Dataplot Vol 2 Vol 1

BTACDF

Name:
BTACDF (LET)
Type:
Library Function
Purpose:
Compute the Borel-Tanner cumulative distribution function.
Description:
Given a single queue with random arrival times of customers at constant rate l, constant service time , and k initial customers, the Borel-Tanner distribution is the distribution of the total number of customers served before the queue vanishes.

The probability mass function of the Borel-Tanner distribution is

with and k denoting the shape parameters. The k shape parameter is a positive integer and = l .

The cumulative distribution function is computed by summing the probability mass function.

Syntax:
LET <y> = BTACDF(<x>,<lambda>,<k>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a positive integer variable, number, or parameter;
<lambda> is a number or parameter in the range (0,1) that specifies the first shape parameter;
<k> is a number or parameter denoting a positive integer that specifies the first shape parameter;
<y> is a variable or a parameter where the computed Borel-Tanner cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = BTACDF(3,0.5,3)
LET Y = BTACDF(X1,0.3,2)
PLOT BTACDF(X,0.3,2) FOR X = 2 1 20
Default:
None
Synonyms:
None
Related Commands:
 BTAPDF = Compute the Borel-Tanner probability mass function. BTAPPF = Compute the Borel-Tanner percent point function. POIPDF = Compute the Poisson probability mass function. HERPDF = Compute the Hermite probability mass function. BINPDF = Compute the binomial probability mass function. NBPDF = Compute the negative binomial mass density function. GEOPDF = Compute the geometric probability mass function.
Reference:
Haight and Breuer (1960), "The Borel-Tanner Distribution", Biometrika, 47, pp. 143-150.

Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 394-396.

Applications:
Distributional Modeling
Implementation Date:
2006/5
Program:
```
title size 3
tic label size 3
label size 3
legend size 3
height 3
multiplot scale factor 1.5
x1label displacement 12
y1label displacement 17
.
multiplot corner coordinates 0 0 100 95
multiplot scale factor 2
label case asis
title case asis
case asis
tic offset units screen
tic offset 3 3
title displacement 2
y1label Probability
x1label X
.
ylimits 0 1
major ytic mark number 6
minor ytic mark number 3
xlimits 0 20
line blank
spike on
.
multiplot 2 2
.
title Lambda = 0.3
plot btacdf(x,0.3,1) for x = 1 1 20
.
title Lambda = 0.5
plot btacdf(x,0.5,1) for x = 1 1 20
.
title Lambda = 0.7
plot btacdf(x,0.7,1) for x = 1 1 20
.
title Lambda = 0.9
plot btacdf(x,0.9,1) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution for Borel-Tanner
```

Date created: 6/5/2006
Last updated: 6/5/2006
Please email comments on this WWW page to alan.heckert@nist.gov.