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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 conver-gence.

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 Laplace-Stieltjes 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 trans-form 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 identi-cally 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 dis-tributions 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 con-ditions 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 ex-pansion (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.