BIOMATHEMATICS TRAlNING PROGRI\M
The Covariance Structure of the Departure Process:
GI/M/l and M/G/l Queues*
by
Nancy Gustafson
ABSTRACT
Formulae, suitable for computation, for the lag n autocorrelation of
departure intervals in the GI/M/I queues are developed for n
=1
to 6.
A
generating function, comparable to the one known for the M/G/I case, is
found for GI/M/I.
For the Ek/M/I and M/Ek/l queues, expressions for arbi-
trary lag are given.
Tables and the APL functions used to produce them are
provided for Ek/M/I, H/M/I, and M/Ek/l.
•
*This research was supported in part by the Office of Naval Research, under
Contract NOOOI4-76-C-0550.
1.
Introduction
Knowledge of the departure process of a queue is essential to the analysis
of networks in which the output of one queue forms the input to another.
particular interest is the covariance structure of the departures.
Of
The work in
this area to date is summarized by Reynolds (1975) and Daley (1976).
The papers serving as points of departure for
Jenkins (1966) and Daley (1968).
this report are those of
In the former, the lag one and two autocorre-
lations of departure intervals are derived for queues with independent Erlangian
service times and Poisson arrivals.
His method may be extended to general
service time distributions, but only with considerable difficulty to lags of
greater than two.
Daley (1968) provides a generating function for the autocovar-
iances of M/G/l departure intervals.
~
He also gives a rather difficult expression
for the lag n covariance in the GI/M/I case, suggesting that it can be used to
derive a generating function.
This derivation, the result of which has a
comparatively simple form, is carried out in section 3 of this paper.
The main purpose here is to convert the GI/M/l and M/G/l generating functions
for the departure interval covariances to explicit formulae which may be used
directly to obtain results for special cases.
Tables of autocorrelations are
given for Ek/M/l, HIM/I, and M/Ek/l queues with various traffic intensities.
The
special Erlang and hypergeometric distributions were chosen because they are so
frequently suggested as approximations to actual arrival and/or service distributions.
•
To match a coefficient of variation less than one, an Erlang is used;
otherwise, the hypergeometric .
-2-
2.
Notation
The queue assumed throughout possesses a single server and infinite waiting
room.
Customers arrive according to a renewal process and are served in order of
arrival.
Service times are i.i.d. and independent of the arrival process.
Notation
is as follows:
l'
n
= interval
between arrivals of customers nand n + 1
W
=
time spent in queue (exclusive of service) by customer n
S
=
service time for customer n
D
n
= interval
n
n
between departures of customers n - 1 and n
Arrival intervals and service times have distribution functions A(x) and B(x),
respectively, with associated Laplace-Stieltjes transforms
-sT
A*(s) = E[e
n] and B*(s)
= E[e
The rate of arrivals is A, the service rate is
-sS
n]
~,
and the traffic intensity
A/~
is
denoted by p.
E stands for the (special) Erlang distribution with k phases, p.d.f.
k
f(t)
t
>
o.
H stands for the hypergeometric distribution with p.d.f.
t ~
where 0 < q < .5.
0,
.
-3-
~
3.
Autocorrelations of Departure Intervals:
Daley (196H) gives as his Theorem 5
GI/M/l.
result that the ueparture intervals
thl~
in a stationary GI/M/l queue have
cov(D ,D )
(3.1)
o
where s
o
n
= ( 1
P
is the root of smallest absolute value of
s
= A*(~[l
- s]),
or alternatively,
(3.2)
cov(D ,D ) = ( 1
o
n
P
!.s ) (!.)
E (W - W IW
~
n
n-l 0
o
= 0).
A method to find the expectation in (3.2) is given in
Taka~s
simpler, however, to use Takacs' generating function for E(W I W
n
0
(1962).
= 0)
It is
to derive a
generating function for the autocovariances, then use the result to obtain explicit
expressions for cov(Do ,Dn ).
Making certain changes in equation (28) on page 121 of Takacs (customer n here
corresponds to his customer n+l), one obtains
00
(3.3)
I
E[e-swn-lIW
=
O]zn =
o
n=l
z_[_l_- s~l(_z~)~]~
(l-z) [1 _ ~sl (z) ]
~ + s
where sl(z) is the root of smallest absolute value of
5
=
z A* (~ [1 -
5])
•
-4-
Evaluating at s = 0 the partial derivative of each side with respect to s results
in
I
n=l
E(W
n-
llw
0
= O)z n
Z sllZ)
=
~(l - z)[l - sl(Z)] ,
and
00
I
n=l
00
E(Wn Iw0
I
= O)zn =
n=O
E(Wn Iw0 = O)zn
00
I
=
n=l
=
~(l
E(Wn_llwo = O)zn-l
Sl (z)
- z) [1 - sl(Z)]
Comhining these yields
00
= ~[l
(3.4)
sl (z)
- sl (z)]
, Izi
< 1
Substituting in (3.2), the desired generating function is found to be
(3.5)
where So and sl(z) are the roots of smallest absolute value of s =
and s =
zA*(~[l
A*(~[l
- s])
- s]), respectively.
To determine the autocovariance lag n for specific n, it is first necessary
to expand sl (z) inz. Takacs (Lemma 1, page 113) provides the formula
~
-5-
e
. 1
00
sl (z) = z
(3.6)
(_~lZ)J-
L
j!
j=l
For compactness of notation, let a. =
J
d j - 1 {A*(s)}j
. 1
]
s=ll
d sJ-
djA*(s)
d sj
IS=ll
.
Then
[sl (z)]
222
= z 0.
0
and so on.
Carrying out these expansions to a number of terms, working out the first six terms
of
00
=
L
[sl(Z)]j
j=l
and multiplying by
(! - ! )(!-)
p
So 112
, one obtains the desired autocovariances.
Daley (1968),
(3.7)
The autocorre1ations, lags 1 through 6, of departure intervals for the
GrlMl1 queue are as follows:
From
-6-
corr(D ,D )
O 1
(3.8)
corr(\)O,D )
2
= Mo. 0
c:::
~11
e
2
('t
() -
I
\lcl.d't I
corr(D ,D )
O 3
= M[o.~
2
2110. 0.
0 1
corr(D ,D )
O 4
= M[0. 40
3
- 3110. 0.
0 1
corr(DO,D )
S
= M[a S0
2T (20. 00. 12
+
2T (60. 00. 1
113
2 3
114
4
- 3! (240.00. 1
+
corr(D ,D )
O 6
4f (240. 00. 1
= M[a60
113
+
1
""""2 [Var TO (p
II
l )
So
5
25
-1
+
3
20. 0. )
0 2
3
0. 0. )]
0 3
+
3 2
2! (120. 00. 1
+
2
0
+
4
20. 0. )
0 3
+
2 2
720. 0. 0.
0 1 2
+
3 2
3
120. 0. + 160. 0. 0.
0 2
0 1 3
450. 40. 0.
0 1 2
+
+
+
4
0. 0. )]
0 4
40.~0.2)
e
S
30. 0. )
0 3
+
3 2
2400. 0. 0.
0 1 2
+
4 2
4
5
300. 0. + 400. 0. 0. + 20. 0. )
0 4
0 1 3
0 2
+
2 3
6000. 0. 0.
0 1 2
+
3
2
3000. 0. 0.
0 1 2
-1
]
4
30. 0. )
0 2
3
240. 0. 0.
0 1 2
114
2 4
4T
(1200. 0.
0 1
llS
1
112
2
0. 0. )]
0 2
+
- 5T (1200. 00. 1
where M =
2 2
+
112
4 2
S
- Sllo.Oo.l + 2! (200. 0. 1 +
0
3 3
-3! (600. 00. 1
+
112
2
90. 0. 0.
0 1 2
+
4
- 4110. 0.
0 1
2
+
3
II
3
3!
(60. 00. 1
~
with
So
+
3 2
2000. 0. 0.
0 1 3
and o.j defined as above.
II (1-50)
There apparently exists an expression for lag of arbitrary n, starting
(3.9)
2
n
n-1
II
corr(DO,Dn ) = M{a 0 - (n-1)1l0.0 0. 1 + 2!
113
- 3!
(n-1)! n-33
(n-4)! 0. 0 0. 1
+
(n-1)!
[ (n-3)!
n-2
3(n-1) (n-3)0.0 0. 10. 2
n-22
n-1
0. 0 0.1 + (n-2)0.0 0. 2
+
n-1
(n-3)0.0 0. 3 ]
+ •••
}
,
-7-
but its complete statement requires further work.
For the Ek/M/l case,
Ct
j =
(k+n-l)! -1 n
kp k+j
(k-l)! (kPlJ) (kP+l)
and inspection of the results of substituting this in (3.8) reveals a general
formula:
~)
corr ( 00,On ) = M( k 1
+
(3.10)
where
nk
n-l
\ (nk.+i)
ken-i)
L
.
i=O
1
(nk+i) (pk)1
1
1
1 -1
2s 0
-1
M = [-2 (-p - -s)
- -l-s ]
kp
0
0
Tables for the autocorrelations of lags one through five for Ek/M/l queues
~
with various k and p, and for H/M/l queues with various q and p, are given in section
5
(Tab Ie I).
The last coltmm of these tables, labelled "TOTAL", contains the values
of
00
(3.11)
L
n=l
corr(OO,On)
=
s
M(_O_)
I-sO
"CV" denotes the coefficient of variation of TO for the given k or q.
To find M requires determination of the root sO.
The method of "false posi-
tion", which was recommended for this problem by Sahin (1970), proved quite fast,
particularly when used with Aitken's method of acceleration.
computer routine used are described in Appendix 1.
are displayed and discussed briefly in Appendix 2.
The algorithm and
The other APL functions required
-8-
4.
Autocorrelations of Departure Intervals:
M/G/l.
For the M/G/l case, Daley (1968) provides the generating function:
l-p
00
I
(4.1)
COV(Do,Dn)Zn
=
n=l
Sl (z)-z
(7){ (l-z) [l-sl (z)]
+
zSl'(z)-sl(z)
Sl (z) Sl ' (z) (l-z) }
Iz I
< 1,
where sl(z) is the root of smallest absolute value of
sl
= zB*(A[l
- s]),
and
=
Sl'(z)
d sl' (x)
dx
The expansion of sl (z) is the same as that given in section 3, with
B.
J
=
djB*(s)
d sj
replacing
Also needed is
<l.
S=A
J
= So -
2zAB OBl +
4Z 3 A3
[6B OB3l
3!
+
Note that So is not involved here.
2 2
2
2!
[2B OBl +
3Z A
9B 2OBl B2
+
3
BO
B3 ]
2
BO
B2 ]
+ •••
However, in other aspects the M/G/l case
is computationally more difficult than the GI/M/l.
The expression in brackets in
(4.1) can be rewritten as
(4.2)
(1
+
z
+
z
2
+ ••• )
{
[sl (z) - z] [1
+
sl (z)
+
2
sl (z)
+ ••• ] +
z
sl (z)
-9-
By methods similar to those of section 3 one obtains
(4.3)
+
3 2
212
z [A (13 0 8 1 + 2 13 0 13 2 )
4 3
332
- z [A (13 0 81 + 2 80 8 182
A(B 0 81
+
1
+
223
- 28 13 ) - 13 + 13 ]
0 1
0
0
3
6 80 83)
The remaining two terms in (4.2) must be expanded in Taylor series about zero
(sl(z) and sl'(z) are non-vanishing in Izi < 1).
The results of this tedious
procedure are
(4.4)
8
z2 A2
z3 A3
z
1 + zA (-.!..)
= 2
!
(8 2 ) + ""3! (38 1 82 + 80 83)
sl (z)
80
80
z4 A4
2
2
2
- 4 ! (128 182 + 68 08Z + 88 0 13 1 8 3 + 8084) + •••
and
(4.5)
1
sl' (z)
=
2
Z 2
28
28 1
Z A
1
_1
)
+ zA ( - ) - 2 ! (38 2
80
80
80
z3 A3
+ '3! (48 083)
z4 A4
-~
2
2
(68 082 + 16 80 8183 + 58 08 ) + •••
4
00
Subtracting (4.5) from (4.4), adding (4.3), then multiplying by
L zj,
j=O
has an expansion for (4.2).
Using this and
one finally
-102
(4.6)
Var (D)
0
= Var (So)
+ ( 1- P2
)
,
A
the autocorrelations for the M/G/l queue are found to be
(4.7)
A13 1
corr(D ,D ) = M[-1 + 13 - - ]
0
O 1
13 0
2
13 1
13 1
2
+
2
)]
corr(D ,D ) = M[-1
13 0 - A (13- + 13 o13 1 ) + A ((32 - 13 0
o 2
0
2
2
80 13 2
81
~\
2
2
2 + 3
-)
corr(D ,D ) = M[-1 + 13 - A ( - + 213 (3 ) + A ( 13 2 - - + (30(31
2
0 1
0
O 3
80
13 0
3 -13 1 (32
- A ( 2
+
80 (33
-)]
2
-11-
e
2
13 1
5
2
4 2
corr(D ,D ) = M[-1 + 80 - A(- + 58 0( 1 ) + A (8 2 - - + 108 o8 1
80
O 6
80
81
6
+
+
+
8013 3
3 -8 18 2
5
+ -28 ( ) - A ( 2
2
0 2
2
4 -~\82
A ( 2
4 2
58 o82
4
+
+
808 183
3
4
58 08183
3
2
8084
+ --+
6
108~13~
2 4
58 o81
1SB~B182
+
8~(~3
+ ---)
2
2
3 2
10 8 1.\8
2
0
+
5
8013 4
+~
2
8013 113 2
4
3
5 -13 113 2
A ( 2
+
+
2
8013 183
6
+
2
808283
4
+
2
7B 013 184
24
+
2 2
313 0 81 13 2
4
3
7 13 8 8
+
+
+
0 1 5
60
where
2
A Var(SO)
M=
with
1-p
[
-1
+
P
+
1]
B. as defined above.
J
As in the E /M/1 case, the expressions for M/E /1 autocorre1ations are
k
k
considerably simpler, and may be generalized to
(4.8)
corr(DO,Dn )
= M[-1
+
n-1 "(k 1) + 1
k
P i+1 k i (k+1)
\'
(1 + . ) (
) ()
(
)
i~O
1
i(k+1)+1 k
k+p
nk n-1
k" k(")
i
k i
\' (n.+1)
n-1 (E.) (
)]
+ ( k )
k+p
.L
1
(nk+i) k
k+p
,
1=0
-12-
where
M=
k(l-p)
2
P (l-k) +k
This leads to the same formulae as given by Jenkins (1966) for lags one and two.
Tables for autocorrelations lags one through five for various M/Ek/l queues
are provided in the following section (Table II).
The last column of each table
gives the sum
2
00
I
(4.9)
n=l
corr(DO,Dn )
"CV" is the coefficient of variation of SO.
(k-l) ]
2k(l-p)
.
= M[ P
APL functions used are described in
Appendix 2.
Acknowledgements
The indispensable guidance of Professor W. L. Smith is gratefully acknowledged.
Thanks are due also to J. O. Kitchen of the UNC Computation Center for his help in
producing the tables.
-13-
REFERENCES
Daley, D. J. (1968), "The correlation structure of the output process of some
single server queueing systems", Annals of Math. Stat. 39, 1007-1019.
Daley, D. J. (1976), "Queueing output processes", Adv. Appl.
Prob.~,
395-415.
Jenkins, .J. H. (1966), "On the correlation structure of the departure process
of the M/E:>J1 queue", J. Roy. Statist. Soc. B 28, 336-344.
Reynolds, J. F. (1975), "The covariance structure of queues and related processesa survey of recent work", Adv. Appl. Probe
2,
383-415.
Sahin, I. (1970), "On the least positive root of the equation z=K(z) in queueing
theory", Can. O. R. Soc. J.
~,
109-115.
Takacs, L. (1962), Introduction to the Theory of Queues, Oxford Univ. Press, New
York.
-14-
Appemli x I.
Finding the Root sO'
So
The APL function used here to obtain values for the root
s
= A*(~[l-s])
is based on a program written by James Kitchen.
of the equation
Modifications were
made to take advantage of the specific situation, and to allow for a vector
arglUllent.
The routine uses the method of "false position", with Aitken's accel-
eration applied on every third iteration.
Algorithm to find the root of f(x)
=0
between iO and
U
o
1.
Choose upper and lower values iO and
2.
Initialize i(the iteration counter) and j(the acceleration flag) to O.
3.
Set i = i
4.
Set x.
=
1
+
1
f(l. )
1
- feu.1 )
5.
Set j = j
6.
If Iy.
7.
If y. > 0, 1et u. 1
I
1
+
1, and z.
J
1
1+
and y.
1
= f(x.1 )
e
If not, go on to (7).
= x.,
i.1+ 1
1
=
L1
= u.,
i.1+ 1 = x.1
1
If j = 3, do acceleration routine; it not, go to (3).
Acceleration routine:
2
(z3 - z2)
1. Let x = z 3
zl - 2z 2 + z3
2.
enclose the root.
= x 1..
< tolerance, stop.
if not, let u.1+ 1
8.
o which
l.
u. f(L) - L f(u.)
1
1
1
U
and y
= f(x).
If i.1+ 1 < x < u.1+ l' set x.1+ 1 = x, y.1+ I = y, and j
= 0,
then go to (6).
If not, reject the accelerated result, setting j = 1, zl
= z3'
and returning
to (3).
The APL function S0 EROOTS K takes as arguments the number of stages K for
the Er1ang distribution desired, and an initial guess S0, a vector of initial upper
bounds on
So
for the n values of p being considered.
The initial lower bounds
~
-15-
~
used are pk
When f(sO)
the vector of roots.
=
So -
A*(l - sO) < TOL for each p, the function returns
Setting TOL
=5
7
x 10- , no more than three or four itera-
tions were necessary.
Warning: this function, as written, requires a minimum of 5 values of p.
-16-
tJEROOTS[[J)'fl
_________ 'Z~SQ_EB.OOIS
[ 1]
._[J;;:]
[3]
"FINDS ROOT FOR EK/MI I:
[jJ
.
m
FOR EACH INTENSI TY IN VECTOR RHO
_
"~HO
AND TOL MUST BE SU?PLIED
.xHD.L.D" ( (N"pRHO) ,3) pB. 5
RHOM .. ~Cp~X..~(2,N)p«RHO*K),SO»pRHO
v .. X_+ LL:t-< I-X) +KxRHOM) *K
IACCEL .. ITER"O
____....-Ll.iL-l.1A. IN: I I Eg .. I TEE + 1
(9)
D; X; Y; XT.;.Y.T-J.-I:-l;...5..G1l.;-RHOM-..
nIiSr.:s METHOD OF FA! SE POSITION lHTH AITKEN'S A.CCEI T:RATI.o.N.-
_L4~_ _
. __ L.6..l..
[7]
to I AceEr.; XHOI
XT"(-/yx$X)+-/Y
___UUJ_Y..'t!:XI.:-.-T-ll + ( 1 -XI) +Kx RHO) *K
[ 11 ]
XHOLD['; IACCEL .. IACCEL+ 1) "XT
___._LL2.J_.....1Illllll..N :"F INAJ.x 1 10I,> fll VI
._------------------------_._---_.--- ------_._-----------._-----,---------------------- ----_._----
X[;SGN"CYTCl»O)+I)"XT
_U.:;:L-Y..Li..S..GrLiJ.:-;...::..YJ..,T..L_
[IS)
"MAINX1IACCEL<3
...L!.6:L_~A-r;;_~ELEH.A
. TED X NOV TBI ED
.
['7)
XT"XHOLDCi3)-ceXHOLDC;3)-XHOLDCil)*2)++/«N,3)p 1 -2 1 )xXHOLD
__.u b L Y_T..o:xr~_±_(.Ltu..:.xT H·KxRHO) *..K.
.
.
...
. _.
..
[ 1')]
.. T.J! TIilNx 1 -I ACC"EL" (XT[ 5] <XC 5 i 1) ) VXTC 5] >XC 5; 2 J
__ L;~.QJ_XliQl...l2U....~HOL!)[ ; 3 J
._--.._--------------------_ .. -------[21)
"MAIN
(13)
[ 2:2 L_E.U1AJ,.--=_~_
':'J
e
..
_
_...---
-
-17-
~
Appendix II.
APL Functions Used
To produce the tables for Ek/M/l and H/M/I queues, an APL function which
could be altered to fit any appropriate arrival distribution was devised.
ECORR takes as arguments the starting vector (R) for the root-finding routine
EROOTS described in Appendix I, and a vector of values (KV) for the Erlang parameter k.
KV must be arranged in increasing size, and contain at least two values.
R is usually set to RHO, the vector containing the various traffic intensities
for which the autocorrelations are to be computed.
The value of FORMAT used for
all the tables is displayed following the functions below.
HCORR takes only one argument, the vector of parameters QV.
To obtain the
values for QV corresponding to a vector of coefficients of variation, use the
function CONVERT.
~
The roots for H/M/I are simply
So
(1-/1- 42(1- p) (1-2q) 2)
Z
= p +
and are provided to HCORR by HROOTS.
To adapt either of these functions for another arrival distribution, make the
following substitutions:
(a)
V + expression for Var(T )
O
(b)
M + expression for M [see (3.8)]
(c)
to (d) expressions for Cl
(e)
Where
e
= a O'
Al
= aI'
AZ
= a Z'
A3
= a 3,
and A4
= a4
is the parameter to be printed, with n decimal places, the
statement should be
0+'
(f)
+
8
=
(n ~ 8)',
number of statement labelled (a).
CV =
3
~
(expression for c.of v.)
-18-
The function for M/Ek/l is designed specifically for Erlangian service times.
A routine alterable for use with any M/G/l queue would be possible, but unwieldy.
CORRE takes the argument KV, the vector of values for k, and uses the pre-set
variables RHO and FORMAT described above.
-
e
~
HC02~
e
QV;QjDl;D2;~ljN2jIjV;N;CT;Cl;C2;C3;C~;C5;Al;A2;A3jA4
___CJ...L__ ~JP,oL~_QLI,.~~J2L4....L5_J';~!)_J-.9_TAL_~1rtP_G.OBB=-J.J~,JIQ~JS_
C2)
__ C3)
Ta)
Q:)_
[Ll)
I~E
QQV IS
FQiLHC[JO
_
RHO A::D FO'·.:-lAT r'IUST 3E ?RO'JIDED
VECTOR C? 'J;'.!..·;E,S Fu:l Q;
1-:1
V~~-1++2x~xI_G.~:)V(lJH-[:;-:O'2
(5J
'1~+(·J~(+:::;:~Cl-+$)-~xS+I-GU-:O---1.i'i:)QTSQ
(0)
CT~~:.;.l-S
_
C I ~ ( (N I "2Y,:,?EOx Q*2 b~l_~_.l.:t.?X?:~Ox "'.t~ (~!2~2XRHO~.Ll,..QliU±..D.a~1 +2xllli.D...~_hQ
A 1 <- - C': 1 ;- [' I "c-: ) 'C f' \ ~ + D<: ,. 2
_____U_J_ _ h2'-2 x_CL±..Dli:.;) +~i2"t~<:.'.I.3._______________
_
CiJ)
A;:,~-·ux,Nl+Dl,,1.l)+N2+D;~.,..1.l
1.9.Llill-a9"~'-:~: (r-u + D1 *:) +;,J,; + n2_2LS.._
(C)
_
(7 J
(c1
(12)
(;3"(
_
(Cl.-2)XA2+2)+(CI-AI ):{C2 .. CIX :CI-Al)
_ _-,r,-I~L_C3_"~~ :t~ C.L~;'. U_+';. <;'0 ?'~~ lY .".~.x u:; L!-;':L...,_!U.l.=-J:_C_b..3)Y~_±.Ji
_
[14]
C5 .. C1X«CI-Ml~~4)+«A~~(3~XA2~-~X~~)XC1*3)+24
_ _ J: 1~) __ ..c_5':C~.±J:;1~LU:..;lUJ..lx£i.;".j-J_L_tl.UL~~,P.~
;~<~2..,_~:jl~Al~f>.22~c.l"'2') + _:.P< CUA21tAUL
_
(1 jJ
1'J:2 p i '
(e )l..J...11_ ~J" ~
L~_..!..__ '-<~'!' Q) L t
LJL-=- _~_...2..i'.....t.:_:.l.±j:2.x_C)LL-:. )::..: .... :i.._
(l.?)
I
i~UTOCO~lf(r:.::.i\TIQN LAG NI
___U-? J__~_.Bl..iQ__ ____.N::.____l
~
J
._A
5. ..
....JQT.A.L!. __
r. ~ ~ J
___ L.;L:!
:'_~E.G?,.::B_.I~.lilli:L.~U6.~~-X~rl1!c(:lJ:::.!.(J.),QG..L._.c.2.-G.3....,..c~.C-S.GI...-------(~~l
...
~4
_
--lR,iL-.!_~ l~+'_!...l
11)
[24)
'-0
I
'"
----------~- H ECORR KV j K j D; I ; V; N i CT j C 1 ; C2 ; C3 j C4; C5; AI; A2 j A3; A4
_ _-'[.........
1 ~]
!tTA3LE Of LAG 1....,.2....3.....4.5 A/"D TOTAi ...AU.T!lCD..B.E..E!..AT-lmJ-S.-FOl'LEA.LH/_.l..-
(2J
_ _-,("",3,-,),(Ll)
-i:Sl
[6]
0KV IS THE VECTOR OF VAL~~S Fon K; ~HO AND FOE~AT
02 IS T;-iE STA!'lT FO:1 E~09TS; i1AY ;jE RIiG.-.lE__ DE-5.IRFD
ll-
~+_(.EE"O~_~~K1l.[
1 -=.5..-.-
-_._-_.-._--- --- _._---_.
_
A4 .. -(~~+~)+D)XA3
_
C4"(C3XCi-AI)+C(Clx2)XA~X(Cl+2)-Al)-CCI*3)XA3+6
---_ .., - - - .----_." -
.--
-._-_."---
(Cl=fd ) *4) + C (."\4+ Ci6X':t2) .,.-.tKb.~xC 1 *3H-;~~
\: 1 J)
C5-_Cl~(
(14]
C~"CS+Clx«(t2xAIXA3+J)+(~.5xA2*2)-~XAlxA2)XCl*~)+3xClXA2xAI*~
(16]
0"
____c15)__1_~_~
-
_
---------_ ... -
....-----t-llL-.cJ~.~,CL,_;.:.)XA~±.<::..J_±.ic..L.,iU2
1LC-":.!:Clx.{Cl_",.AU
(12J
_
BE PROVIDED
~
;l1L--C~.s+
(i~J
O_~__ n~____
';',;-;c.
-L!...1L_'_'
(2uJ
___[3 JJ .
,,=
_
1
_
_
n .
CV
K
,CUlflO, I
£lrTOG.Q.:mE.LLTlDNLP.G !'),
G
3
4
_
='
,3'l';{",-(j.S
s
TOTP.L
Q~
I .. I+cr.:1"S
__ ~ ;2
. __..
7
u
n
_
I
------- -----
J .. FD~MAT'~HC,~C(0,N)p~)X(6,N.. PRHO)PCI,C2,C3.C4,C5,CT
__ --l£~:t~L=Oi~'l):)1
[~.::::]
= ,
,
[lJL_~
( 1 (;)
----- ----- --
_
_
(,,)
H 1 ~- (K+D) xC 1" U\x?hO+ [0"1 +':x RHO) *K
----C.i_L--.A3 .. -:. ( ~_:i:t.£.l.±D~" - ( (c + U±.D.2.2:...ti1.
__
~UST
---~-----
r"l
M.. +(~+(+!'lHO)-+S)-2xS+l-S .. R EROOTS
I
......
~(I=oQV)oIDC
_
-20-
[lJ
£.2]
V CORRE KY;K;I;~;N;KRO;KRHO;T;~K
ATA8LE OF LAG 1-5 AND TOTAL AUTOCORRELATIONS FOR M/EK/l:
ARHO MUST BE SET TO THE VECTOR OF INTENSITIES.
en
C~«N~,RHO),5),I~1
[4]
~~2
[5]
M~KX(1-RHO)+«RHO*2)Xl-K)+K~KY[I]
[6J
[7J
CT~(RHO*2)X(K-l)+(1-RHO)X2XK
FOR K IN KY
C(;l]~-l+KRHO~RHOXKRO~K+K+RHO
[S]
C[;~J~C[;~-IJ+«~-l)~JK)x(K+~K)X«RHO+K)*~)XKRO*~K~~+Kx~-l
[9]
~8X\(6)~~~+1)
Cl0]
C[;lJ~C[;lJ+KRo*K
[11]
[12]
[13]
C[;2J~C(;2J+(KRo*2XK)xl+KRHO
[14]
[15]
(16]
[17]
.
C[;3J~C[;3J+(KRo*3xK)Xl+(2XKRHO)+(1+3XK)X(KRHo*2)+2XK
C[;4J~C[;4J+(KRO*4XK)xl+(3XKRHO)+«T~1+4xK)X(KRHo*2)+K)
C[;4J~C[;4J+(KRO*4XK)XTX(T+l)X(KRHO*3)+6xK*2
C[;5J~C[;5]+(KRo*5xK)Xl+(4xKRHO)+(3X(T~1+5XK)X(KRHO*2)+2x
[IS]
10 2 ,
[19J
[20]
[21]
[22J
(23J
[24]
D~
)
I
,
K == '
,(O.,K), I
AUTOCORRELATION LAG N'
I
I":HO
3
2
1
D~FORMAT.,RHO,(~(6,N)rM)XC,CT
~4x«,KY)lI~I+l)
v
VCQNV~RIrOJV
I
v Q~CONVEnT CV
I
_..._LLl.._--.fl-o:::.U: { (-1 + CJljr2J ; 1 + C v* 2 l..~lL.....5..L~~
v
I
I
FORMAT
4 2
K
C[;5]~C[;5J+(KRO*5XK)X(Tx(T+l)x(KRHO*3)+3xK*2)
C[;5]~C[;5]+(KRO*5xK)XTx(T+l)X(T+2)x(KRHO*4)+24XK*3
14 6 10
6
III
6
10 6
IG
6
III
6
4
5
TCITAL
I
4.
21
TABLES
I (a). Ek/M/l
e
1<
....
•...
r)
CV
~~UT()COI:;:Fo:E:.J...AT
r-:HCl
l'~ ::::
..... 004063
....
• 013:t.O:I.
....
• 023690
..... 0:3;3f:l1f:l
..... 04~.~42f:l
..... 0490~:j</
..... O~:;~5:~j97
....
• O~:;6 :1. 24
..... O~::;6f:ll f:l
..... O~:j5902
..... O~:j360:;~
..... O~:j013f:l
..... O4~)71:1.
..... 04049'7
• 0 ~:j
.:1.0
• 1. ~:j
.20
1")1::'
• ~:.. \.J
.30
• 3~.:j
.40
• 4":'
,.J
• ~:,:; 0
• ~:) ~:5
.60
, ,oo.
.0:.)
.'70
• ?:j
.f:lO
• H~5
• </0
• 9~.:j
.....
r)
1
0341.)5~~
..... 028312
..... 021 ~j8f.l
..... 014~:j71.>
..... 00735B
..... 0 000 n:j
• OOO9?O
..... OO~5202
..... O06?04
....
• 011,000
..... O:I.::.'j~:;23
..... 01,9'1'7B
...• 023404
..... 026 :1. Ml
....
• O2?951
-.02B'714
"·.0213478
- • 02?;'502
.... 02~:j264
-'. 02245~:;
.- • 0 Hl96</
..... 014f.l97
..... 010321.>
..... 005336
...
J... ~~c;
:3
10',.
....
J. ()N
..... 000003
..... 0OOO9~5
'.'
.000552
..... 00 :1.1.)'74
..... 003:550
·".006048
.... OOBB9B
..... 0 :1. :1. 7B7
..... 014429
..... 016596
··.01B127
"·.01B926
..... 01B95~?
'.'
.0:t.B203
..... 01,6707
..·.014~H1
-'.011.1.)'1'7
... • 008268
..... 004354
.?07
N
4
...
• 000000
.... 000010
.... 000107
....
• 000467
·".001.274
- .00260'7
..... 004406
... • 001.)503
..... 008676
.... 0:1. ()'700
..... 012376
..... 01,3553
..... 0141.25
"'·.O:t.4034
.... 01~52:::j8
..... 011,805
.... 009705
-.007001
..... OO~~747
I::"
TOT~~I...
••J
.... 000000
....
• 000001
..... 000022
..... 0001:39
....
• 000487
..... 001. :1. 9~5
.... 002:'U 0
- .00:3789
.... OO~5495
..... 00'724'7
..... 008B~):'j
·".010145
.•.• 0109BO
..... 011260
..... 010927
''''.009954
- • 008:345
....
• 006123
- • 00332~5
....
• 004:1.61
'·'.0141 ?6
.•.• 02 ?::.'jf:l0
.....
_. 042B6·/
• 05907~5
... • 0 '7:'.'j ~:; f.> f:l
••• 091.944
-.10'793'7
-.1233f.l6
''''. L3fH 9?
.- .152320
..... 165'740
..... 1784~:;<j>
- .190494
-.2018'71
.•.
_. • 21.2f.>20
....
....
• 222'1'74
• 2:3236'7
• 241434
e
K
1:;:HCl
• O~:j
.lO
• 1 ~~j
.20
")1::'
•
/I,'_ ...
J
.30
• 3:~j
.40
• 4~:;
• ~:; 0
• ~)5
.60
• f.> ~::;
e:;g
.HO
• f:l ~.:j
• <j> ()
.9::';
l'~
::::
cv
~~
AU T Cl C () Fo: F.: E: I... ~~ T J. ON
:3
'1
.:..
1
..... 00'702:3
..... 02411.>3
·.... 044'7'71
..... 0(.>404'7
..... O?9~:j3?
..... O</041~:;
....
• O96?73
..... O<j>9:1. 22
..... 09B1 1,9
...
• 09441'7
..... OB81.>06
.... • OB:1. :1.9:t.
....
.... • O?2~:j86
• 063:1.27
..... O~,'j3081
....
.042660
....
• 03202B
..... 021312
....
• 010I.>:I.~:'
....
....
• OOOO~:;6
.... 000982
..... 004:1.07
_.• 009?:I. 0
....
• 011.)994
·"'.024?f.l3
..... 0320~5~5
...• 0~~8021
.... .04234B
• 044B67
...
.... • 04:~j602
• 0441.>B1
-. • 042285
...
• 0~5861,:5
..... 033872
...
• 028244
....
....
• 021 <102
• 0:1.499'7
.... 007659
....
• 000001,
..... OOOO~j~~
....
• 000491,
..... 0011390
..... 004~.'j96
.... 008491,
-.013098
- • 0:1. '7820
-'.022:1.09
.- • O~~5550
....
• 027880
...
• 028974
"·.02880B
027430
.......024934
- .021438
·.... 01,7072
....
.. .01:1.963
• 006235
....
L~~G
.577
l-l
4
..... 000000
..... 000003
..... 000066
....
• 00041,4
....
• 001:391
..... 003236
....
.... • 005924
.... • 009:1.91
• 01,2638
-.015856
....
.018494
....
• 020300
....
• 021,121
- .020891
- .019610
"".0:1. '7329
- .0:1.4:1.27
... • 010101
- .005358
~::;
.- • 000000
..... 000000
..... 000010
.... 00009'7
.... 000450
''''.001314
·".002B4b
..... OO:';iO 19
"".00'7626
-.010357
....
• 012078
.•.• 01,4B91
-.016173
_.
• 01,1.>578
..... 011.>034
....
• 014531
-.012103
.- .008f:l14
...
• 004749
l't1TAL
....
• 0070130
.... • 02~5202
• 049446
-.0761,91
..... 103204
- .1292:39
....
.... •
15~51.>76
..... 176260
- .196950
- .215lH4
... .232971.>
- .248581
- • :M2'7?B
.- • 2n.'j71, 1
-.2B7:'j1,2
-. 298~~02
.•.
• 308189
.... 317269
....
• 32562'7
22
1<
1':1--10
·.10:::,
()
• I ::.'i
.20
·
~.~ ~:.:;
.30
.3::5
.40
• 4~:i
• ::.:iO
1::'1::'
• ...J,,}
.60
• 6~5
.'70
.'7::.:i
.80
• 13 ::.:j
• (lO
.95
101::;:
....
• O~'i:l.2:1.6
..··.03fH29
...• 02:7i:1. 76
·.... 0:1.:~446
4
('~l.lT
ClCClI:;:I:;:EI... (.~ T:I: Cll·l
,.>
:1.
..... O()l'lfl30
.OJ40:I.O
..... 0634Bl
..... 090;J30
..... :1.:1. 0/86
..... 12;39'7El
....
• :1.30489
..... :1.3:1.448
..... :l.2B05:7i
..... 1.21391.
..... :I. :1.2~~46
·..·.:1.0:1.632
..... 089794
..... O'7'72~.50
..... 0643:1. 4
cv ....
....
;'5
A..
..... 000032
..... OOOU/4
..... OO43()l4
..... 01.135'7
.... • 020T72
..... 03092f.l
..... 0402<;>:1.
..... 047B~'j3
..... 0:)~H14
..... 0~:i~'j9~50
..... O~:i6472
..··.0:54917
0 ~'j 1. ~.) '7 ::.5
m' •
·..·.04c>'742
..... 040698
"".0;336<16
..... 02~i<)l~i3
..... 01'7 c>56
..... 00896~?
K
....
• 000000
• OOO()30
·..·.000401
... • 001852
••• 004977
..... 00<;>'722
....
.... • 015478
• 021420
..... 026'794
....
• 0310~)O
....
... • 033862
.... • 0~5~5094
• 034'752
....
.... • 0:~29:~1
...
• 029T79
.... 02:7i466
....
• 020169
".
• 014057
..... 00'72El8
J::'
CV
~J
• 0~5
.:1.0
• :I. ::.:j
.20
• 2~:':;
.30
.3::.:j
.40
• 4~.5
• ::.:i 0
• ~:5 ~:.:j
.60
• 6~5
.70
• '7~5
• flO
• B~5
.90
• 9~j
1·1::::
,.>
:1.
..... 0:1.2:1.03
·.... 04:H03
....
-.. • OEl03'7:~
• :I. :I. :~348
..... :1.3'726:1.
....
.... • :I.~H4n5
.... • :I. ~5'7~!.4~'j
.... • 1 ~:i63:55
• :I. :::iO:::i:1.4
....
• :1.41 :1.46
..... :1.29:-562
..... :1.16002
....
• :L01684
....
.... • OEl6B59
.0'7:1.B49
....
•
O::'i6El84
....
• 042124
·.... 02'7680
....
• 013623
A..
....
• 0000:1.9
•.• 00O'7c>9
..··.004482
.... 0:1. 2;'59'7
.... • 02340'7
"•• 03~5336
.... • 046246
..... 0~:i4907
..... 060·764
m' • 063'735
·..·.064016
....
• 061 <jl3El
..... 057El73
.... 0~52:J. 92
... • 04::.5229
....
• 03'72'79
..... 02EI~'j92
•... 0193'75
..... 00979El
• ~.'j 0 0
N
I::'
,.I
4
....
('~l.l T<:lCf.)F;:F;:EI... (.~
1';:1--10
I...('~G
T:I: fJN
....
• OOOOO()
..... OO()OO:l.
....
• 000042
..... 000341
..... O(H339
..... 0034:1. 1
..... 006~'j91.l
....
• 01.0~:iT7
..... 01.48~53
..... 01El8:1.6
..... 0220c>7
m' .024266
..... 0:~5232
.m.024902
- • 023~~O 1
..... 020510
....
• 01c>64<7
....
• 01 HJ5~~
..... 006257
....
I... ('~G
;3
....
.000000
....
• OOO()19
....
.000332
...
.... • 0 (H '7 '7 1
.005131
..• 0:1. 044:1.
....
.0:1.699f.l
....
.... • 02~5ElOO
• 029937
....
.034
'75::'i
....
.037886
....
.039195
....
.03f.l'713
....
• 0365'7:3
....
.... • O:~~.~96::'j
.... .02ElO96
• 0221'78
....
... • 01~i40'7
• 007962
.... • ()O()()OO
....• O()OO()()
..... OOOOO:7i
..... OOOOI.l8
·.... 000386
....• 00:1. 2'7'7
..... 002990
..... O():)~5;3EI
..... 0086'79
···.()1,2015
..... O:l.5:L:J.0
..... 01n581
..... 0:1.9:1.46
....• ()1, 9631
..... 01.f~963
..... 01'7:1.43
..... O1423~~
..... (>:I. 0328
m·.00:'.'i543
T<:)T('~I...
.... .0091.16::.)
..... 0:549.1. (,)
..... OI.l13~~2()l
"".103966
"".138430
..... 170 1f.>:I.
..... 1.9864B
..... 22390~:i
··... 2461ElO
....
• 265803
··... 283:1. :1.0
..... 29841:5
- • ;31199::.)
.... • 3240<)lO
....
• :~:~4904
....
• 3446:1.2
..... ~5:)3360
• ~5612'7;3
..... :~684~;;6
-
'm
.447
N
4
....
.... • 000000
• 000001
....
• 000028
....
.... • 000287
• OO:J.266
·..·.003450
·..·.006943
·..·.0:1.1400
....
• 0:1.6208
....
•
.... 020'720
.024394
....
• 026Elc>2
....
• 02'7925
....
• 027526
..... 02::j'707
....
.0225'76
....
.... • 0:1.82'79
... •.0:1.29'78
006833
T(JT('~I...
~:j
....
• 000000
..... 000000
'm .000003
000050
....
• OOO~5~55
··... 00:1.218
..... 00:3020
..... 00::5794
..... 009282
..... ()13()::~5
..·.016::.513
..... 019;:~00
....
• 02:1.0c)::~
..... ()2:1. 60:5
....
• 020856
..... O:l.B!330
.m
....
•
• 01. ~560~'j
..·.011300
....
• 0060:';iO
....
• 0:1.2:1.23
• 043139:1.
.m.OEl5219
..... 127864
• 16n529
..... 2026~52
"".23:3064
..... 259:1.67
....
• 2El 1 ~7j39
'm • ~~007::5B
....
• 31 '7~54:1.
....
'm
....
...
• 331 '72c>
... • :3442'7::5
• :355~~B6
- • 3c)~.'j()02
..... 3'7362~!.
.•.
.38:1.309
....
• 388196
- • :394396
e
23
K
e:
....
cv
6
,(.'LJTOC:~Oj';:j';:r:::L.AT:[ON
HO
• () ::.:j
.:1.0
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-.037629
-.025183
- .012589
"M
K
-
4
3
~
_.
-.000000
-.000000
.000003
.- .000111
_.
• 000863
• 003172
-.007543
-.013623
-.020456
-.026957
-.032238
- .035729
- .037168
-.036526
-.033931
- .029599
-.023786
- .016754
-.008750
• 000000
.- .000001
-.000114
....
• 001261
-.005135
-.012379
- .021969
-.032117
-.041192
-.048115
-.052368
-.053856
- .052748
- .049351
-.044025
- .037133
- .029009
-.019951
-.010210
N"
.M •
cv
~~
LJ
-
TOTAL
5
-.000000
-.000000
- .000000
-.000011
-.000156
- .000872
-.002767
-.006150
-.010773
-.015962
-.020906
-.024892
- .027421
- .028226
-.027232
-.024508
- .020213
- .014549
-.007739
_.
e
-.•• 045454
142580
-.234538
- .301713
-.347482
-.378820
- .400892
-.416956
-.429011
-.438306
-.445645
-.451557
- .456403
-.460437
-.463837
- .466739
-.469238
-.471412
-.473318
.200
AUTOCORRELATION LAG N
RHO
• 05
.10
.15
.20
'1 1: '
.~.J
.30
.35
.40
.45
• 50
I:'~
.JJ
.60
.65
.70
.75
.80
.85
.90
.95
N=
1
····.055555
-.166142
-.258248
.310127
- .328784
····.325757
.309885
....
• 286915
-.26038a
_.
• 232440
-.204340
- .176826
-.150309
-.125004
-.101003
-- .078323
- .056939
....
• 036801
-.017845
'M,
"M
3
~
L
- .000000
- .000301
- .004388
- .016797
-.035781
- .056179
...
• 073831
- .086665
- .094194
- .096793
-.095206
- .090259
-- .082724
-- .073266
- .062433
- .050663
.- .038301
- .025610
- .012792
.•.
.000000
.- .000001
- .000102
- .001217
- .005111
.- .012494
- .022324
- .032743
-.042055
- .049144
- .053483
-.054979
.0.053817
-.050317
- .044855
-.037805
- .029514
- .020284
.010374
M"
4
- .000000
-.000000
- .000003
- .000101
-.000828
- .003130
-.007550
-.013746
- .020735
- .027395
_-. .032806
• 036381
- .037851
-.037192
- .034539
-_. .030117
• 024191
- .017030
.- .008890
TOTAL
~
~
- .000000
- .000000
-.000000
-.000009
- .000145
- .000841
.- .002728
- .006143
-.010844
- .016142
- .021201
-.025284
- .027878
- .028706
- .027697
- .024924
- .020549
- .014786
- .007861
.-.
• 055555
- .166444
- .262741
- .328252
-.370681
- .398736
-.418023
-.431816
....
.442035
.- .449838
_.
• 455951
-.460846
- .464839
-.468149
- .470931
- .473296
-.475330
- .477095
-.478639
e
27
K
....
50
e:;:HO
• 0 ~.:i
.:LO
• :L ~5
.20
• 2~:5
.30
• 3~:i
.40
• 4~:.)
• ~::i 0
1::'1::'
• d ...1
.60
• 6~:.:j
.70
• 7~::i
.80
• f.l ~::i
.90
• 9~)
1·1 :":
,.)
1
....
• :LOOOOO
..... ~?49t>20
....
• 340405
....• ~P7975
..... 38:;~2 :1. ~.:i
·..·.3t>?430
..... 342429
....
• 3:1.~.~42::.~
..... 2B04:39
....
• 24B219
·..·.2:J.6739
..... :L 86~.:i24
..... 1 ~57f.l2B
..... 1. ~~O75::'~
....
• :1.O:'.'i29EI
..... 0f.l142:1.
.....
....
0~i9044
• 0~5ElO78
..... 01f.l42B
• 000000
.- • 0OO2~:i6
..•
• OO44Bc)
.... • 0 :1. ? El ~5 ~.:i
.... • 03fB~32
• 060:L40
...·.OlElllB
....
• 092:1.:1.2
·.... 099722
....
• 10209t>
..... 10008 :1.
- • O(14~)88
....
• 08644f.l
"".076370
.06492EI
....
no'
no'
•
O~'j2~5'78
..... 0396 '72
..... 0264130
....
• 013206
1<
I:':HD
·
1·1 ::::
1
....
.3:'.'i
• 166c)6'7
• 3~530:L 7
..... 403?32
..... 42~H:I.~)
"".414855
..... ~~9:1. ~.)9 :1.
..... 3c)06c)7
.40
• 4~:;
• ~50
1::'1::'
• 326~388
..... 29123~5
..... 2~::;661.0
..... ::'~232'70
() ~:)
• :1. 0
• :L ~)
.20
+
r)I::'
A'. ....
J
.30
+ ... J..J
• 60
• 6~.:i
.70
• '7 ~:)
.00
e:~~;
.95
....
....
....
....
.:J.91.~:i93
• :1.617~~:=:i
..... :I. ~~~5'723
.... • :l.On'HO
..... • 0f.l~50:l. :I.
• 060:1.21
..... 03B?30
"".018'725
...
• 000000
....
... • 000000
• 000081
····.001.:1.32
..... 005060
..... 0 1.2?2~::i
....
• 02~504B
..... O~34026
....
• 043B29
..... O:'.'i:I.26:3
....
• 055T75
..... O~57288
..... 056009
..... 052296
..... 04t>552
"".039180
..... 03054~5
....
.... • 020963
• 010707
:J.OO
(.~ l.J T
DC D Fo: Fo: r::1... (.~ T :r (J I-!
• 0'7791.)7
• 066209
- .053558
....
• O403?3
....
• 026924
.- .013417
4
....
• OO()OOO
..... 000000
..... 0000'72
....
.... • 00:1.093
.... • 00503~:;
• 012841
..
• 0234:1. '7
..... 034684
....
• 044741
"".052353
....
• 056954
....
• O~i84 73
"". 0~j71.35
....
.... • 0~)3~HO
• 04742:L
....
.0:39882
....
... • 031069
.... • 021 :~O9
• O1087?
~j
....
• 000000
• 000000
.•.
• OOO()O2
....
• OOOOB2
..... OOO?~:'j9
..... OO~5035
....
• OOn'j49
..... 0 :1. 3978
....
• 021288
..... O~.~El27 6
033957
..... 037707
'''' • 039245
....
.038553
..... 03~ilB 1
·"·.031:L'74
"".025015
"".017592
....
• 0091'73
....
'no
•
L.. Ar,;
3
")
....
• 000000
....
• 000244
..... 00460El
..... O:J. f.l~i~~9
....
.... • O~WEIO?
.... • 0623~30
• 08145:J.
..... 0<15013
- .10263B
....
• 104B'73
- .102620
.- • 09683~?
....
• OB8:371.
....
.141
I-!
cv ....
....
.:..
....
I...(~G
3
.:..
...
...
t~V
AtJTelCrJf;:I:;:F.::I...AT:r ell-!
....
• 000000
....
..• • 000000
• ()()OOOO
.OO()OO6
....
• OOO:L ~~3
..... 000 l Ti'
....
• 00264:3
..... O()t>115
"".010(171
....
• 016492
....
.... • O:;.~ 1l8i3
... • 026074
• O::'~B802
..... 029683
.no.02f.l644
....
.025/69
.... 021235
..... 015268
....
.OOB111
'm
....
• 100000
no' .249B76
-
• ~3449?4
- • 397()~.:i2
-.426~H3
....
• 444396
..... 4~560:L ~~i
.46400?
..... 469?64
..... 4'7406EI
·"·.4773B7
.480010
- • 4821 ~.~<1
···.483f.l70
-. 485~~24
-'. 486~)53
-.48'7604
....
• 48B51. :;~
-- .4B9304
no'
-
• :1. 00
N
~)
4
.... • OOO()()O
..... 000000
....
• 000001
..... 0000'74
....
000'724
... •• 002984
... • 00'7540
_.
• 014086
....
• 021560
.....
_. 028'718
• 034540
038382
""
• 039n:i6
...
• O~~924?
''''.036414
..... 031?:I.3
....
_. • 0254:36
• Ol?8?9
-'.009318
'm •
TOTAL
..... OOO()OO
....
• OOOO()O
""
.000000
..... 000005
....
.000 :1.12
..... 000744
..... 002:7j96
.006093
....
• 011026
....
• 016661
no' .0220'79
....
• 026470
....
• 029269
....
.... .030 :1. 78
• 029:1. ~~6
....
•
.... 026200
-
....
-
.021~:i84
• O:L5~:H3
.OOB238
TOT('~I...
..... :1.66667
''''. ~~~·5326:1.
.... 40B4:1. ~::i
·..·.442B27
-·.460:'.'i:7i4
• 470l5?
....
-.477170
-.4814136
-.48454'7
-- .486810
....
• 488~540
.- .4fJ9899
.- • 490990
....
• 49188~5
.492626
.- • 49~5252
....
• 493?8'7
.- • 49424?
·"·.49464f.l
'no
28
K
-
cv
10000
H_
.010
AUTOCORRELATION LAG N
RHO
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
~c:·
.JJ
.60
.65
.70
.75
.80
.85
.90
.95
N=
2
3
-.000000
-.000249
-.004828
....
• 019375
.041442
- .064665
-.084246
-.098013
-.105632
-.107710
- .105203
-.099108
-.090317
- .079580
-.067499
-.054544
-.041077
- .027370
- .013628
-.000000
-.000000
-.000065
-.001056
-.005012
-.012958
-.023787
-.035347
-.045661
-.053452
-.058143
-.059669
-.058268
-.054331
-.048295
-.040588
-.031596
-.021656
-.011047
1
-.490196
-.497262
-.493989
-.478856
-.452310
-.418181
-.380226
-.341109
-.302480
-.265273
-.229967
-.196765
-.165704
-.136731
-.109743
-.084612
-.061203
-.039383
-.019022
UN
K
H_
- .000000
-.000000
_.
• 000001
-.000066
-.000689
- .002930
-.007526
-.014187
-.021826
-.029157
-.035122
- .039057
-.040668
-.039943
-.037050
-+032253
-.025857
-.018166
....
.009463
cv
1000000
AUTOCORRELATION LAG N
RHO
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
C:'I::'
.~JJ
.60
.65
.70
.75
.80
.85
.90
.95
N=
....
1
• 499900
-.499725
-.495092
.._. .479483
• 452715
-.418462
-.380430
...341261
-.302595
-.265361
-.230036
.196818
..165745
-.136762
-.109765
- .084628
-.061214
- .039390
-.019025
.ON
,
3
- .000000
-.000000
-- .000000
- .000065
-.001056
- .005012
- .012959
- .023791
-.035353
-.045670
-.053463
- .058155
- .059681
- .058279
- .054341
-.048304
- .040595
-.031602
- .021659
-.011048
~
-.000250
....
• 004831
- .019385
- .041459
- .064689
-.084275
- .098044
-.105662
-- .107738
- .105230
....
• 099131
- .090337
- .079597
....
• 067512
- .054554
-- .041084
.- .027374
- .013630
5
4
-
4
-.000000
-.000000
-.000000
-.000004
-.000102
-.000711
.
• 002546
-.006066
-.011075
-.016823
-.022366
-.026864
-.029735
-.030674
-.029609
- .026631
-.021933
-.015759
-.008365
~
~
J
- .000000
.- .000066
- .000000
- .000004
- .000000
-.000689
- .000102
-.007526
-.014187
-.021829
-.029161
-.035128
-.039064
.- .040676
- .039950
-.037056
-.032259
- .025861
-.018169
- .009464
- .002546
- .006066
- .002930
e
-.490196
- .497511
-.498883
-.499358
-.499574
-.499692
-.499763
-.499809
-.499842
-.499866
-.499884
-.499898
-.499909
-.499918
-.499926
-.499932
-.499938
-.499942
-.499946
.001
- .000000
- .000000
-- .000001
TOTAL
-.000711
-.011075
- .016825
-.022369
- .026868
- .029740
-.030679
- .029613
- .026635
- .021937
-.015762
- .008366
TOTAL
- .499900
- .499975
-.499989
-.499994
- .499996
- .499997
- .499998
- .499998
- .499998
- .499999
.- .499999
-.499999
-.499999
- .499999
-.499999
- .499999
- .499999
- .499999
- .499999
e
29
I (b) . H/M/l
.:HD
G
..-
cv .... 1.10
• 34~:)9
ALJT()COI";r.;E:L.. (.~T:[ Cll-!
LAG
I-!
:L
~.~
3
4
r.:-
.:/.o
• 0 ~:j
.000 :1. 7B
• 0OO64~)
• :/. ~5
.O()13:1.~)
.20
• O()::~:/. :1.9
.OO299B
.00390 :L
.004'7(16
.00:56:1. 4
• 0 0 I.) :3 ~.:j :L
.0(>1.>964
.007423
• 0 0 00 :1. El
.000:1. ~.~O
.OOO:H1
.OOO6El3
.OO11:H
.0():L660
.002240
.OO2f.l40
• OO:~427
.0()3969
.004434
.004?90
.OO500?
• 005053
.OO4El99
.0045:1,4
.00:386::'j
.002920
.001644
.000002
.OOOO2?
.000 :1. 0'7
• ()()0264
.OO050El
.OOOEl35
• 00 12:~2
.OO:L6'77
.002146
.002610
• 0030~59
.00:340:1.
.003666
• 003f.lO:l.
.003'772
• 003~:;48
.003094
.0023'76
• 00:l.~55El
.000000
.00000'7
• ()0003'7
.0001 :1.2
.000250
.0004::'j8
.000'734
.00:1.0'70
.00144'7
.OO:l.f.l42
.002229
.0025'77
.002857
.003034
.0030'76
.00294'7
.002612
.002035
.0011'78
.OOO()OO
• OOO()O~.~
.000014
.OOOO::'jO
.000 :1. ~?9
.000264
.000460
.OOO/1:::j
.001020
.OO:L357
.O01?03
.002031
• 002:H 1
.002510
• 002:~j95
• 002:~j2El
.0022'75
.001795
.001051
I-!::::
{')I::'
• ",',. ,J
.30
.3::5
.40
.4::.:j
• ::'jO
1::'1::"
• d ...J
.60
• 6~:j
.70
.OOT701
.OOT7'70
• OO'760~.~
.OO'7:L69
.006444
.00::'j39'7
.00:3996
.002209
·.ElO
/~.)
• B~5
.90
.n)
l~
_.
• 0 ~5
.:/.O
• :I. ~.:j
.20
• 2~:;
• ~5 0
.3::5
.40
• 4~.)
• ~:;O
1:=-1::'
• ..J ...J
.60
/, 1::'
~ \.)d
.lO
.n-:;
e~:~~
• t:
J
.90
.9:':,
I·!::::
LAG
2
3
4
5
.0002El4
.00 1 0~50
.OO2:1.0El
.OO:H:l.O
.OO4El4'7
.006343
.OO/f.l32
• 0092~.:;4
.0:1. O~:j52
.0:1.16'73
• O:/.2~.)64
.013:1.7:3
• 0:1.:~444
• 01~53:1.l3
.0:1.2'73:1.
• OOOO:~O
.000203
.OOO::'j'75
.00:1.:1,50
.001904
.002'79El
.003'787
.004f.l20
.000004
.000049
.OO01El9
.000464
.000ElElEl
.001454
.00214:1.
.00291'7
• 003'7:~9
.004564
• 005:~40
.0060:L6
.006535
• 0068:~'7
.006859
.00653:1.
.0057'75
.004504
.002618
.000001
.000013
.000069
.000205
.()00451
.000El21
.000000
.000004
.000026
.000096
• 000~.~42
.00048/
.000f.l40
• 001~,~9B
.00 Hl43
.00244'7
.003072
• 00:36 7~~
.004197
• 004~588
.004'781.
.004'706
.004284
.003428
.002039
.01161:~
.009883
.007448
.004:L97
.OO6El:l.2
.00'766::';
.008351
.OO8B13
.00E1992
• 0088~~5
• 008240
• 00'7 U) 1.
• OO~j499
.003153
.OOO19El
• OOOBO:l.
.OOHI2::~
• 00 :'5 ~.~ n:j
• 005:1, '76
.00'7543
.0:1.0:396
• 013n:j~'j
.01764:3
.0220E!'7
.027:J,14
.032756
.039045
.046021
.053'724
.06:;~20:1.
.0'7150::'j
.08169~:j
.09283::'j
I-!
1
.OO::'j846
TIJTAL..
cv ,- 1.20
.2ElT7
(..)LJT ClCOF;;F;:E::L_ AT:J: Cll'l
I":HO
~J
.001~510
.001902
.002569
.003273
.00397:1.
.004613
.005:1,44
.005506
• 0056:~4
.00545'7
.004900
.0038'73
.002279
TOT('~I...
• OOO:H 9
.001300
• 0029El~.:j
.005420
.0086::'j6
• 01~,~n'.'iO
.017769
• 0237B~5
• 0~50Ell6
.0391~5B
.048676
.05960B
.0'720'72
.08622'7
.102261
• :1.20:39:1.
.:1.40881
.16404~'j
.190262
30
C~
.246El
....
A U TO C 0 F;: j';: r:~ L ~~ T
I~:HD
N:""
• O~:j
:1.
r)
• :I. ~.:j
.20
• 2~:.:j
• ()O~:)939
.OO/BOE!
.009691
• 0 :L 1 ~'i:L /
• O:l.~5220
.30
• ()()()()39
• () () () () () ~.:j
• ()()()2~:j9
• ()O()O6~:)
• 0002~:; :1.
• 0()()6 :1.1
.OO:L:l.64
.00:l.90:L
.002796
.OO3f.l09
.004891
.()O5987
.00/035
.007970
• OOfl7 :1. 7
.009:1.97
.0093:1."7
.OOB974
.008042
.0063"70
.003"771
.OO()()():1.
.OOO():l.El
.000095
.()002BO
.000610
.00110:1.
.001/50
• ()()()OOO
• ()0()()()~5
.OOOO3B
• OO():1. 3~:5
• ()OO:3:~7
.000670
.001 :1.4'7
.00176:1.
.002492
.003303
.004:1.47
.004969
.005/0J.
.006267
.O024:1.l
.O:l.5<j>8El
.0:1.69:1.6
.017437
.0174b7
.0:1.6906
.BO
• ()1. :'.'j6~5~j
.01:L2~50
• f.l ~.:j
• <yO
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.005327
.006896
.008556
.010263
.011967
.013612
.015129
.016430
.017390
.017817
.017388
.015499
.010849
Q
3
4
.000008
.000091
.000338
.000802
.001499
.002420
.003542
.004833
.006253
.007760
.009306
.010837
.012288
.013574
.014580
.015130
.014933
.013443
.009493
.000002
.000032
.000156
.000439
.000921
.001616
.002518
.003604
.004843
.006195
.007617
.009054
.010443
.011705
.012730
.013352
.013301
.012072
.008586
cv
.0745
MO'
M ••
~
~
.000000
.000012
.000076
.000252
.000593
.001127
.001863
.002790
.003883
.005109
.006424
.007779
.009112
.010346
.011376
.012044
.012096
.011055
.007912
TOTAL
e
.000355
.001529
.003727
.007221
.012374
.019668
.029732
.043379
.061628
.085741
.117268
.158177
.211103
.279829
.370148
.491507
.660433
.908543
1.304606
2.50
AUTOCORRELATION LAG N
RHO
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
I='~'
.J~
.60
.65
.70
.75
.80
.85
.90
.95
N=
1
2
3
4
5
.000263
.000963
.001993
.003276
.004752
.006373
.008099
.009894
.011723
.013549
.015332
.017024
.018566
.019878
.020841
.021265
.020806
.018777
.013543
.000040
.000259
.000721
.001431
.002368
.003500
.004794
.006214
.007727
.009292
.010872
.012418
.013876
.015173
.016203
.016802
.016677
.015244
.011121
.000007
.000084
.000311
.000737
.001375
.002218
.003246
.004430
.005737
.007132
.008574
.010019
.011413
.012685
.013738
.014421
.014467
.013348
.009819
.000002
.000030
.000147
.000412
.000862
.001509
.002347
.003357
.004510
.005773
.007108
.008471
.009808
.011052
.012111
.012842
.012995
.012082
.008947
.000000
.000011
.000073
.000242
.000566
.001072
.001766
.002638
.003667
.004822
.006067
.007358
.008645
.009860
.010916
.011676
.011904
.011139
.008295
TOTAL
.000312
.001355
.003328
.006508
.011271
.018140
.027821
.041257
.059662
.084543
.117744
.161572
.219124
.294925
.396131
.534871
.733388
1.037054
1.554436
e
3S
C~
e
HO
N::::
• O~;
.to
• :I. ~5
.20
'1 c::'
• ~:.....J
.30
• 3~:j
• 40
• 4 ~::;
• ~:;O
I··· ..••
• . '.."J ....)
• t)o
• t) ~~;
.'70
• '7~:j
cv 'N'
• Ot)2:~
AUTC)CC)F;:F;:ErLAT J: ON LAG
~~ • 7~)
N
:I.
:~
:3
4
~:)
• OOO:~~30
.OOOB43
.00:1.745
.0028'71
.004:1.69
• 005~:i9El
.00'7:1.26
.OOEl'72:1.
• O:l.O:~~56
.O:l.200()
.0:1.3621
• O:l.~H81
.000036
.OO02:U
.000644
.00:L27'7
• 0 0 2 :1.:1. ~5
.003:1.24
.0042El3
• 005:':;~;8
.006921
.00El341
.009783
• 0 :1.1 ~~ :1. 1
.000001
.OOOO:W
.000 1 :~6
• 000~5ElO
.000'794
.OOO()OO
.00001:1.
.000Ob9
• OOO~~27
• OOO~:;29
.00:1.000
.001643
.000274
.001 :1. In'
• 0029~59
• OO~)8~~9
• 0 10 :Lf.r7
.O024~:)1
.03E)791
.057063
.082324
.:1.16684
.0:1.66~B
.0:1. 2~~j'79
.0:1. '791:1.
.0:1.8921
.0:1.3828
.0:1.4872
• O:l.~)::j72
.000007
.000077
.000283
.000669
.001247
.0020 :1.1
.002942
.0040:1.6
• 00:':; ::.~ 0 ~5
.0064'79
.00'7804
.009:1. 4~5
.0:1. 04~5~.~
.0:1. :1.6'7:3
.0:1.2'727
.0:1.3483
.0:1.37:1.0
.012921
.80
.O:l.9~501
• 8~:)
.0:1.9355
.01'7854
.013:3'74
.90
• 9~:;
....
• 01~i67:'~
.0:1.4638
.01:1.08'7
C~
• 00986~i
('~LJTOCC)Fo:r.:EL(..) T
I~:HO
·.:1.0
() ~:;
• :L ~:)
.20
...) I::'
fo ...•..... 1
.30
• :3~.:j
.40
.4::';
• ~.'i0
+ 5~.:;
.60
• 6~.:j
e
.'70
• '7 ~:;
.BO
e
+ 8 ...:'J
• Ii> 0
.95
1'1::::
:L
""
.000202
• OOO'7:~S)
.00:1. ~5:~2
.002:::;22
.003665
.00492b
.0062'77
• 00"769:~
.OO9:L50
.010624
.0:1.2089
.013512
.01485'7
.0:1.60'73
.017080
.0:1.7'748
.0 :1. 7824
.016"751
.012982
• 0000~52
.000206
• OOO~5'7 4
.00:1.:1. ::Hl
• 00:I.B8~5
• 002"78~)
.003819
.004961
.00bHl4
.007464
• 00877:';~
.0:1.00'78
.0:1.:1.344
~)
J: C)N
LAC';
• 00~:j6:~4
.OO6B42
• OOElO~jEl
.009225
.0102'70
.01:1.0'7t)
.O:l.:L436
.01092:1.
• 0084~54
.16:'~'7:34
• :';~2~59:30
.30540El
• 415~)0:3
• ~j6El796
.79306:1.
1 • :I. 4ElOO:':i
1 • 7887~.):L
• 00
N
4
c:'
,J
.000006
.000069
.000001
.000026
.000000
.000010
.OOOOf.)4
.00021:1.
.000490
• 0OO9~!3
• 001~514
.002256
.003:L31
.004:1.16
.005:1.83
• 006~502
.007436
• 008~:'.i40
.009550
.010370
.0:1.0821
.000242
.00 1 O~5B
.002629
• OO~:)212
.009:1.8:L
.0:1.5084
.023'724
.036245
.05421:1.
.079603
.:1.14784
.16257::';
.226'707
• :U2'797
.430:L90
.010::'i:l.5
1.243~H3
.008392
2.006050
.OO02::'i6
.00O:I.2~)
.000604
.00:1. :1.26
.OOHl:l.4
.000:348
.000'726
.00126'7
.001966
.002808
.0037"73
.004835
.005966
.007:1.36
.008307
.009434
• 0026:':i:~
.O:L 429:':;
.0:1.4554
.O:l.:~845
.012:~04
.0:1.084'7
.009714
.O:l.2~:;23
.0:1. 02~.)9
.01131:L
.0:1.2099
.012405
.0117'75
.009046
.00:3404
.0044'76
.01t)5l6
.02::'i'760
:~
• 00:~b22
.004698
• OO~58:':j4
.007062
.008292
• 00950~)
• (>:I. 0657
.OU,6Ell
.0:1.2469
.012822
• 013~i46
• 00906~.:j
cv .... ~5
• O~:j28
N"
.001~588
.002156
.00308:1.
.004:1.40
.005303
• 0 0 t) ~:) :3 '7
.OO'7l306
TC)TAI...
.01045~5
.0112(,)2
.0116'73
.0:1.1279
• 0089~:;'7
TC)T('~L..
• ~.jn:;6~jO
.B42112
36
C~
-
cv ....
.045:'5
AUTC)COf::fi:F.::L.. ~~T:r. ON
I:;:HO
1-1::::
.O::.:j
.10
• :I. ~.:j
.20
• ::.~ ~:.:j
.30
• 3~:j
.40
• 4:~j
• ::.:i 0
I::".::'
+ ..J ,.1
.60
• 6~5
4
~:i
.000 :l.lEl
.000651
.O(H350
.002224
• 0 0 ::~ ::.! :3 ~5
.004:'549
• 00 ::5 ::'i 4 '7
.OO6B06
.OOEl:l.06
.00942'7
.0:1.0'746
• 0 :/.:W~59
.OOOO2B
.000:1. El4
• OOO~:;:I. 2
.000001
.000024
.000:1. :1.4
.000:'5:1. 7
.000660
• 0 0 :1.:1. ~5 2
.00:1. '7El6
• 002::.'i~:; 1
.000000
.000009
• 01.:~27::'i
.0:1.02:1.5
• O:J. 1.:3:1.6
.0:1.2296
.01.3060
.000006
• 0OOO6~5
.0002:31
• OOO~7i4~'i
.0010:1.4
.00:l.c>34
• OO~,!390
• 003~?.63
.0042:34
.OO:'.'i2Bl
.006 :3'79
.007501.
.0086:1.9
.009694
.O:l.Ob?:I.
.0:1. :I.4c>2
.O:l.:l.B9B
.O:1.5:~91
.BO
• El::.:j
.90
.O:Lb:I.OO
• O:l.6:~2El
• O:I.5~'i90
.0:1.2454
• 0010:I.~i
.OO1.C>79
.OO24B4
.003408
.0044:30
• 0 0 ~,:i ::'i :;! El
.OO66BO
.OOlEl63
• 0090~'i2
.0:1. 342::'i
• O:l.::'!9l~?
.O:L04'15
(~
....
• 0 :~j
.:/.O
• :I. ~:j
.20
"')1::"
• ..... \.oj
.30
• 3~.i
.40
• 4~.:j
• ~50
'::'1::'
• ...Jd
.60
• 6~.:j
.lO
• 7~5
.HO
• El :::j
.90
• (7 :::j
1·1::::
.OO:342l
.004394
.005427
.00649(i!
.00'7::'iB1
• 008C>~53
.OO9bO::'i
.0:1.04:1.0
• (HOB89
.010682
.008740
.(H:I.~59:5
.009432
cv
.0393
ALJTC)CC)):;:I:;:r:i:I...AT :r.l11·1
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N
3
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1
.0:1.4413
.lO
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....
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N
.00OO::'j9
.000194
.OOO4:7iO
.000846
.00:1.:387
.002065
.002IM::i
.OO:·P6·7
.O04?46
• OO~'il?6
.006B26
.OO'1B59
.00BB22
.0096:34
.010:L42
.010003
.008223
~~
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4
~)
.000:1.5'1
• OOO~'i??
.OO:l.:I.(i!6
.00:1. (i!'1O
.002866
.OO3B::i7
.004923
.006045
.OO?207
.00B:392
• 009~m:l.
.O:I.O?53
.0:1. :1. BEl::'j
.0:1.2942
• O:I.:~El75
• 0OOO2~'i
.000 1 c>~i
• 0OO4~'iB
.OOO(i!07
.001501
.002222
• OO:~O49
.003965
• OOOOO~"j
.OOOO5?
.OOO20(i!
• 00049:~
• 00000:1.
.000022
.000104
• 00O:~89
.OOOc>OO
.001046
.001621
.OO::'!315
.0031 :1.:1.
.003990
• 0049:31
.005912
.006(i!07
.OO?BEl4
• 008ElO :1.
.OOn'iB6
.0:1.0:1.04
.0:1.0042
• 0084~513
.000000
.OOO()O9
• OOOO~)5
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.0004:1.2
.OOO7?4
• O():J. 26B
.OO1B87
.0026:1.7
.003440
• 004~5~56
• 0052131
• 0062~5()
.0072:1.1
.008:1.20
.OOEl910
.009449
.OO(i!441
.OO?969
.O:l.4::'i94
.0:1.4922
• O:1.444~.!
.O:l.:l.B53
.0049:':i2
.005990
.007059
.008139
• 009204
.0:1.0224
.0:l.:J. :J.~.'i2
.O:J.190El
.012340
.O:l.20El:7j
.O:J.OO24
.00091::'i
.002:1.54
• OO~?942
• 003fJ:l. 9
.004766
.OO576:?
.OO6?85
.00'713:1. :1.
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.009?3:J.
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• (>:!.O99:1.
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.0009313
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• 0046c>~:i
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.021796
• 0~537~.'jB
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.Ol66??
.1124~H
.161649
.228193
.:U80IB
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• B826(i!6
1. 325l0:~
2.205913
• 50
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• 000Hl9
• 0OO8:3~5
.002090
.0041 B::'i
.007464
.01246:?
.0200:1.3
.0313913
.048482
• 0737:1.~?
• :I.0992?
.16029:1.
• 22BB~:i8
.3~.!19:1.~'i
• 450~5?4
.634611
• 9:J.6~5:J.~5
1.396529
2. 3B86~5'1
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37
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.0:'544
AU T()C()fi:Fi:r:::I... ('~T I ON
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• ::.~ ~.:j
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• :'.'jO
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.85
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2
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4
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.001754
• 00 2 ~:j::j:~
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.004389
• OO~):394
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.009637
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.002736
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• 00~53B5
• 00635~5
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• 008~509
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.00005 :I.
.000:1.89
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• 0026~37
.003450
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.0052:1.3
• 00614~:j
.00708:'.'j
.OOEI006
.00EIE172
.009620
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• 0 :I. 0 1 HI
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• 000~.~6~5
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.00210:i
.00282::'j
.OO362t"j
.004482
.005379
.006293
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• 0:\.:1.326
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.002;'589
.003141
.003960
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• 00886~::;
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4.00
2
3
4
5
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.008668
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.000021
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.001795
.002464
.003207
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.004857
• 0057~55
.006628
.007519
.008388
.009202
.009906
.010394
.010399
.009025
.000004
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.0001'11
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• 00390:~
.004726
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.007288
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.009331
.009411
.008226
.000001
.000018
.000086
.000239
.000497
.000865
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.000000
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.000150
.000345
.000648
• 0010~3(1
• 001~:i74
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.OO2B70
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• 00523~j
.006060
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• 007~:'i80
.008:1.33
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.000746
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.00376::5
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• 045T7~5
• 07080:~~
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.158712
.229000
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.944901
1.457719
2.5550613
N
1
.0:1.2030
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• O:l.O~:jB2
l'Dl·AL..
.OO191:~
.002::'j70
.003298
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T(JT('~I...
• 0 0 0 :1. ~5:J.
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• 003~599
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1 • 51 O?5~5
2.706204
38
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.O0202l
• 00263::'~
• 00:~~.~92
.003(Y92
.O04?20
• OO~)464
.OOf.>212
• OOf.>S)~'j1
.00?f.)5f.>
.008292
.0087'79
.00892:7j
.008029
.000003
.OOOO3(Y
.000 :1. 42
• OOO~B3
.0006:1.9
.OO09n'j
.00l455
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• 002::=jEl~5
.OO323l
• 0 0 ~5 (Y :I.e)
.OO4f.>2El
• 0053~:'i4
.OOf.>OT7
.00677El
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.008123
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.00000:1.
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.0000'72
.000200
• 000415
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.00 1. ~593
.002:1.41
.002'749
.003404
.004093
.OO4B02
• OO~j5:1.f.>
.006213
.006B5?
.OO73?9
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• 0()69~!7
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.00:50'74
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.0:1.4363
• 0235~:;6
• ():n~~.)74
.0621300
.1000:1.~5
• 1~Bf.>94
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• 32(y:::j?~:j
.4'71543
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:I.• 00f.>B?:I.
:1..59'7184
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.019f.>
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AUTOC<JFi:r';:E::I...(..) T:r ClI·l
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.0032::j4
• OO~5El92
• 004~)49
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• 00::jEl92
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.00'7El4:1.
• OOEl40~:j
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.000014
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.002'74:1.
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.004562
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• 0064~57
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.007464
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.0O'7:L:l.O
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.000:5 :l.El
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.001. 21El
.0016f.>5
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.002'706
.003283
.003B84
.004499
.005 :1.1 El
.005'724
.006292
.006'769
.007020
.OOf.>543
.000001
.0000:1.3
.00006:1.
.000169
.000349
.000607
.000939
.00:1.341
.001802
.002314
.00286'7
.0034:'.'jO
.0040:7;4
.004666
.00:7;269
.005840
• 006;3::.~f.>
.OObbOl
.006:1.83
.000000
.OOOOO:7j
• OOOO~53
.000:1.0'7
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• 0020~56
.002571
.00314:1.
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.00434:1.
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.00b299
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• 9~:;
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4
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• 00 0 ~:j~~ ~:j
• 000882
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.002218
.002731
• OOOO:L2
.000077
.OOO2:L3
.00042 :I.
.000697
.001.032
.O(H417
.00Hl4"7
.002312
.002807
• 00:~:'524
• 003EI~jb
.004:397
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.005469
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• OOOOO~?
.000027
.00010:1.
• 00O:?:~6
.OOO43B
.00O"70:7j
.001031
.001409
.OOHl3:3
.002293
• OO::'~78~5
.00329:7i
• 00:~821.
.0043:7i4
.004880
.005382
• 005El:L 9
• 00609:~
• 00~:j810
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• OOO(H :L
• OOOO~:j2
• OOO:L 4~5
.00029"7
• OOO:7j 16
.000'799
.OO:L140
• OO:l.~B2
.00:1.968
.002439
.002<"38
.003455
• OO:~982
• 004~:j07
• OO~)O:l.1
.00::)456
• 00~j"746
• 005507
.000000
• OOOOO~)
.000028
.000092
.000211
.000394
.OO064::'!
• 0009~j4
.001322
.00:1. "738
.002195
.002683
.00:3194
.0037:1."7
.004241
.004747
.00c)396
.006644
• 00629:3
G
....
cv ....
• (H3"7
AUTDCDF;:Fo:ELAT I ON
I:;:HO
.0::5
.:1.0
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.20
r)r.:'
• ...... "J
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• ~5~)
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• OO~5B23
• 004:~89
.004960
.005::)3 :L
.006094
.006637
.007:1.38
.007552
.OO775f.l
.007:?73
1::'1::"
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+ ...J"J
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.00~j:l.97
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1
4
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.000219
.000454
.000"750
.001092
• OO:L47:~
.OOOO:!.O
.OOO06:7j
.000:1.81
.000359
.000594
.000880
.001209
.001576
.0019"73
• 002:~96
.002B39
.003::'!96
.003'762
.()O4231
.004693
• 0051 :~::=j
• 005~'j2:::i
.OO5"7f.13
.0055f.l0
.000002
.000024
.000086
.000202
• OOO~575
.000604
.0008f.13
.001206
.OOl::'i69
.00:1.964
• 00::'!:~B4
•oO::'!B2:7j
.003279
.()03740
.004200
.000000
.000009
.000045
.00(H23
.000255
.00044:'5
.OO06B6
.000979
.001316
.001690
.00::'!096
.000000
.000004
.O()OO24
.000079
.000182
• 00O:~39
.000553
.OOOB::'!2
.001:L38
.001.49'7
.001891
.002:3:1. 3
.002755
.003210
.003668
.004c)43
.005040
.005317
.005165
.3~521.45
.481.6:39
.702207
1. 0~:j6617
1. '71547:1.
3.3585"78
N
3
.OO:l.8EI~)
.OOOO8:::j
.000:38 :L
.OOO96?
.0019'74
.00:3614
• 006~!.6:1.
.01060B
• ()1 8 0 0 [)
.O3106[)
.054087
.0918:50
• 14779~5
.225539
6.00
2
.002322
.OO27BO
• 00325~3
.003"73"7
.00422'7
.00471f.l
• 0052()~:;
.005679
.006:1.24
.006506
.006733
.006432
TOT~~I...
.002525
.00297:3
.003430
.003889
.004334
.004737
.005026
.004907
.00411~)
.004523
.004821
.004'725
TOT~~I...
.0000"73
.000324
.OOO82::'j
.001.l)88
.003:1.00
.005398
.009222
.0158"78
.028050
.0504"7B
.0884"7:3
.14534"7
.224404
.332736
.4B47B2
.709767
1.073330
1.756577
~5. 504648
40
0
RHO
N=
.05
.10
.15
.20
'lC~
.L~J
.30
.35
.40
.45
.50
.55
.60
i~
.~)~J
.70
.75
.80
.85
.90
.95
-
e
cv _.. 7.00
.0101
AUTOCORRELATION LAG N
1
,
3
4
5
.000045
.000163
.000339
.000559
.000815
.001099
.001407
.001734
.002077
.002432
.002796
.003166
.003540
.003912
.004280
.004632
.004950
.005179
.005087
.000008
.000049
.000136
.000269
.000446
.000660
.000907
.001182
.001482
.001800
.002134
.002480
.002835
.003194
.003552
.003900
.004221
.004466
.004431
.000002
.000018
.000065
.000153
.000283
.000455
.000665
.000910
.001183
.001481
.001800
.002134
.002481
.002835
.003190
.003540
.003865
.004121
.004116
.000000
.000007
.000034
.000094
.000193
.000336
.000520
.000741
.000996
.001281
.001589
.001916
.002258
.002609
.002965
.003316
.003645
.003909
.003924
.000000
.000003
.000019
.000060
.000138
.000258
.000421
.000625
.000866
.001139
.001439
.001761
.002100
.002451
.002807
.003160
.003493
.003762
.003791
~
TOTAL
.000054
.000242
.000619
.001270
.002345
.004115
.007122
.012552
.023144
.044444
.082886
.141361
.222462
.333333
.489050
.720373
1.097191
1.816511
3.728008
e
0
RHO
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
I::'~
.J~J
.60
.65
.70
.75
.80
.85
.90
.95
N=
-
cv
.0078
-
8.00
AUTOCORRELATION LAG N
1
2
3
4
~.
.000034
.000126
.000262
.000432
.000630
.000850
.001089
.001342
.001608
.001883
.002166
.002455
.002747
.003040
.003331
.003614
.003877
.004086
.004092
.000006
.000038
.000106
.000209
.000346
.000512
.000704
.000918
.001150
.001398
.001659
.001929
.002206
.002489
.002772
.003051
.003315
.003533
.003574
.000001
.000014
.000051
.000119
.000220
.000354
.000518
.000708
.000921
.001154
.001403
.001664
.001936
.002215
.002497
.002776
.003043
.003268
.003328
.000000
.000006
.000026
.000073
.000151
.000262
.000406
.000579
.000778
.001000
.001241
.001498
.001766
.002044
.002326
.002607
.002877
.003108
.003180
.000000
.000002
.000015
.000047
.000108
.000202
.000330
.000489
.000678
.000892
.001127
.001380
.001647
.001924
.002207
.002490
.002763
.002998
.003079
~
TOTAL
.000042
.000188
.000480
.000988
.001830
.003229
.005642
.010125
.019374
.039637
.078530
.138344
.220950
.333566
.491739
.727287
1.113018
1.857107
3.886786
e
41
e
l1
_.
cv .... 9.00
.0061
~\LJTOCClf':f':F.::L..(:!IT
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.20
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.30
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• O~:j
.10
• :I. ~:j
.20
•:? ~:5
.30
• 3~:i
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• 4~:i
• ~5 0
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• ...J ... J
.60
• 6~5
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• B~5
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4
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.0002"76
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.OOO:::j62
• 0OO7~52
.000918
.00:1. :1. :1.6
• 0 0 :1. :5 ;.~ 4
.001 :'.'i40
.OO:t. 7 f.l~~
.OO:l.li90
• 0022;.~O
.002447
.0021.>66
• OO~?B~5f.1
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.000011
.000040
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.0004:1.4
• ()O()O()()
.000004
.00002:1.
.000059
.000121
.000210
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.000002
.000012
.000038
.00008"7
.000162
.OOO:=:i66
• 0007~r7
.OO()464
.000623
.OOOB01
.000995
.00:1. :;~()O
.0O:t.4:t."7
.OO:t.1.140
.OO:t.869
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....
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• 0 0 :t. ~.:j :5 0
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.00200:3
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• 0024:=:j:1.
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.002"732
• 00032~i
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.002520
• 00;,~6:J. 5
.OOO26:'.'i
• 004~i6 7
• 00831 ~5
• O:l.642~:i
• 03~7j736
.0"75090
.:L36040
• 00090~:i
.00:1.108
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• OO:I.~:i47
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.00200B
.00223:3
.002435
• 002:'5:~6
I•• (-) Co
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4
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.OOOO2:?
.OOOOB2
.000 :1. 70
.000280
.OOO40B
• OOO~5~i:t.
.000705
.OOOB?O
.001042
.OO:J.22:I,
.001.406
.000004
.OOOO2:=.'j
.000069
.000:1.:36
• 0OO22~:';
.000333
.OO04:5D
• 00000:1,
.000009
.000033
.OOOO?B
.000144
.000231
• OOO~~:3B
.000462
.000602
.000000
.000004
.OOO01?
.000048
.000099
.000172
.00021.16
.OOO3?9
• OOO~"j:J.O
• 0006~55
.OOOB14
.0009El2
.001160
.000000
.OOO()02
.0000:1,0
.00003:1.
.OOOO?l
.000:L33
.OO021?
.000322
.000446
• OOO~586
.000741
.000908
.00:l.~544
.OOl269
• OO:L 45El
.001650
.001838
.002012
.00:U17
.001:i94
.OO:L lB6
.00 :1. 980
.002:1."74
• 002:36~5
.002::549
.002711
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.OO2:35:J.
.002436
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• :B3649
.49354:?
.73203 11'
1 • :I. ;!402B
1 • El1357~54
4.()O2704
N
:5
.OOOl4B
.000910
.0010ElO
.001257
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.00UI;?6
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.002004
• OOOO~53
.000:1. :=:jO
.OO03El3
.OOO·78 1i
.O()1466
• 002~596
.000393
.000544
.0007:1.6
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• OO~50
AUTClC OF;:F;:E~I... ('~T To ClN
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• 000027
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.OOO20El
.000:344
• OOO~)O:l.
• OOOf.)"77
.OOOB6f.1
.OO:l.06B
.00 :1. 2BO
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.001726
• OO:t. 956
.002:1.91
.002427
.0021.>62
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.00:'51 :J. 2
.003297
• OO:~34B
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..
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:1.
.OO07:=:i4
.OO09:l6
.0010BB
.OO:t.26?
.00:1.452
.001640
.OOlEl29
.0020:1. :~
.002 :1.131
.0022?6
.OO:L:532
• 00:1. 7~?2
.00 :1. 909
.0020ElO
.002181
.00:1. 08~i
TClT('~1...
.000027
.000 :1. 22
.000312
.000644
.001199
.002129
.003766
.006932
• O:L 40BO
.032517
.O?2337
• 1342::-j6
.2:1.8ElB2
.33366?
.494B:l.1
.'735444
1.131983
1.90666"7
4.089444
42
II.
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1-1:=:
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.10
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.20
• O()0~i6c)
• 0020~:;:I.
.004 Hl4
.001.)747
• O()9~i~:;EI
.0:1.2470
.015359
.O:L8:1.16
'")1::•
1
A..... J
.30
• 3:~j
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• ~:iO
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• 02466~i
.60
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.0:?4010
.020E189
.O:L618'7
.009449
• l:':'i
.80
• B:~j
.90
.9::)
M/Ek/l
r 01-1
.-
e
.707
LAG
1-1
2
3
4
~:;
• OOOO~:i4
.000006
.000073
.00030 :1.
.000776
.001548
.002630
.003999
.005603
.007371
.009212
.0:1.1024
.012696
.014100
.015097
.015524
• 015:1.8El
.013E149
• O:L 1:1.90
.006781
.00000 :L
.OOOO:L6
.000091
.000295
.000000
.000004
.000029
.000 :1.1 B
• 0003~!6
.()OO711
.001:315
.00216:1.
.003239
• 004:H 3
.00::5919
.007:368
.008746
.009917
.O:l.07:1.rJ
.0:1.0945
.010362
• 0086~)2
.00539'7
.OOO36l:1
• OO:J.069
.002:L El5
.003684
.00550:1.
.007548
.009729
.0:1.1941
.0:1.4080
.016037
.0:1.7701
.018950
.019652
• 0:1. 96::.'i 1
.018'760
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.001338
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• 009~.:;21
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5
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.000006
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.003308
.005148
.007385
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.023425
.024966
.025444
.024332
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.000017
.000101
.000336
.000815
.001615
.002788
.004351
.006286
.008:538
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.013603
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.018412
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.021054
.020S62
.017923
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.000000
.000004
.0000:30
.000128
.00036'7
.00082'7
.001580
.002678
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.000006
.000087
.000369
.000986
.002039
.003590
.005657
.008218
.011216
.014562
.018138
.021795
.025345
.028549
.031085
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.032062
.028566
.019653
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.000017
.000102
.000:347
.000852
.001714
• 00:~002
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.006972
.009613
.012602
.015820
.019105
.022236
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.026697
.026924
.024454
.017113
.000000
.000003
.000030
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.000860
.001672
.002881
• 004~)28
.006618
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--
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2
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.003420
.007132
.011764
.017070
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.067887
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.003196
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.008615
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.000017
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.003111
.004972
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.255287
.322368
.409091
.524590
.684834
.920455
1.298561
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K
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.029974
.034470
.038461
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.000016
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.000874
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.003174
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.007601
.010651
.014198
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.003040
.004863
.007236
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1.038462
1.516807
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ALJTOCClF::f;:El..AT r ON
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.044978
.000006
.000088
.000381
.001033
.002173
.003889
.006230
.009205
.012785
.016907
.021476
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.031390
.036329
.040845
.044419
.046147
.044200
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.000016
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.000350
.000877
.001799
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.007769
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.014654
.018829
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.009832
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.058659
.079470
.105013
.136364
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.362069
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.813321
1.135514
1.708202
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AUTOCOr.:r.:ELAT I ON
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3
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.001591
.003392
.005971
.009312
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.027162
.032481
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.000349
.000879
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.024075
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.033792
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.484615
.636364
.859388
1.21673B
1.877415
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AUTOCOr.:r.:ELAT I ON
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.35
.40
.45
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LAG
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1
2
3
4
5
.001033
.003848
.008078
.013419
.01961E1
.026450
.033756
.04134'7
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.064423
.071720
.07E1520
.OE14558
.089436
.092490
.092454
.086528
.066892
.000074
.000529
.001604
.003425
.006039
.009435
.013564
.010350
.023695
.029486
.035592
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.044245
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.041178
.000001
.000016
.000100
.000348
.000879
.001815
.003262
.005302
.007902
.011311
.015258
.019751
.024668
.029825
.034940
.039561
.042868
.043116
.035551
.000000
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.000028
.000124
.000372
.000878
.001758
.003121
.005053
.007613
.010817
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.023729
.028604
.033200
.036761
.037679
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l·OTAL..
.001114
.004484
.010204
.0184:33
.029412
.043478
.061097
.082902
.109756
.142857
.183891
.235294
.300712
.385827
.500000
.659794
.897516
1.285714
2.028090
46
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AUTOCOfi:fi:ELAT:r ON
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.066013
.073629
.080789
.087239
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.049234
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• 06e:~71
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.053805
.000006
.000088
.000382
.001045
.002211
.003985
.006428
.009565
.013384
.017837
.022848
.02B302
.034051
.039884
.045499
.050410
.053730
.053513
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.000001
.000016
.000100
.000348
.000880
.001819
.003277
.005338
.008054
.011439
.015469
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.030508
.035885
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.000;370
.00087'7
.001761
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.005088
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.034247
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.67924::-;
.929593
1.345018
2.:1.63116
tit
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AUTOCOtO:fi:EL.. AT I ON
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N
1
2
3
4
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.108366
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.000075
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.006249
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.031484
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.073168
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.000006
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.000382
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.006569
.009830
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.023925
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.000099
.000345
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.005439
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.000874
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.003169
.005:1.83
.007890
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.025847
.031703
.037619
.042951
.046199
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TOTAl...
.001169
.004711
.010725
.01939:1.
.0;30973
.045852
.064546
.087774
.116523
.:l52174
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.253012
.325537
.421376
.552632
.741722
1.035312
1.549180
2.671247
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K
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20
AlITClCOF::F::ELAT I ON
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1-1::::
• ()~'j
.10
.15
.20
'")1::• ...._ ""J
.30
.35
.40
.45
.50
2
3
4
5
.001109
.004148
.008749
.014606
.021466
.000075
.000541
.00:1.654
.003560
.006323
.009956
.014430
.019688
.025656
.032242
.039343
.046840
.054592
.062423
.070082
.077166
.082885
.085303
.077527
.000006
.000087
.000382
.001053
.002246
.004079
.006634
.009956
• 01405~j
.018908
.024466
.030647
.037342
.044398
.051594
.058570
.064644
.068161
.063305
.000001
.000015
.000098
.000344
.000878
.001833
.003334
.005484
.008359
.011998
.016409
.021558
.027375
.033739
.040459
.047214
.053391
.057514
.054438
.000000
.000003
.000027
.000119
• 000:364
.000871
.001769
.0031.83
.005227
.007988
.011522
.015844
.020924
.026678
.032947
.039446
.045615
.050108
.048245
.02911~)
c:-=:"
.60
.65
.70
.75
.80
• 8~j
.90
9"'·hI
•
• 2;'~4
N
1
• 037:~72
.046081
.055103
.064312
.073585
.082796
.091800
.10040B
.108340
.115111
.119731
.119691
.10594·7
• ""J;J
L.~~G
TOTAL
.001190
.004796
.010921
.019751
.03156:1.
.046747
.065851
.0896=?:3
.119099
.155738
.201631
.259878
.335247
.435454
.573826
.775510
1.094261
1.669197
3.005697
e
K
..-
cv
25
ALlTOCOF::I':ELAT I ON
I':HO
• O~.'i
.10
.:1.5
.20
,.,.,.
t..:..J
.30
.35
.40
.45
.50
.55
.60
.65
_.70
.75
.80
.8::)
.90
.95
1-1:::
..LAG
.200
N
1
2
3
4
5
.001121
.004198
.008861
.014805
.021776
.029564
.037987
.046890
.056139
.065610
.075186
.084751
.094170
.103278
.111827
.119393
.12510:5
.126699
.115581
.000075
.000543
.001662
.003579
.006366
.010037
.014567
.019904
.025977
.032700
.039975
.047691
.055718
.063897
.072005
.079685
.086244
.089939
.084256
.000006
.000087
.000382
.001053
.002251
.004095
.006672
.010031
.014185
.019120
.024791
.031126
.038026
.045352
.052909
.060379
.067159
.071764
.068711
.000001
.000015
.000097
.000343
.000876
.001834
.003343
.005510
.008416
.012106
.016595
.021857
.027833
.034417
.041441
.048622
.055418
.060508
.059048
.000000
.000003
.000026
.000118
.000362
.000869
.001769
.003190
.005251
.008045
.011634
.016041
.021248
.027186
.033717
.040592
.047318
.052689
.052310
TOTAL
.001203
.004847
.011038
.019967
.031915
.047285
.066636
.090737
.120655
.157895
.204622
.264059
.341184
.444109
.586957
.796680
1.131854
1.748201
3.242515
48
K
=
cv
50
AUTOCOf;:f;:ELAT I ON
Fi:HfJ
N:::
.05
.10
.:1.5
1
.001146
.004297
.009085
.015205
.022404
.030473
• 0:~9235
• ()48~539
• 058~!5::=j
.068270
.078481
.088789
.099093 .109275
.119176
.128533
• 1 ~56806
.142543
.139546
• '1r
~?o
• ~ ...J
.30
• ~'5~j
.40
• 4~:j
• ~jO
·
1:.,"1::"
~'-,
.60
.65
.70
.75
.80
.85
.90
• 9~'j
_.
LAG
e
.141
N
2
3
4
5
.000075
.000546
.001675
.003618
.006450
.010196
.014838
.020333
.026619
.033620
.041252
.049422
.058026
.066943
.0'76019
.085021
.093510
.100367
.100942
.000006
.000086
.000381
.001054
.002260
.004126
.006744
.010174
.014441
.019540
.025443
.032094
.039419
.047315
.055645
.064198
.072587
.079857
.082107
.000001
.000015
.000096
.000340
.000874
.001835
.003358
.005559
.008525
.012318
.016964
.022458
.028763
.035809
.043480
.051589
.059789
.067226
.070466
.000003
.000026
.000117
.000358
.000865
.001768
.003203
.005297
.008156
.011854
.016435
.021904
.028225
.035:312
.043004
.050985
.058478
.062372
.OOO()OO
TOTAL..
.001228
.004948
.011274
.020400
.032623
.048366
.068214
.092979
.123791,
.162252
.210682
.272559
.353315
.461908
.614206
.841202
1.212622
1.9248:30
3.82'7131
e
K
=
cv =
100
AUTOCORf;:EL.AT I ON
Fi:HO
• O~:j
.10
.15
.20
rIo::'
• .:..
~J
.30
.35
.40
.45
.50
.55
.60
6C:
•
..J
.70
.75
.80
.85
.90
.95
N::
L.AG
.100
N
1
2
3
4
5
.001:1.58
.004347
.009198
.015406
.022'721
.030934
.039869
.049378
.059336
.069634
.080176
.090875
.101649
.112409
.123051
.133413
.143185
.151531
.154677
.000075
.000548
.001682
.003636
.006491
.010274
.014972
.020546
.026939
.034082
.041898
.050303
.059208
.068517
.078115
.087846
.097445
.106254
.111447
.000006
.000086
.000380
.001054
.002264
.004140
.006778
.010244
.014566
.019749
.025769
.032583
.040128
.048325
.057069
.066215
.075521
.084419
.090534
.000001
.000015
.000096
.000339
.000872
.001835
.003365
.005581
.008578
.012422
.017147
.022759
.029235
.036522
.044538
.053154
.062149
.071011
.077646
.000000
.000003
.000026
.000116
.000356
.000862
.001766
.003208
.005319
.008208
.011962
.016632
.022235
.028756
.036138
'.044274
.052963
.061738
.068699
TOTAL.
.001241
.004999
.011391
.020616
.032978
.048908
.069006
.094106
.125371
.164452
.213750
.276880
.359513
.471062
.628350
.864629
1.256080
2.023978
4.193734
e
49
•
e
K
-
10000
AUTOCOf;:f;:ELAT I ON
f;:HO
N::::
• ()~j
.:1.0
• :I.:'.')
.20
"')I:~
• ......J
.30
.35
.40
.45
.50
.55
.60
6""~)
.70
.75
.80
• 8~5
.90
.95
·
_
.05
.10
.15
.20
'')1:::
t '"- ...J
.30
3""~
.40
.4:5
• :50
.55
.60
.65
·
•
_.70
.'75
.80
8""~
.90
.95
·
LAG
N
.010
2
3
4
5
.001:1.71
• 004:~9'7
.009310
.015607
.023037
.031394
.040503
.050220
.060422
.071006
.081886
.092986
.104244
.115606
• :1.27026
• :1.38464
.149884
.161250
.172491
.000076
.000549
.001688
.003653
.006530
.010349
.015103
.020'755
.027256
• 0:~4541
.042542
.051185
.060398
.070110
.080251
.090755
.101560
.112600
.123791
.000006
.000086
.000380
.001054
.002267
.004152
.006811
.0:1.0311
.014689
.019954
.026092
.033069
.040839
• 04934~5
.058516
.068288
.078585
.089332
.100432
.000001
.000O:L5
.000095
.000337
.000870
.001835
.003371
.005602
.008629
.012522
.017327
.023058
.029706
.037240
.045611
.054759
.064611
.075085
.086078
.000000
.000003
.000025
.000115
.000354
.000859
.001'764
.00;3213
.005338
.008259
.012068
.016826
.022565
.029290
.036975
.045576
.055026
.065246
.076126
::::
cv
1000000
AUTC)CORf;:ELAT J: ON
N::::
_.
1
K
f;:HCl
cv
L.AG
::::
r)
3
4
5
.001171
.00439El
.009311
.015609
.023041
.031399
.040510
• 000076
.000549
.001688
.003653
.006531
.010350
.015104
.020758
.027259
.034546
.042548
.051194
.060410
.070126
.080273
.090785
.101602
.112667
.123926
.000006
.000086
.000380
.001054
.002267
.004153
.006811
.010312
.014690
.019956
.026095
.033074
.040846
.049353
.058531
.068309
.078617
.089384
.100540
.000001
.000015
.000095
.000337
.000870
.001835
.003371
.005602
.008629
.012523
.017329
.023061
.029711
.037247
.045622
.054776
.064637
.075128
.086170
.000000
.000003
.000025
.000115
.000354
.000859
.001764
.003213
.005339
.008260
.012069
.016828
.022569
.029295
.036984
.045589
.055048
.065282
.076208
.14995:~
.161351
.1'72686
.011~)()8
.020831
.0333;30
.049445
.069793
.095227
.126943
.166644
.216815
.281206
.365738
.480298
.642710
.888642
1.301333
2.130458
4.623463
.001
...
.060433
.071020
.081903
.093007
.104270
.115638
.127066
.138516
.001253
• 0050~.)()
N
1
• 0~'j0229
TOT~~L
TOTAl...
.001253
.005050
.011509
.020833
.033333
.049450
.069800
.095238
.126959
.166666
.216846
.281250
.365800
.480391
.642856
.888886
1.301797
2.131568
4.628158
lJNC LASS IFIED
SECURITY CLA!lSIFICATlON OF THIS PAGE (When DeteBntered)
r
READ INSTRUCTIONS
BEFORE COMPLETING FORM
REPORT DOCUMENTATION PAGE
~
I"
I
REPORT NUM!!!!:R
30VT ACCESSION NO.
.. The Covariance Structure of the Departure
TI TL E (end Subtitle)
I
Process:
GI/M/l and M/G/l Queues
••
RECIPIENT'S CATALOG NUMBE"
5. TYPE OF REPORT 6 PE"IOD COVERED
ffECHNICAL
6.
PERFORMING ORG. REPORT NUMBER
Mimeo Series No. 1144
;, 7. AUTHOR(a)
~
8. CONTRACT OR GRANT NUMIIER(a)
NOOO14-76-C-0550
Nancy Gustafson
I)
1
~
rg~RF6-RMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT. PROJECT. TASK
j IJcp;Jrtm(~l)t of Slalistics
~ IIJliv{'rsit\" nF Nurth Carollna
,~ I:hapf' i 111'11, North Carolina 27514
AREA 6 WORK UNIT NUMBERS
-'-----CONTROLLING OFFICE NAME AND ADDRESS
12. REPORT DATE
',II.
ii
October, 1977
Office of Naval Research
Arlington, Virginia
~4.
--
II. NUMBER OF PAGES
50
MONITORING AGENCY NAME 6 ADORESS(II dll/ar. .t lrom Contromn" Olllea)
15. SECURITY CLASS. (01 thla raport)
~
UNCLASSIFIED
L
I Sa. DECLASSIFICATION/DOWNGRADING
SCHEDULE
DISTRIBUTION STATEMENT (01 thla Report)
116.
I
r
Approved for Public Release: Distribution Unlimited
\1
~
I
DISTRIBUTION STATEMENT (01 tha abatr.et enter.d In Sloelr :lO, II dlll.rent lrom R.po,t)
f"
p
I
~
SUPPL EMENTARY NOTES
18
l
1
I
i
~KE:Y
I
WORDS (Continue on ,eve,. . • Id. II n.e. . ..", end Identity by bloelr numb.,)
Queueing Theory, Outputs from Queus, Autocorrelation
I
I
~
,~
. 20.
ABSTRACT (Contlnu. on reve," .Id. II n.e....ry . .d Id.ntlly by bloelr numb.,)
Formulae, suitable for computation, for the lag n autocorrelation of
departure intervals in the GI/M/l queues are developed for n=l to 6. A generating function, comparable to the one known for the M/G/l case, is found for
GI/M/l. For the Ek/ M/ l and M/Ek/l queues, expressions for arbitrary lag are
given. Tables and the APL functions used to produce them are provided for
Ek/M/l, H/M/l, and M/Ek/l .
DD
.,ORM
1 JAN 73
1473
EDITlO~ OF I NOV illS OBSOL.ETE
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE ("".n D.t. Bnt.,ed)
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS
PAG~(It'h. .
D.t. Bnt.,.d)
I
'I
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE(When Dete Entered)