On the Convergence of Global Rational Approximants for ... - CiteSeerX
On the Convergence of Global Rational Approximants for Stochastic Discrete Event Systems 1 Wei-Bo Gong2
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Tel: (413) 545-0384, Fax: (413) 545-1993 Email: [email protected]
Hong Yang
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Email: [email protected]
Hanzhong Hu
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Email: [email protected]
Abstract
We study the convergence properties of some rational approximants for stochastic discrete event systems. Examples of the systems considered include computer and communication systems and general distributed and parallel processing systems. Diculties often arise in the analysis of such systems due to the so-called \curse of dimensionality" in calculating some integer-parameterized functions, where the integer parameter represents the system size or dimension. Our basic idea is to develop global approximants for such functions by exploring the properties of the systems. Various examples in a previous paper have demonstrated the application of the approach. In this paper we analyze the convergence rates of these approximants.
This work was supported in part by the National Science Foundation under Grant ECS-9110090 and Grant EID 9212122, by the U.S. Army Research Oce under Contract DAAL03-91-G-0194 and Contract DAAH04-95-0148, by AFOSR under Contract F49620-93-1-0229DEF, and by Air Force Rome Laboratory under Contract F30602-94-C-0109. 2 Preferred address for correspondence. 2
1
1 Introduction One essential diculty in analyzing large size stochastic discrete event systems (DES) is the computational complexity. When the system is small, we can often get accurate results. As the system size grows, the computational burdens become overwhelming. On the other hand, an impressive amount of knowledge has been accumulated in the past decade regarding the qualitative behavior such as monotonicity, convexity, boundedness and the asymptotic properties of the performance functions for very general stochastic DES (see, for instance, [6] and the references therein). It is now clear that performance functions for many stochastic DES have provable nice qualitative properties. We propose to take advantage of such properties to help evaluating the quantitative behavior of the performance functions. A Newton-Pade type of rational approximant was introduced in [8] for calculating integer-parameterized performance functions of various stochastic DES. In this paper, we analyze the convergence properties of the rational approximants described in [8], (named as Type-2 rational approximants in this paper), as well as another type of rational approximants called Type-1 rational approximants. The rest of the paper is organized as follows. Section 2 de nes the two types of rational approximants used in global approximation for stochastic DES. Section 3 discusses the convergence of Type-1 rational approximants. Section 4 establishes convergence conditions for Type-2 rational approximants. Section 5 is the conclusion.
2 Global Rational Approximants We use f (n) to denote the system performance function with n representing the system size or the model dimension. Our goal is to predict the performance function values for large values of n, based on function values at small n's and the asymptotic behavior of f (n). To achieve this goal, we want to develop a sequence of approximants that converges uniformly to the performance function f (x) over an in nite interval, say [1; 1). f (x) is an \expansion" of f (n) which we will discuss later. Polynomial approximants are not suitable here, as pointed out in [17] (P.11): \Since no sequence of the polynomials (with the exception of a stationary sequence) can be uniformly convergent in an in nite interval, not a single function other than a polynomial can be the limit in an in nite interval of 2
a uniformly convergent sequence of such polynomials." On the other hand, for rational function sequences we have the following well-known theorem.
Theorem 1 (Timan [17]) For any function f (x) continuous on the whole real axis and
having the nite limit limx!1 f (x) = c, there exists a sequence of continuous rational functions of type n, fRnn (x)g, such that
sup jf (x) ? Rnn (x)j ! 0: nlim !1 0<x< 1
(1)
In the above theorem the rational function of type n refers to a rational function with both the denominator and the numerator being degree n polynomials. We can largely enrich the class of the functions that are amenable to uniform rational approximations on (0; 1) ( [1; 1)) by including the summation of polynomials and rational functions. In other words, we allow the degrees of the numerator and the denominator to be different. Moreover, we can apply some carefully chosen non-linear transformations to the performance functions before developing the rational approximants (see [8] for examples). With such pre-processing transformations we can develop accurate approximants for many integer-parameterized performance functions encountered in stochastic discrete event systems.
2.1 Gregory-Newton Series We are concerned with integer-parameterized functions. The rational approximants we want to develop are determined only by the function values at the integers. However, for the convergence study of such approximants, we need to \expand" the performance functions to the complex plane. The so-called Gregory-Newton series (GN series) serves this purpose. In general, for a function f () with given values f (i) = fi at integer points i = 1; 2; , we have its Gregory-Newton series of the form (see [13], [20]):
F (z) = a0 +
1
X
k=1
ak (z ? 1)(z ? 2) (z ? k)
(2)
where the coecients ak 's are divided di erences, i.e.,
a0 = f1; ak = k f1=k!; for k = 1; 2; ; 3
(3)
with
1fm = fm+1 ? fm ; k+1fm = k fm+1 ? k fm; k = 1; 2; :
(4)
Note that the GN series takes the same values at integer points as the original performance function f (n). Any real function f (x) de ned on [0; 1) has its GN series, which may or may not converge to f (x) at non-integer points (see [13]). Convergence of GN series has been well studied [13]. It has been shown that the GN series converges on a half complex plane fz : g, where is called the abscissa of convergence. In our case, n (5) = lim sup log j f0j ; log n n!1 where f0 is an arbitrary constant introduced in front of the concerned sequence ffn g1 1 (see [13], P.310). The following results exemplify convergence properties of certain classes of GN series with desirable convergence regions.
Theorem 2 (Whittaker, 1964, [20]) Let fdng10 be a sequence such that lim sup jdn j1=n = k < 1:
(6)
d0 + zd1 + z(z2!? 1) d2 +
(7)
n!1
Then the series
converges uniformly in any nite region of the complex plane to a holomorphic function.
Note that sequence fe?n g1 0 satis es the conditions in the above theorem and its GN series converges in the whole complex plane (i.e., its convergence abscissa is ?1) to e?x. In general, for the so-called \completely monotonic sequence", we have the following theorem.
Theorem 3 Suppose the sequence fang10 is completely monotonic, then its GN series F (z) converges on half plane fz : 0g to a holomorphic function f (x). For the de nition of the completely monotonic sequence and the proof of the above theorem, see the Appendix. 4
An example of completely monotonic sequence is again exponential function fa?ng (a > 1). It is easily seen that the GN series of the linear combination of completely monotonic sequences converges on the half plane fz : 0g. As an example of stochastic DES, the normalization constant in the closed Jackson network is a linear combination of exponential functions (see [9], [23]). Therefore, its GN series converges on the half plane fz : 0g. It is also known that if a Gregory-Newton series converges in a certain region (convergence half plane), it will converge to a holomorphic (analytic) function in the region [13]. As we will see in the following sections, this analyticity property is important in establishing convergence and convergence rates for our rational approximants. In the following, we introduce the two types of rational approximants (RAs) that we propose to use in analyzing stochastic discrete event systems. We call these types of rational approximants global rational approximants to emphasize that we are concerned with the uniform approximation over the whole positive real axis.
2.2 Two Types of Rational Approximants
De nition 1 (Type-1 RA) Given initial n points of f (x), f (i) (i = 1; 2; ; n), rational function
R1n (x) =
where
n
f (j )B (x) ; 0 j =1 (x ? j )B (j ) X
(8)
n (x ? j ) B (x) = (j=1 x + 1)n ; Q
(9)
is called a Type-1 rational approximant.
Note that as its variations, B (x) can also be de ned as
B (x) =
Q
n (x ? j ) j =1 xn ;
or, B (x) =
x ? j: j =1 x + j n
Y
(10)
It is easily seen that the Type-1 RA interpolates f (x) at x = 1; 2; ; n, that is,
R1n (j ) = f (j );
for j = 1; 2; ; n: 5
(11)
De nition 2 (Type-2 RA) Let f (x) be a real-valued function on (0; 1). The rational function
a + a x + + a p 0 1 l xl l (x) 2 = R (x) = q (x) b + b x + + b xm ; m 0 1 m
is called an [l=m] Type-2 rational approximant of f (x), if pl (i) = f (i) for i = 1; 2; ; l + m + 1: qm(i)
(12) (13)
Note that the above R2(x) is sometimes abbreviated as R2l;m. Further, R2n;n is shortened as Rn. We now compare the Type-2 RA with the well-known Pade approximant, which has been shown to be very useful in performance analysis of queueing systems [7]. For function f (x) which has its MacLaurin series P1j=0 cj xj , we know that the Pade approximant [l=m] is de ned by (14) [l=m] = QPl((xx)) m
where Pl (x) is a polynomial of degree at most l and Qm(x) is a polynomial of degree at most m. The coecients of Pl(x) and Qm(x) are determined by the equation f (x) ? QPl((xx)) = O(xl+m+1 ); (15) m which means that the rst l + m + 1 coecients of the MacLaurin expression for the Pade approximant match those for f (x). Similarly, the rst l + m + 1 coecients (ai, i = 0; 1; ; l +m in Equation (2)) of the [l=m] Type-2 RA for f (x) match the rst l +m+1 coecients of the GN series of the concerned function. Moreover, both the Type-2 RA and the Pade approximant are special cases of the Newton-Pade approximants [5]. Since we are approximating the performance index function over all positive integers, it is helpful to have information regarding the asymptotic behavior of the function when the parameter n goes to in nity. Such information is often obtainable through analyzing the systems themselves. To e ectively incorporate the asymptotic results we often need to do some transformations before carrying out the rational approximation ([8]). Calculation for the Type-1 RA is straightforward. For Type-2 RA, various rational interpolation algorithms can be employed (see [2]). The one we choose is the algorithm proposed by Werner. See [19] for the details of the Werner algorithm. 6
3 Convergence of Type-1 Rational Approximants We generate a sequence of rational approximants, fR1n(x); n = 1; 2; g, with increasing orders. It is clear that the Global Rational Approximation approach is e ective when the rational approximant sequence converges to the GN series of the original integerparameterized function. In the following, instead of discussing integer-parameterized functions (or sequences), we consider functions (e.g., Gregory-Newton series) de ned on the complex plane or on the real axis. To study the convergence rates of the Type-1 RA sequence, we rst introduce a general error estimation formula for interpolation. For the Lagrange's interpolant
p(z) =
nX +1 k=1
f (zk ) (z ? !z ()z!) 0(z ) ; k
k
!(z) = (z ? z1)(z ? z2) (z ? zn ? 1);
(16)
we have the Hermite formula (see, e.g., [18], P.50) I 1 f (z) ? p(z) = 2i !!((tz)()ft(?t)dt z) ; (z 2 D): @D More generally, for the rational interpolants, we have the following result.
(17)
Theorem 4 (see, for example, [16]) Let D be a simply-connected domain (see [10]) in the complex plane. Assume that f (z) and B (z ) are analytic in D. Let Z denote the collection of all the zeros of B (z) in D, i.e.,
Z = fz : z 2 D; B (z) = 0g:
(18)
If Z is nite and all zeros are simple, then for any 2 D,
f () ?
f (x)B () = B () 0 2i x2Z ( ? x)B (x) X
f (z) dz: @D (z ? )B (z )
I
Proof:
Applying the residue theorem we have
B () 2i
f (z) dz = B () f (z) (z ? ) (z ? )B (z) @D (z ? )B (z ) z!
I
7
(19)
+ B () (z ?f(z)B) (z) (z ? x) x2Z z!x f ( ) = f () ? B () B (z)?B (x) x2Z ( ? x)f z?x j z!x g f (x)B () = f () ? 0 x2Z ( ? x)B (x)
X
X
X
The above theorem provides a formula for evaluating the error of the rational approximants. We now use this theorem to establish the convergence rate of the Type-1 RA sequence.
Theorem 5 Suppose the Gregory-Newton series of f (n) converges to f (z) and suppose f (z) satis es the following conditions: 1. f (z) is analytic in D = fz 2 C : jArg(z)j < g, (where 0 < =2). 2. jf (z)j ! 0, for jzj ! 1 and Z 2 D . 3. 01 (jf (rei )j2 + jf (re?i )j2)dr < C1, where C1 is a constant. R
Then for Type-1 rational approximant n
X R1n(x) = (xf?(j )jB)B(x0 ()j ) ; j =1
we have, for 8x > n,
where B (x) =
n (x ? j ) j =1 (x + 1)n ;
Q
jf (x) ? R1n (x)j < Cn?1=2;
(20) (21)
where C < 1 is a positive constant.
Proof:
It follows from (19) that
B (x) f (z) dz 2i @D (z ? x)B (z) f (z) dz : = jB2(x)j @D (z ? x)B (z )
jf (x) ? R1n (x)j =
I
I
Since for any x > n we have
0 < B (x) < 1: 8
(22)
Thus
Z Z f ( z ) f (z) dzg 1 1 f z =rei ;re?i dz + jf (x) ? Rn(x)j < 2 Rlim i z=Re !1 2(?;) (z ? x)B (z ) r2(0;R) (z ? x)B (z ) Z Z 1 f ( z ) f ( z ) 1 2 Rlim z =rei ;re?i dz + lim z =Rei dz !1 r2(0;R) (z ? x)B (z ) 2 R!1 2(?;) (z ? x)B (z ) (23)
We now consider the two integrals in (23). The second integral 1 lim Z f (z) dz 1 lim Z jf (Rei )j R d i 2 R!1 z2=(Re?; 2 R!1 ? j(Rei ? x)B (Rei )j ) (z ? x)B (z ) q Z (R2 + 1 + 2R cos )n R 1 i )j lim q 2 Rlim d : max j f ( Re !1 2[?;] R!1 ? (R2 + x2 ? 2Rx cos ) Qn (R2 + j 2 ? 2jR cos ) j =1 Interchanging the integration and the limit gives lim
Z
R!1 ?
q
R (R2 + 1 + 2R cos )n d < 1 (R2 + x2 ? 2Rx cos ) nj=1(R2 + j 2 ? 2jR cos ) Q
(24)
lim max jf (Rei )j = 0:
(25)
q
From Condition 2,
R!1 2[?;]
Therefore,
1 lim Z f (z) dz = 0: i 2 R!1 z2=(Re?; ) (z ? x)B (z ) The rst integral in (23)
(26)
f (z) dz 1 Z 1 Z f (z) dr i ? i i ? i z = re ;re z = re ;re 2 r2(0;1) (z ? x)B (z) 2 r2(0;1) (z ? x)B (z) Z Z 1 2 1 = 2 2 ( z=rei;re?i jf (z)j dr) ( z=rei ;re?i j(z ? xdr)B (z)j2 )1=2 r2(0;1) r2(0;1) (27)
Note that the last inequality is due to the Schwartz inequality. From Condition 3, Z 1 2 1 = 2 ( z=rei ;re?i jf (z)j dr) = ( (jf (rei )j2 + jf (re?i )j2)dr) C1: 0 r2(0;1) Z
9
(28)
We then have
(29) < 2C1 ( z=rei ;re?i j(z ? x1)B (z)j2 dr)1=2: n r2(0;1) For the integral on the right-hand side, we claim that performing the same integration but on the imaginary axis of the Z plane would yield the same result; i.e., we want to prove 1 1 dr = dr: (30) z=ri z=rei ;re?i 2 r2(?1;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j To do so, we consider domain Q = fz : < Arg(z) < =2g and function (31) g(z) = j(z ? x1)B (z)j2 ; which does not have singularity in domain Q. Applying Cauchy Theorem, we get 1 1 1 dz ? dz + lim dz = 0: z=ri z =Rei z=rei 2 2 R!1 2(;=2) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j r2(0;1) j(z ? x)B (z )j (32) Clearly, 1 1 dr = (33) z=rei z=rei 2 j(z ? x)B (z)j j(z ? x)B (z)j2 dz ;
jf (x) ? R1 (x)j
Z
Z
Z
Z
Z
Z
r2(0;1)
and
Z
Z
r2(0;1)
Z 1 1 dr = dz : z =ri z=ri r 2(0;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j2 And the third integral in (32) Z 1 lim dz Rei R!1 2z=(;= 2) j(z ? x)B (z )j2 Z
= Rlim !1
=2
Z
We then have
(R2 + 1 + 2R cos )n Rei id (R2 + x2 ? 2Rx cos ) Qnj=1(R2 + j 2 ? 2jR cos ) = 0:
(35)
Z 1 1 dr = dr: z=ri 2 r2(0;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j
(36)
Z 1 1 dr = dr: z=ri z=re?i 2 r2(?1;0) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j
(37)
Z
z=rei
Similarly,
(34)
Z
10
Now we turn to bound the right-hand-side of equation (30). On the imaginary axis, jB (z)j 1. So Z 1 Z 1 dr 1 dr z=ri 2 ?1 j(ri ? x)j2 r2(?1;1) j(z ? x)B (z )j Z 1 1 dr = 2 ?1 r + x2 (38) = x Finally, we have r C 1 1 jf (x) ? Rn (x)j < 2 x r = C1 x?1=2 2 r < C1 2 n?1=2 Corollary 1 For the Type-1 rational approximant R1n (x) to function e?ax (a > 0), we have ?ax e ? R1n (x) < Cn?1=2; for x > n; (39) where C is a positive constant.
Proof:
For any satisfying 0 < < =2, it is clear that e?ax (a > 0) satis es the conditions in Theorem 5. In particular, for Condition 3, Z 1 ?arei 2 1 (je j + je?are?i j2)dr = a cos (40) r=0 is bounded.
Corollary 2 If functions f1(x), f2(x), , fK (x) satisfy the three conditions in Theorem
5, then
K
X
k=1
ak fk (x) ? R1n(x) < Cn?1=2; for x > n;
where R1n(x) is the Type-1 RA to 1; 2; ; K , are real constants.
P
K a f (x), k=1 k k
11
(41)
C is a positive constant, and ak , k =
Proof:
It is easily seen that PKk=1 ak fk (x) satis es the rst two conditions in Theorem 5. Furthermore, for any x 2 C , y 2 C , we have (see [10])
jx + yj2 2(jxj2 + jyj2): From this inequality, we have
j
K
X
k=1
ak fk (x)j2 2dlog2 Ke
K
X
k=1
jak j2jfk (x)j2;
(42) (43)
which implies that PKk=1 ak fk (x) satis es Condition 3 in Theorem 5. Note that linear combinations of exponential functions are frequently encountered in stochastic discrete event systems. Moreover, many functions can be approximated accurately by exponential sums (see [3]).
4 Convergence of Type-2 Rational Approximants In this section, we study the convergence of the Type-2 rational approximant sequence. Like the Pade approximants or the best rational approximants [14], the [l=m] Type2 approximants form a two-dimension table with l; m taking all the values of integers. Various sequences (such as row sequence, ray sequence, diagonal sequence, paradiagonal sequence, for details, see [1]) can be formed from the two-dimension approximant table. We will mainly discuss the convergence of the diagonal approximant sequence, namely, the [n=n] Type-2 approximant sequence. Our idea is to start with functions that can be approximated with precision by some rational approximants. Since these rational approximants are not necessarily Type-2; i.e., they do not interpolate the functions at the integers, their convergence rates deviate from the rates associated with the Type-2 rational approximant sequence. On the other hand, continuity argument will guarantee that the deviation does not explode. Our strategy is then to show the convergence of the Type-2 rational approximants via an estimation of this deviation.
4.1 Existence of Low Degree Rational Approximants We rst de ne the functions we want to study. 12
De nition 3 Consider a real valued function f (x) de ned on [1; 1): If there exists an [n=n] rational function sequence fRbn ()g such that
p
sup f (x) ? Rbn (x) Cn e? n ;
x2[1;1)
(44)
where is a constant, C , are positive constants, we say that f (x) has low degree rational approximants.
Since the error is bounded by a term dominated by an exponential factor, Rbn(x) provides a good approximation to f (x) with a relatively small value of n. A theorem by Frank Stenger gives conditions for the existence of low degree rational approximants.
Theorem 6 (Stenger, 1983, [15]) Let 0 < < 1, and 0 < d =2. Let f () be analytic in the region D = fZ 2 C : jarg(z)j < dg and for all z 2 D, jf (z)j Ajzj j1 + zj?2 ;
(45)
Y
z ? qj ; j j =?n z + q
(46)
q = e? p2 n :
(47)
p f (x) ? Rbn (x) Cn =2e?d 2 n;
(48)
where A is a constant. Let
B (z) = 1 +z z
and Then, for all x 2 (0; 1),
n
where C is a constant, and
Rbn (x) =
N
f (qj )B (x) : j 0 j j =?N (x ? q )B (q ) X
(49)
Stenger's theorem guarantees that functions satisfying condition (45) have low degree rational approximants. This condition is not necessary, though. For example, the exponential function e?x does not satisfy condition (45) but allows low degree rational approximation as shown by the following stronger result (see [3, 14]). 13
Theorem 7 (Opitz and Scherer 1985; Schonhage 1982) De ne sup je?x ? r(x)j: nn = r2inf Rnn
(50)
1 < lim 1=n < 1 : 12:93 n!1 nn 9:037
(51)
x2[0;1)
We have
4.2 Convergence Rates of
n=n]
[
Type-2 Rational Approximants
In the following, we focus on functions having low-degree rational approximants. Our task is to nd convergence rates for the [n=n] Type-2 rational approximant sequence for such functions. We rst give an intuitive idea about the GRA convergence result to be introduced in this section. Consider a function f (x) and suppose that R2m (x) is a Type-2 RA to f (x): Then, R2m(i) = f (i); 8i = 1; 2; ; 2m + 1: (52)
On the other hand, let fRbn(x)g denote low degree rational approximants to f (x). Clearly, for a given n, Rbn(x) is a rational function. Suppose that the Type-2 RA sequence fR~ 1m(x) : m = n; n + 1; g interpolates Rbn(x) at integers; i.e.,
R~1m(i) = Rbn(i); 8i = 1; 2; ; 2m + 1: Then,
(53)
(54) R~1m(x) = Rbn(x) since the rational interpolant is unique (under certain conditions, for details, see [5]). Notice that R~1m (i) = Rbn(i) = f (i) + i; 8i = 1; 2; ; 2m + 1 (55) where i = Rbn(i) ? f (i): When n is relatively large, it follows from equation (44) that f (i) ? Rbn(i) is small, or, R2m (i) ? Rbm(i) is small. Therefore, R2m (x) can be viewed as an interpolant to Rbn (x) but with perturbed function values.
14
With this observation in mind, we establish convergence rates for the Type-2 RA sequence by studying the relationship between the change of R2m(x) and the perturbation of function values for the interpolation. We rst introduce a transformation of variables, which maps the in nite interval [1; 1) onto the nite interval (0; 1=2]: For the function f (x); de ned on [1; 1), we de ne another function g(x) on (0; 1=2] as (56) g(x) = f ( x1 ? 1): Clearly, if Rbn(x) is the low-degree rational approximant to f (x), Rbn( x1 ? 1) is a low-degree rational approximant to g(x). Let rn(x) denote the [n/n] rational interpolant for g(x) which interpolates g(x) at pairs f(xi = i+11 ; f (i)); i = 1; 2; ; 2n + 1g. Then R2n (x), de ned as
R2n(x) = rn( 1 +1 x );
x 2 [1; 1)
(57)
is the Type-2 RA to f (x): Moreover,
sup f (x) ? R2n (x) = sup jg(x) ? rn(x)j :
x2[1;1)
x2(0;1=2]
(58)
Now express 2 n rn(x) = pqn((xx)) = an10++ban1xx++banx2x2 ++ ++bannxnx : n
n1
n2
nn
(59)
The interpolation conditions are
rn(xi) = g(xi);
xi = i +1 1 ; i = 1; 2; ; 2n + 1:
(60)
Let rnb (x) be a low degree rational approximant to g(x): Similarly, we express the [n=n] rational function rnb (x) as 2 n (61) rnb (x) = ^an0 +^^an1x +^a^n2x2 + +^^annnx : 1 + bn1x + bn2x + + bnn x As we discussed above, rnb (x) can be considered as an interpolant to g(x) with perturbed function values.
15
Let gi = g(xi) = f (i), i = 1; 2; , be given. From Equation (60), when the solution to the set of linear equations
an0 + an1xj + an2x2j + + annxnj ? gj (bn1xj + bn2x2j + + bnnxnj) = gj ; for j = 1; 2; ; 2n + 1; exists and is unique, rn(x) is uniquely de ned. Now we introduce 2 1 x1 x21 : : : xn1 ?g1x1 ?g1x21 6 2 n x2 x2 : : : x2 ?g2x2 ?g2x22 6 1 6 An = 6 .. .. ... ... ... . . . ... 4 . . 1 x2n+1 x22n+1 : : : xn2n+1 ?g2n+1x2n+1 ?g2n+1x22n+1 and
an = [an0 an1 : : : ann bn1 bnn ]T ; gn = [g1 g2 : : :g2n+1 ]T ;
to write equation (62) as
An an = g n :
(62)
: : : ?g1xn1 : : : ?g2xn2 ; ... ... : : : ?g2n+1 xn2n+1 (63) 3 7 7 7 7 5
(64) (65) (66)
We then have
Theorem 8 Assume that
(a) f (x) > 0, and it is bounded above for all x 2 [1; 1); (b) f (x) has low degree rational approximants Rbn () satisfying
sup f (x) ? Rbn (x) Cbn 1 e? 1
x2[1;1)
pn
(67)
where Cb and 1 are positive constants, 1 is a constant; (c) the condition number of matrix An, k(An), satis es
k(An) Ck n 2 e 2n1=4 ;
(68)
where Ck and 2 are positive constants, 2 is a constant, and
2 < 1: 16
(69)
If the Type-2 RA to f (x), R2n(x) = QPnn((xx)) , satis es
0 < jR2n (x)j < Cr < 1; jQn(x)j > Cq > 0
(70)
(8x 2 [1; 1); 8n > N ) where N (an integer), Cr and Cq are constants, then there exist constants C > 0 and such that p (71) sup jR2n (x) ? f (x)j < Cn e? n; x2[1;1)
for suciently large n and = 1 ? 2 (> 0).
Proof: Let
kyk = kyk1 = maxfjy1j; ; jynjg; (72) denote the norm of n-dimension vector y. The induced matrix norm is then de ned as kjAkj = kjAkj1 = 1max in for an n n matrix A = [aij ]. Recall Equation (58)
n
X
j =1
jaij j;
sup f (x) ? R2n (x) = sup jg(x) ? rn(x)j :
x2[1;1)
x2(0;1=2]
(73)
(74)
We then consider the pair (g(x); rn(x)): From Condition (b), we can deduce that g(x) has low-degree rational approximants, say rnb (x); i.e., p sup g(x) ? rnb (x) Cbn 1 e? 1 n: (75) On the other hand,
x2(0;1=2]
jg(x) ? rn(x)j jg(x) ? rnb (x)j + jrn(x) ? rnb (x)j: We then proceed to estimate supx2(0;1=2] jrn (x) ? rnb (x)j. Let rn(x) = rn (x) ? rnb (x) 17
(76)
(77)
and
dgi = rnb (xi) ? g(xi); De ne the perturbation vector as
i = 1; 2; ; 2n + 1:
(78)
dgn = [dg1; ; dg2n+1]T :
(79)
Let
b = kdgn k = i=1;max 2;;2n+1 jrn (xi ) ? g (xi )j: From Equation (75), there exists an integer N such that for n > N
(80)
p
Cbn 1 e? 1 n :
(81)
It has been shown that in order for rn(x) to be a rational interpolant to g(x), the coecients must satisfy the equation
An an = g n :
(82)
Similarly, after we introduce small perturbation dgn to gn, we have (An + En )a^n = gn + dgn ;
(83)
where
::: 0 ?dg1x1 ?dg1x21 : : : ?dg1xn1 ::: 0 ?dg2x2 ?dg2x22 : : : ?dg2xn2 (84) ... ... ... ... . . . ... : : : 0 ?dg2n+1 x2n+1 ?dg2n+1 x22n+1 : : : ?dg2n+1 xn2n+1 and a^n, the solution to the perturbed linear equations (83), must agree with the coecients of rnb (x) (refer to Equation (61)) due to the uniqueness of rational interpolation. The non-singularity of An is guaranteed by the existence of rn (x). Since 1 ? xnj +1 kjEn kj = max fjdgj j 1 ? x g < 2kdgn k = 2; (85) j j kjEnkj can be made arbitrary small for suciently large n (refer to Equation (81)). It is easily seen that kjAnkj 1: (86) 0 6 0 6 En = 66 .. 4 . 0 2
0 0 ... 0
0 0 ... 0
3 7 7 7 7 5
18
Therefore, (An + En )?1 exists for suciently large n (see [12]). We then have an = a^n ? an = (An + En )?1 ? A?n 1 gn + (An + En )?1dgn : i
h
(87)
Using the properties of the induced norm, we have
kank = ka^n ? an k kj(An + En)?1 ? A?n 1kj kgnk + kj(An + En )?1kj kdgn k It follows from the conditions of the theorem that 0 < kgnk Cg ;
(88)
where Cg is a positive constant. Furthermore, kjEnkj kjA?n 1kj < 2 kjk(AAnkj) n p < 2Ck Cbn 1+ 2 e?( 1 n? 2 n1=4 ):
(89)
It is now clear that for large enough n, (and recall that 2 < 1),
kjEnkj kjA?n 1kj < 1:
(90)
Hence, applying results in [12] (P.336), we have kj(An + En )?1 ? A?n 1kj 1 ? kjEk(kjAn )kjA?1kj kjkjEAnkjkj kjA?n 1kj n n n ? 1 k ( A ) kj A kj n n < kjA kj kjEn kj
0 is a constant. Thus, for suciently large n, p
kank < C1n +2 e?( ? ) n + C2n + e?( p C3n +2 e?( ? ) n 1
2
1
2
1
2
1
2
1
2
1
pn? n1=4 ) 2
(97)
where C3 > 0 is a constant. Now, for a given x 2 (0; 1=2], if kank is suciently small, then n
@rn a + n @rn b + o(ka k) nj nj n j =0 @anj j =1 @bnj n xj n xj p (x) n = bnj + o(kank) a nj ? 2 j =0 qn (x) j =1 qn (x)
rn(x) =
X
X
X
X
(98)
It follows from (57) and (70) that 0 < jrn (x)j < Cr < 1; jqn(x)j > Cq > 0:
(99)
Therefore, there exists a constant C4, such that for 8x 2 (0; 1=2]
j xqp2 n(x(x) ) j < C4:
j q x(x) j < C4;
j
j
n
n
20
(100)
For suciently large n and therefore suciently small , (which implies suciently small anj ; bnj , j = 1; 2; ; 2n + 1), we have n
n
jrn(x)j < C4janj j + C4jbnj j j =1 j =0 (2n + 1)C4kank 3nC4kan k X
X
It follows form inequations (97) and (101) that
jrn(x)j < 3C3C4n +2 +1e?( ? ) 1
Finally, for suciently large n,
2
1
2
pn p
sup jR2n (x) ? f (x)j < 3C3C4n 1+2 2+1e?( 1? 2 ) n + Cbn 1 e? 1
x2[1;1)
< Cn e?
(101) (102) pn
pn
where is a constant, C and are positive constants. This theorem says that if a function can be approximated accurately by low degree rational functions, it can be approximated accurately by Type-2 rational approximants, p and the approximation error is of order O(n e? n ). The key assumption is about the condition number of An. In practice, the algorithms computing rational approximants check the condition of An and will give warning if it is poor conditioned. If An is well conditioned, the accuracy is likely to be good as the above theorem states. Our task therefore is to establish the existence of low-degree rational approximants for integer parameterized performance functions of stochastic discrete event systems. To this end, there has been some characterization of functions that is good for rational approximation; for example, see Stenger [16] and Popov and Petrushev [14]. Exponential function a?x (a > 1) often describes the asymptotic behavior of stochastic discrete event systems. One example is the normalization constant in the closed Jackson network (see [8]). As we mentioned before, the Type-2 RA is e ective when there exist rational functions that accurately approximates the concerned function. For the particular case of exponential function, a result regarding the fast convergence of the Type-2 RA was presented in [8]. 21
5 Conclusion Integer parameter functions are often encountered in the performance evaluation and analysis of computer systems, communication networks and general distributed and parallel processing systems. To tackle the \curse of dimensionality" in calculating some integer-parameterized functions, we have developed global approximants by exploring the qualitative properties of the systems. Applications to several example systems have demonstrated the power of this rational approximant approach. The convergence and convergence rates of the global rational approximant sequences are established for certain classes of performance functions. Two results are obtained. First, a convergence rate of p order O(1= n) is obtained for the Type-1 rational approximant sequence. Secondly, we established conditions under which the sequence of [n=n] Type-2 rational approximants pn ? has a convergence rate of order O(n e ). Further results regarding the convergence properties for these and other types of global approximant sequences are under investigation.
Appendix:
Completely Monotonic Sequences
De nition 4 (completely monotonic sequence) The sequence fn g1 0 is completely monotonic if (?1)k k n 0 where
k
n
For example, the sequences f n +1 1 g10 are completely monotonic.
=
k
X
(n; k = 0; 1; 2; )
(?1)m
m=0
and
!
k m n+k?m :
(104)
fcng10 (0 < c 1)
(105)
De nition 5 (completely monotonic function) 22
(103)
A function f (x) is completely monotonic in a < x < b if
(?1)k f (k)(x) 0 for (a < x < b; k = 0; 1; 2; )
(106)
For example, f (x) = c?x (c > 1) is completely monotonic in (?1; 1).
De nition 6 (minimal completely monotonic sequence) A completely monotonic sequence fang1 0 is minimal if it ceases to be completely monotonic when a0 is decreased.
Theorem 9 (Widder, 1946, [21])
A necessary and sucient condition that there should exist a function f (x) completely monotonic on [0; 1) such that
(n = 0; 1; 2; )
f (n) = an
(107)
is that fang1 0 should be a minimal completely monotonic sequence.
Proof of Theorem 3:
It is easy to show that
jna0j jn?1a0j a0;
(108)
simply applying mathematical induction and the fact that (?1)k k an 0
(n; k = 0; 1; 2; ):
(109)
Thus, the convergence abscissa of the GN series n = lim sup log j a0j 0; (110) n!1 log n which implies that the GN series for the sequence fang1 0 converges on the half plane fz : 0g.
23
References [1] G.A. Baker, and P. Graves-Morris, Pade Approximants, Part I: Basic Theory. New York: Addison-Wesley, 1981. [2] G.A. Baker, and P. Graves-Morris, Pade Approximants, Part II: Extensions and Applications. Reading, MA: Addison-Wesley, 1981. [3] D. Braess, Nonlinear Approximation Theory. Springer-Verlag, 1986. [4] C. Brezinski, and M. Redivo Zaglia, Extrapolation Methods { theory and practice. New York: North-Holland, 1991. [5] M.A. Gallucci, W.B. Jones, \Rational Approximations Corresponding to Newton Series" J. of Approxi. Theory, 17, pp. 366-392. 1976. [6] P. Glasserman, and D. Yao, Monotone Structure in Discrete-Event Systems. Wiley, New York, 1994. [7] W.B. Gong, S. Nananukul, and A. Yan, \Pade Approximation for Stochastic Discrete Event Systems", IEEE trans. on Automatic Control, to appear, August, 1995. [8] W.B. Gong, H. Yang, \Rational Approximants for Some Performance Analysis Problems", to appear in IEEE trans. on Computers, 1995. [9] J.J. Gordon, \The Evaluation of Normalizing Constants in closed queueing networks", Opns. Res., vol. 38, pp. 863-869, 1990. [10] E., Hille, Analytic Function Theory I, Chelsea Publishing Co., New York. 1959. [11] E. Hille, Analytic Function Theory II, Chelsea Publishing Co., New York. 1962. [12] R.A. Horn, and C.R. Johnson, Matrix Analysis. Cambridge University Press, Cambridge, UK. 1985. [13] L.M. Milne-Thomson, The Calculus of Finite Di erences, London: MacMillan And Co., 1951. 24
[14] P.P. Petrushev, and V.A. Popov, Rational Approximation of Real Functions, London: Cambridge University Press, 1987. [15] F. Stenger, \Polynomial, Sinc, and Rational Function Methods for Approximating Analytic Functions", Proceedings of the 1983 Tampa Conference on National Approximation and Interpolation, Springer-Verlag Lecture Notes in Math., #1105, Springer-Verlag, New York. 1984. [16] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. SpringerVerlag. 1993. [17] A.F. Timan, Theory of Approximation of Functions of a Real Variable, New York: Macmillan Company, 1963. [18] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Baltimore, Maryland: American Mathematical Society, 1956. [19] H. Werner, \A Reliable Method for Rational Interpolation", Pade Approximation and its Applications, L. Wuytack (Ed.), Springer-Verlag, pp. 257-277, 1979. [20] J.M. Whittaker, Interpolatory Function Theory, New York: Stechert-Hafner Service Agency, 1964. [21] D.V. Widder, The Laplace Transform, Princeton University Press, 1946. [22] L. Wuytack, \A New Technique for Rational Extrapolation to the Limit", Numer. Math., vol. 17, pp. 215-221, 1971. [23] H. Yang, W.B. Gong, \On Calculating the Normalization Constants in Queueing Networks", Proc. of the 33rd CDC, pp. 2075-2076, 1994.
25
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Tel: (413) 545-0384, Fax: (413) 545-1993 Email: [email protected]
Hong Yang
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Email: [email protected]
Hanzhong Hu
Department of Electrical & Computer Engineering University of Massachusetts, Amherst, MA 01003 Email: [email protected]
Abstract
We study the convergence properties of some rational approximants for stochastic discrete event systems. Examples of the systems considered include computer and communication systems and general distributed and parallel processing systems. Diculties often arise in the analysis of such systems due to the so-called \curse of dimensionality" in calculating some integer-parameterized functions, where the integer parameter represents the system size or dimension. Our basic idea is to develop global approximants for such functions by exploring the properties of the systems. Various examples in a previous paper have demonstrated the application of the approach. In this paper we analyze the convergence rates of these approximants.
This work was supported in part by the National Science Foundation under Grant ECS-9110090 and Grant EID 9212122, by the U.S. Army Research Oce under Contract DAAL03-91-G-0194 and Contract DAAH04-95-0148, by AFOSR under Contract F49620-93-1-0229DEF, and by Air Force Rome Laboratory under Contract F30602-94-C-0109. 2 Preferred address for correspondence. 2
1
1 Introduction One essential diculty in analyzing large size stochastic discrete event systems (DES) is the computational complexity. When the system is small, we can often get accurate results. As the system size grows, the computational burdens become overwhelming. On the other hand, an impressive amount of knowledge has been accumulated in the past decade regarding the qualitative behavior such as monotonicity, convexity, boundedness and the asymptotic properties of the performance functions for very general stochastic DES (see, for instance, [6] and the references therein). It is now clear that performance functions for many stochastic DES have provable nice qualitative properties. We propose to take advantage of such properties to help evaluating the quantitative behavior of the performance functions. A Newton-Pade type of rational approximant was introduced in [8] for calculating integer-parameterized performance functions of various stochastic DES. In this paper, we analyze the convergence properties of the rational approximants described in [8], (named as Type-2 rational approximants in this paper), as well as another type of rational approximants called Type-1 rational approximants. The rest of the paper is organized as follows. Section 2 de nes the two types of rational approximants used in global approximation for stochastic DES. Section 3 discusses the convergence of Type-1 rational approximants. Section 4 establishes convergence conditions for Type-2 rational approximants. Section 5 is the conclusion.
2 Global Rational Approximants We use f (n) to denote the system performance function with n representing the system size or the model dimension. Our goal is to predict the performance function values for large values of n, based on function values at small n's and the asymptotic behavior of f (n). To achieve this goal, we want to develop a sequence of approximants that converges uniformly to the performance function f (x) over an in nite interval, say [1; 1). f (x) is an \expansion" of f (n) which we will discuss later. Polynomial approximants are not suitable here, as pointed out in [17] (P.11): \Since no sequence of the polynomials (with the exception of a stationary sequence) can be uniformly convergent in an in nite interval, not a single function other than a polynomial can be the limit in an in nite interval of 2
a uniformly convergent sequence of such polynomials." On the other hand, for rational function sequences we have the following well-known theorem.
Theorem 1 (Timan [17]) For any function f (x) continuous on the whole real axis and
having the nite limit limx!1 f (x) = c, there exists a sequence of continuous rational functions of type n, fRnn (x)g, such that
sup jf (x) ? Rnn (x)j ! 0: nlim !1 0<x< 1
(1)
In the above theorem the rational function of type n refers to a rational function with both the denominator and the numerator being degree n polynomials. We can largely enrich the class of the functions that are amenable to uniform rational approximations on (0; 1) ( [1; 1)) by including the summation of polynomials and rational functions. In other words, we allow the degrees of the numerator and the denominator to be different. Moreover, we can apply some carefully chosen non-linear transformations to the performance functions before developing the rational approximants (see [8] for examples). With such pre-processing transformations we can develop accurate approximants for many integer-parameterized performance functions encountered in stochastic discrete event systems.
2.1 Gregory-Newton Series We are concerned with integer-parameterized functions. The rational approximants we want to develop are determined only by the function values at the integers. However, for the convergence study of such approximants, we need to \expand" the performance functions to the complex plane. The so-called Gregory-Newton series (GN series) serves this purpose. In general, for a function f () with given values f (i) = fi at integer points i = 1; 2; , we have its Gregory-Newton series of the form (see [13], [20]):
F (z) = a0 +
1
X
k=1
ak (z ? 1)(z ? 2) (z ? k)
(2)
where the coecients ak 's are divided di erences, i.e.,
a0 = f1; ak = k f1=k!; for k = 1; 2; ; 3
(3)
with
1fm = fm+1 ? fm ; k+1fm = k fm+1 ? k fm; k = 1; 2; :
(4)
Note that the GN series takes the same values at integer points as the original performance function f (n). Any real function f (x) de ned on [0; 1) has its GN series, which may or may not converge to f (x) at non-integer points (see [13]). Convergence of GN series has been well studied [13]. It has been shown that the GN series converges on a half complex plane fz : g, where is called the abscissa of convergence. In our case, n (5) = lim sup log j f0j ; log n n!1 where f0 is an arbitrary constant introduced in front of the concerned sequence ffn g1 1 (see [13], P.310). The following results exemplify convergence properties of certain classes of GN series with desirable convergence regions.
Theorem 2 (Whittaker, 1964, [20]) Let fdng10 be a sequence such that lim sup jdn j1=n = k < 1:
(6)
d0 + zd1 + z(z2!? 1) d2 +
(7)
n!1
Then the series
converges uniformly in any nite region of the complex plane to a holomorphic function.
Note that sequence fe?n g1 0 satis es the conditions in the above theorem and its GN series converges in the whole complex plane (i.e., its convergence abscissa is ?1) to e?x. In general, for the so-called \completely monotonic sequence", we have the following theorem.
Theorem 3 Suppose the sequence fang10 is completely monotonic, then its GN series F (z) converges on half plane fz : 0g to a holomorphic function f (x). For the de nition of the completely monotonic sequence and the proof of the above theorem, see the Appendix. 4
An example of completely monotonic sequence is again exponential function fa?ng (a > 1). It is easily seen that the GN series of the linear combination of completely monotonic sequences converges on the half plane fz : 0g. As an example of stochastic DES, the normalization constant in the closed Jackson network is a linear combination of exponential functions (see [9], [23]). Therefore, its GN series converges on the half plane fz : 0g. It is also known that if a Gregory-Newton series converges in a certain region (convergence half plane), it will converge to a holomorphic (analytic) function in the region [13]. As we will see in the following sections, this analyticity property is important in establishing convergence and convergence rates for our rational approximants. In the following, we introduce the two types of rational approximants (RAs) that we propose to use in analyzing stochastic discrete event systems. We call these types of rational approximants global rational approximants to emphasize that we are concerned with the uniform approximation over the whole positive real axis.
2.2 Two Types of Rational Approximants
De nition 1 (Type-1 RA) Given initial n points of f (x), f (i) (i = 1; 2; ; n), rational function
R1n (x) =
where
n
f (j )B (x) ; 0 j =1 (x ? j )B (j ) X
(8)
n (x ? j ) B (x) = (j=1 x + 1)n ; Q
(9)
is called a Type-1 rational approximant.
Note that as its variations, B (x) can also be de ned as
B (x) =
Q
n (x ? j ) j =1 xn ;
or, B (x) =
x ? j: j =1 x + j n
Y
(10)
It is easily seen that the Type-1 RA interpolates f (x) at x = 1; 2; ; n, that is,
R1n (j ) = f (j );
for j = 1; 2; ; n: 5
(11)
De nition 2 (Type-2 RA) Let f (x) be a real-valued function on (0; 1). The rational function
a + a x + + a p 0 1 l xl l (x) 2 = R (x) = q (x) b + b x + + b xm ; m 0 1 m
is called an [l=m] Type-2 rational approximant of f (x), if pl (i) = f (i) for i = 1; 2; ; l + m + 1: qm(i)
(12) (13)
Note that the above R2(x) is sometimes abbreviated as R2l;m. Further, R2n;n is shortened as Rn. We now compare the Type-2 RA with the well-known Pade approximant, which has been shown to be very useful in performance analysis of queueing systems [7]. For function f (x) which has its MacLaurin series P1j=0 cj xj , we know that the Pade approximant [l=m] is de ned by (14) [l=m] = QPl((xx)) m
where Pl (x) is a polynomial of degree at most l and Qm(x) is a polynomial of degree at most m. The coecients of Pl(x) and Qm(x) are determined by the equation f (x) ? QPl((xx)) = O(xl+m+1 ); (15) m which means that the rst l + m + 1 coecients of the MacLaurin expression for the Pade approximant match those for f (x). Similarly, the rst l + m + 1 coecients (ai, i = 0; 1; ; l +m in Equation (2)) of the [l=m] Type-2 RA for f (x) match the rst l +m+1 coecients of the GN series of the concerned function. Moreover, both the Type-2 RA and the Pade approximant are special cases of the Newton-Pade approximants [5]. Since we are approximating the performance index function over all positive integers, it is helpful to have information regarding the asymptotic behavior of the function when the parameter n goes to in nity. Such information is often obtainable through analyzing the systems themselves. To e ectively incorporate the asymptotic results we often need to do some transformations before carrying out the rational approximation ([8]). Calculation for the Type-1 RA is straightforward. For Type-2 RA, various rational interpolation algorithms can be employed (see [2]). The one we choose is the algorithm proposed by Werner. See [19] for the details of the Werner algorithm. 6
3 Convergence of Type-1 Rational Approximants We generate a sequence of rational approximants, fR1n(x); n = 1; 2; g, with increasing orders. It is clear that the Global Rational Approximation approach is e ective when the rational approximant sequence converges to the GN series of the original integerparameterized function. In the following, instead of discussing integer-parameterized functions (or sequences), we consider functions (e.g., Gregory-Newton series) de ned on the complex plane or on the real axis. To study the convergence rates of the Type-1 RA sequence, we rst introduce a general error estimation formula for interpolation. For the Lagrange's interpolant
p(z) =
nX +1 k=1
f (zk ) (z ? !z ()z!) 0(z ) ; k
k
!(z) = (z ? z1)(z ? z2) (z ? zn ? 1);
(16)
we have the Hermite formula (see, e.g., [18], P.50) I 1 f (z) ? p(z) = 2i !!((tz)()ft(?t)dt z) ; (z 2 D): @D More generally, for the rational interpolants, we have the following result.
(17)
Theorem 4 (see, for example, [16]) Let D be a simply-connected domain (see [10]) in the complex plane. Assume that f (z) and B (z ) are analytic in D. Let Z denote the collection of all the zeros of B (z) in D, i.e.,
Z = fz : z 2 D; B (z) = 0g:
(18)
If Z is nite and all zeros are simple, then for any 2 D,
f () ?
f (x)B () = B () 0 2i x2Z ( ? x)B (x) X
f (z) dz: @D (z ? )B (z )
I
Proof:
Applying the residue theorem we have
B () 2i
f (z) dz = B () f (z) (z ? ) (z ? )B (z) @D (z ? )B (z ) z!
I
7
(19)
+ B () (z ?f(z)B) (z) (z ? x) x2Z z!x f ( ) = f () ? B () B (z)?B (x) x2Z ( ? x)f z?x j z!x g f (x)B () = f () ? 0 x2Z ( ? x)B (x)
X
X
X
The above theorem provides a formula for evaluating the error of the rational approximants. We now use this theorem to establish the convergence rate of the Type-1 RA sequence.
Theorem 5 Suppose the Gregory-Newton series of f (n) converges to f (z) and suppose f (z) satis es the following conditions: 1. f (z) is analytic in D = fz 2 C : jArg(z)j < g, (where 0 < =2). 2. jf (z)j ! 0, for jzj ! 1 and Z 2 D . 3. 01 (jf (rei )j2 + jf (re?i )j2)dr < C1, where C1 is a constant. R
Then for Type-1 rational approximant n
X R1n(x) = (xf?(j )jB)B(x0 ()j ) ; j =1
we have, for 8x > n,
where B (x) =
n (x ? j ) j =1 (x + 1)n ;
Q
jf (x) ? R1n (x)j < Cn?1=2;
(20) (21)
where C < 1 is a positive constant.
Proof:
It follows from (19) that
B (x) f (z) dz 2i @D (z ? x)B (z) f (z) dz : = jB2(x)j @D (z ? x)B (z )
jf (x) ? R1n (x)j =
I
I
Since for any x > n we have
0 < B (x) < 1: 8
(22)
Thus
Z Z f ( z ) f (z) dzg 1 1 f z =rei ;re?i dz + jf (x) ? Rn(x)j < 2 Rlim i z=Re !1 2(?;) (z ? x)B (z ) r2(0;R) (z ? x)B (z ) Z Z 1 f ( z ) f ( z ) 1 2 Rlim z =rei ;re?i dz + lim z =Rei dz !1 r2(0;R) (z ? x)B (z ) 2 R!1 2(?;) (z ? x)B (z ) (23)
We now consider the two integrals in (23). The second integral 1 lim Z f (z) dz 1 lim Z jf (Rei )j R d i 2 R!1 z2=(Re?; 2 R!1 ? j(Rei ? x)B (Rei )j ) (z ? x)B (z ) q Z (R2 + 1 + 2R cos )n R 1 i )j lim q 2 Rlim d : max j f ( Re !1 2[?;] R!1 ? (R2 + x2 ? 2Rx cos ) Qn (R2 + j 2 ? 2jR cos ) j =1 Interchanging the integration and the limit gives lim
Z
R!1 ?
q
R (R2 + 1 + 2R cos )n d < 1 (R2 + x2 ? 2Rx cos ) nj=1(R2 + j 2 ? 2jR cos ) Q
(24)
lim max jf (Rei )j = 0:
(25)
q
From Condition 2,
R!1 2[?;]
Therefore,
1 lim Z f (z) dz = 0: i 2 R!1 z2=(Re?; ) (z ? x)B (z ) The rst integral in (23)
(26)
f (z) dz 1 Z 1 Z f (z) dr i ? i i ? i z = re ;re z = re ;re 2 r2(0;1) (z ? x)B (z) 2 r2(0;1) (z ? x)B (z) Z Z 1 2 1 = 2 2 ( z=rei;re?i jf (z)j dr) ( z=rei ;re?i j(z ? xdr)B (z)j2 )1=2 r2(0;1) r2(0;1) (27)
Note that the last inequality is due to the Schwartz inequality. From Condition 3, Z 1 2 1 = 2 ( z=rei ;re?i jf (z)j dr) = ( (jf (rei )j2 + jf (re?i )j2)dr) C1: 0 r2(0;1) Z
9
(28)
We then have
(29) < 2C1 ( z=rei ;re?i j(z ? x1)B (z)j2 dr)1=2: n r2(0;1) For the integral on the right-hand side, we claim that performing the same integration but on the imaginary axis of the Z plane would yield the same result; i.e., we want to prove 1 1 dr = dr: (30) z=ri z=rei ;re?i 2 r2(?1;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j To do so, we consider domain Q = fz : < Arg(z) < =2g and function (31) g(z) = j(z ? x1)B (z)j2 ; which does not have singularity in domain Q. Applying Cauchy Theorem, we get 1 1 1 dz ? dz + lim dz = 0: z=ri z =Rei z=rei 2 2 R!1 2(;=2) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j r2(0;1) j(z ? x)B (z )j (32) Clearly, 1 1 dr = (33) z=rei z=rei 2 j(z ? x)B (z)j j(z ? x)B (z)j2 dz ;
jf (x) ? R1 (x)j
Z
Z
Z
Z
Z
Z
r2(0;1)
and
Z
Z
r2(0;1)
Z 1 1 dr = dz : z =ri z=ri r 2(0;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j2 And the third integral in (32) Z 1 lim dz Rei R!1 2z=(;= 2) j(z ? x)B (z )j2 Z
= Rlim !1
=2
Z
We then have
(R2 + 1 + 2R cos )n Rei id (R2 + x2 ? 2Rx cos ) Qnj=1(R2 + j 2 ? 2jR cos ) = 0:
(35)
Z 1 1 dr = dr: z=ri 2 r2(0;1) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j
(36)
Z 1 1 dr = dr: z=ri z=re?i 2 r2(?1;0) j(z ? x)B (z )j2 r2(0;1) j(z ? x)B (z )j
(37)
Z
z=rei
Similarly,
(34)
Z
10
Now we turn to bound the right-hand-side of equation (30). On the imaginary axis, jB (z)j 1. So Z 1 Z 1 dr 1 dr z=ri 2 ?1 j(ri ? x)j2 r2(?1;1) j(z ? x)B (z )j Z 1 1 dr = 2 ?1 r + x2 (38) = x Finally, we have r C 1 1 jf (x) ? Rn (x)j < 2 x r = C1 x?1=2 2 r < C1 2 n?1=2 Corollary 1 For the Type-1 rational approximant R1n (x) to function e?ax (a > 0), we have ?ax e ? R1n (x) < Cn?1=2; for x > n; (39) where C is a positive constant.
Proof:
For any satisfying 0 < < =2, it is clear that e?ax (a > 0) satis es the conditions in Theorem 5. In particular, for Condition 3, Z 1 ?arei 2 1 (je j + je?are?i j2)dr = a cos (40) r=0 is bounded.
Corollary 2 If functions f1(x), f2(x), , fK (x) satisfy the three conditions in Theorem
5, then
K
X
k=1
ak fk (x) ? R1n(x) < Cn?1=2; for x > n;
where R1n(x) is the Type-1 RA to 1; 2; ; K , are real constants.
P
K a f (x), k=1 k k
11
(41)
C is a positive constant, and ak , k =
Proof:
It is easily seen that PKk=1 ak fk (x) satis es the rst two conditions in Theorem 5. Furthermore, for any x 2 C , y 2 C , we have (see [10])
jx + yj2 2(jxj2 + jyj2): From this inequality, we have
j
K
X
k=1
ak fk (x)j2 2dlog2 Ke
K
X
k=1
jak j2jfk (x)j2;
(42) (43)
which implies that PKk=1 ak fk (x) satis es Condition 3 in Theorem 5. Note that linear combinations of exponential functions are frequently encountered in stochastic discrete event systems. Moreover, many functions can be approximated accurately by exponential sums (see [3]).
4 Convergence of Type-2 Rational Approximants In this section, we study the convergence of the Type-2 rational approximant sequence. Like the Pade approximants or the best rational approximants [14], the [l=m] Type2 approximants form a two-dimension table with l; m taking all the values of integers. Various sequences (such as row sequence, ray sequence, diagonal sequence, paradiagonal sequence, for details, see [1]) can be formed from the two-dimension approximant table. We will mainly discuss the convergence of the diagonal approximant sequence, namely, the [n=n] Type-2 approximant sequence. Our idea is to start with functions that can be approximated with precision by some rational approximants. Since these rational approximants are not necessarily Type-2; i.e., they do not interpolate the functions at the integers, their convergence rates deviate from the rates associated with the Type-2 rational approximant sequence. On the other hand, continuity argument will guarantee that the deviation does not explode. Our strategy is then to show the convergence of the Type-2 rational approximants via an estimation of this deviation.
4.1 Existence of Low Degree Rational Approximants We rst de ne the functions we want to study. 12
De nition 3 Consider a real valued function f (x) de ned on [1; 1): If there exists an [n=n] rational function sequence fRbn ()g such that
p
sup f (x) ? Rbn (x) Cn e? n ;
x2[1;1)
(44)
where is a constant, C , are positive constants, we say that f (x) has low degree rational approximants.
Since the error is bounded by a term dominated by an exponential factor, Rbn(x) provides a good approximation to f (x) with a relatively small value of n. A theorem by Frank Stenger gives conditions for the existence of low degree rational approximants.
Theorem 6 (Stenger, 1983, [15]) Let 0 < < 1, and 0 < d =2. Let f () be analytic in the region D = fZ 2 C : jarg(z)j < dg and for all z 2 D, jf (z)j Ajzj j1 + zj?2 ;
(45)
Y
z ? qj ; j j =?n z + q
(46)
q = e? p2 n :
(47)
p f (x) ? Rbn (x) Cn =2e?d 2 n;
(48)
where A is a constant. Let
B (z) = 1 +z z
and Then, for all x 2 (0; 1),
n
where C is a constant, and
Rbn (x) =
N
f (qj )B (x) : j 0 j j =?N (x ? q )B (q ) X
(49)
Stenger's theorem guarantees that functions satisfying condition (45) have low degree rational approximants. This condition is not necessary, though. For example, the exponential function e?x does not satisfy condition (45) but allows low degree rational approximation as shown by the following stronger result (see [3, 14]). 13
Theorem 7 (Opitz and Scherer 1985; Schonhage 1982) De ne sup je?x ? r(x)j: nn = r2inf Rnn
(50)
1 < lim 1=n < 1 : 12:93 n!1 nn 9:037
(51)
x2[0;1)
We have
4.2 Convergence Rates of
n=n]
[
Type-2 Rational Approximants
In the following, we focus on functions having low-degree rational approximants. Our task is to nd convergence rates for the [n=n] Type-2 rational approximant sequence for such functions. We rst give an intuitive idea about the GRA convergence result to be introduced in this section. Consider a function f (x) and suppose that R2m (x) is a Type-2 RA to f (x): Then, R2m(i) = f (i); 8i = 1; 2; ; 2m + 1: (52)
On the other hand, let fRbn(x)g denote low degree rational approximants to f (x). Clearly, for a given n, Rbn(x) is a rational function. Suppose that the Type-2 RA sequence fR~ 1m(x) : m = n; n + 1; g interpolates Rbn(x) at integers; i.e.,
R~1m(i) = Rbn(i); 8i = 1; 2; ; 2m + 1: Then,
(53)
(54) R~1m(x) = Rbn(x) since the rational interpolant is unique (under certain conditions, for details, see [5]). Notice that R~1m (i) = Rbn(i) = f (i) + i; 8i = 1; 2; ; 2m + 1 (55) where i = Rbn(i) ? f (i): When n is relatively large, it follows from equation (44) that f (i) ? Rbn(i) is small, or, R2m (i) ? Rbm(i) is small. Therefore, R2m (x) can be viewed as an interpolant to Rbn (x) but with perturbed function values.
14
With this observation in mind, we establish convergence rates for the Type-2 RA sequence by studying the relationship between the change of R2m(x) and the perturbation of function values for the interpolation. We rst introduce a transformation of variables, which maps the in nite interval [1; 1) onto the nite interval (0; 1=2]: For the function f (x); de ned on [1; 1), we de ne another function g(x) on (0; 1=2] as (56) g(x) = f ( x1 ? 1): Clearly, if Rbn(x) is the low-degree rational approximant to f (x), Rbn( x1 ? 1) is a low-degree rational approximant to g(x). Let rn(x) denote the [n/n] rational interpolant for g(x) which interpolates g(x) at pairs f(xi = i+11 ; f (i)); i = 1; 2; ; 2n + 1g. Then R2n (x), de ned as
R2n(x) = rn( 1 +1 x );
x 2 [1; 1)
(57)
is the Type-2 RA to f (x): Moreover,
sup f (x) ? R2n (x) = sup jg(x) ? rn(x)j :
x2[1;1)
x2(0;1=2]
(58)
Now express 2 n rn(x) = pqn((xx)) = an10++ban1xx++banx2x2 ++ ++bannxnx : n
n1
n2
nn
(59)
The interpolation conditions are
rn(xi) = g(xi);
xi = i +1 1 ; i = 1; 2; ; 2n + 1:
(60)
Let rnb (x) be a low degree rational approximant to g(x): Similarly, we express the [n=n] rational function rnb (x) as 2 n (61) rnb (x) = ^an0 +^^an1x +^a^n2x2 + +^^annnx : 1 + bn1x + bn2x + + bnn x As we discussed above, rnb (x) can be considered as an interpolant to g(x) with perturbed function values.
15
Let gi = g(xi) = f (i), i = 1; 2; , be given. From Equation (60), when the solution to the set of linear equations
an0 + an1xj + an2x2j + + annxnj ? gj (bn1xj + bn2x2j + + bnnxnj) = gj ; for j = 1; 2; ; 2n + 1; exists and is unique, rn(x) is uniquely de ned. Now we introduce 2 1 x1 x21 : : : xn1 ?g1x1 ?g1x21 6 2 n x2 x2 : : : x2 ?g2x2 ?g2x22 6 1 6 An = 6 .. .. ... ... ... . . . ... 4 . . 1 x2n+1 x22n+1 : : : xn2n+1 ?g2n+1x2n+1 ?g2n+1x22n+1 and
an = [an0 an1 : : : ann bn1 bnn ]T ; gn = [g1 g2 : : :g2n+1 ]T ;
to write equation (62) as
An an = g n :
(62)
: : : ?g1xn1 : : : ?g2xn2 ; ... ... : : : ?g2n+1 xn2n+1 (63) 3 7 7 7 7 5
(64) (65) (66)
We then have
Theorem 8 Assume that
(a) f (x) > 0, and it is bounded above for all x 2 [1; 1); (b) f (x) has low degree rational approximants Rbn () satisfying
sup f (x) ? Rbn (x) Cbn 1 e? 1
x2[1;1)
pn
(67)
where Cb and 1 are positive constants, 1 is a constant; (c) the condition number of matrix An, k(An), satis es
k(An) Ck n 2 e 2n1=4 ;
(68)
where Ck and 2 are positive constants, 2 is a constant, and
2 < 1: 16
(69)
If the Type-2 RA to f (x), R2n(x) = QPnn((xx)) , satis es
0 < jR2n (x)j < Cr < 1; jQn(x)j > Cq > 0
(70)
(8x 2 [1; 1); 8n > N ) where N (an integer), Cr and Cq are constants, then there exist constants C > 0 and such that p (71) sup jR2n (x) ? f (x)j < Cn e? n; x2[1;1)
for suciently large n and = 1 ? 2 (> 0).
Proof: Let
kyk = kyk1 = maxfjy1j; ; jynjg; (72) denote the norm of n-dimension vector y. The induced matrix norm is then de ned as kjAkj = kjAkj1 = 1max in for an n n matrix A = [aij ]. Recall Equation (58)
n
X
j =1
jaij j;
sup f (x) ? R2n (x) = sup jg(x) ? rn(x)j :
x2[1;1)
x2(0;1=2]
(73)
(74)
We then consider the pair (g(x); rn(x)): From Condition (b), we can deduce that g(x) has low-degree rational approximants, say rnb (x); i.e., p sup g(x) ? rnb (x) Cbn 1 e? 1 n: (75) On the other hand,
x2(0;1=2]
jg(x) ? rn(x)j jg(x) ? rnb (x)j + jrn(x) ? rnb (x)j: We then proceed to estimate supx2(0;1=2] jrn (x) ? rnb (x)j. Let rn(x) = rn (x) ? rnb (x) 17
(76)
(77)
and
dgi = rnb (xi) ? g(xi); De ne the perturbation vector as
i = 1; 2; ; 2n + 1:
(78)
dgn = [dg1; ; dg2n+1]T :
(79)
Let
b = kdgn k = i=1;max 2;;2n+1 jrn (xi ) ? g (xi )j: From Equation (75), there exists an integer N such that for n > N
(80)
p
Cbn 1 e? 1 n :
(81)
It has been shown that in order for rn(x) to be a rational interpolant to g(x), the coecients must satisfy the equation
An an = g n :
(82)
Similarly, after we introduce small perturbation dgn to gn, we have (An + En )a^n = gn + dgn ;
(83)
where
::: 0 ?dg1x1 ?dg1x21 : : : ?dg1xn1 ::: 0 ?dg2x2 ?dg2x22 : : : ?dg2xn2 (84) ... ... ... ... . . . ... : : : 0 ?dg2n+1 x2n+1 ?dg2n+1 x22n+1 : : : ?dg2n+1 xn2n+1 and a^n, the solution to the perturbed linear equations (83), must agree with the coecients of rnb (x) (refer to Equation (61)) due to the uniqueness of rational interpolation. The non-singularity of An is guaranteed by the existence of rn (x). Since 1 ? xnj +1 kjEn kj = max fjdgj j 1 ? x g < 2kdgn k = 2; (85) j j kjEnkj can be made arbitrary small for suciently large n (refer to Equation (81)). It is easily seen that kjAnkj 1: (86) 0 6 0 6 En = 66 .. 4 . 0 2
0 0 ... 0
0 0 ... 0
3 7 7 7 7 5
18
Therefore, (An + En )?1 exists for suciently large n (see [12]). We then have an = a^n ? an = (An + En )?1 ? A?n 1 gn + (An + En )?1dgn : i
h
(87)
Using the properties of the induced norm, we have
kank = ka^n ? an k kj(An + En)?1 ? A?n 1kj kgnk + kj(An + En )?1kj kdgn k It follows from the conditions of the theorem that 0 < kgnk Cg ;
(88)
where Cg is a positive constant. Furthermore, kjEnkj kjA?n 1kj < 2 kjk(AAnkj) n p < 2Ck Cbn 1+ 2 e?( 1 n? 2 n1=4 ):
(89)
It is now clear that for large enough n, (and recall that 2 < 1),
kjEnkj kjA?n 1kj < 1:
(90)
Hence, applying results in [12] (P.336), we have kj(An + En )?1 ? A?n 1kj 1 ? kjEk(kjAn )kjA?1kj kjkjEAnkjkj kjA?n 1kj n n n ? 1 k ( A ) kj A kj n n < kjA kj kjEn kj
0 is a constant. Thus, for suciently large n, p
kank < C1n +2 e?( ? ) n + C2n + e?( p C3n +2 e?( ? ) n 1
2
1
2
1
2
1
2
1
2
1
pn? n1=4 ) 2
(97)
where C3 > 0 is a constant. Now, for a given x 2 (0; 1=2], if kank is suciently small, then n
@rn a + n @rn b + o(ka k) nj nj n j =0 @anj j =1 @bnj n xj n xj p (x) n = bnj + o(kank) a nj ? 2 j =0 qn (x) j =1 qn (x)
rn(x) =
X
X
X
X
(98)
It follows from (57) and (70) that 0 < jrn (x)j < Cr < 1; jqn(x)j > Cq > 0:
(99)
Therefore, there exists a constant C4, such that for 8x 2 (0; 1=2]
j xqp2 n(x(x) ) j < C4:
j q x(x) j < C4;
j
j
n
n
20
(100)
For suciently large n and therefore suciently small , (which implies suciently small anj ; bnj , j = 1; 2; ; 2n + 1), we have n
n
jrn(x)j < C4janj j + C4jbnj j j =1 j =0 (2n + 1)C4kank 3nC4kan k X
X
It follows form inequations (97) and (101) that
jrn(x)j < 3C3C4n +2 +1e?( ? ) 1
Finally, for suciently large n,
2
1
2
pn p
sup jR2n (x) ? f (x)j < 3C3C4n 1+2 2+1e?( 1? 2 ) n + Cbn 1 e? 1
x2[1;1)
< Cn e?
(101) (102) pn
pn
where is a constant, C and are positive constants. This theorem says that if a function can be approximated accurately by low degree rational functions, it can be approximated accurately by Type-2 rational approximants, p and the approximation error is of order O(n e? n ). The key assumption is about the condition number of An. In practice, the algorithms computing rational approximants check the condition of An and will give warning if it is poor conditioned. If An is well conditioned, the accuracy is likely to be good as the above theorem states. Our task therefore is to establish the existence of low-degree rational approximants for integer parameterized performance functions of stochastic discrete event systems. To this end, there has been some characterization of functions that is good for rational approximation; for example, see Stenger [16] and Popov and Petrushev [14]. Exponential function a?x (a > 1) often describes the asymptotic behavior of stochastic discrete event systems. One example is the normalization constant in the closed Jackson network (see [8]). As we mentioned before, the Type-2 RA is e ective when there exist rational functions that accurately approximates the concerned function. For the particular case of exponential function, a result regarding the fast convergence of the Type-2 RA was presented in [8]. 21
5 Conclusion Integer parameter functions are often encountered in the performance evaluation and analysis of computer systems, communication networks and general distributed and parallel processing systems. To tackle the \curse of dimensionality" in calculating some integer-parameterized functions, we have developed global approximants by exploring the qualitative properties of the systems. Applications to several example systems have demonstrated the power of this rational approximant approach. The convergence and convergence rates of the global rational approximant sequences are established for certain classes of performance functions. Two results are obtained. First, a convergence rate of p order O(1= n) is obtained for the Type-1 rational approximant sequence. Secondly, we established conditions under which the sequence of [n=n] Type-2 rational approximants pn ? has a convergence rate of order O(n e ). Further results regarding the convergence properties for these and other types of global approximant sequences are under investigation.
Appendix:
Completely Monotonic Sequences
De nition 4 (completely monotonic sequence) The sequence fn g1 0 is completely monotonic if (?1)k k n 0 where
k
n
For example, the sequences f n +1 1 g10 are completely monotonic.
=
k
X
(n; k = 0; 1; 2; )
(?1)m
m=0
and
!
k m n+k?m :
(104)
fcng10 (0 < c 1)
(105)
De nition 5 (completely monotonic function) 22
(103)
A function f (x) is completely monotonic in a < x < b if
(?1)k f (k)(x) 0 for (a < x < b; k = 0; 1; 2; )
(106)
For example, f (x) = c?x (c > 1) is completely monotonic in (?1; 1).
De nition 6 (minimal completely monotonic sequence) A completely monotonic sequence fang1 0 is minimal if it ceases to be completely monotonic when a0 is decreased.
Theorem 9 (Widder, 1946, [21])
A necessary and sucient condition that there should exist a function f (x) completely monotonic on [0; 1) such that
(n = 0; 1; 2; )
f (n) = an
(107)
is that fang1 0 should be a minimal completely monotonic sequence.
Proof of Theorem 3:
It is easy to show that
jna0j jn?1a0j a0;
(108)
simply applying mathematical induction and the fact that (?1)k k an 0
(n; k = 0; 1; 2; ):
(109)
Thus, the convergence abscissa of the GN series n = lim sup log j a0j 0; (110) n!1 log n which implies that the GN series for the sequence fang1 0 converges on the half plane fz : 0g.
23
References [1] G.A. Baker, and P. Graves-Morris, Pade Approximants, Part I: Basic Theory. New York: Addison-Wesley, 1981. [2] G.A. Baker, and P. Graves-Morris, Pade Approximants, Part II: Extensions and Applications. Reading, MA: Addison-Wesley, 1981. [3] D. Braess, Nonlinear Approximation Theory. Springer-Verlag, 1986. [4] C. Brezinski, and M. Redivo Zaglia, Extrapolation Methods { theory and practice. New York: North-Holland, 1991. [5] M.A. Gallucci, W.B. Jones, \Rational Approximations Corresponding to Newton Series" J. of Approxi. Theory, 17, pp. 366-392. 1976. [6] P. Glasserman, and D. Yao, Monotone Structure in Discrete-Event Systems. Wiley, New York, 1994. [7] W.B. Gong, S. Nananukul, and A. Yan, \Pade Approximation for Stochastic Discrete Event Systems", IEEE trans. on Automatic Control, to appear, August, 1995. [8] W.B. Gong, H. Yang, \Rational Approximants for Some Performance Analysis Problems", to appear in IEEE trans. on Computers, 1995. [9] J.J. Gordon, \The Evaluation of Normalizing Constants in closed queueing networks", Opns. Res., vol. 38, pp. 863-869, 1990. [10] E., Hille, Analytic Function Theory I, Chelsea Publishing Co., New York. 1959. [11] E. Hille, Analytic Function Theory II, Chelsea Publishing Co., New York. 1962. [12] R.A. Horn, and C.R. Johnson, Matrix Analysis. Cambridge University Press, Cambridge, UK. 1985. [13] L.M. Milne-Thomson, The Calculus of Finite Di erences, London: MacMillan And Co., 1951. 24
[14] P.P. Petrushev, and V.A. Popov, Rational Approximation of Real Functions, London: Cambridge University Press, 1987. [15] F. Stenger, \Polynomial, Sinc, and Rational Function Methods for Approximating Analytic Functions", Proceedings of the 1983 Tampa Conference on National Approximation and Interpolation, Springer-Verlag Lecture Notes in Math., #1105, Springer-Verlag, New York. 1984. [16] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. SpringerVerlag. 1993. [17] A.F. Timan, Theory of Approximation of Functions of a Real Variable, New York: Macmillan Company, 1963. [18] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Baltimore, Maryland: American Mathematical Society, 1956. [19] H. Werner, \A Reliable Method for Rational Interpolation", Pade Approximation and its Applications, L. Wuytack (Ed.), Springer-Verlag, pp. 257-277, 1979. [20] J.M. Whittaker, Interpolatory Function Theory, New York: Stechert-Hafner Service Agency, 1964. [21] D.V. Widder, The Laplace Transform, Princeton University Press, 1946. [22] L. Wuytack, \A New Technique for Rational Extrapolation to the Limit", Numer. Math., vol. 17, pp. 215-221, 1971. [23] H. Yang, W.B. Gong, \On Calculating the Normalization Constants in Queueing Networks", Proc. of the 33rd CDC, pp. 2075-2076, 1994.
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