On a characterization of the exponential distribution

On a characterization of the exponential distribution

Citation for published version (APA):

Harn, van, K., & Steutel, F. W. (1990). On a characterization of the exponential distribution. (Memorandum COSOR; Vol. 9021). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1990

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 90-21 On a charcterization of the

exponential distribution K.vanHarn F.W. Steutel

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box513

5600 MB Eindhoven The Netherlands

Eindhoven,July 1990 The Netherlands

On a characterization of the exponential distribution

K. van Ham!, Free University of Amsterdam F.W. Steutel2_{, Eindhoven University of Technology}

ABSTRACT

The characterization of the exponential distribution as given in [2] is improved by sub-stantially weakening the conditions.

CHARACTERIZATION THEOREM, EXPONENTIAL DISTRIBUTION, LAPLACE-STIELTJES TRANSFORM

1. Introduction

In [2] Dimitrov and Khalil consider the 'blocking time' Z in a certain queueing system. This random variable can equivalently be defined as follows:

(1) Z = YI

+ ... +

YN-I

+

XN ,

where X}, YI , X2 , Y2 , ••• are independent, the X n are nonnegative and Ld. with P(XI = 0) < 1, the Yn are Ld. and exponential withEYI = 1/A, and N is defined by

N=min{nEIN;Xn<Yn} .

The main theorem formulated in [2], which is also the theorem we shall prove here, can now be phrased as follows:

Theorem: Z

~

Xl, if and only if Xl has an exponential distribution.

In view of the proof in [2] this formulation is somewhat misleading; the authors of [2] use the condition that Z, whose distribution depends on A, has the same distribu-tion as Xl for all A

>

o.

We shall show that a single value ofAsuffices. This makes the result more attractive and more natural.

As it is easily verified, and shown in [2], that Z

~

Xl ifXl has an exponential distri-bution, we shall only consider the converse, Le., the 'only if' part of the Theorem. 2. Proof of the Theorem

It is shown in [2], and not hard to prove directly from (1), that the following relation exists between b(s) = Ee-· z and g(s) = Ee-·X1 :

(2) b(s) = (s

+

A)g(S

+

A) s

+

Ag(S

+

A) .

Ipostal Address: Department of Mathematics and Computer Science, Free University, De Boelelaan 1081, 1081 MC Amsterdam, The Netherlands.

2postal address: Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

IfZ

~

Xl,Le., ifb= g,then from (2) it follows that (3) if we put (4) g(s)

=

{I

+

~(

(

1 A) - l)}-l

=

{I

+

8h(s

+

A)}-l , 8+",gS+ 1 1 h(s) = -( - () - 1) . s 9 s

Now (3) implies that h(s) = h(s

+

A), and we obtain (5) g(s)

=

1h()

(s

>

0) ,

1

+

s s

where h is positive and periodic with period A. Clearly,ifwe can prove that h is

con-stant, Le., that 9 has the form g( s) = (1

+

cs)-1, then the Theorem is proved. We state this result as a lemma.

Lemma IT Z is a nonnegative random variable with P(Z

=

0)

<

1and Ee-·

z

=

g(s), where 9 has the form (5) and h is periodic (with period A),then Z has an exponential distribution.

Proof: From (4) it follows that h has a continuous derivative on (0,

(0),

which is also periodic with period A. So for 8

>

0 we have

0::;

-g'(s)= (sh'(s)

+

h(s))(1

+

sh(S))-2 ,

Le., sh'( s) ~ -he8). By the periodicity ofh' it then follows that

h'(s) = h'(s

+

nA) = limh'(s

+

nA)

~

_ lim h(s

+

nA) n ... oo n ... oo S

+

nA

= _ lim h(s) = 0 n ... oo8

+

nA

Therefore h is nondecreasing, periodic and, by (5), positive. So h(s)

==

c>

0, and the Lemma is proved.

3. Concluding remarks

Added in proof, in [2] the relation h(s

+

A) = h(s) for all A

>

0 is used for s

1

0 to obtain h(A}= h(O) for all A

>

O. In fact, for a simple proof of the constancy ofhtwo values ofAare sufficient: ifh is periodic with periods Al and A2 with AllA2irrational, then by its continuity h must be constant.

A simple proof of the fact that one value ofAsuffices would be possible ifit were known that limsg( s) exists. Then, since g( s) -+ P( Xl = 0) as 8 -+ 00 we would have

hs _ 1-g(s+n.\) ( ) - (s

+

n.\)g(s

+

n.\)

lim 1 - g(s

+

n.\) = c n-+oo (s

+

n.\)g(s

+

n.\)

However, lim sg(s) does not exist for arbitrary Laplace-Stieltjes transforms g; a fairly n-+oo

complicated counterexample was shown to me by Sl2iren Asmussen. This limit does exist if the distribution corresponding to9 has a density with a right-hand limit at zero (see e.g. [3], p. 182).

References

[1 ] Asmussen, S. (1990) (personal communication).

[2 ] Dimitrov, B. and Khalil, Z. (1990) On a new characterization of the exponen-tial distribution related to a queueing system with an unreliable server, J. Appl. Prob., 27, 221-226.

[3 ] Widder, D. V. (1972) The Laplace transform, Princeton University Press.

~INDHOVENUNIVERSITY OF TECHNOLOGY )epartment of Mathematics and Computing Science

lROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH AND SYSTEMS

rHEORY

>.0.Box 513

)600 MB Eindhoven - The Netherlands iecretariate: Dommelbuilding 0.03 felephone: 040 - 47 3130

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