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Appendix A
Laplace-Stieltjes Transforms
A.1 Laplace-Stieltjes Transforms
Let F(t} be a well-defined function of t specified for t ~ 0 and s be a complex number. If the following Stieltjes integral:
(ALI)
converges on some So. the Stieltjes integral (A.LI) converges on s such that ~(s) > ~(so). The integral (A.LI) is called the Laplace-Stieltjes transform of F(t). If the real function F(t) can be expressed in terms of the following integral:
F(t) = lot dF(x) = lot f(x)dx, (AL2)
then
(AL3)
which is called the Laplace transform of f (t). Noting that F(t) is in one-t~one correspondence with F*(s) (ref. Theorem
2.2.2 and Table 2.2.2), F(t) can be uniquely specified by F*(s). The inversion formula for obtaining F{t) from F*{s) can be given by
1 l b+ic eat F(t) = lim -. -F*(s)ds, c-oo 211"z b-ic S
(AI.4)
where i = A is an imaginary unit, b > ma.x(u,O} and u is a radius of convergence.
The following two theorems are well-known and of great use as the limit theorems for the Lapla.ce-Stieltjes transform F*(s) of F(t).
242 APPENDIX A. LAPLACE-STIELT JES TRANSFORMS
Theorem A.l (An Abelian Theorem) If for some non-negative number a,
lim F(t) = C t-+oo to. r(a+1)'
(A.l.5)
then
lim so. F*(s) = C, 8-++0
(A.l.6)
where r(k) = 1000 e-xxk-1dx is a gamma function of order k defined in Eq.(2.4.13).
Theorem A.2 (A Tauberian Theorem) If F(t) is non-decreasing and the LaplaceStieltjes transform
F*(s) = 1000 e-8tdF(t) (A.l.7)
converges for ~(s) > 0, and if for some non-negative number a,
lim so. F*(s) = C, 8-++0
(A.l.8)
then
lim F(t) = C . t-+oo to. r(a+1)
(A.l.9)
Exampe A.l.l (Elementery Renewal Theorem) The Laplace-Stieltjes transform of the renewal function is given in Eq.(4.2.15):
M*( ) = F*(s) s 1 - F*(s)'
where F*(s) is the Laplace-Stieltjes transform of F(t). Let us apply Theorem A.2.
We have
lim sM*(s) = lim F*(s) 8-++0 8-++0 [1 - F*(s)]/s
since
and
lim F*(s) = l. 8-+0
1 J.£'
That is, from the Tauberian Theorem, we have
lim M(t) = .!:.. t-+oo t J.£
A.2. PROPERTIES OF LAPLACE-STIELT JES TRANSFORMS 243
A.2 Properties of Laplace-Stieltjes Transforms
For the Laplace-Stieltjes transform, we have the following relationship:
(A.2.1)
That is, the Laplace-Stieltjes transform F* (s) can be obtained by s times the Laplace transform of F(t). We can easily obtain the Laplace-Stieltjes transforms from the corresponding Laplace transforms, since most textbooks only discuss the latter.
Table A.l shows the the general properties of the Laplace-Stieltjes transforms. The general properties in Table A.l can be applied in practice. Besides the general properties in Table A.l, Table A.2 shows the important formulas for the Laplace-Stieltjes transforms. Such formulas in Table A.2 can be applied to derive the Laplace-Stieltjes transform F*(s) from F(t), and vice versa.
Table A.l General properties of the Laplace-Stieltjes transforms.
F(t) F*(s) = 1~ e-stdF(t)
Fl(t) + F2 (t) F;(s) + F;(s)
aF(t) aF*(s)
F(t - a) (a> 0) e-saF*(s)
F(at) (a> 0) F*(s/a)
e-at F(t) (a> 0) s
--F*(s+a)
F'(t) = dF(t) s+a s[F*(s) - F(O)]
dt tF'(t) dF·(s) -s ds
l F(x)dx !F*(s) tOt s
10 ···10 F(t)(dtt ~F*(s) sn lim F(t) t-++O lim F*(s)
s-+oo
lim F(t) t-+oo
lim F*(s) s-++O
244 APPENDIX A. LAPLACFrSTIELT JES TRANSFORMS
Table A.2 Formulas of the Laplace-Stieltjes Transforms
F(t) F·(s) = LX) e-8tdF(t)
6(t - a)t (a > 0) se-SO
l(t - a)* (a> 0) e-SO
l(t) 1
t 1 -s
tn (n : a positive integer) n! -sn
ta (a> -1) r(a + 1)
sa
e-at (a> 0) s --
s+a te-at (a> 0)
s (s + a)2
tne-at (a> 0) n!s
(s + a)n+1
tfje-at (a> 0,{3 > -1) sr({3 + 1) (s + a)fj+1
S2 (!R(s) >1 a I) cos at
S2 +a2
sin at sa
(!R(8) >1 a I) 82 +a2
cosh at 82
(!R(8) >1 a I) S2 - a 2
sinh at sa
(!R(8) >1 a I) 82 - a 2
logt -"'( -logs §
t Dirac's delta function. + Heaviside's unit function. § "'( = 0.57721 ... , Euler's constant.
A.3. APPLICATIONS TO DISTRiBUTIONS 245
A.3 Applications to Distributions
We discuss the Laplace-Stieltjes transforms of the distributions. As shown in Section 2.3, we have introduced six common discrete distributions. We derive the Laplace-Stieltjes transforms for a few discrete distributions.
Example A.3.1 (Binomial distribution)
Fi(s} = ~ e-s:c (:) p:cqn-:c = (pe-s + qt . (A.3.1)
Applying the formulas for the moments, we have
E[X] = (_I}dFx(s) I = np, ds s=o
(A.3.2)
cPF*(S}1 E[X2] = (_1)2 x2 = n(n - l}p2 + np, ds s=o
(A.3.3)
which imply
Var(X) = E[X2] - E[X]2 = npq. (A.3.4)
Example A.3.2 (Geometric distribution)
(A.3.5)
Example A.3.3 (Negative binomial distribution)
F*(s}=~e-s:c(X-l) r :c-r= [ pe-s ]r X ~ X - r p q 1 - qe-S
(A.3.6)
As shown in Example 2.3.2, if Xl, X 2 , ••• ,Xr are independent and identically distributed random variables with Xi '" GEO(p) , the random variable Sr = Xl + X 2 + ... + Xr is distributed with the negative binomial distribution Sr '" N B(p, r). Here we have verified this fact by using the Laplace-Stieltjes transforms.
We next show the Laplace-Stieltjes transforms for the continuous time distributions in Section 2.4.
Example A.3.4 (Exponential distribution)
F.i(s} = t JO e-stdFx(t} = roo e-st ~e->'tdt = _~_. k k s+~
(A.3.7)
Applying the formulas for the moments, we have
E[X] = (_I}dFx(s) I = l/~, ds s=O
(A.3.8)
246 APPENDIX A. LAPLACE-STIELT JES TRANSFORMS
which imply
2 1 1 Var(X) = >.2 - >.2 = >.2'
Example A.3.5 (Gamma distribution)
Applying the formulas for the moments, we have
k Var(X) = >.2'
Example A.3.6 (Equilibrium distribution)
(A.3.9)
(A.3.1O)
(A.3.11)
(A.3.12)
(A.3.13)
In Chapter 4, we have introduced the equilibrium distribution in Eq.( 4.3.44):
1 lot Fe(t) = - [1 - F(y)] dy, J.t 0
(A. 3.14)
where J.t is the mean of F(t). The Laplace-Stieltjes transform of Fe(t) is given by
F;(s) = fo'X) e-st dFe(t)
= ~ t'" e-st d [1 - F(t)] (see Table A.l) J.tS io
=~[I-F*(s)]. J.tS
(A.3.15)
A.4 Applications to Differential Equations
Example A.4.1 (Example 6.4.5) We have discussed a two-state Markov chain whose Kolmogorov's forward equations are given by
P~o(t) = ->.Poo(t) + J.tP01(t),
P~l(t) = >.Poo(t) - J.tPOl(t),
(A.4.1)
(A.4.2)
AA. APPLICATIONS TO DIFFERENTIAL EQUATIONS
with the initial conditions that Poo(O) = 1 and POI (0) = o. Let
P;is) = 1000 e-st dPoj(t)
247
(A.4.3)
be the Laplace-Stieltjes transforms of POj(t) (j = 0,1). Noting the initial conditions and using Table A.l, we have the Laplace-Stieltjes transform expressions for Eqs.(A.4.1) and (A.4.2):
sP;O(s) - s = -AP;o(s) + JLP;1(S),
SP;1(S) = AP;o(s) - JLP;1(S),
whose solutions are given by
P.* ( ) s + JL s + _JL_. A + JL 00 s = s+A+JL s+A+JL A+JL s+A+JL'
P.*() A A A+JL 01 s = S + A + JL - A + JL . s + A + JL .
(A.4.4)
(A.4.5)
(A.4.6)
(A.4.7)
Applying the formulas in Table A.2, we have the following inversions of the Laplace-Stieltjes transforms:
POO(t) = _JL_ + _A_e-(>'+I')t, A+JL A+JL
(A.4.8)
( ) A A -(>.+)t POI t = -- - --e 1', A+JL A+JL
(A.4.9)
which have been given in Example 6.4.5.
Example A.4.2 (M/M/2/2/2 queueing modeD We have discussed an M/M/c/c/c queueing model with finite population in Section 9.4 in general. We restrict ourselves to a case of c = 2 (i.e., 2 machines). Let QOj(t) be the probabilities that (2 - j) machines are operating at time t given that 2 machines are operating at t = 0, where j = 0,1,2. Kolmogorov's forward equations are given by
Q~O(t) = -2AQoo(t) + JLQ01 (t),
Q~1(t) = 2AQoo(t) - (A + JL)Q01(t) + 2JLQ02(t) ,
Q~2(t) = AQOl(t) - 2JLQ02(t),
with the initial conditions Qoo(O) = 1 and QOj(O) = 0 (j = 1,2). Let
Q~j(s) = fooo e-st dQoj(t)
(A.4.1O)
(A.4.11)
(A.4.12)
(A.4.13)
be the Laplace-Stieltjes transforms of QOj(t) (j = 0,1,2). The Laplace-Stieltjes transform expressions for Eqs.(A.4.lO), (A.4.l1), and (A.4.12) are given by
sQ~(s) - s = -2AQ~(s) + JLQ~1(S), (A.4.14)
248 APPENDIX A. LAPLACE-STIELT JES TRANSFORMS
SQ~l (s)
SQ~2(S)
= 2AQ~O(S) - (A + /-l)Q~l(S) + 2/-lQ~2(S),
= AQ~l(S) - 2/-lQ~2(S).
Solving with respect to Qoo(s), we have
* S2 + (A + 3/-l)s + 2/-l2 Qoo(s) = (s + A + /-l) [s + 2(A + /-l)]
= 1- 2A/-l A + /-l (A + /-l)2 s + A + /-l
A2 2(A+/-l) (A + /-l)2 . S + 2(A + JL)"
(A.4.15)
(A.4.16)
(A.4.17)
where the last equation has been derived by applying the partial fraction expansion (or decomposition). Applying the formulas in Table A.2, we have the following inversion of the Laplace-Stieltjes transform in Eq.(A.4.17):
(A.4.18)
where Poo(t) has been given in Eq.(A.4.8). Of course, Eq.(A.4.18) can be easily derived by considering two MIMl11111 queueing models in parallel (cf. Section 9.4). Similarly, we have the following Laplace-Stieltjes transforms:
(A.4.19)
2A2 Q~2(S) = (s + A + JL) [s + 2(A + JL)]' (A.4.20)
which imply
Q01(t) = 2POO (t)P01 (t), (A.4.21)
(A.4.22)
where Poo(t) and P01 (t) have been given in Eqs.(A.4.8) and (A.4.9), respectively.
A.5. APPLICATIONS TO RENEWAL FUNCTIONS 249
A.5 Applications to Renewal Functions
In Chapter 4 we have developed renewal processes. Let {N(t), t ~ O} be a renewal process with interarrival distribution F(t). The renewal function is given by
00
M(t) = E [N(t)] = L F(n)(t). (A.5.I) n=l
The Laplace-Stieltjes transform of MCt) is given by
M*(s) = F*(s) 1 - F*(s) , (A.5.2)
where F*Cs) is the Laplace-Stieltjes transform of F(t).
Example A.5.1 If we assume that F(t) obeys a gamma distribution, i.e., Xi '" GAM('x, k) in general (k is a positive integer), we have the LaplaceStieltjes transforms of F(t) and M(t), respectively:
F*(s) = c~'xr, (A.5.3)
M * ( ) _ (sh) k _ 1 s - k - k·
1 - (8~>') (1 + f) - 1 (A.5.4)
Let €r = e¥ be all the distinct roots of an equation Sk = 1, where r = 0,1,2,···, k - 1. By applying the partial fraction expansion, we have
M*Cs) = .! E 'x€r . kr=os+,X(I-€r)
By inversion, we have
,Xt 1 k-l € MCt) = - + - L _r_ [1- e->.t(l-er )] ,
k k r=l 1 - €r
which has been given in Example 4.2.3.
Example A.5.2 If we assume that
F(t) = 1 - ~ (e->.t + e-2>'t) ,
we have the Laplace-Stieltjes transform of F(t):
rcs) = 1 _ .! (_S_ + _S_) . 2 s +,X s + 2,X
CA.5.5)
(A.5.6)
(A.5.7)
CA.5.8)
250 APPENDIX A. LAPLACE-STIELT JES TRANSFORMS
The Laplace-Stieltjes transform of M(t) is given by
M*(s) = F*(s) 1 - F*(s)
3>.s + 4>.2 s(2s + 3>')
= 4>. + !. 3>./2 3s 9 s + 3>./2
(A.5.9)
by applying the partial fraction expansion. By inversion (i.e., applying the formulas in Table A.2), we have
4>.t 1 ( a) M(t) = """3 + 9 1 - e-"2 At . (A.5.1O)
Example A.5.3 If we assume that F(x) obeys a negative binomial distribution of order 2 (refer to Section 2.3), we have the following Laplace-Stieltjes transform:
F* (s) = (--'pe=---_S_) 2
1 - qe-S (A.5.11)
The Laplace-Stieltjes transform M*(s) of the renewal function M(x) is given by
M*(s) = F*(s) 1 - F*(s)
= pe2-s [l_le_s - 1- (1 ~ 2P)e-s ]
= f: !!. [1 - (1 - 2ptl e-s(n+l), n=12
which implies the (discrete) renewal density
m(x) = ~ [1- (1- 2p)X-l] (x = 2,3,·· .).
That is, the (discrete) renewal function is given by
00
M(x) = Lm(j) j=2
= p; - ~ + (1-42P)X (x = 2,3,···)
(see Problems 4.4 and 4.5).
(A.5.12)
(A.5.13)
(A.5.14)
Appendix B
Answers to Selected Problems
Problems 1
1.2
(i) ABccc, (ii) Au B u C, (iii) AB u AC U BC,
(iv) ABC, (v)ABCCC U AC BCc uN BCC,
(vi) ABc U ABcC U AC BC = (AB U AC U BC) - ABC,
(vii) NWCc, (viii) (ABC)c.
1.5
(i) P{A I B} = 1/3, (ii) P{B I A} = 1/4, (iii) P{A - B} = 1/4,
(iv) P{B - A} = 1/6.
1.9
(i) (3x2 - 2y)3 = 27x6 - 54x4 y + 36x2y2 - 8y3,
(ii) (4x + 3y2)3 = 64x3 + 144x2y2 + 108xy4 + 27y6.
1.10
(51°) (~) = 921,200 ways.
252 APPENDIX B. ANSWERS TO SELECTED PROBLEMS
1.12
(ii) 27 ways.
1.13
(ii) 20 ways.
1.14
n r ( distinguisable ),
1.15
Problems 2
2.7
2.14
n 1 2 3 4 5 6 7 8 9 10 11
(1') E[X] 1 = \ \' 1\1 + 1\12
( n+sS-l) (indistinguisable) .
Theoretical values Data 18.27 14 13.81 12 10.44 12 7.89 8 5.96 7 4.51 4 3.41 2 2.57 4 1.95 4 1.47 3 1.11 2
Problems 3
3.3
3.4
3.10
3.11
(i) t = 0 (9: 00 a.m.), t = 9 (6: 00 p.m.),
E[N(9)] = At = 90 persons,
Var(N{9)) = At = 90 persons2 •
(U) P{N(O.5) = O} = e-5 = 0.00674.
(i) P{N(30) = O} = e-5 ,
P{N(30) = 2} = 25e-5 2 .
P{N(30) = I} = 5e-5 ,
(ii) Upper limit 90 + 3V90 = 118.3 persons.
Lower limit 90 - 3V90 = 71.7 persons.
m(lO) = 440 persons, Variance = 440 persons2 .
F{Xl ~ t} = 1 - e-(at)/l (Weibull distribution).
Problems 4
4.3
4.4
Var(N(t)) = 2M * M{t) + M(t) - [M(t)]2
(i) F*(s) = pe- s /(1 - qe-S ),
(ii) M*(s) = pe-s /(1 - e-S ), M{m) = pm (m = 1,2, ... ).
253
254 APPENDIX B. ANSWERS TO SELECTED PROBLEMS
4.5
M(m) = pm _ ~ + (1 - 2p)m 2 4 4
(m = 2,3,·· .).
Problems 5
5.1
(i) 11'(2) = [0.68 0.32], 11'(4) = [0.6672 0.3328], 11'(8) = [0.6667 0.3333].
(ii) 11'(2) = [0.66 0.34], 11'( 4) = [0.6664 0.3336], 11'(8) = [0.6667 0.3333].
(iii) 1I'(n) = [2/3 1/3] (n = 2,4,8).
5.2
5.4
5.5
5.8
(iii) pn = [1- hr an
where
(n = 1,2,3,· .. ).
(iii) One class {O, 1, 2}, recurrent and aperiodic.
(ii) i 2
Pi,i-i = N2' i(N - i)
Pii = 2 N2 ' (N - i)2
Pi,i+1 = N
(i=O,1,2,···,N), Pi,i-i + Pii + pi,i+1 = 1.
(iii) One class {O, 1,2,···, N}, recurrent and aperiodic.
Pi: {O, 1, 2} recurrent.
P 2 : {I} recurrent (absorbing), {O,2} transient.
P 3 : {O, I} recurrent, {2} recurrent (absorbing), {3} transient.
255
5.9
PI: {O, 1, 2} recurrent, periodic with period 3.
P 2 : {O, 1, 2} recurrent, aperiodic.
P 3 : {O, 1,2,3, 4} recurrent, aperiodic.
5.11
(iii) PI: 11'0 = 11'1 = 1/2.
P 2 : 11'0 = 11'1 = 1/2.
P 3 : 11'0 = 11'1 = 1/2.
(iv) r pn [1/2 n.!..rr;., i = 1/2 1/2] 1/2
(i = 1,2).
(v) lim .! fpi = [1/2 1/2] n-HX) n . 3 1/2 1/2 .
0=1
5.12
[1/3 2/3 0 0
o 1 1/3 2/3 0 0 0 lim pn = 0 0 1 0 o . n-+oo
3/5 2/5 0 0 0 0 0 0 3/5 2/5
5.13
2' --+ 0, 4' --+ 1, 0' --+ 4, l' --+ 5, 3' --+ 2, 5' --+ 3.
0 1/3 2/3 0 0 0 0 1 1/2 1/2 0 0 0 0
p=2 0 0 2/3 1/3 0 0 3 0 0 1/2 1/2 0 0 4 0 1/4 0 0 1/4 1/2 5 0 0 3/4 0 1/4 0
256 APPENDIX B. ANSWERS TO SELECTED PROBLEMS
A state transition diagram of Problem 5.13.
0 3/7 4/7 0 0 0 0 1 3/7 4/7 0 0 0 0
r pn 2 0 0 3/5 2/5 0 0 1m = 0 0 3/5 3/5 0 n-+oo 3 0
4 6/35 8/35 9/25 6/25 0 0 5 3/70 4/70 27/50 18/50 0 0
5.15
o [0 p 0
~l p = 1 q 0 P 2 0 q 0 3 p 0 q
Doubly stochastic (irreducible, recurrent, periodic with period 2).
5.16
1 0 ~ 0 ; 0 q
0 1 0 ~ 0 ~ lim p2n = (1 - ~)
q q q
1 0 ~ 0 ; 0 n-+oo q q
0 1 0 ~ 0 ~ q q q
257
0 !. 0 ~ 0 ~ q q
1 0 ~ 0 ; 0 q
lim p2n+1 = (1 - p) 0 !. 0 ~ 0 ~ q q q n--+oo q
1 0 ~ 0 ~ 0 q4
Problems 6
6.1
M(t) = e>.t.
6.2 1 1 1
E[N(t) = n) = ~ + 2,\ + ... + n'\)
1 1 1 Var(N(t) = n) = ,\2 + 22,\2 + ... + n2,\2'
6.3
1 1 1 E[X(t) = kJ = n'\ + (n - 1)'\ + ... + (n - k + 1)'\)
1 1 1 Var(X(t) = k) = n2,\2 + (n _ 1)2,\2 + ... + (n _ k + 1)2,\2·
6.4
V · . 2(>,-,,)t ,\ + J.I. [1 e-(>'-/J.)t] anance = ze .... -- - . ,\-J.I.
6.5
{I (,\ < J.L)
(iv) lim PlO(t) = r (,\ > J.L) t--+oo 0 (,\ = J.L).
258 APPENDIX B. ANSWERS TO SELECTED P8,OBLEMS
6.8
6.9
(i) M'(t) = a + (A - p,)M(t), M(O) = i.
{ ie(.~-I')t + I'~~ (A I: p,)
(ii) M(t) = at + i (A == p,).
{ I'~~ (A < p,)
(iii) lim M(t) = t--+oo
00 (A ~ p,),
6.10
a(a + A)(a + 2A) , " [a + (j - I)A] (J' = 1,2", ,), Pi = j! p.i Po
= [1 ~ a(a + A)(a + 2A) , " [a + (j - I)A]]_l Po +~ " .
;=1 J, p.l
6.11
(i) ap~,s) +p,(S_I)ap~:,s) =A(s-I)P(t,s),
00 [..\ (1 - e-I't)]i (iii) P(t ) =" I' -t(1-e-I'I). j ,s ~ "e s,
j=O J,
[ ~ (1 e-I't)]i n (t) ;; - -4(I-e-I'I) ( 1 2 ) rOj = " e I' j = 0, , , ....
J,
( ~ . ( ') }' n (t) ;;}' _4 (P . d' 'b ' ) IV 1m rOj = -,-e I' Olsson lStn utlOn ,
t--+oo J!
6.12
1 (A)i _4 Pi = j! ~ e I' (j = 0,1,2", .), Poisson distribution,
6.13
6.14
[-2A
(i) P/(t) = P(t)A = pet) ~
2
[ " + Ae-(~+#')t] (ii) R (t) ,-
00 = A+f.t '
2A -(A + f.t)
2f.t o 1 A "
-2f.t
lJi. + Ae-(~+#')t] [A - Ae-(~+#')t] P01 (t) =2 , \+' , + f.t 1\ f.t
[ A - Ae-(H#')t1 2 P02 (t) = ---.,.\--
I\+f.t
(iii) lim Poo(t) = (~) 2 , t-+oo 1\ + f.t
lim POl(t) = 2 (~) (~), t-+oo 1\ + f.t 1\ + f.t
lim P02 (t) = (~)2 t-+oo 1\ + J.t
(1"1") Po -_ (1 + 2A + 2A2)-1 2A f.t f.t2 ,PI = /iPo,
Problems 7
7.1
(i) Q(oo) = [Oi4 0"6] o "
(ii) 5f.t 3A Po = 3A+5f.t' PI = 3A + 5f.t "
7.2
(ii) f.t A Po= A+f.t' PI =--"
A+f.t
259
260
7.3
7.5
APPENDIX B. ANSWERS TO SELECTED PROBLEMS
(ii) 11"0 = 11"1 = 11"2 = 1/3.
(iii) Po = (1 + ~ + ~) -1 ,
J.Ll J.L2
(ii) M*(s) = 1 _ F*tS) G*(S) l G*~~fs;(S) F*7s;~~(s)]' (iii) G~(s) = Gil(S) = G*(s)F*(s),
G~l(S) = G*(s), Gio(s) = F*(s),
P*(t) = [Poo(S) Pi1(S)] Pio(s) Pll(s)
1 [1 -G*(s) = 1 - F*(s)G*(s) F*(s)[l - G*(s)]
G*(s)[l - F*(S)]] 1- F*(s) .
Problems 8
8.2
8.3
(i)
(ii)
P{min(X1, X 2 ,'" Xn) > t} = F 1(t)F2 (t) .. · F n(t)
= exp [- t ri(t)] . • =1
1 1 E[X] = -n-' Var(X) = )2' EAi (~Ai
8.6
8.7
261
{ ~ [e-A(HZ) _ e-(A+I')t] + ~ [e-AZ - e-A(HZ)] (A :f J.t)
R1(x,t) = Ate-A(Hz) + le-AZ -le-A(t+z) (A = J.t).
lim Ro(x,t) = ~e-AZ, t-+oo 1\ + J.t
Problems 9
9.1
9.2
9.5
(i)
(ii)
(iii)
(iv)
(i)
(ii)
(iii) (iv)
(i)
(ii)
(iii)
P{The attendant is busy} = p = 3/4.
p2 1 Lq = -- = 2-4 persons.
1-p
Wq = P = ~ hour = 3 min. J.t(1- p) 20
P { U > 115} = ~e-l = 0.276.
P{The secretary is busy} = p = 0.8.
L=~ =4jobs. -p W = L/ A = 40 min.
P{U + V ~ 40} = 1 - e-1 = 0.632.
P{The dentist is busy} = 1 - Po = 0.6346.
L = 1.423 patients.
Wq = 23.64 min.
262 APPENDIX B. ANSWERS TO SELECTED PROBLEMS
9.8
A U = - = 1.0, 2.0 (0.1)
f..£
U P4 1.0 0.0154 1.1 0.0201 1.2 0.0262 1.3 0.0324 1.4 0.0396 1.5 0.0479 1.6 0.0563 1.7 0.0658 1.8 0.0747 1.9 0.0848 2.0 0.0950
A U = - ~ 1.5 ===> P4 = 0.0479 < 0.05 (5%).
f..£
Appendix C
The Bibliography
[1] A. O. Allen, Probability, Statistics, and Queueing Theory with Computer Science Applications, Academic Press, New York, 1978.
[2] L. J. Boon, Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods, Marcel Dekker, New York, 1978.
[3] R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965.
[4] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1975.
[5] U. N. Bhat, Elements of Applied Stochastic Processes, Wiley, New York, 1972.
[6] K. L. Chung, Elementary Probability Theory with Stochastic Processes, Springer-Verlag, New York, 1974.
[7] E. Qinlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.
[8] D. R. Cox, Renewal Theory, Methuen, London, 1962.
[9] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York, 1968.
[10] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., Wiley, New York, 1971.
[11] B. V. Gnedenko, Y. K. Belyayev, and A. D. Solovyev, Mathematical Methods of Reliability Theory, Academic Press, New York, 1969.
[12] P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Stochastic Processes, Houghton Mifflin, New York, 1971.
264 APPENDIX C. THE BIBLIOGRAPHY
[13] S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic Press, New York, 1975.
[14] S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Ac&demic Press, New York, 1981.
[15] J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York,1976.
[16] L. Kleinrock, Queueing Systems, Vol. I: Theory, Wiley, New York, 1975.
[17] L. Kleinrock, Queueing Systems, Vol. II: Computer Applications, Wiley, New York, 1976.
[18] J. Kohlas, Stochastic Methods of Operations Research, Cambridge University Press, Cambridge, 1982.
[19] R. G. Laha and V. K. Rohatgi, Probability Theory, Wiley, New York, 1979.
[20] N. R. Mann, R. E. Schafer, and N. D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York, 1974.
[21] S. Osaki, Stochastic System Reliability Modeling, World Scientific, Sing&pore, 1985.
[22] E. Parzen, Modem Probability Theory and Its Applications, Wiley, New York, 1960.
[23] S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.
[24] S. M. Ross, Stochastic Processes, Wiley, New York, 1983.
[25] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, AC8r demic Press, Orlando, 1984.
[26] W. A. Thompson, Jr., Point Process Models with Applications to Safety and Reliability, Chapman and Hall, New York, 1988.
[27] K. S. Trivedi, Probability & Statistics with Queueing, and Computer Science Applications, Prentice-Hall, Englewood Cliffs, New Jersey, 1982.
Index
A Abelian theorem, ................ 242 Actual arrival rate, ......... 226, 228 Age replacement models, .... 200-203 Age, .......................... 73,98 Arrival rate, ..................... 217 Arriving time, .................... 68 Availability, ................. 194-195
average, ....................... 194 joint, .......................... 195 limiting, ....................... 194 limiting average, ............... 194
Availability theory, .......... 192-199
B Elath-tub curve, ............. 187-188 Elayes'theorem, ................... 12 Elernoulli distribution, ......... 34-35 Eleta distribution, .............. 47-48 Elinomial coefficient, .............. 13 Elinomial distribution, . 35-36, 56, 245 Elirth and death process, ..... 143-155
definition, ..................... 144 linear growth process, ......... 148 linear growth with immigration, 163
Elirthday problem, ................ 17 Elivariate exponential distribution, 53 Elivariate joint distribution, ....... 49 Elivariate normal distribution, ..... 52 Ellackwell's theorem, .............. 94 Ellock diagram, .................. 147 Ellock replacement models, ......... .
200, 203-207
C Central limit theorem, ............ 58 Channel, ........................ 215 Chapman-Kolmogorovequation, .... .
108,136 Characteristic function, ........... 31
Chebyshev's inequality, ........... 57 Coincidences
(See Matches) Combinations, .................... 14 Combinatorial analysis, ........... 13 Communication relation, ......... 112 Conditional density, ............... 50 Conditional distribution, .......... 50 Conditional expectation, .......... 50 Conditional probability mass function,
50 Conditional probability,
definition, ....................... 8 Convolution, ...................... 54 Correlation coefficient, ............ 52 Counting process, ................. 63 Covariance, ....................... 52 Cumulative hazard, .............. 187 Current life, ................... 73, 98 Customers, ...................... 215
D Degenerate distribution, ..... 218-219 Density, .......................... 29 Dependability, ................... 195 DFR, ............................ 187 Directly Riemann integrable, ...... 95 Distribution,
continuous, ..................... 40 definition, ...................... 29 discrete, ...................... :. 33
Doubly stochastic matrix, ........ 133 Down time, ...................... 193
E Elementary renewal theorem, ..... 92 Embedded Markov chain, ........ 168 Empty set
(See Null set) Equally likely, ..................... 6
266
Equilibrium distribution, 98, 101, 246 Erlang's B formula, ......... 231, 234 Erlang's C formula, .............. 230 Erlang's loss formula
(See Erlang's B formula) Event, ............................. 1 Eventual transition probability, ..... .
114, 168 Excess life, .................... 73, 97 Expectation, ...................... 30 Expected cost rate, .............. 200 Expedited order, ................. 207 Exponential distribution, ........... .
42-44, 188, 245-246
F Failure rate, ................ 186, 233 Failure, .......................... 186
random, ....................... 186 First passage probability, ........ 114 Furry process, ................... 139
G Gamma distribution, ............... .
44-45, 56, 188, 246 Gamma function, ................. 44
Geometric distribution, ............ . 36-37, 191-192, 245
H Hazard function, ................. 187 Hazard rate,
(See Failure rate) Holding time, .................... 216 Hypergeometric distribution, ...... 21
I IFR, ............................. 187 Impossible set
(See Null set) Inclusion and exclusion formula
(See Poincare's theorem) Independent event
(See Mutually independent event) Independent increments, .......... 65 Independent trials, ................ 10 Indicator, ......................... 34 Infant phase, .................... 187 Infinitestimal generator, ..... 154, 157
INDEX
Initial distribution, .............. 107 Initial probability, ............... 107 Intensity function, ................ 77 Interarrival time, ............. 68,217 Interoccurrence times
(See Interarrival times) Inventory cost ................... 207 Item, ............................ 186
J Joint density, ..................... 49
Joint probability mass function, ... 49
K k-Erlang distribution, ........ 217-218 k-out-of-n system, ............... 212
Kendall's notation, .............. 220 Key renewal theorem, .......... 95-97 Kolmogorov's forward equation, .....
138, 142, 146, 153, 155 Kolmogorov's backward equation, ...
154, 155 Kurtosis, ......................... 32
L Laplace transform, ............... 241 Laplace-Stieltjes transforms, ........ .
32, 172, 241 Lifetime, ........................ 186 Little's formulas, ....... 225, 228, 236 Lognormal distribution, .. 47, 190-191
M Maintainability, .................. 192 Maintenance, .................... 192
corrective, ..................... 192 preventive, .................... 192 scheduled, ..................... 193
Marginal density, ................. 50 Marginal distribution, ............. 49 Marginal probability mass function, 49
INDEX
Markov chain, absorbing, ..................... 116 continuous-time, ............... 135 continuous-time finite-state, ... 155 discrete-time, .................. 105 ergodic, ....................... 119 finite-state, ................ 123-128 irreducible .................... 113 spatially homogeneous, ........ 111
Markov property, ................ 105 Markov renewal function, ........ 170 Markov renewal process,
alternating, ............... 17~181 definition, ..................... 167 stationary, .................... 177
Markov's inequality, .............. 56 Mass function, ................... 167 Matches, ......................... 19 Maximum queueing system capacity, .
219 MDT, ........................... 194 Mean
(See Expectation) Mean recurrence time, ........... 118 Mean value function, .............. 78 Moment generating function, ...... 32 MTBF, .......................... 194 MTBM, ......................... 194 Multinomial coefficient, ........... 20 Multinomial expansion, ........... 21 Multiplication principle, .......... 14 Multivariate distributions, ..... 48-56 MUT, ........................... 194 Mutually independent event, ... 9, 10
N ~fold (Stieltjes) convolution, .. 54, 84 ~step distribution, .............. 109 Negative binomial distribution, ..... .
37-39, 192, 245 Normal distribution, ...... 46-47, 190
standardized, ................... 47 nth moment, ...................... 31
about the mean, ................ 31 about the origin, ............... 31
Null set, ........................... 3 Number of service channels, ...... 219
267
o One-step transition probability, .. 167
Order statistics, ................. 212
Ordering models, ............ 207-211
p
Parallel system, .................. 212
Partition, ......................... 11
Pascal distribution
(See Negative binomial distribution)
Permutation, ..................... 14
Poincare's theorem, ............... 17
Point
(See Sample point)
Poisson arrivals, ................. 217
Poisson distribution, ... 39-40, 67, 192
Poisson process, ............... 65-68
decomposition, ................. 75
definition, ............... 66, 67, 69
nonhomogeneous, ............ 77-79
nonstationary, ............... 77-79
superposition, .................. 75
Population, .................. 216-217
finite, ................ 217, 233-238
infinite, ....................... 216
Probability distribution
(See Distribution)
Probability mass function,
definition, ...................... 29
Probability,
definition, ....................... 5
Pure birth process, .......... 136-141
definition, ..................... 137
Pure death process, .......... 141-143
with linear death rate, ......... 142
268
Q Queueing discipline, ......... 219-220 Queueing models, ............ 215-220
Infinite server Poisson queue, ... 72 M/M/1, ....................... 150 M/M/1/K/K, ............. 233-234 M/M/1/N, ........... 152,226-228 M/M/1/oo, ............... 221-226 M/M/1/oo queue with impatient cus-
tomers, ..................... . 164
M/M/2/2/2, .............. 247-248 M/M/c/c, ................. 230-231 M/M/c/c/c, ............... 237-238 M/M/c/K/K, ............. 235-236 M/M/c/oo, ............... 229-230 M/M/oo, ...................... 151 M/M/oo/oo, .............. 231-232 multiple server, ....... 219, 228-232 single server, ......... 219, 221-228
Queueing models with finite population, 233-238
R Random trial, ...................... 1 Random variable,
arithmetic, ..................... 94 continuous, ..................... 29 definition, ...................... 29 discrete, ........................ 29 independent, ................... 49 lattice, ......................... 94 standardized, ................... 31
Random walk, on a circle, .................... 133 one-dimensional symmetric, .... 116 two-dimensional symmetric, ... 131 with reflecting barrier, .... 121, 134
Randomization, .................. 159 Rayleigh distribution, ............. 46 Regeneration point, .............. 169 Regular order, ................... 207 Reliability, .................. 193-194
interval, ....................... 194 limiting interval, .............. 194
Reliability function, (See Reliability)
Reliability models, ........... 185-199
INDEX
Renewal density, .................. 95 Renewal equation, ........... 88, 104
in matrix form, ................ 170 Renewal function, ............ 85, 249 Renewal process,
definition, ...................... 83 delayed, ........................ 99 stationary, ................ 101-103
Renewal-type equation, ........... 96 Repair rate, ..................... 233 Repairman problems, ............ 233 Repeated trials
(See Independent trials) Replacement models, ........ 200-207 Residual life, .................. 73, 97
S Sample function, .................. 65 Sample path, ..................... 65 Sample point, ...................... 1 Sample space, ...................... 1 Sample with replacement, ......... 14 Sample without replacement, ...... 14 Sampling inspection, .............. 21 Second moment, .................. 30 Semi-Markov kernel, ............. 167 Semi-Markov process,
definition, ..................... 167 Series system, ................... 212 Server, .......................... 215 Service, .......................... 215 Service rate, ..................... 219 Service time, .................... 219 Shortage cost, ................... 207 Skewness, ......................... 32 Standard deviation, ............... 31 State classification, .......... 112-117
absorbing, ..................... 116 aperiodic, ..................... 113 non-null recurrent, ............ 118 null recurrent, ................. 118 period, ........................ 113 periodic, ...................... 113 positive recurrent, ............. 118 recurrent, ..................... 114 transient, ...................... 114
State space, ...................... 65 State transition diagram, ........ 106
INDEX 269
Stationary distribution, .......... 119 Stationary increments, ............ 65 Stationary independent increments, 65 Stationary probabilities, ..... 176-177 Stieltjes convolution, .............. 54 Stieltjes integrals, ................. 30 Stirling's formula, ............... 115 Stochastic process, ................ 64
continuous-time, ................ 64 discrete-time, ................... 64
Strong law of large numbers, ... 57-58
T Tauberian theorem, .............. 242 Telephone line, .................. 216 Total event, ........................ 3 Total life, ......................... 73 Total probability formula, ......... 11 Traffic intensity, ................. 222 Transition probability matrix, .... 106 Transition probability, ...... 105, 135
n-step, ........................ 105 Truncated normal distribution, ... 190 Type I counter, .................. 182
U Unconditional distribution, ...... 170 Unconditional mean, ............. 170 Uniform distribution, 33, 71, 188, 191
continuous, .................. 40-42 discrete, ........................ 34 standard, ....................... 41
Uniformization, .................. 159 Up time, ........................ 193 Useful life phase, ................ 187 Utilization factor, ................ 223
V Variance, ......................... 30
W Waiting time, ..................... 68 Wearout phase, .................. 187 Weibull distribution, . 45-46, 190, 212
y Yule process, 139-140
