ML20149E710

Applicant Exhibit A-19,consisting of Undated Article on Definitions,Relationships & Distributions & Traffic Flow Theories

ML20149E710
Person / Time
Site:	Seabrook
Issue date:	11/05/1987
From:	AFFILIATION NOT ASSIGNED
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References
OL-A-019, OL-A-19, NUDOCS 8802110246
Download: ML20149E710 (4)

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11. De6nitions, Relationships, and Distributions 21 i

g g ,q gi,fgttlic data are found to conform to a binomial or a Poisson distribution.

7P the implication is that the tratlie esents measured, for example, the inter.

" * " " '# " " #b" " E' '

6 0FFICE OF SE < f .it,significant underlying situation daiating from a basic probabilistic occurrence s

00CMEimG ), S rV% events.

W'N C *' in addition to gising insight mto the nature of an underlying process, N,

theoretical distributions permit !arge amounts of data to be summarized into

,7 a few parameters that can become part of a stochastic theory of trallic tiow.

(- r l

Headway Distributions x ,

We can find the distribution of headways by observing the times of arrisal

,- of successise schicles in a eisen lane at a point along the lane. As noted q

--- i presiously, we define the headway of the ith vehicle as g - t. see i h , = t , .. i - r,,

where t,_, is the artis al time of the s chicle ahead and I, that of the ith s chicle.

If we observe many schicles, we can obtain the distribution of h, and insesti-i gate whether this sariable can be described by some theoretical distribution

function. i I Some of the characteristics of such distributions are well established. For example, under licht trat'ie conditions, the headway distribution function ,

twing tryectories i reduces to a nega'tise esponential or displaced negatise exponential ds. '

j tribution. These functions de3cribe the distribution of certain random esents,

}

such as the intervals between radioactise atomic fissions obsersed on a 1 Geiger counter.

t has e been considered It was first succested by Adams $ that the number of schicles in licht r, their insersions, as tratTic passing a p'oint in equal intervals of time follow a Poisson distribution.

speed has also been if this is the case, the distribution of headways can be show'.i to be described that not only are the by the negatise exponential distribution. The Poisson distribution gises the y theories, but also the p'robability, or proportion of a number of equal tinte interval <, durine which

~

iTratlic tiow is a very any numb'er of vehicles, n, will arrise at a point as

!ask of studying it is to ris and study these in p'g , "T4ffe n!

ped and the likelihood f  ;

cased. Some theorists '

where qr is the mean number of schicles arrising during a tinie intersal of, f and spacino, are suth- say, t sec. Thus i pn.

P (t) = probabihty of no sehicle in t sec = e"',

e the tools with which 2

, underlying complex P (t) = probability of I schicle in t sec = 9ted',

' I t

tia are found to fit a P (t) == probability of 2 vehicles in t sec = 1(qt)'ed',

, e assumptions of the process. For example, P3 (t) = probability of 3 schicles in t sec = (9 t )'c"'.

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22 Flow Theories With the Poisson counting distribution there is no upper limit on how many vehicles might arrise in a given time intersal. Sometimes this is not the case. g For example, suppose a parking lot with Nochiele capacity empties in a time T. If any schiele selected at random has a probability p ofleasing in any small time intersal, the counting distribution can be gnen by the binomial expression:

Fi n ) = C( N n )p"( I - p P "

1 P(ni = probability that n schicles lease in a small time interval At if p = St/T = probability that any sehicle leaes in a par-ticular interwl Ar, N = number of sehicles leasing during T, i = any small part of time T, j I

C(N, n) = combination of N things taken n at a time.

N!

nl (N - n)! '

If N = 10, T = 60 min, and 3r = 1 min, then p = , t he as erage departure rate would be m = Np = 0.167 schiele/intersal, and the sariance would be Np(I - p) = 0.164. The probability density would be the following:

P(0) = 0.845.

P(1) = 0.143, P(2) = 0.012.

1.000 and the probability of three or more exiting at once would be small, less than once in 20 hr.

When N is small, the computations are not tediou,. As N gets larger, tables may be used, but esen this becomes tedious with increasing salues for N. Howeser,if X is sery large and p is small, it has been shown by Molina i that the Poisson distribution may be used as an increasingly close approsi-mation to the binomial.

Also, when the mean value exceeds a salue of about 5, the normal distri-d buuon becomes a close approximation. Table I shows the fit of both the Poisson and normal density functions to data giving the number of schicles arrising in bmin intervals when the rate of flow is ME700 sehicles./hr for a total of 450 intervals. The mean rate of tiow is 5.46 schieles' interval, and the variance is 2.73 schicles.

11. Definitions, Relationships, and Distributions 23 TAHLE I sper limit on how many CostPARisON Of POISSON AND NORst AL COUNrlNG f UNCTIONS TO mes this is not the case. OnstRvro otNsity capacity empties in a mlity p of leasing in any Frequencies

' gisen by the binomial Occurrences Actual Poisson Normal l n f );. fx

! o 6 1.3 0.4 1.11 a small time 1 I8 4.0 2.3 2.77 2 35 7.8 6.3 5.7 I le leases in a par- 3 52 11.6 11.5 9.81 4 68 15.1 15.7 14.04

. 5 7I I 5.8 17.2 16.75 7* 6 55 12.2 15.6 16.62 7 45 I b.0 12.2 13.73 8 39 8.7 8.3 9.45 it a time- 9 26 5.8 5.1 5.42 10 14 3.1 2.5 2.58 11 8 1.8 1.4 1.0 12 6 1.3 06 0.4

,' , the aserage departure i 13 4 0.9 0.3 0.1 d the sariance weuld tc j 14 2 0.4 0.1 0.02 0.2 0.04 0.01 he th.e following: 15 1 1

i 450 100.0 100.0 100.0 yJ. - 0.74.

7,i = 0.46. ,

i For these data, the normal fits slightly better titan the Poisson, but both j ce would be small, less gise excellent fits. i Jious. As N gets larger, The probability function for headways when the arrival or counting sith increasine values for function is Poisson can be obtained as follows: We treat the time intersal t

~

L been shown by Molina as a variable in the Poisson expression for P(0), which gives creasingly close approsi-P,(t) = e"' or Pd t) = r",

sout 5. the normal distri-hows the fit of both the where h is the mean headway. This function gises the probability that no ,

); the number of schicles sehicle artises in an interval t. The probability that some schicle arrises in t MX)-700 schicleslhr for a is the same as the probability that a headw ay h, is equal or less than t, namely, Is chicles! interval, and the j P(h, g t) = 1 - c-'" I i

o i

l

24 Flow Theories This is a probabil;ty distribution function, and we can obtain the probability density function by ditTerentiation:

1l n

P'(h, s t) = P(h, = t) = Ie -' '

h While the Poisson and negatise exponential distributions are sery useful in trallic flow theories for light tratlic, especially because of their mathe-matical simplicity, they base shortcomings in not taking into account the limitations imposed by the physical size of schicles and finite selocities. These w===#M limitations make it less realistic for a single lane without passing than for Mf . $ muttipic Ianes.

h .f N h :.,.y;;g:c . {).:.t.. @h One .rIO. way. to deal with the etTects of these limitations is to replace the neg tise exponential headway distribution by its "displaced" analog:

-[$V ...h..

. . t . .,y. .

.g ..  ?.- .^ lc; A., k, b.h_.

1.,. .3

.; 1. V ^ty

• Pt h, = t > = 0 0 $ t f 1,

_g .

_ : - ..:..;y - ) _ L _ ; Q**In n.,

. l--  :([i~ ! l  : .. ; ?:ll , ,

e

-r h go g g,

. ; .y.y t,;q('{:[

_.s.,z: ,

..a .g3 . -s -t6 '<J." ~V'

'i Yd-9 N. .g.. i;M ' P(h, s ti = 0 0 s t g to

.i t-; 6M.Yy.W O..j = 1 - e" ^

fL:{kO.Mff';.MvL'Ql.q,: to S t.

. [, a s ?s 9 , a

.u #-e. ...

^-

3  : where to is the minimum allowable headway This is equisalent to saying the

+p > $. 2.

. .:e.'.. ~;s- .

w mj.. .. . '. A . . .

5. following:

'f f Sr[-h. . .b'j  ?,

• &(.,;j,: ' y_ The probability that a hcodway. starting at t = 0, ends in the interval k'f h;,.'.{!.:'C6.f

_l

..yff;;;.b;.yllf Q iYQ2.;Y?$

3 (t, t + dr), provided that it did not end before time t is

+z.  :  :. Sy if (t < to)

' 0: 'lf.y;. O

s, .Pllf.;.y )$..N. .3&

' N l] $ or

.:lts_/]*

.. g..g df. $;.

gre, . a :& y ?5 *.'.kf*

R " : ?' W

.,x' Ijh if (to f I).

3 g, R g '. . : s .s ---s.' ay F4-j .?Mr '.N/-([m-: 4.

• The counting distribution for the displaced exponential headway

.h;..h.-![ khb.g 3 distribution has been derh ed by Oliver.'It is given by the following equations:

FUL;/v<t  ; +-

P,(t) = 0

~

0 $ t < nt,

. :;.s. W. fig:)M.y:

-- w X.n.;l';;s = ;e

- '[n;qtt - nr)]o nt, $ t < (n + 1)t, gg (n - 1)!

y[n; q(t - nr o)]

=

(n - 1)!

- y[n + 1: q(t - (n + lito)] (n + 1)t, f t, n!

- . . _