·
..
by
Walter L. smith
This research was supported in part
by the United states Air Force through
the Office of Scientific Research of
the Air Research and Development Command •
. . 1JI!t:.
....~
•
•
Institute of Sta.tistics
Mimeograph Series No. 144
January, 1956
,j
.-,
.
...
ON REl'JEWi-lL UlliORY, COUNTER PROBLEMS, A.NlJ QUASI-POISSON PROC.l!;SSBS
1
By walter L. Smith
Introduction.
j..
for the solution
by Feller
L-7-7,
o~
'rhe power and appropriateness of renewal theory as a tool
general problems concerning counters has been amply
who considered a variety of counter problems and reduced them to
special renewal processes.
The use of what may be called "renewal-type" arguments
§, of Domb L-3_7) but
had certainly been used previous to Feller's work (e.g. in
it was only in
demonstr~ted
L-7-7
that the simplicity of the renewal approach to counter problems
was recognised and systematically applied.
More recently, Hammersley
L-8_7
was
concerned with the generalisation of a counter problem previously studied by Domb
This problem may be introduced, mathematically, as follows.
Let
1Xi ~
,
be two independent sequences of independent non-negative random variables
which are not zero with probability one (i.e. two independent renewal processes);
the
~
Xi}
are distributed in a negative-exponential distribution with mean A-I,
we write E;A (x) for their distribution function and say" I
process" to imply this special property of.:f ; the
function B(x)2
nt to be
with mean b l ~
th~~ integer
00.
(:;
i yi ~
have a distribution
Form the partial sums X
n
k such that Xk
~
t, taking Xo
=0
~ xi~ ) is a Poisson
= L~
and n
Xi' and define
t
=0
if Xl
> t.
Then define the stochastic process
(1.1)
= 1,
otherwise.
1. This research was supported in part by the United States ~1r Force through
the Office of Scientific Research of the Air Research and Development Command.
~
2.
All distribution functions are to be considered as continuous to the right.
..
2
Hammersley's counter problem concerns the stochastic process
~t
Nt
=
1:
¢(Xn )
n=l
(1.2)
=
o,
,
if
> 0,
nt
otherwise.
not
We shall/elaborate the physical interpretation of the Nt process because Hammersley
has already given an adequate account of this in the paper cited
the
1xnl
length
if ¢(t)
Briefly,
are coordinates of the left-hand end-points of closed intervals of
3Yn ~
= 0,
L-8_7.
placed on the real axis.
or "free" i f ¢(t)
= 1,
Points t on the axis are either "covered"
and the number of left-hand closure points of
the resulting covered stretches in (O,t-! is Nt.
As a matter of fact our formula-
tion differs somewhat from Hammersley's in that we suppose that a covered stretch
always starts at t
= 0;
also Hammersley supposed his intervals to lie on the circum-
ference of a circle instead of the infinite real axis.
The formulation given here
is more convenient for the renewal theory approach and makes no difference to the
asymptotic results in which we are interested.
Domb in
L-2_7
considered the particular case in which B(x)
with probability one.
~
y
L-7..1.
Hammersley, essentially, considered the case in which
equation B(x)
= 1,
- T), where
l assume the constant value
i
This particular case was also treated by Feller
U(.) is the Heaviside unit function, i.e., the
T
= U(x
T,
is finite; i.e. he merely supposed the
the least solution of the
~Y ~
i
to be bounded.
Using special methods of considerable compleXity, which at not less than one point
depend critically on the boundedness condition, Hammersley deduced the following
asymptotic results which hold as t
L
---> +
00.
,
e
.
(A)
eN
(B)
Var Nt
t
-}"b
I"V
}..t e
"'-J
}..t e
1
~ lA-e-~b 1_
-}..b
1
{
1 - 2
h(ZU dz } J
0
where
h(z)
(0)
= >Jl(z
- 0) exp
{-~
j
[1 - B(YU dy
1
J
Nt is a.symptotioa,lly normal with the a.symptotio mean and variance given in
(A) and (B).
Let us write Zo
= 0,
Zl for the lea.st Xn suoh that ¢(Xn )
sma,llest X suoh that ¢(x )
n
n
zn = Zn - Zn_l' n
.~ 1,
= 1,
and so on.
the sequence
~ =
= 1,
Z2 for the second
Then it follows tha,t if we write
i 1
Zn
is a. renewa.l process.
This
result is obviouB physically, but the rea.der with no fa.ith in intuition Oan derive
it swiftly from (1.1) and (1.2).
of the
~ Zn
1 ' and ~,
Let us write F(·) for the distribution function
2 for the first and second moments of F( .), whioh mayor
11
may not be infinite.
Our primary obJeot in this note 1s to show that, apart from certain minor
a.dditions to the general theory (a.dditiona which we develop in
§ 2),
the standard
ma.ohinery of renewa.l theory leads direotly to results (A), (B) and (e) with no
recourse to unnatural boundedness coOOitiona.
In faot, onoe we notice that Nt is
the number of renewa.ls in the ~ prooess up to time t, (A) is merely the wellknown renewa.l theorem
1
(1.3)
~
( 0 , if
first proved rigorously by Feller
particular, Doob
£4_7,
!1. = 00),
1- 5_7,
and later by many authors inoluding, in
who proved (1.3) directly fran the strong la~ of large
·.
4
numbers.
81m1larly (B) is an expression of the renewal theorem 6 of smith L-10_7,
=
(1.4)
2
,
a
say.
The asymptotic norma.lity result (e) is but a. restatement of the centra.l limit
theorem for renewal processes, Feller's proof
elegant.
of
~
L- 6_7
of which is both brief and
Thus Hammersley's results (A), (B), and (e) will be proved once the values
and 112 are determined.
The manner in which renewal theory simplifies this
ca.lcula.tion is the subject of
f 2.'
In
§
of (A), (B) and (e) for the counter process
3 we briefly indicate the derivation
'2J .
Apart from. the grea.ter genera.lity
of the renewa.l a.pproa.ch, this method a.lso ena.bles one to a.void the ra.ther fussy
discussion of differentia.ls, and is noticea.b11 shorter and simpler.
Results (A) and (B) are not stated above in quite so precise a. form a.s the
ones obta.ined by Hammersley 1.-8_7, who showed tha.t the rv
sign could be repla,ced
by equa.lity • Hammersley wa.s able to do this partly because of his a.ssumption tha.t
the Interva.ls were on the circumference of a. circle and partly beca,use of the
boundedness assumption
T
<
00.
In the genera.l ca,se the asymptotic sign is the
most tha.t can be hoped for, but a,n examina.tion of the consequences of the a.ssumption
T
<
00
he.s led us to an interesting specia.l renews.l process which we cell a
Qua,si-Poisson Process beea,use of its similarity to the Poisson process.
Poisson Process is discussed a.t some length in
T
<
00
we can repla.ce the
N
§4
The Qua.si-
and it is shown that when
sign by equa.lity for all sufficiently large t.
La.stly we consider two methods of "censoring" a renews.l process to yield a
new renewa.l process.
e
The first method is introduced a.t the end of
be cs.lled the guarantee-censor for rea.sons we shell there expla.in.
f
4 and ma.y
General results
are obta,ined by renewal theory, and it is shown that when the renews.l process is
Qua.s1-Poisson these become particularly simple.
;
The second method is discussed in
5 a.nd may be called the para,lysis-censor beca.use it corresponds to the effect
5
of an automatic self-para,lysis mechanism in a, counter, of the type described by
Hammersley 1n
§
3 of
L-8_7.
Under certa,in conditions, involving particula.rly the
assumption tha,t the renews,l process 1s Q.ua.si-Poisson, we show tha.t the properties
of the para,lysis-process ma.y be very ea,sily obtained; an opera.tional form of Bolut10n is offered for the study of the ca.se 1n which these conditiona do not hold, and
is applied to a, few examples.
to
§3
It transpires tha.t, while a.pperently (i.e. a,ccording
of £8_iJproposing to study the effect of a para.lysis-censor on
Hammersley ha.s in fa.ct dea,lt with the effect of a. guarantee-censor.
~
,
Moreover, a
formula. for the variance of the number of recorded points which Hammersley gives
for this guarantee-censor ea,se is erroneous.
the paralysis-censor are given in
§
The correct formula.e for the ca.se of
5 (particularly (5.4)).
We close this introduction by mentioning that Polla.czek
1.- 9_7 has
derived,
under certa.in analyticity conditions, some genera.l formula.e a.pproprie.te to the
counter problem for which ~ is not necessarily a Poisson process; he ha,s obtained
from these genera.l formulae specific solutions for two specia.l ca.ses.
Whilst our
calcula.tiona will not cover the more genera.l circumstances enVisaged by Pollaczek,
it should be clear that much of the genera,l discussion of this
§ ,
at lea,st, does
so~ Lastly we mention tha,t an ela.borate trea.tment of a general class of stocha,stic
processes derived from a, Poisson process ha,s recently been given by Ta.ke,cs
L-12_7.
Takacs makes no use of renewa.l theory.
2.
Renewal Theory.
Let ~!!!
l t1}
be a, renewal process,
a,sBocia.ted partia.l sums, Nt th#l."f:f k auch tha.t Tk
function of the
infinite).
i ti.s
~
~ Tn
1 the
t, F(t) the distribution
, and ~, ~2 the first two moments of F(t) (they ma.y be
Write
3. Pollaczek does not give simple formula.e for
Sf>Elcia.l cases mentioned.
~
2
and (J , even for the two
See, however, our footnote to Example 1 in § 4.
6
(2.1)
= (;
R(t)
Nt
,
00
F* (s)
(2.2)
=
r
e
r
e- st dH(t) •
-st
dF(t) ,
0-
R* (a)
(2.3)
Then it is well-known
=
£10_7
0-
tha,t R(t) ea,tiafies the renewa.l equation:
t
(2.4)
R(t)
=
F(t) +
S
R(t - u) dF(u)
,
0-
and tha.t if Fm(t) repreaents the distribution function of T then
m
00
R(t)
=
I:
1
Fm(t) •
An ee.sy and well-known consequence of the la.st four equa.tiona a.re the following
rela.tiona .
R* (a)
(2.6)
F* (e)
=
=
F* (s)
1 -
F* (s)
R* (a)
1 + R* (s)
,.
,
(2.1)
(The relations between transforms are va.lid provided the real pa.rt of s is suffi-
e
ciently large) •
The following lemma. represents the necessity part of a. theorem, of which the
7
sufficiency part is the following renewa.1 theorem (Theorem 5 of smith £10_7):
if 112
<
00
then a,s t
->
00,
=
R(t}
(2.8)
Lemma. 1.
<
If III
co then a. necessary condition for the lim! t
to exist and be finite is tha.t J.L2
Proof.
e
<
00.
When this is so, 112
Suppoae the limit f3 exists and ia finite.
= 2~2 (1
+ (3) •
Then by (2.6) and the
Abelian Theorem for Lap1a.ce-StieltJes tranaforms (Widder £13_7 p. 182) we deduce
that
*
F (a)
lim
s=O+
Since ~
<
00, a.s 8
1 - F* (s)
->
- -~s1
= 13 •
*
0+ we ha,ve 1 - F (a)
=
~a + o(s}; and if we combine this
fa,ct with (2.9) it is fa,irly easy to deduce that
(2.10)
*
F (s)
=
222
1 - ~B + ~(l-$)S + o(s ) ,
from Which it follows tha.t
(2.11)
lim
8=0+
But
F* (O)-2F* ~8)+F* (2s)
s
= 2~(1
+ (3) ,
8
00
F* (0)-2F* (a)+F* (2a)
(2.12)
a
f
-ax} 2
1-e
ax
= )
2
0-
x2dF(x) ,
2
x dF(x) ,
where
€
is chosen so tha.t for a.11 positive y
we a.110w s
->
<€
we may ha.ve 1 - ·e-y
0+, (2.11) and (2.13) combine to prove tha.t 1J.
2
the fa.ct tha.t 1J.
2
= 2~2 (1
fa.ct tha.t we now know 1J.
2
right of (2.12).
lim
< 00.
> yo.
If
We can deduce
+ ~) either from the sufficiency theorem (2.8) or use the
<
00
to a.pp1y domina.ted convergence to the integra.1 on the
If we do this we observe that
CD
F* (0)-2F* (s}+F* (2s)
s=O+
s
=
2
5
x
2
dF(x) ,
0-
=
The lemma. now follows from (2.11).
Notice that as a consequence of our results, if 1J.2
= CD
then the limit 13
either fails to exist or is infinite.
We now introduce a. renewa.1 process of a specia.1 type.
two independent renewa.l processes; the
and the
tvn~
have E",( .) i.e.
structiona,l property:
process.
t
n
= un
For a. ",-type process
1vn ~
n
i IJ.n1 , i vn ~
is a Poisson process.
If ~ has the conis a. ",-type renewa.1
,F(t} is obviously a.bso1ute1y continuous, with
a. frequency function f(t), sa.y; and :fI(t) a.dmits of a. renewa.1 density function
h(t)
= :fII(t).
Let us define TO
'!ret)
be
ha.ve a. distribution function G( •) ,
for a.11 n, we shall sa.y ~
+ v
'J
f un ~
Let
= 0,
and consider the function
=
0
if
=
1
otherwise.
t
~ TN + u.., 1
t
l~t +
'
9
Thus, as the renewal process
'J
develops we can imagine 1jr(t) a.ssuming a.lternately
L·Tn ,
the va.lues zero in the closed interva,ls
vening open interva,ls (Tn + un+ l' Tn+ 1)'
if 1jr (t)
=1
and "covered" otherwise.
T +
n
U
1 7 and unity in the intern+-
It is convenient to say that t is "free"
Similarly it is convenient to speak of free
or covered intervals.
Write rr(tl.
clear that h(t)
=g1jr(t) for the proba.bil1ty the,t t is free.
= A.1f(t). This may be proved a.s followB
rr(t)
=
J
5
t-v
t
t
e-A.(t-u) dG(u) +
j h(v)
1
o
0-
Then it is intuitively
e-A.(t-v-u) dG(u) } dv ,
0-
t
=
1
1
~ ret) + ~
1( h(v)
f(t-v)dv ,
o
= I1
h(t) ,
by the integra,l equation for renewal density functions which is ana.logoua to (2.4)
(Smith
£10_7>.
We can now prove:
Theorem 1. For any A.-tyPe renewa.l process ~
11m
t=oo
1f(t)
which is to be taken a,s zero if
J..1
l
there exists the limit
J..1
l
= 00.
If this limit is non-zero, so that
< 00, and if
t3
=
du
11m
t=oo
exists and is finite, then
J..1
2
< 00
~ J..1
2
= 2~2 (1
However, if ~
+ t3).
the limit t3 either does not exist or is infinite, then
J..1
2
= 00.
< 00 and.
10
Furthermore, if
~2
< 00 then
00
fo
Proof.
'1f(u) -
A~
,
du
<
00.
The fa.ct tha.t f(x) is the convolution of a nega.tive exponentia.l fre-
quencY' function and G(x):
r
.t
=
rex)
Ae-A(x-z) dG(Z) ,
0-
involves, a.s an ea.s11Y' deduced consequence, the fa.ct tha.t rex)
Also, by HBlder I s inequa.l1tY', for 1 < P
where p
-1
+ p'
-1
= 1.
->
0 a.s x ->
00.
<2,
Hence
x
S APe-Ap(x-z) dG(z),
0-
from which it is ea.sy to see tha.t f(x)
(A~l )
-1
€
L (0, 00).
p
The convergence of :rr(t) to
is now a. consequence of known theorems on the convergence of renewa.l densitY'
(smith L-IO_7, 1-11_7) •
The part of Theorem 1 which concerns the limit t:3 follows directlY' from Lemma.
1 once it is noticed tha.t
H(t)
=
l
o
f
t
h(u) du
=
A
o
:rr(u) du •
11
The result concerning the a.bsolute convergence of the infinite integra.l is not
needed in the present trea.tment and is merely sta.ted for completeness in the trea.tment of A.-type processes.
It is e consequence of Corollary 8.1 of Smith L-10_7.
Theorem 1, which is little more than a. pulling-together of known renewal
theorems, ha,s obvious a.pplice,tions to the counter process} • We develop these
in the next section.
the present pe,per.
However, its field of epplica.tion is wider than the scope of
It is of grea,t use in dee.ling with queueing theory, in perticu-
ler with the busy interva,l distribution; this e,pplice.tion will e.ppear elsewhere.
3 • The Counter Process. As the counter process
¢(t) beha.ves exa.ctly like the function
rzJ'
develops, the funotion
'i( t) in the last section.
It a.ssumes the
va.lue zero on the closed covered interva.ls, and unity on the open free interva,ls.
e
Evidently, ea.ch z
n
is the sum of u , the length of the n-th covered interva.l, and
n
v ' the length of the n-th free interva,l.
n
Further, by a, well-known property of the
EA.( .) distribution function, the ~ v } will be distributed in e,ocordance with
n
EA.( .) (since the vn mea.sure the distance from the end of e, olosed covered interva.l
to the next renewal in a, "Poisson process).
renewa.l prooess.
It is thus evident the,t
11
is e, A.-type
Theorem 1 shows the,t once 1t(t) is determined the caloula.tion of
2
1J.1 and 1J.2 (and hence a ) is simple.
A particularly ea.sy example is afforded by the counter process
B(y)
= U(y
- T), where
°< T<
00.
As stated in
f
1zf
for which
1, this problem has been
ta.ckled by Domb, Feller and Hammersley (in various wa.ys 4nd with varying degrees of
complexity).
But it is obvious tha~ for this particular process 1t(t)
since no free instant can ocour in
if no Xn fa.lls in
f
t- T , t_7.
L- 0,
t_7 and an instant t
(3.1)
2
= A.
e
-t-. T (
1 - 2A.
e
-A.T
)
U{t-T),
is free if and only
Th us we deduce a.t once from Theorem 1 tha.t
-1 t-.T
A.
e
a
>T
= e-~T
12
e
From (; .1) the results (A), (B) and (0) follow ea,s1ly v1a. (1.;) and (1.4).
Moreover,
we see that for this particular process (a.dopt1ng the nota.tion used for ~ ),
(; .2)
and so, by (2.7), we he.ve
*
(; .;)
F (s)
e
=
-kST
s
k
-as
It follows from (; .;), by a routine 1nterpreta.t1on of e . as a. "shift" opera.tor and
of s -n a.s the La.pla.ce- Stieltjes transform of t~( t) /n '., tha.t
(; .4)
=
F(t)
00
(_1)
E
k=l
k+1
A.ke-kP-.T (t-k T )k
U(t-kT).
k~
The right-hand-side of (; .4) is very simply rela.ted to the "ta,pered exponentia.l
function" which a,rose in the course of Hammersley's 1nvestige.t1on
For~
L-8_7.
the ca.se of e, qUite genera,l B(y) the determine,tion of net) is hardly eny
more difficu1t then it was in the specie.1 CB.se we ha.ve just discussed.
appa,rent tha.t if n
t
=k
we ha.ve k "intervals" with left-hand closure points inde-
pendently and uniformly distributed over (0, t), so that
f
t
prob
{¢(t)
= 1Ft = k} = :B(t-C)
i~
o
whence
h(t)
= A.1f(t)
For it is
B(g) dg
13
e
Theorem 2.
~, ~
For the counter-process
1-1 - B( • t7 € L1(0,00). 1!
~ is finite, it is given by the equa,tion
A.b
(3.6)
~
is finite if and only if
=
III
e
1
,
11 is finite if end only if
2
~ f £1 - B(G)_7d9]
;. B(t) exp
belongs to L (0,00) •
1
.!!
\-1
2
is finite, then
;. e-;'b 1 {1 + 2
(3.8)
- i\.b
e
-;.
r
(j
2
1
is given by the expression
£h(t)
o
~
h(t) is given in
~.
By Theorem 1 end
~:~
1s not zero.
III
<
00
(3.5).
(3.5)
we see that III is finite if and only if
~
B(t-O) exp [-;.
£1 - B(GU d9i
The fa,ct that B( .) is non-decree,sing a,nd bounded by unity shows tha,t
if a,nd only if
1-1 - B(Q)_7
L (O,oo).
1
€
This in turn implies that B(oo)
and if we observe tha.t
00
b1
f1
=
1 - B( 9 )
1
d9 ,
o
then (3.6) follows.
The rest of Theorem 2 is an immediate consequence of (3.5),
Theorem 1, and the simple relation
r:J
2
=
-1
III (1 + ~).
= 1,
14
From (3.6) and (3.8), the results (A), (B) and (0) follow
sent theorem provides some insight into the
fa.il and is clearly va,lid when
constant for all t
~ T
T
= 00.
W8.y
a,s before.
The pre-
in which results (A), (B) and (o) may
Notice however that if
(this fe,ct we,g used by Hammersley).
T
<
then h(t) is
00
We shall make use of this
remark in the next section.
The argument lea,ding to (3.5) ma,y be compared with that lea.ding to equa,tion (5)
4. Qua.si-Poisson Processes. We return for this section to the discussion of
the quite general renewal process
~ of
§ 2,
not necessarily supposing it to be of
the A.-type.
In £1o_7 the residual life-time a.t y is defined a,s TN +1 - Y (this is
y
the "forward delay" of ["l_nand
( 4.1)
Y(y; x)
= prob
{ TN +l-Y
:s x
Y
When
'J
I.
ha,ppens to be a Poisson process, a, number of plea,sant propm-t1es hold.
particular, h(z} is constant for all
Zj
and Y(y;x} is independent of y.
In
In
§ 3 we
sa.w how to construct arbitrarily many renewa,l processes for which h(z} is constant
for all z exceeding some fixed
T
< 00,
to whether Y(y;x) is independent of
and the question na.turally presents itself as
y~ T
for such processes.
The next theorem answers
this question in the a,ffirma,tive, and for this rea.son we call these specia.l renews,l
processes:
Qua.si-Poisson processes.
Note that in our discussion
Theorem 3.
y for all y
~.
~ T
1\-1
We ca,ll
T
the index.
is to be interpreted as zero if
lJ.
l
= 00.
A necessary and sufficient condition for Y(YiX) to be independent of
is tha,t H(x) be linear in x for all x
~ T.
EVident1y,from (2.1) and (1.3), if H(x) is linear for a.ll large x then
it must be of the form
-1
~
x + 'Y, where 'Y is some constant.
We observe that
15
~ F(x+y)
Y(y;x) =
- F(y)
1
+
~
f
E
k=l 0-
i F(x+y) - F(y} ~
=
y
00
+
!t
~
F(x+y-z) - F(y-z)
dFk(z) ,
F(x+y-z) - F(y-Z)} dH(z),
0-
x+y
=
I
H(x+y) - H(y) -
F(x+y-z) dH(z) ,
y-O
x
~0- 1 1 -
=
( 4.2)
F(x-z}
~
d
Z
H(y+z)
(We deduced the third step in this cha.in of equa.tiona from the renewal equa.tion (2.4).)
2:
To prove the sufficiency part of the theorem we suppose y
d
e
z
H(y+z)
=
-1
~
dz.
so that
Thus (4.2) gives at once
x
(4.;)
T ,
Y(y;x)
= 1- i
III ~
o
f
1 - F(z)
1
dz,
and shows Y(y;x) to be correctly independent of y ~
Notice that (4.;) agrees
T.
with the known result
x
lim
(4.4) .
Jt
Y(y;x)
y=oo
1 - F( z}
~
dz,
o
which holds for any renewal process (this result is Theorem 4 of £10_7) .
To prove the neceesity part of the theorem we choose any y
(4.2)
and
(4.4)
2:
T
and. deduce from
that
x
j
°
t
1 - F(x-z)
~
x
dz
=
f
0-
~1
- F(x-z)
t
d z H(y + z) •
16
Let
0:
be a:ny number such that F(o:)
<
1 (there must be such an 0:
> 0) .
Then a straight-
forward measure-theoretic argu:nent ba.sed on (4.5) shows tha.t H(y+z) is a,bsolutely
continuous for a.ll 0
this range to
-1
~
<
z
< 0:,
and possesses a deriva.tive equa.l a.lmost everywhere in
• The arbitrariness of y completes the proof.
We notice tha.t, if
~
< 00,
a,s a consequence of Theorem 3 the limit 8 of Lemma
1 will always exist and be finite, since it will be given by H(T) -
fini te for all t and alJY renewa.l process.
Thus 11
2
< 00.
ll~lT, and H(t) is
In fa.ct, rather more is
true for a Quasi-Poisson "Process, a,s the next theorem shows.
However, before we prove
this theorem we must first shoW' that it is impossible to ha.ve III
= 00
for a Quasi-
Poisson process.
Lemma 2.
F(x)
= F(O) <
'Proof.
finite x.
1 fo:c all finite x.
Let us notice first the meaning of the sta.tement F(x)
We cannot ha.ve F(O)
=1
= F(O)
by definition of a. renewal process.
are zero with probability F(O) and infinite with proba.bllity 1 - F(O).
for all
Thus the ti
Such a, Qua.si-
Poisson process we sha.ll call degenerate.
Suppose there exists an x such that F(x)
> F(O).
both strictly positive and such that F(b) - F(e)
H(nb) - H(na.)
>
ciently la.rge x.
>
O.
Then there exists an e"
b,
From this fa.ct we see that
0 for a.ll integers n, so that H(x) ca.nnot be constant for all suffiBut if III
= 00
and the process is Qua.si-Poisson then it follows
from Theorem 3 that E.{x) is constant for all large x.
This contraaiction proves the
lemma.
Theorem 4 •
.!!
~ is a (non-degenerate) Quasi-Poisson process with index
then there exists a. 5
Ra
>
-0.
>
T,
0 auch tha.t the integra.l F* (a) of' (2 .2) is convergent for a,ll
Consequently all the moments of F(x) are finite.
17
e
Proof.
by
-1
~x
By Lemma 2 we ma.y suppose ~
~
+ '1 for a.ll x
T ,
< 00;
and since H(x), by Theorem 3, is given
we observe tha.t
T
H* (s)
(4.6)
=
f
e- sx dH(x)
0-
Hence, from (2.6), F* (s) is the following ratio of integra.l functions,
T
*
~s}
F (s)
(4.7)
e-ex dH(x) + e- TS
= - - .o-. ; . .T- - - - - - - - - ills + ~e
f
e- s"dH(x) + e-
TS
0-
Now it follows from Widder (p. 58, (13)) that the singularity, if any, of F* (s) which
=0
is nearest the point s
in the complex s-plane must be loca.ted on the real axis
and is, in fact, where the real axis is cut by the axis of convergence of (2.2).
It
CI.
cannot be in the open ha.lf-plane Rs> 0; and must be
in the denominator of
I
at~.zero
of the integral function
(4.7). This zero is obviously not at s = 0 and so must be
loca,ted at some point s
= - 8,
where 8 >
0 • This proves the theorem.
A further plea.sant feature of the Poisson process, in a,ddition to those a.lrea.dy
referred to, is that the cumula,nts of the distribution of Nt are directly proportional
to t.
An a,na,logous result holds for the Q.ua,si-Poisson process, a.lthough it is not so
simple.
In the following theorem we consider only the mean and variance of Nt.
Theorem 5.
then for B.ll
t
while for B.ll t
If ~
~ T
is a (non-degenerate)Qua,si-Poisson process with index
,
G 2T
,
= (J2t
-
where ~ is the third moment (necessarily finite) of F(·).
,
T
18
e
~.
3 that all t ~
We know already from Theorem
T,
B(t)
= ~l
t + '1, and,
since all the moments of F(') are finite, we can now identify the constant '1 with
the limit f3 of Lemma 1.
e
(4.8)
Next we remark that by equation (7.6) of L-10_7,
t
5 B(t - z)dR(z).
tNtJ 2 = B(t) + 2
0-
It 1s e simple matter to verify that the right-hand-side of (4.8) is a. qua.dra.tic
function of t for all t
~
2
T.
To determine the a.ppropriate coefficients we observe
that the Laplace transform of (4.8) is
*
ro
\ e-st ~ {Nt ~ 2 dt =
~
. t
F
s
is)
Ll-F (8»)
•
where a(l)
-:->
0 as s
->
0+.
Thus, knm4'ing that for large t
quadratic function of t, we may infer from
(4.9) 'that for t
~
e. tNt ~
2 is a
2T,
(4.10)
If we combine (4.10) with the result we have already obtained for B(t)
= g Nt'
the
variance of Nt turns out to be as announced in the theorem.
We complete our present disoU/:it"lion of f'luasi-Foisson processes with the following
Theorem 6.
!
"l\
>0
If
and a p, 0
eJ i~Q.1'-~~~~-P2.isl3on
~ p
< 1,
F(x)
~~£~~ tl.!:~t
= (1
pr02..~~s_'Wit:tL~o index then there exists
for x
~ 0,
- p)(l _ e-"l\x) + p.
·
.
19
-1
Suppose H(x) = III
Proof.
x + 7 for all x> O.
Clearly 12= 0, since H(x) must
be non-nega.tive • Then
H* (s)
and so, by (2.6) end some elementary manipula.tion,
F* (s)
=
(1+1) •
Inverting this Lapla.ce-Stieltjes transform yields
F(x)
( 1+')') ,
and this completes our proof.
From any renewe.l process
a constant a>
> O.
A
Define TO
8
= 0,
A
A
A
= ATn
A
as follows.
A
and T a.s the least T such that T - T 1
n
n
nl
Then define T a.s the second smallest Tn such that Tn - Tn- 1
2
evident that if t n
~
we may derive a new process
A
- Tn- l' then ~::
~A
tn
~
> as,
Choose
> Cfi.
and so on.
It is
is a new renewal process (we
A
she.ll identify all functions etc. associated with ~
by the sign " ).
The process
A
~ would arise if, in studying a. physical realisation of ~ we refuse to count renewals if they are within a. distance (on the timescs.le)
(whether this was oounted or not).
More Vividly,
A
~
~
of the preceding renewal
would arise if the renews.ls
carried a. "ma.ker's guarantee" to replace Without charge the renewal if it needs
replacement in less tham
A
"
a>
renewal instants T of 0C5
n
time units from its moment of instslla.tion.
represent only those renewa.ls in
2
Then the
which cost anything,
A
and Nt will be the number of renewals actua.lly paid for in the period (O,t_7.
"
The simplest a.pproa.ch to an understanding of ~
"
appears to 11e in the ca.lcula-
tion of H(t), and this is rendered straightforward by the follOWing observation.
At
· .
20
A
moat only one Tn can fall in an interval (t, t+h_7 if h ~ (/) and t has any non-negative
A
value.
Thus the probability that one Tn does fall in (t, t+h 7 can be equated to the
A
A
Let us take t ~rn.
expectation expression H(t+h) - H(t).
Then
t-~
A
A
H(t+h) - R(t)
00
= F(t+h)
- F(t) + ~
k=l
fi
F(t+h-z) - F(t-zll
dFk(t)
0-
t+h-a>
~
+
)
k=l t-ct>-O
~F(t+h-Z)-F(a;)}
t-ro
= F(t+h) - F(t) +
)
i F(t+h-z)
dFk(Z),
- F(t-z)
~
dH(z)
0-
t+h-a>
~
+
1
F(t+h-z) - F(m)
5
dH(z),
t-w-O
t+h-ro
(4 .If)
= F(t+h)
J
+
F(t+h-z) dH(z) - F(w)R(t+h-ffi)
o
t-ffi
- F(t) -
f
F(t-z) dH(z) + F(w) R(t-a»
o
It will be noticed that the right-hand-side of (4.11) is of the form }(t+h) - :J(t),
from which we deduce immediately that (4.11) is va.lid for a,l1 h
>
O.
If we use the
A
fa,ct that H(ro)
=0
we can now deduce from (4.11) tha.t for all t~ 05 ,
t-a\
(4.12)
;(t) = F(t) +
~
F(t-z) dR(z) - F(w) H(t-ro) - F(6)) - F(a) R(O) +F(ffi) R(O)
0t
= H(t)
-
\
t-~-O
F(t-z) dR(z) -
F(~)
- F(w) H(t-m) .
21
1\
Theorem 7.
onl~
<J
In the renewal process
if lil is finite and F(CI)
< 1.
1\
,
~1
is finite if and
1\
III
~
and 1J.2 is finite if and only if
~
If ~ is finite then it is given bl
1\
(4.13 )
derived from
,.
=
1 - F(liS)
,
}12 is fin! te • If
1\
1i
2
is finite, we have
to
r
(4.14)
zdF(z) .
o
Proof.
We will use the consequence of Theorem
lim
(4.15)
t
= 00
f
1 of
Smith
1-10_7,
that
m
t
F(t - z)dH(z)
f
= .l..
III
t-lb-O
F(z)dz,
o
where the right-hand-side of (4.15) is to be ta.ken as zero if III
= 00.
Thus, from (4.12) and (1.3),
1\
(4.16)
.L =
tl
lim
R(t)
t=oo t
=
lim
R(t) - F(~)R(t - ill)
=
1 - F(ro}
t
t = 00
1J.1
1\
Equation (4.16) proves that part of our theorem which refers to III and (4.13).
1\
now suppose ~
1\
(4.17)
R(t)
< 00,
we observe next .that by (4.12) and for all t ~
t
i
-- =
H( t) -
~l ~
- F(&\)
~
H( t - 11»
~
- t
~:if,}
+ ill
1" ,
~~~) -
t
F(t - z)dR(z).
t-oo-O
From (4.17) we deduce via. Lemma 1 and (4.15), that
1\
(4.18)
f3
=
F(z)dz;
F(ilI)
If we
}\
A
so that 13 is finite if and only if 13 is finite. Hence ~2 is finite if and only if
A2
A -1
A
~2 is finite.
Also, since cr = ~l (1 + 2 ~), and since
C6
.!~l
I
F(z)dz
~l
-
o
as
fo
zdF(z),
the formula (4.14) is derivable from (4.18) by elementary manipulations.
Corollary 7.
with index
T
Proof:
+
If
~
is quasi-Poisson with index
T
J
~
then
is Quasi-Poisson
ill.
>
If R(t) is linear in t for all t
have from (4.12) that for all t ~
say R(t)
r
=
-1
~l
+ 7, then we
+ ~,
,...
ill
~il
A
R(t) =
T
T ,
F(z)dz - F(w) - F(a))
~
o
A
i.e., R(t) is linear.
This proves the Crrollary.
It should now be clear that, if ~2
< 00, results (A), (B), and (C) will hold
A
for the
'J
process, and that (A) and (B) will be replaceable by the exact results
of Theorem '5 i f
c:J
is Qua.si-Poisson.
We close this section by giving two examples.
Example 1.
we find that
~
If ~
is Poisson with a distribution function F(') = E~( .), then
-1
2
="1\ , cr
A
~
1
A
cr
2
=~,
="1\
F(liS) = 1 - e
-1
="1\e
e
-Ni5
and Theorem 7 yields the results:
ND
-M)
,.,
(1 - 2~ille
-N:6
)
which are identica.l with the equations <:3.1).
Reflection will show that when the
u~derlYing ~
~
counter process
process is Poisson the process
Z5
for which Bey)
= u(y
-
as).
is identical to the special
Thus the present example represents
another method of considering. the example considered in
§
3. 4
tr =
A
In fact, i f we take l: 5 CCJ , with general F(.), then
2. is the more
general counter process ~nviBaged by Pollaczek, for the case B(Y) = U(y-cD). Theorem
7 thus provides ~ and cr for a useful part of ~ollaczek's wider class of counter
processes.
4.
23
Example 2.
of
f
~
Let us next suppose
•
'?f , where '7f
3 with B(y) = U(y - T). It should be obvious that if
and 80 we ma.y ima.g1ne (j)
of Ill' a2 and F(t ) in
easily from
>
For
T.
1?
is the counter process
1\
as ~
T then ~
;;
if
,
we have a.lrea,dy given the appropriate values
(3.1) and (3.4). Thus the calcula.t10n of
1\
~ and
1\2
a follows
(4.13) and (4.14) by elementary manipulations. However, the ca.lculation
of the integral
...
CD
f
(4.19)
zdF(z),
o
which is required by
ing way.
(4.14), can be simplified by the use of transforms in the follow-
By the convolution property of Lapla.ce-Stieltjes transforms, the La.place-
St1eltJes transform of
t
S (t -
(4.20)
z)dF(z)
o
~ is simply S-~*(8), or, by (3.3) it is
(4.21)
But we recognise
(4.21) as the transform of
From the equivalence of
(4.20) and (4.2:=') we therefore deduce that
......
f
CD
(4.23)
zdF(z)
= A-I e~ T F(li5
+ T) - ID['1 - F(ID)_7
o
Elementary computa.tion ba.sed on
(4.23) then g1ves
24
t 1 - F(w) 5
/\
lJ.
1
=
Let us s8,y that we ha,ve derived
censor.
J
/\
C'1
from Q)
by application of a guarantee-
§ 4 of ['8_7 is actually studying the effect of a,
counter process 15 , and not, a,s he a,pparently supposes,
Then Hammersley in
guarahtee-censor on the
effect of paralysis.
the
The Justification for this contention lies in the critical
four lines of text following equation (47) of
1-8_7.
It is therefore to be expected
that our (4.24) will agree with Hammersley's (59) and (60).
Unfortunately, while
A
2
they a.gree with respect to ~, they indicate quite different values for 'd . A simple
A
2
test revea Is that Hammersley's value for a must be incorrect.
then)'
!!:
r,
For if we put ,. = 0,
the basic Poisson process (since 8,11 the "intervals" involved in the
construction of the counter process
3
have shrunk to points).
Thus we reduce our
A
2
problem to the easier one considered in Example 1 above, for which a is simply
~e
-~ ( 1-2~e
'" -f$.))
1 - F(lli) = e
-NI>
= 1\2
0 ,
0
say.
If in (4 .2 4) we put,.
'-2
, then it is ee.sy to see tha,t 'c1
fa.ils to pa,ss this test, it gives a. va,lue for
=
/\2
0
=0
2
fro'
and notice that for this case
correctly.
Hammersley'S (60)
1'2.
O
which is quite different from a
and we must therefore conclude that his formula is incorrect. 5
5. Paralysis. 6 Let ~ be the renewal process of
Poisson, and let
W ::
i <.oi ~
§
2, not necessarily Quasi-
be en independent renewal process with associated
5. In comparing the results of the present wOik with those of Hammersley
it 1s useful to observe that e (-a,b) = 1 - F(m), e (-8., b+l) = 1 - F(ilS + T).
1-8_7,
6. In this section we consider the effect of the censoring procedure discussed
by Hammersley in ~:3 of
1-8_7.
25
distribution function C(.) and moments c ' c 2 '
l
... etc.
Let us write
~
TO = 0, Tl
for
~
the least Tn which exceeds (01' T2 for the lea.st Tn which exceeds T + (02" and, in a
l
N
N
general way, T for the least T which exceeds T 1 + (0.
n
n
nn
N
write t l
....
r..J
,..;
= Tl ,
= Tn
tn
""
- T _ (n
n l
C>f
..
is a renewal process
~
(L)
is Qua.si-"Poisson,
Theorem 8.
If ~
etc. with regard to ~ , a,nd say
gj is derived
~ can be particularly ea.sy to study.
is Q.u8si-f'oisson with index T, and if C(T)
""*
F (e)
and so, for n
= 1,
(5.2) may be infinite.
.....
Let
K:
then
2, ••• ,
rig,.~t-hand-side of
Proof.
= 0,
,
=
=
N + 1.
COl
with proba,bility one, COl ~
T
Then it is evident that t
~ = TK:
Thus
•
l
= T K: = (TK-col )
+
c.L].. But,
- C01' the "residua,l lifetime lt in the
~
process at co1' has the distribution function
x
= )
o
F (1) (x)
which is independent of c.L]..
Hence t
l
1 - F( z)
III
dz ,
ma.y be considered the sum of two independent
random variables ~ a,nd co1' with distribution functions F(l)(·) and C(.) respectively.
Equation (5.1) then follows from the two equations
00
F*(1) (e)
e
(5.3 )
=
i
e
-ex
( )
dF(l) x
0-
I'J*
F (s)
=
*
C* (s) F(l)(s)
=
1 - F* (8)
~s
•
/V
by s. para.lysis-censor.
When ~
where the
f tn~
the sequence
F( .), 1;,
We shall employ the nota.tions
from ~
> 1),
Then it follows that if we
"-'
26
l"rom ('5.3) we can find the moments of F (1) ( .) by expansion a.s a Taylor series in s.
we find that the r-th moment of F (1) ( .) is Il~l/(r+l)~.
e (~
+
~)n =
n
!: (n)
r
0
(e ~
r)(
8
m~-r)
n
=
Il +
E (n) r l
0
r
c
(~l) III
Hence
,
n-r
as was to be proved.
As
an example of the application of Theorem 8, let us take
= U(y-T)
B(y)
as > T.
vided
= U(m-(6).
and C(m)
cl = 2J
, with
Then the conditions of Theorem 8 are sa.tisfied, pro-
AI
11
In fact
is the counter process with paralysis appropria.te to the
blood-cell counter problem considered in
£8_7.
Ca.lculations based on the values of
*
2 and F (a) which are given (or implied) by equations
~l' 11
e
""'
F * (a)
(3.1)
and
(3.3)
then yield
- (kT + 8))8
s
k+l
whence
00
".;
F(x) :; E
o
~l
(_l)k ("e -":)k+l(X-kT-~)
'(k+l)~
II
(t-kT-ll)) •
We a.lao find that
tV....
III :; ro-
-l:\T
e
T +]1.
('5.4)
rJ
From (5.4) the results (A), (B), and (C) follow at once.
However,
'0
Quasi-Poisson process, so that (A) and (B) are not exact for all large t.
is not a
27
Vhen the specia.l conditions of Theorem 8 fail to hold the calculation of
'V
iJ.
1
a.nd
1\12
(J
becomes much more complicated.
F( '),
He conclude this note with the development
of some opera.tiona.l formula.e which ma.y be used in the more difficult circumstances.
To simplify our calculations we shall assume 8.11 distribution functions to be absolutely continuous (a,lthough we shall be qUite prepared to apply our results to ca.ses
A(')
where they are not ~)
If C( .) = E
i
x
(5.5)
tv
f(x) =
~
Ae -)y.z
0
then we have
z
(
~
0
r(x) +
f(X-Z ) h(zl) dZ )
l
1
dZ
1
which simplifies after an integra.tion by parts to the equation
x
(5.6)
f(x)
= f(x)
- e-
Ax
r
hex) +
f(x-z)
L-e-~z
h(z)_7 dz
o
e
(we have a,lso made use of the renewal integral equation connecting f(x) and h(x» .
If we write
00
f
* (a)
=
e
\
-sx
f(x) dx ,
o
(.10
h * (a)
=\
e- ax hex) dx ,
o
for Laplace transforms, then
~*
f (s)
(~.6)
gives quiCkly
= f * (s) - h* (s+ ~ ) + f * (s) h* (s+ ~ )
=
*
*
h (s ) - h (s+]I. )
*
1 + h (s)
since f * (8) = h* (s)/
L- 1
*
-
+ h (s)_7.
If c(x)
= C'(X)
were quite general, (5.5) would be replaced by
x
("5.8)
rex) =
J
z
~ rex)
c(z)
o
5 r(x-z l ) h(Zl) dZ l ~
+
.
dz •
0
Let ue suppose that we can represent c(x) in the form
c (x)
=
f
d¢(~)
Aa -":x
E
where E is some set in the complex open half-plane {
A > 0 and ¢( .) is seme measure
defined on E for which the integral (5.9) has a mea.ning.
the:lntegral (5.9) is absolutely convergent at x
order of certain multiple integrations.
(5.10)
1
=
f
= 0,
We
shall suppose also that
to Justify the changing of the
Thus we have from (5.9) that
d
¢ (t-.)
E
a.nd we may infer from (5.7) and the linearity of (5.8) in c(x) that in the "general"
caee
(5.11)
f* (s) =
*() - h-* (e)
he
1 + h* (e)
ii* (5) =
f
where
(5.12)
h* (s+ t-. ) d
¢ (t-.) •
E
We consider some illustrative examples of the use of (5.11).
Example 1.
If
00
C*(8)
= [
o
a- 8X c(x) dx ,
29
r > 0,
then under certain conditions, for some
(5.·13)
c(x)
absolutely convergent).
1
= 2:d
~ in (5.1~) we deduce a representation
=-
If we put s
(5.9) for c(x) in which E is the line joining r - ioo to r + ioo and
= C* (-~)
(5.14)
2]'(i )..
d~
Hence
)'+ioo
6*(s)
*
h (s+~)
f
1
= 21CT
C
* (-~)
~
d~
r-ioo
Example 2.
As a special case and a check on the last example, let us suppose
that ~ is Quasi-Poisson with index
h * (s)
Where
r* (s)
= I * (8)
- 'T"B
+ _e__
1J.1S
is an integral function which is O( e-
Let us suppose that c(x)
Thus c* (s)
Then (c.f. (4.6»,
'T".
= e -ws ,
and
=0
(x-tIS); where
(5.15)
as>
T
TS)
in the half-plane
1t
B
:s O.
and 0 (x) is the Dirac delta. function.
reduces to
)'+ioo
h*(s)
1
= 21ti
S
d~
•
)'-ioo
The integrand of this contour integra.l is meromorphic, having only two poles (both
e
single); one is at ""
plane
51
A.
<
O.
way by allowing 'Y
=0
and the other at ""
= -s,
which may be supposed in the half
The integral may be evaluated by contour integra.tion in a familiar
->
0+ (and providing a,n indentation a.t 0), and by then s,dding to
,.
30
the contour aiJ. infinitely large semi-circle joining
tQl..
"
S
5 0•
-1
00
to +i
00
in the half-plane
We deduce tha.t
e -em
-*
h (s) = I * (s)
= h* (s)
If we insert this determination of h* (s) into (5.10) we discover that
1*(s)
=
e
-sal
N
= e -SID
~s
i
I-f*(s)
I.l.1 s
~
'
in agreement with Theorem 8.
Example 3.
Puppose that we can represent c(x) in the form
(5.16)
c(x)
=
where the summation may be finite, or infinite, where the
such that
51 A
V
> 0,
and where E lc)
represented in the form of
<
cy
=E
-:;:-
'II
'\I
Thus
Clearly (5.16) could (but will not) be
00.
*
h (s+ A
*
f
are complex numbers
(5.9), and we infer tha.t
_*
h (s)
N*
A'\I
J.
Cu
*
h (s) - E -:;:- h (S+AJ
v
v
(s) = ---~*--'----1 + h (s)
,
and provided an ana.lytic form for h* (s) is known, the calculation of the lower
moments of f * (s) follows from a straightforward expansion as a power series in s.
It is apparent that these moments will involve the derivatives of h* (8) at the
points AV '
As a closing remark, let us notice that (5.11) can be thrown into the following
rather more vivid form
'f*(s)
=
f*(s) \" 1 _ li.:(s) '(
Z
h (s)
S
,.
31
References
fI'
1-1_7 cox,
r .
e
D. R. and SMITH, W. L., On the superposition of renewal processes.
Biometrika, 41 (195 4 ), 91-99.
L-2.:1
D01.ffi , C.
L-3_7
DOME, C. The statistics of correlated events. I. Phil. Mag. (7), 41 (1950),
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L-4_7
DOOB, J. L. Renewal theory from the point of view of the theory of probability
Trans. Amer. Math. Soc. 63 (1948), 422-38.
L-5_7
FELLER, W. On the integral equation of renewal theory.
L-6_7
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L-7_7
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versary Volume (1948), 105-15.
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L-9...7
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(1941), 243-67.
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Courant Anni-
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SMITH, W. L.
C. R.
Proc. Roy. Soc. Edinb. A, 64 (1954).
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TAKACS, L. On secondary processes generated by a Poisson process and their
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