)
}
}else{
t<-tD; n<-n-1
if (n==0){
tD<-Inf
}else{
r<-runif(1); tD<-t-1/mu*log(r)
}
}
}else{
ws[k]<-if(tD==Inf) 0 else tD-t
if (n>0){
t<-tD; n<-n-1
if (n>0){
r<-runif(1); tD<-t-1/mu*log(r)
}
}else
tp<-1
10.4 (0���7y�$ftw 591
}
if (tp==1) break
}
data.frame(Ls=sum(ws*wn)/t, Ws=sum(ws*wn)/nA,
Pwait=sum(ws[wn>=1])/t)
}
� 10.14 �V�$
u*�x�A�8|ezU�%7|ae��3V�$�
% �(�T=U�%Æur ae�x�2�B%��[h3V�$�% �
��U Poisson Y�!� 4 H / �r�V�r℄a���\g�!��2 6 \
%��P~�$OL:eoD$$B{ (Ls) �!�9Wr℄ (Ws) ��%%�
$fa (oDMkfa)(Pwait).
`�;�:<,tO queue1.R, �O+�,R#bA�ar 1000 7y,~
Qt�_,1G1�
> source("queue1.R")
> queue1(lambda=4, mu=10, T=1000)
Ls Ws Pwait
1 0.6938313 0.1685005 0.4118629
�
�`y Ls = 0.6666667(E), Ws = 0.1666667(7y). Pwait = 0.4.
� 10.15 ub5$
h�8. ATM F�[h3=1$�%!�o\% 0.6
u�Gou�%$!�=1$r℄U 1.25 \%��P~�$OL:e ATM F
$B{ (Ls) �!�9Wr℄ (Ws) ��%%�$fa (Pwait).
`�ar 10000 kp,~Qt�_,1G1�
> queue1(lambda=0.6, mu=0.8, T=10000)
Ls Ws Pwait
1 2.949336 4.895917 0.7577775
�
�`y Ls = 5(E), Ws = 5(kp). Pwait = 0.75.
Æ_X����otg{�ar`�
�`` E0C,�
2. S > 1 fJ} (M/M/S/∞)
5�4�
592 4wC �8rar
R� S > 1 ,1�>�wzq1_� S = 1 ,1+℄�d �y, tD
y#��=�l��E>� SS, �8�_,�E1�
(0�� II
(1) z�L�h t = NA = 0, ao$r&��_,z�y� T0, h tA =
T0, tD(i) = ∞, i = 1, 2, · · · , S(�y�_o�$r). SS(i) = 0, i = 1, 2, · · · , S +
1(SS(1) �8�_!#�E,$r#� SS(2 ∼ S +1) �8 S �tÆC,���
E� 0 yu"� 1 y��), h k = 0.
(2) N2 SS(1) = 0, ;h t1 = ∞, i1 = 1; p;h t1 = min(tD), i1 =
argmin(tD).
(3) �8�_�E�h k = k + 1, wt(k) = t, wn(k) = n. N2 tA < T , ;h
ws(k) = min(tA, t1)− t,
BM� (4); p;h
ws(k) =
{
0, t1 =∞,
t1 − t, t1 <∞,
BM� (11).
(4) N2 tA < t1, ;h t = tA, NA = NA + 1($r&��# +1), ao�l$
r&��_,y� TA. h n = SS(1), SS(1) = n+ 1(�_o$r# +1).
(5) R� i = 1, 2, · · · , S, N2 SS(1 + i) = 0(4 i �tÆCu"), ;h
SS(1 + i) = 1(%�_o,$rk��4 i �tÆC�f�tÆ), ao4 i �t
ÆC_$r�f,y� TD(i), BMo
W_�
(6) N2 tA ≥ t1, ;h t = t1, n = SS(1), SS(1) = n− 1(�_o$r# −1).
(7) N2 n = 1(�_o�$r), h SS(1 + i) = 0, tD(i) =∞, i = 1, 2, · · · , S.
(8) N2 n ≤ S, h SS(1 + i1) = 0, tD(i1) =∞(4 i1 �tÆCu").
(9) N2 n > S, ao$r�f4 i1 �tÆC,y� TD(i1).
(10) � (2).
(11) (�y tA ≥ T , J40�?$r�dnp�_o$r,tÆ). h n =
SS(1). N2 n > 0, ;h t = tD, SS(1) = n− 1(�_o$r# −1), BM� (7);
p;� (12).
10.4 ~�OLu�Bd$$KP 593
(12) �8Qf (Ls) ��aE0y� (Ws) �$r.�,�= (Pwait).
Ls =
1
t
∑
k
ws(k) · wn(k),
Ws =
1
NA
∑
k
ws(k) · wn(k),
Pwait =
1
t
∑
wn(k)≥S
ws(k),
Y
�8��{ Ls � Ws � Pwait.
R NS(tO_� queue2.R)
queue2<-function(lambda, mu, T, S=2){
k<-0; wt<-0; wn<-0; ws<-0
tp<-0; nA<-0; t<-0
r<-runif(1); tA<--1/lambda*log(r)
tD<-rep(Inf, S); SS<-rep(0, S+1)
repeat{
t1<-if(SS[1]==0) Inf else min(tD)
i1<-if(SS[1]==0) 1 else which.min(tD)
k<-k+1; wt[k]<-t; wn[k]<-SS[1]
if (tA < T){
ws[k]<-min(tA, t1)-t
if (tA < t1){
t<-tA; nA<-nA+1
r<-runif(1); tA<-t-1/lambda*log(r)
n<-SS[1]; SS[1]<-n+1
for (i in 1:S){
if (SS[1+i]==0){
SS[1+i]<-1
r<-runif(1); tD[i]<-t-1/mu*log(r)
break
}
594 (q� R#F~�
}
}else{
t<-t1; n<-SS[1]; SS[1]<-n-1
if (n==1){
SS[2:(S+1)]<-0; tD[1:S]<-Inf
}else if (n<=S){
SS[1+i1]<-0; tD[i1]<-Inf
}else{
r<-runif(1); tD[i1]<-t-1/mu*log(r)
}
}
}else{
ws[k]<- if( t1==Inf) 0 else t1-t
n<-SS[1]
if (n>0){
t<-t1; SS[1]<-n-1;
if (n==1){
SS[2:(S+1)]<-0; tD[1:S]<-Inf
}else if (n<=S){
SS[1+i1]<-0; tD[i1]<-Inf
}else{
r<-runif(1); tD[i1]<-t-1/mu*log(r)
}
}else
tp<-1
}
if (tp==1) break
}
data.frame(Ls=sum(ws*wn)/t, Ws=sum(ws*wn)/nA,
Pwait=sum(ws[wn>=S])/t)
}
� 10.16 h�I�U 3 |��e�!�ou\f$�Ir℄U 10 \%�G\f
10.4 ar
[6~Q�o,�� 595
$ �aUo�r 15 f��P~�$OL:e�I�\f$B{ (Ls) �\f$
!�9Wr℄ (Ws) �\f%�$fa (Pwait).
`�;�:<,tO queue2.R, �O+�,R#bA�ar 1000 7y,~
Qt�_,1G1�
> source("queue2.R")
> queue2(lambda=15, mu=6, T=1000, S=3)
Ls Ws Pwait
1 5.980315 0.4010408 0.7002678
�
�`y Ls = 6.011236( ), Ws = 0.4007491(7y). Pwait = 0.7022472.
10.4.4 �h��7�7y(L
=ui~QaE[e�y M/M/S/S, ! S �t�0��M�$r
B�
;�
iBi~QaE[e�y M/M/S/K, |� S �tÆCmtÆ%��_u
�L�y K(K ≥ S), ! K �~hq0$r��y�?&,$r
B�;�!�
_o�u~hy�?&,$rBO�_~Q.��! K = S y�iBi~Qa
ESj[p=ui~QaE�
M�d�{iBi~QaE,ar1�~y! K = S y�S =ui~
QaE,1�6#X�{.�iaE,arM�iBi~QaE,arS��
T!�dLR#X,tO�7,J��6!#�_$r#�& K y�;?&,
$r
B�f��AtOJ>�
�X�{+�,8[�tO�w�R�=ui�iBi~QaE�|'A
Qf (Ls) ��a.�y� (Ws) m�`h'A�_,$r=u= (Plost).
1. S = 1 fJ} (M/M/1/K)
(0�� III
(1) z�L�h t = NA = 0, ao$r&��_,z�y� T0, h tA = T0,
tD =∞(�y�_o�$r). h k = 0.
(2) �8�_�E�h k = k + 1, wt(k) = t, wn(k) = n. N2 tA ≤ T , ;h
ws(k) = min(tA, tD)− t,
596 (q� R#F~�
BM� (3); p;h
ws(k) =
{
0, tD =∞,
tD − t, tD <∞,
BM� (9).
(3)N2 tA < tD, ;h t = tA, NA = NA+1($r&��# +1), n = n+1(�
_o$r# +1), ao�l$r&��_,y� tA.
(4) N2 n = 1, aotÆC_$r,�fy� tD.
(5) N2 n = K(!#$r�&�_L�), �N����
R tA < tD(?$r6q0tÆ,$r�f#&�), ;ao�l$r&�
�_,y� tA(~yM�?$rLh�f), ℄f tA ≥ tD y
�
(6) N2 tA ≥ tD, ;h t = tD, n = n− 1(�_o$r# −1).
(7) N2 n = 0(�_o�$r), h tD = ∞; p;aotÆC_$r,�f
y� TD.
(8) � (2).
(9) (�y tA ≥ T , J40�?$r�dnp�_o$r,tÆ). N2 n >
0(�_o`�$r), Fh t = tD, n = n − 1(�_o$r# −1). N2 n > 0, a
otÆC_$r,�fy� TD, BM� (2); p;� (10).
(10) �8Qf (Ls) ��aE0y� (Ws) ��_,$r=u= (Plost).
Ls =
1
t
∑
k
ws(k) · wn(k),
Ws =
1
NA
∑
k
ws(k) · wn(k),
Plost =
1
t
∑
wn(k)≥K
ws(k),
Y
�8��{ Ls � Ws � Plost.
R NS(tO_� queue3.R)
queue3<-function(lambda, mu, T, K=1){
k<-0; wt<-0; wn<-0; ws<-0
tp<-0; nA<-0; n<-0; t<-0
r<-runif(1); tA<--1/lambda*log(r); tD<-Inf
10.4 ~�OLu�Bd$$KP 597
repeat{
k<-k+1; wt[k]<-t; wn[k]<-n
if (tA < T){
ws[k]<-min(tA, tD)-t
if (tA<=tD){
t<-tA; n<-n+1; nA<-nA+1
r<-runif(1); tA<-tA-1/lambda*log(r)
if (n==1){
r<-runif(1); tD<-t-1/mu*log(r)
}
if (n==K){
while (tA < tD){
r<-runif(1); tA<-tA-1/lambda*log(r)
}
}
}else{
t<-tD; n<-n-1
if (n==0){
tD<-Inf
}else{
r<-runif(1); tD<-t-1/mu*log(r)
}
}
}else{
ws[k]<-if(tD==Inf) 0 else tD-t
if (n>0){
t<-tD; n<-n-1
if (n>0){
r<-runif(1); tD<-t-1/mu*log(r)
}
}else
598 4wC �8rar
tp<-1
}
if (tp==1) break
}
data.frame(Ls=sum(ws*wn)/t, Ws=sum(ws*wn)/nA,
Plost=sum(ws[wn>=K])/t)
}
� 10.17 h�;+2{�!�o\%U 0.6 �)7�To�A2r℄!�U 1.25
\%��P~�$OL:eoD$B{ (Ls) �!�9Wr℄ (Ws) �oD$'
na (Plost).
`�;�:<,tO queue3.R, �O+�,R#bA�ar 10000 kp,~
Qt�_,1G1�
> source("queue3.R")
> queue3(lambda=0.6, mu=0.8, T=10000)
Ls Ws Plost
1 0.4289211 1.259454 0.4289211
�
�`y Ls = 0.4285714(�), Ws = 1.25(kp). Plost = 0.4285714.
� 10.18 �>K,�U 1 |>Ke�Ey)Uz�,?GE"N� 4 |�%�
[h3>K$�%B Poisson �� ��!� �aUo�r 6 H�>Kr℄a
���\g�!� 12 \%"U 1 |�%>K��P~�$OL:eoD$B{
(Ls) �!�9Wr℄ (Ws) �oD$'na (Plost).
`�ar 1000 7y,~Qt�_,1G1�
> queue3(lambda=6, mu=5, T=1000, K=4)
Ls Ws Plost
1 2.364356 0.5412132 0.2718579
�
�`y Ls = 2.359493(E), Ws = 0.5451565(7y). Plost = 0.2786498.
2. S > 1 fJ} (M/M/S/K)
(0�� IV
(1) z�L�h t = NA = 0, ao$r&��_,z�y� T0, h tA =
T0, tD(i) = ∞, i = 1, 2, · · · , S(�y�_o�$r). SS(i) = 0, i = 1, 2, · · · , S +
10.4 ar
[6~Q�o,�� 599
1(SS(1) �8�_!#�E,$r#� SS(2 ∼ S +1) �8 S �tÆC,���
E� 0 yu"� 1 y��), h k = 0.
(2) N2 SS(1) = 0, ;h t1 = ∞, i1 = 1; p;h t1 = min(tD), i1 =
argmin(tD).
(3) �8�_�E�h k = k + 1, wt(k) = t, wn(k) = n. N2 tA < T , ;h
ws(k) = min(tA, t1)− t,
BM� (4); p;h
ws(k) =
{
0, t1 =∞,
t1 − t, t1 <∞,
BM� (12).
(4) N2 tA < t1, ;h t = tA, NA = NA + 1($r&��# +1), ao�l$
r&��_,y� TA. h n = SS(1), SS(1) = n+ 1(�_o$r# +1).
(5) R� i = 1, 2, · · · , S, N2 SS(1 + i) = 0(4 i �tÆCu"), ;h
SS(1 + i) = 1(%�_o,$rk��4 i �tÆC�f�tÆ), ao4 i �t
ÆC_$r�f,y� TD(i), BMo
W_�
(6) N2 SS(1) = K(!#$r�&�_L�), �N����
h t1 = min(tD). R tA < t1(?$r6q0tÆ,$r�f#&�), ;a
o�l$r&��_,y� tA(~yM�?$rLh�f), ℄f tA ≥ t1 y
�
(7) N2 tA ≥ t1, ;h t = t1, n = SS(1), SS(1) = n− 1(�_o$r# −1).
(8) N2 n = 1(�_o�$r), h SS(1 + i) = 0, tD(i) =∞, i = 1, 2, · · · , S.
(9) N2 n ≤ S, h SS(1 + i1) = 0, tD(i1) =∞(4 i1 �tÆCu").
(10) N2 n > S, ao$r�f4 i1 �tÆC,y� TD(i1).
(11) � (2).
(12) (�y tA ≥ T , J40�?$r�dnp�_o$r,tÆ). h n =
SS(1). N2 n > 0, ;h t = tD, SS(1) = n− 1(�_o$r# −1), BM� (8);
p;� (13).
600 (q� R#F~�
(13) �8Qf (Ls) ��aE0y� (Ws) �$r.�,�= (Plost).
Ls =
1
t
∑
k
ws(k) · wn(k),
Ws =
1
NA
∑
k
ws(k) · wn(k),
Plost =
1
t
∑
wn(k)≥K
ws(k),
Y
�8��{ Ls � Ws � Plost.
R NS(tO_� queue4.R)
queue4<-function(lambda, mu, T, S=1, K=1){
if (K0){
t<-t1; SS[1]<-n-1;
if (n==1){
SS[2:(S+1)]<-0; tD[1:S]<-Inf
}else if (n<=S){
SS[1+i1]<-0; tD[i1]<-Inf
}else{
r<-runif(1); tD[i1]<-t-1/mu*log(r)
}
}else
tp<-1
602 (q� R#F~�
}
if (tp==1) break
}
data.frame(Ls=sum(ws*wn)/t, Ws=sum(ws*wn)/nA,
Plost=sum(ws[wn>=K])/t)
}
� 10.19 �z|$F-V�$
U 9 |V�z�EUy'z �$
GE"
?N� 12 .�2V�$hV�[h��$hVB Poisson �� ��!�o:
4 .�V�hVa���\g�o.hV!��2 2 :r℄��P~�$OL:
eoD$B{ (Ls) �!�9Wr℄ (Ws) �oD$'na (Plost).
`�;�:<,tO queue4.R, �O+�,R#bA�ar 1000 S,~Q
t�_,1G1�
> source("queue4.R")
> queue4(lambda=4, mu=1/2, T=1000, S=9, K=12)
Ls Ws Plost
1 7.736918 2.148876 0.08801383
�
�`y Ls = 7.872193(C), Ws = 2.153466(kp). Plost = 0.08610186.
4�k
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x$Æ/\g��%�g 1 ∼ 4 fb�$faU
1 f� 0.5, 2 f� 0.2, 3 f� 0.2, 4 f� 0.1.
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10.5 BrO;f~�>K,oDzJ8/80�
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b 10:00 �p��pr`�%%��
(2) v�%~_3Æ,>KM r3Æ,>K\*H`��5IBIu�%
�$r℄℄ta���U 4 \%$��\g�
(3) �%$U 60% $H{bK� 40% $�2nK�bK��K�
(4) aeeZU8Y�%bK$r℄a� 6 \\%$��\g�nK�bK
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g���\`U 5 \%� 7.5 \%�
(5) ��% �r�OKx=U 6 Y�%Æu�B%�ae�xS/%�
=
?��
(6) oYaeef℄�'U 4 Y�%ae(:2�k 1 \%�
(7) >K,O 8:00 (���%|P�uN�, $�%ae(�p�
�~�u8:$qL$�3,�%$B{ (Ls) �!�9Wr℄ (Ws) �>K,
$'na (Plost).
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