UN!VERSrI'Y OF NOmH CAROLINA
Department of Statistics
Chapel Hill, N. C.
,
,.
A SECOND RENEWAL THEOREM FOR NONlDEm'ICALLY DISTRIBUTED
VARIA:BIES AND ITS APPLICATION TO THE THEORY OF QUEUES
'e
by
C. R. Heathcote
June 1963
This research was supported b,y the Office of Naval
Research under contract No. Nonr-855(09) for research
in probability and statistics at the Uni'Versity of
North Carolina, Chapel Hill, N. C. Reproduction in
whole or in part is permitted for any purpose of the
United States Government.
Institute of Statistics
Mimeo Series No. 371
,
.~
,:...
ERRATA SHE:m'
tar
A
S~OND
RDEWAL TBEORl!.H FOR NONIDmtl'ICAL.LY DISTRIIllTED VAllIABU5
AND l'l'S APPLICATION TO THE THEORY OF 'lJl!lJES
by
CoR Heathcote
0
(103) should read
co
t~
CD
qttR(t)
J
-.l..
s
loll
q(u) au
0
o
Secondly, Lemma 1 (bottan ot page 4) anitted the bn>atheses ACO)
B(0)
lIil
0
q(t)
Suppose that RCt) is the renewal tunction given by (102) coand 'that
i! a non-nesative function tor vb1ch (103) is trueo It loll -=
1
2
x dB(X)
1oL2 ==
are both tinite, and A(O)
inequality
0, B(O)
g
IS
J
xdB(x)"
0, then the 0
2t
qJ+R(t)
holds
and
0
It should read
Q
Lemma 10
!!II
<
•
~~
1
tor t > i12/~1f} unitOl'Dl.l¥ in
Proof': Since RCO)
s
J
~~2
q(u) du
o
eo
0, 1ntegrat1<:m by parts shows that
r.7L
Fran (102)
~~~U(t) ~ B(t:r
Integrating this expression f'ran 0
(104)
to t
i!B
~Bti'R{t)
iB
~B(t) ..
~A(t)o
and "then 1ntegra.t1Dg the 1"esultiDg len""
band side by parts shows that
i1~l{~U ..
B)"
~ "[~R(t?J
t
t
B
11;:1
J ~A(u) au ~ ~il J
The rest at the proof is as betoreo
o
0
q(u} au
0
A Second Renewal
~eorem
for nonidentically
distributed variables and its application to
the 'fheory of Queues.
by
c.
R. Heathcote.
Summary.
Let {Xi} bea sequence of independent, positive, non...lattice
variables with renewal function
H( t)
= ~Pr' {
.1=1
Xl
•. ,
+ ~ + .•• + Xj.
l!!
t}.. •
If the Xi are identically distributed with finite first two. moments III and 112
the second renewal theorem states that as t
+
00
Provided certain conditions are satisfied an analogous result
is derived for nonidentlcally distributed variables.
(~eorem
2)
An upper bound for
the quantity H(t) .. tllll and a limiting result concerning the distribution
of the forward delay are established in
examples.
a3.
a4
contains two illustrative
The remainder of the paper treats the renewal sequence formed
by the first passage times from queue length .1 to qq.eue length .1+1 , .1
= 0,1, ••• ,
ot the single se'rve'r queue with renewal input and negative exponential
service time.
-
It is shown that, provided the queue is transient, the expected
value of maximum queue length in (O,t] is a renewal function satisfying
the second renewal theorem derived earlier.
1.
Introduction
X2, ...
Let Xl'
be independent, positive, non-lattice
random variables with distribution functions Fl(X), F2 (x), •••
and finite expectations ml' ~, ••• respectively.
We assume that
the sequence (Xi) does not degenerate.
If
, j
= 1,
2, .•. ,
the renewal function associated with the sequence (Xi) is
00
R(t)
That is, for each t
>0
=
(1.1)
E Gj(t) •
j=l
, R(t) is the expectation of the maximum n such
that the inequality
is satisfied.
If the Xi are identically distributed the sequence (Xi) is
called a renewal process, and for renewal processes one has available the
following three principal results;
lim
t~oo
!!ttl. =
t
1
I!l
t
;;:
~
(X)
q(t-u) dH(U)
~
- 1,
Q( u) du ,
~l ~
.. ,
2.
The function q(x)
,Here, 1J. 1 and 1J.2are the first two moments of Xi'
has been variously defined but we will take it to be of bounded
variation, zero for negative argument,approaches zero as
integrable over (O,~).
~,
and
10r the widest definition of the class K
of which q(x) is a typical member, see Smith (1961" page 469).
An
equivalent statement to the second result is Blackwell 1 s Theorem,
lim H( tia) -H( t)
= ex1J.-
"b+'oo
•
1
The above three results are known as the elementary renewal
theorem, the key renewal theorem, and the second renewal theorem,
One may ask whether similar theorems hold for sequences
respectively.
Smith (1963, 1961) has obtained an elementary renewal theorem
and a key renewal theorem for general aperiodic (i.e., not all Xi
are lattice) sequences under mild restrictions on the random variables
of the sequence.
One such restriction is a weak convergence property,
namely that
lim
~~
This defines the constant IJ. l appearing in the two
To date there is no analogue for sequences to the second
uniformly in n.
theorems.
renewal theorem, and the main purpose of the first part of this paper
is to derive such a result, (Theorem 2).
The second renewal theorem for renewal processes isa
simple consequence of the key rene~l theorem (by a suitable choice of
q(x) ) and the fact that the renewal function satisfies the integral
equation
t
R(t)
= F(t)
+
J
o
F(t-u) dH(u) ,
where F(t) is the common distribution function of the Xi' (Smith, 1954
page 29).
However this approach tails in the case of sequences since
the renewal function does not, in general, satisfy an integral equation
of the above type.
The approach adopted here is described in the
In § 5 this
next section where our main result is also derived.
theoreDi is applied to a problem in queueing theory in which the
rather stringent conditions of the theorem are satisfied.
Convolutions will be written as-follows:
... are distribution functions of independent positive random
variables then
t
=jo
... -.
The j fold iterated convolution of a distribution function F(x) with
itself will be written
t
F*C1)(t)
= !,F*(j-l)(t_u) dF(u)
,
j
= 2, 3, ... .
o
where U(t) is the Hesviside function
U( t)
.'
={
'.
0 if t
1 if t
<
~
0 .
0
•
t
On occasion we. will use t*B( t)
,;"J B( u)
duo
o
To close this section we establish a preliminary result which
will be needed later..
Let (Zi' i
= l, 2, •.• ), be a sequence of
4.
independent 1 pos1tive l non-lattice 1 random variables with distribution
functions
The (Zi)
Pr (~ ~ x )
= A(x)
Pr(Zi ~ x )
= B(x)
constitute a so-called delayed renewal process and the
associated renewal function
R(t)
= ~
j=o
A*B*(j)(t)
'
satisfies the integral equation
R(t)
= A(t)
+B*R(t).
If we wish to draw attention to the fact that A(t)
= A(tj8)
depends on
a parameter 8 1 the integral equation will be written as
R(t)
= A(tj8) + B*R(t)
(1.2)
Suppose q(x) is a member of the class of functions for which the key
renewal theorem holdsj
co
lim
t~
R*q( t)
=1...J q( u) du •
J.!.l
o
The result we require relates to distribution functions B(x) with
finite coefficient of variation and is as follows:
Lemma 1.
Suppose that R(t) is the renewal function given by (1.2) and
that q(t)
is a non-negative function for which (1.3) is true.
If
00
1J.1=
J
00
x dB(x) and
~2 =
J
o
x
2
dB(x)
0
are both finite, then the inequality
2t
R*q(t)
J
(~~~~2)
> 1l2/~1
holds for t
Proof:
:i!
(1.4)
q(u) du
o
uniformly in
e •
From the integral equation satisfied by R(t),
R*q(t) = A*q(t) + B*R*Q(t),
and hence, for all t,
;§
Il~l
t*q( t)
= Il~l
t
J
q( u) du .
o
The function
B(l)(t)
= Il~l
t* (U*B)
= Il~l
t
J
[l*B(u)] du
o
is an honest distribution function with finite expectation 1J.2/~1'
Il~l
t
t* (U*B)*R*q(2t)
=
J
B(1)(2t*u) d{R*q(u) }
o
2t
+
J
t
B(1)(2t-u) d (R*q(u)}.
Thus
6.
Combining the above two inequalities we have
R*q(t)
But
00
~
J
Y dB(l) (y)
t
so that
for
and (1.4) follows from (1.5)
1J.
t
> -2j..Ll2
.
A Second Renewal Theorem
~2.
Let Y be a positive random variable with distribution
function F(x).
Let
be the ordinary renewal function associated with F(t).
Suppose the
sequence of independent random variables {Xj.l is formed from the
(Xi) of &1 by the truncation procedure
, 'i = 1, 2, ••• , n ,
, i = n+l, n+2, •••
For a fixed integer n, the renewal function of (Xi ) is
~
+
j=n+l
=
n
E
.1=1
G ( t)
j
+
G *K( t)
G *F*(j-n)(t)
n
(2.1) .
n
It is evident that for each fixed n, renewal theorems for
Bn(t) can be deduced without difficulty from the corresponding theorems
for K(t).
In fact (Xi) is a simple extension of a so-called delayed
renewal process.
Furthermore, since the Xi do not degenerate, we ha.ve
for all finite t,
so that
lim
woo Hn(t) = BCt) •
8.
To obtain asymptotic formulae for R(t) from those valid
for each member of the sequence (Hn(t)}, one requires
=
lim lim H (t)
t+oo %1+00 n
lim lim H (t)
n+oo
t+oo
n
(2.2)
•
Clearly this imposes restrictions on the choice of the random variable
Y used to form the truncated sequence
(Xj.l.
Our procedure is to
')
derive renewal theorems for Rn(t), valid for general Y, and subsequently
choose Y so that the interchange of limits (2.2) holds for the formulae
of interest.
1
(i)
+
-
loLl
J
00
(ii)
q*Hn(t)
+~
q(u) du •
o
(iii)
If loLl <
00 ,
1oL2
= ~<
00,
then
t
Hn(t) - loLl
+
The class Q of which q(x) of (ii) is a member has been defined
in ~l.
Parts (i) and (ii) of the Lemma state that the elementary
renewal theorem and key renewal theorem valid tor Hn(t) are those
valid for K(t)" and this is izmnediatelyapparent from the definition (2.1).
9.
To establish (iii), note that
n
H (t) - Ie( t) =
n
G ( t) .1
I:
.1=1
(U - G ) *K( t)
n
00
+
n -
J
~l
1
[l-Gn(U)] du
=
o
-
n
I:
\-Lli=l
(\-L -m )
1 i
by the key renewal theorem for K(t). Thus, by the second renewal
theorem for K(t),
H (t) -
n
L = [H (t) n
\-Ll
\-L2
+ -
",.2
"+"1
IC( t) ] + [ K( t) _
1
- 1
+-
pro
~l
1-. ]
\-L l
n
I:
i=l
(\-Ll - mi ).
A deduction from (iii) of Lemma. 2 is that if in fact the
interchange of limits (2.2) is pe~issible, then the fo~ of a second
renewal theorem for R(t) is
00
Apparently we require the series !: (\-Ll-m ) to converge, implying
i
i=l
lim
woo mn
=. \-Ll '
so that F(x) is related to the sequence (Fn(x)} by
10.
J
00
= ;;;:
[Fn(x)-F(x)] dx
=
O.
o
The conditions 0 stated below, which are sufficient fore 2.2) to hold,
impose a stronger mode of convergence on (Fn(X)} and uniquel.y determine
the constant 1-1- 2 ,
We will say that the sequence of random variables {Xi} satisfies
condition 0 if it is true that there exists a positive non-lattice
random variabl.e Y with distribution function F(x) and finite first
two moments 1-1- 1 and 1-1-2 such that
00
e(ii)t (I-I-l-mn )
n=l
converges.
An immediate consequence is
Theorem 1.
Suppose {Xi) satisfies condition 0, with the exception
that 1-1- 2 need not be finite.
11m
Fn (x)
n+oo
=
Then
F(x) almost everywhere.
Proof:
Without loss of generality suppose Fn (x)
Then both
00
Fn(x)-F(x) and I-I-l-mn
=
f [Fn(x) -F(x)]
iii:
F(x) for all nand x.
dx
o
are non-negative.
Thus
~1]'n(x)-F(X)} is a sequence of non-negative
integrable functions for which, by C(il),
11.
00
dx
E (Ill-mn)
n=l
By
the
Lebes~ue
(l(l
•
monotone convergence theorem
00
J n~l
<
00
[Fn(x)-F(x)] dx
o
= n~lJ [Fn(x)-F(x)]
dx
9
0
00
Thus
E
converges almost everyhwere (Titchmarsh, 1960,
[Fn(x) -F(x)]
n=l
page 347) , implying
Fn(X) -F(x)
-+-
0 a1t@ost everywhere.
The random variable Y is thus specified as the almost
everywhere limit in distribution of the sequence {Xi}'
moment the constant 11 2 is uniquely determined.
As its second
Note that 112 need not
bear any relation to the sequence of second moments (~), and in fact
it is not necessary that the latter exist (although this will be the
if Fn(x) ~ F(x) ).
c~se
The finiteness of 11 2 is not required by Theorem 1
but is essential for the main result (2.3).
Theorem 2. If (Xi) satisfies conditione then (2.2) is true and
~;: [
H(t) ..
I~
1""1
2
]
= 11 2 - 1 +.l ; (fl . - m )
~l
fl1 1.1 1
i
12.
Proof:
Suppose again that Fn(X) ~ F(x) for all nand x.
Hj(t) is a non-decreasing function of both
= (G j _l
j
Then
and t.
+ Gj_l*K) * (Fj(t) - F(t».
I
t
The function
is a delayed renewal function.
Also, Fj(t)-F(t) is non-negative, of
bounded variation, Ll(O,oo), and approaches zero as t
-+
Thus Lenu:na 1
00.
applies and
2t
Hj ( t) -H j _l ( t)
2t
<
iii
~l t"1J. 2
(1 - ~lt)
'fhus for t ~ t
Hj(t) - Hj_l(t)
iii
:1
J
2t
+a
[F j ( u) - F(u)] du
o
-1
c
=
<
00
![Fj(U) - F(u)] du
o
for
=
00
•
c(lJ.l-m j )
1J. 1
t
>
1J.2
~l
•
13.
and
(2.4)
Since E (~l-mj) is convergent by C(ii), the sequence (Hn(t) - t/~l)
j
.
is uniformly convergent to (H( t) - t I~l) for t ~ .t
>
~2
b.
f~l, which
(2.3) follows on applying this result to (iii) of Lemma. 2.
proves (2.2).
The proof when Fn6c) ~ F(x) is similar.
It was mentioned earlier than an elementary renewal theorem
and a key renewal theorem have been established by Smith for general
aperiodic sequences (Xi)'
If (Xi) satisfies condition C, simple proofs
of these theorems can be obtained and for completeness we state without proof
Theorem 3:
If (Xi) satisfies condition C then
lim
=
t+oo
1
~l
J.
,
00
lim
t+oo q*H(t)
=
L
III
q(u) du,
o
where q(x) is a non-negative function satisfying the key renewal theorem
for K(t).
14.
13.
Some further properties of the renewal sequence
The first result is an upper bound for the renewal function
H(t), valid for large but finite values of the argument.
Lemma. 3:
If 00R(t) is the delayed
renewal function defined in (1.2)
.
00,
and if Vl
J
=
xdA(x) and V2
=
J
o
for t
> J.L 2 / ~
2
x dA(x) and both finite, then
0
1
R( t) _.::- :Ii
loLl-
[l~~
t]-l
..... 1 ,
Proof: _Take q(t) = 1 - B(l)(t) in Lemma. 1.
q*R(t) = R(t) and, for t > 1oL 2 /2J.L
1
~ +!...
loLl
loLl
Then
t
J[l -
A(u)] du
o
'
2t
~J
o
J -
t
1
[l-B(l)(Ul]dU
-J.L1J[o - A(u) ] du.
t
1
But v;:-
[1
o
~ A(u) ] du is a distribution function with finite
V
expectation - 2/;av •
Hence by the Chebyshev type inequality used in
1
the proof of Lemma. 1
1
t
-- J
f.L l
.
0
[1 - A( u) ]
'.
15·
I
The Lemma follows.
Note that the right hand side of (3.1) is a
decreasing function of t and has the same limit as the left hand side
as t ..
00.
Sharper upper bounds for H( t) - t l ll1 can be obtained if the
finiteness of higher moments is assumed, and one can obtain lower bounds
by similar arguments.
We do not pursue this question further at the
moment but state for sequences (Xi) that are bounded below
Theorem
4: If Fn(X)
~
F(x) for alln and x and if (Xi) satisfies
> 1l2/~
condition C then for t
,
1
.
J-1 {
t
[
11 2
H(t) - -;! 1 - -
1l1~1t
11 2
~ - 1 +
=
1
~llll
00
}.
E (1l1-mi)
i=l
11 2
+-
.I
~l t
e(t),
where
e(t)
[1 -
J
~
J~lt
l
,
00
n4
1
) =
x
2
dF1 (x)
o
Proof:
finite
Since Fn(x) ~F(x) and' 1JI 2 is as·sumed finite, each Xi has a·
seco~d moment,
and in particular
of Theorem 2 it was shown that for t
n41 )
is finite.
> 1l2/~1
112
J-
Hj ( t) - Hj -1( t);! [ 1 - 2IJ. t ·
1
l
(1l 1 - m)
Ill'
In the proof
16.
It follows that for these values of t
t
toL2
]-1 1
H (t) - ~l
- - t
[
n
toLl
at l
t
--.
n
-
~l
E (~l-mj). + Hl(t)
j=2
,
and hence
t
] -1 1
toL2
[
CIO
H(t) - -
~ 1 - o.,t.
But Hl(t)
= Fl(t)
applies.
(3.2) follows from (3.1) and (3.3) on rearranging terms.
.
~l
<+"1
E
~l jff2
(~l-mj) + Hl(t)
t
.
~l
+ F*Hi(~) is a delayed renewal function to which Lemma 3
Consider now the forward and backward delay' of the sequence (Xi)'
J
If N(t) denotes the maximum n for which
is true, the forward delay
~(t)
~(t)is
= SN(t)+l
defined as
- t •
That is, set) is the time from t until the next renewal occurs.
backward delay
~(t)
is the 'agel of the process at time t,
~(t)
If (Xi)
The
=t
- SN(t)'
is a renewal process with renewal function K(t) and forward delay
'let) it is known (Smith, 1958, page 246 and also Takac's discussion to the
paper) that
t + ET1 ( t)
= ~ 1 [ K( t) + 1 ] •
17.
Furthermore,
lim
E1l( t)
t+co
1J.2
= 2j.L 1
and
x
~: Pr[q(t) ~ xl
-- 1J.1
±...
J
[1 - F( u) ] du
•
o
The backward delay of a renewal process has the same limiting
distribution function as 'let),
Although its expectation does not have a
simple relation to the renewal function K:(t).
In the case of a sequence (Xi) we have (with the same notation
as before)
Theorem 5:
If 1J. 1 <
then
co
co
t+ E~(t)
=~
+
~ mj +l Gj(t) .
j=l
If (Xi) satisfies condition C then
~:
Pr[
~(t) ~
xl
=
~1
x
J
[1 - F(u) ] du
o
and
lim
t-+co
Proof:
Ef(t) = ~
1
(3.4) is a case of Wald1 s .equation for independent but not
identically distributed random variables;
~(t) = j~O E [
•
get)
I
j~ (~oj."'2 +
Sj < t " Sj+l ] Pr [Sj < t " Sj+l ]
... + mj +1 - t) [Gj(t) - Gj+l(t) ] ,
(3.5)
18.
where Go(t) is the Reaviside function U(t).
A simple manipulation
yields
00
EHt) = ~l (R(t) + 1] - t - 1: (~l - mj +l ) Gj(t)
j=o
.
which is the same as (3.4).
An appeal to Theorem 2 shows that
lim
when condition C is satisfied t :+00 Eg(t) has the required value.
To prove (3.5) proceed as follows.
For fixed n let
S (t)be the forward delay of the sequence (XiI), whose renewal
n
function is Hn(t).
Then
00
Pr (t < Sj ~ t+x < Sj+l]
1:
j=l
where Sj =
Xl + X2
+ ..• +
n-l
Pr(Sn(t) ~ x] = 1:
j=l
Xj'
It follows that
t+x
J
(l-Fj+l(t+x-u)] dGj(U) +
t
..
~j +;1 - F(t+x-u)]
j=
=
d (G
t
n
*F*(j-n~u)J
I
..
t-x
(F(t+x-u) - Fj+l(t+x-U)]dGj(u) +
n~lJ
j=l
t
t+x
+
J
(1 - F(t+X-U)]dHn(U)
t
For fixed n the sum on the right hand side has the limit zero as
t
+
00.
The function
q(y) = 1 - Fey)
,
t
~
Y
~
t+x ,
!
=0
(
otherwise
(3.7)
19.
satisfies the conditions of the key renewal theorem.for H (t),
n
and hence
t+x
lim Pr [~ (t) ~ x)
n
t+oo
=
lim
t+oo
J
[1 - F(t+x-u») dHn(p)
t
x
= l...
1!1
J
[1 - F(u») duo
o
We have then that
x
lim lim Pr [~ (t) ~ xl
11+00
n
t+oo
= !I!l
J
[1 - F(u») du,
o
and (3.5) is established if we can justify the interchange of limits.
Note that the uniform convergence of H (t) to H(t)
n
for sufficiently
large t does net necessarily imply uniform convergence of (3.7).
Suppose first that F (x) ~ F(x) for all nand
X.
n
Then
wn(t,x) = Pr[~ n (t) ~ x) - Pr [~ n- l'(t) ~ xl
t+x
t+x
=J
[1 - F(t+x-u)l d [Hn (u)-Hn-.....
,(u)]
t
-J
[Fn(t+x-u) -F( t+x-u) dGn _1 (u)
t
The first member on mhe right hand side is dominated by
for t
> 1!2/~.
Since Fn(Y) - F(y) is non.negative, the absolute
1
value of the second term on the right is dominated by
20.
t+x
J
[Fn(t+x-u) - F(t+x-u)] du
~
III - mn
t
Recalling that Z (1l1-mn) is
for large enough t, uniformly in x.
n
convergent we can repeat the argument used in Theorem 2 to justify
intercblnging limits in (3.8).
The proof when Fn (x)
<
F(x) is
<=
similar.
The corresponding result for the backward delay is
x
;:: Pr [~(t) ~
xl =
1
III
J
[1 - F(u)] duo
(3.9)
o
The proof is based on the same principle as the proof of (3.5), but
is slightly longer because of the fact that ~(t) is bounded above by t,
the distribution function Pr [~(t) ~ x] having a positive jump into
the value unity when x
= t.
We do not give the proof of (3.9) .
Another quantity of interest is the variance of the number
of renewals in { 0, t].
To date we have had no success in studying this
aspect of the sequence (Xi)'
21.
e4•
Examples
The following example illustrates the point made in
second moments ~
~2
that the
are not required to converge for Theorem 2 to hold.
For any 5> 0 and n
= 2,3, ...
suppose the sequence (Xn ) is
defined by the distribution functions
, 0<
X
< 1,
.. i - :.1+a
n
.
.. 1 -
2
:3ii'
n
.. 1
2
"n :; x
The sequence bas the following properties:
If'J?-..
~2 - ~
+ n~ -1
n
n
'
00
E (mn - !-Ll) converges"
n=2
F(x) is the two point distribution function with equal jumps at 1 andJ2,
and whose mean is !-L l and second moment !-L 2 =
3/2
.
22.
Since Fn (x) ~ F(x) for all n and x) Theorem 2 holds and
in fact
H(t)
-
2t
1+.[2-
.
-2.4..2
(1+ ../2)2
" 1
+ 2 t .............
n=2 n 3...a
at
- (1+ ./2) n",,2
At
I:
1
~
n
However)
if 0<5 < 1)
at
l~
11+'00
2.+
2
me2n =
2
~
2
so that only for 8
>1
if 5
=1
)
if 5 > 1)
does E~ .. 1J2.
It may appear from the result of Theorem 2 that the convergence
of I: (1J1-mi) is essential for a second renewal theorem to hold.
This
i
seems a plausible induction from the fact that if only a finite number of
the Xi differ (the rest being identically distributed) then indeed a
second renewal theorem takes the form given by Lemma 2(iii).
On the other
hand the boundedness condition e(i) does appear too restrictive and one
suspects that it can be weakened considerably.
The next example shows that
a second renewal theorem exists for a sequence (Xi) for which condition e
is not fulfilled in any respect.
In fact) this example is such that it is
difficult to find any distribution function F(x) to utilise the truncation
method adopted in
~ 2•
Suppose (Yt ) and {Zt) are two independent renewal processes with
Pr [ Yi
=i
x
J
= A(x),
Pr [Zi
=i
x ]
= Hex)
,
and first two moments (assumed finite)
Form a. renewal sequence {Xi) by taking
if i is odd,
if i is even.
If
Pr [Yi + Zi :; x ]
= A*B(x) = C(x)
and
then for j= 1,2, ••• ,
G _ (x) = A*C*( j-l)(x),
2j l
00
The renewal function H(t)
= E Gj(t) is absolutely convergent so we can
11=1
rearrange terms to write
24.
Bo(t) is a delayed renewal function consisting of the odd terms in
the sum EGj( t) and satisfies the integral equation
j
He(t) consists of the even tel'1l1s of ~Gj(t) and satisfies
He (t) = C(t) + C*H(t)
•
e
If
00
Vl =
J
xdC(x) = a l + bl '
o
and
00
"2
=
J
2
x dC(x)
= a 2 + 2a1b1 + b2
'
o
then
H (t)
_. t
0
-l>
Vl
He (t)
-
t
-l>
- V-1
,
V2
2\12
-
.
-
V1
a
V
2
2
2V
1
_
1
1
1
The second renewal theorem for H(t) is then
Het) - V
2t.
1
= [H0 (t) _:L.
V J + [.
1
He (t) _:L
V1
J.
that is,
(4.1)
If one takes F(x)
= [ A(x) + B(x) ] 2-1 , so that
J
GO
III
=
xdF(x)
= (al + bl )2-1
o
J
GO
11 2
=
2
x dF(x)
= (a2 + b 2 )2-l,
and assumes that the method of t2 is valid, the result is
2t
I(t) - - -
(4.2)
only if b l = ale
It is in fact not ,hard to
show for general a l and b l and the above choice of F(x) that the
(4.2) agrees with
(4.1)
interchange of limits (2.2) is not valid.
Furthermore there seem~ to
be no obvious choice of a distribution function F(x) with the required first
two moments that will yield (4.1) by the truncation method.
If F(x) is
+
b
_7
-1
1
'chosen so that its first moment is III III
2 . l III 2 v, the truncated
.
£a
renewal functions (Hn(t») have the following property:
H
_ (t) -
2n l
2t
~2
---to -
Vl
vi
..
1
+
(bl-al )
,
(bl+al )
(4.4)
It is apparent that not even Cesaro limits of the right hand sides of
(4.3)
and (4.4) will yield the required result.
The second example is a special case of a semi-Markov renewal
sequence.
Suppose there is given a matrix of transition probabilities
26.
(Pij), i,j
= 1,2, •••
, 0 ~ Pij ~ 1,
~ Pij
=1
for every i;
an initial
probability distribution (a j ); and a renewal sequence {Yilt
We form a
new renewal sequence (Xi) by the following rule: if at any instant the
current 'lifetime' is distributed as Yi then the next 'lifetime' will be
Thus (Xn ) is described by
distributed as Yj with probability Pij'
,
The example above results if a l
= 1,
P12
n
= 2,3, ..• ,
= P21 = 1
.
, P
tj
=0
otherwise.
Conditions ensuring a second renewal theorem for these sequences are not
yet known~
However, the following is not without interest.
Let
Pi(t) be the probability that at time t a lifetime distributed as Yi
is in progress and suppose
~:: Pi(t) = Pi' In the example, it is not
hard to show that
Thus
1
(4.1)
can be rewritten as
.
given renewal sequence is, of course, a semi-Markov sequence ''lith
Pii+l=l, 1 = 1,2,... • Sane results for the general case are available
and will be discussed elsewhere.
'Any
The maximum of the queue
The renewal theorems developed in
GI/M/l.
~~1-3
27.
will now be applied to a
problem in the theory of queues, namely, given that the system is not ergodic
how fast does the maximum size of the queue grow?
We consider the single
server queueing process commonly characterised by the symbols GI/M/l.
The intervals between the arrival of successive customers are of 'General
Independent' type, that is, constitute a renewal process, the common distribution
function being, say, A(x) with
J
00'
A(O+)
=0
and a l
=
xdA(x) <
00.
o
In most of what follows we will also assume that the second moment
J
00
a2
=
2
x dA(x) is finite.
The symbol 'M' indicates that the service times
of cus~omers are independently and identically distributed with common
negative exponential distribution, say
=
c( x)
1 - e~x •
Also, the sequence of interarriva1 times is assumed to be independent of
In general the queue discipline must also be specified,
the service times.
but this is not relevant to the present work.
One further point is that we
are here dealing with the' reflecting barrier ' situatlon, the server.:remaining
idle until the next customer arrives so that zero queue length constitutes
a reflecting barrier.
Let Q(t)
= 0,1,2, •••
denote the length of the queue at time t
(including the customer in service, if any), and let
M( t)
=
max
O<'1"~t
Q
('!)
28.
be the largest length attained by the queue in CO,t] •
Q( 0)
=
Suppose that
°
so that successive maxima are attained when the queue first reaches the
lengths 1,2,3, .•••
Let Tij denote the
size i to queue size j
the value i), j
passage time from queue
(measured from the instant queue size first attained
= itl,i+2,
Furthermore, since A(O+)
Then, given Q(O)
fi~st
= 0,1, •••
•••• , i
, and write
= 0,
= 0,
and
00
EM( t)
= .r.
F0 j ( t)
J=l
The last expression shows that EM(t) is the renewal function of the
sequence
(T
i
i+l)' i
= 0,1,2, ••• ,
EM(t) is given by the results of
so that the asymptotic behaviour of
ss2-3if'it can be shown that (Ti i+l )
satisfies condition C.
Firstly note that the assumption of negative exponential
service times ensures the mutual independence of the Tii +l •
For other
distributions of service time this independence would not hold and it
is for this reason that the present remarks apply only to the queue
GI/M/l.
only.
The restriction to one server is for convenience in calculation
If explicit expressions for the FOj(t) are available then M(t)
can be studied directly.
However, this seems to be the case only
when the queueing process is also a birth and death process and even then
direct calculations can be very lengthy.
The first result we require
is
Lemma
~:
4:
The sequence (Fi_li(x») is monotone non-increasing for all x.
For fixed i, first passage from queue length i to queue length i+l
can be achieved in the following mutually exclusive ways.
If k
= 0,1,2, ••• ,i,
service completions are achieved before the next arrival occurs then,
consequent on this arrival, we must have first passage from i-k+l to i+l.
Suppose this next arrival occurs at time u, the probability of which is
CA(u) •
The probability of k service completions in (O,u ] is
if k
i-l
1 -
'='
LA
r=o
Thus
x
e
.
i.J.U
!" .. \ r
~
i-l
=J
e i.J.u CA( u) + ~
r=l
o
i-l
~
o
r=o
e i.J.u
rl
= 0,1, ••• ,i-l,
if k = i.
x
J
e i.J.u (Ilu)r
r I.
( ) .:lA( )
Fi+l-r i+l x-u uti. u
o
(5.1)
30.
The Lemma is proved if we can show
for all x, and we do this by induction on i.
We will assume that the
time from the origin until the first arrival occurs has the distribution
function Fo1{X) = A{x).
Now
x
=
J
x
e"1J.
u
dA{u} +
J
[l_e"1J. u ] F {x-u} dA{u}
12
o
o
~
x
J
[e"1J. u + 1_e "1J. u ] dA{u}
= A{x}.
o
Assume that
,
i-2
j
= 2,3, ... ,i-1.
{} r
Writing 1- .E e "1J. u~~
r=o
find from {5.1} on rearranging te~s that
i-1JX
F. l' (x) - Fi-l+l(x) =.E
e "1J.u
~- ~
~
r=l
~[
u
F . . (x-u) -F '+1
r1
~-r~
.
"+l(x-u)
~-r ~
o
i-l
()~[
~
;..L up.
t!:
F (
) F
(
) ] M()
- r:o e
r I.
l~ x-u - 1i+l x-u
\l
independence of the (T _ ),
i li
we
have
•
]
M( u)
31.
J
the inequality following from the induction hypothesis and the fact that
Fli(x) ~ F i _r +l i(x) for r
= 1,2, ••• ,i-l.
Hence for all x,
X
=
J
(l_e'1i
U
>[ Fi - li
- F11+1 ] *Fli(x-u)dA(u)
(5.2)
o
But the right hand side of (5.2) is less in absolute value than
IFi_li(x) - Fii+l(X)I
so that (5.2) can hold for all X ~ 0 only when
Fi_li(x) - Fii+l(x) ~ O.
A consequence of Lemma 4 is that the expectations E T +l = mii +l
ii
form a monotone non-decreasing sequence.
The next result concerns these
quantities.
Note that the traffic intensity of the queue is
(~al)-l.
~2.
If Ilal < 1 the queue is not ergodic and an equilibriwn distribution of
queue length does not exist.
Lemma
5:
Iflla l < 1, then
lim
<
i~ mi i +l
00
If in addition, a 2
=
J
2
x dA(x)
<
00
,
then
o
00
(
1:
i=o
1
)
al
- mii+
converges.
a
l
"1J. l
lim
Since (mii+l ) is a non-decreasing sequence, i+oo mi i+l certainly
exists, although it may be infinite. (5.~) is established by the following
Proof:
argument.
For a fixed queue length i, the probability of k service
completions in an interarrival interval is, for k = O,l, ..• ,i-l,
J
00
(lk
=
e "1J.u
(it~) k
dA( u) ,
o
and when k = i, is
-
i-l
1:
k=o
e "1J.u
-U-
·(lJ.u)k ]
i-l
dA(u) = 1 - 1: (lk •
k=o
If in fact k service completions eventuate before the first arrival after
the queue reaches length i, then
Unconditionally,
i-t
= a l + :tz· mi _k+l i+l a k +
k=l
~
+ I: a
~ i l k=i k'
nl..
i
2 3
= , , •.. ,
~
(5.5)
•
But
mi _k+l i+l = mi _k+l i-k+2 + mi _u+2 i-k+3 + .•• + mi i+l
and
~i+l
(5.4)
thus yields the inequality
;a;
1-1
~
a l + mi1 +1 I: ka k + mi i+l I:
k=l
k=i
:ka
k •
Noting that
~
I:
k=l
ka
k
we find
=
~
I: k
k=l
J~
o
(.
)k
e 1J.u~.!.U¥
kl·
J
~
dA( u)
=
o
IJ.UdA(u)
=
so that, 1f
11m
1~
~al
m11+1
=
Suppose m1i+l -+
< 1,
t.
00.
Then the generating :£'unct10n
1
cio
(t -
z
E
ID
i=o
t-I
converges for at least
IG( z) I ~ I
From
<
11m sup m
1-+00
11+1
U +l )
= G(z)
< 1, s1nce
00
E z1 (t - al' I = (t - al' 1(1 - z) -11 •
1=0
(5.4) - (5.6), calculations show that
t
G( z)
= r:z
alA*(~;..L~)
- A*(~ ;..Lz) -z '
where
A*(~ ;u) =
00
J
(5.7)
e -u(~;..L z) dA( u)
o
By virtue of the convergence of
t
-m +l to zero, a standard Abelian argument
ii
yields
o. = lim
(t - mii+l ) = 11m
(l-z) G(z)
1-+''0
z+l-
which proves
that for
(5.3). To prove the second part of the lemma we recognise
~al
< 1,
r
00
=
Q( z)
I:
r=o
(l-z)A*(Il-j.lz)(l-j.lal )
z qr =
A*(Il-j.lz) - z
( 5.8)
is the generating function of an honest probability distribution {qr}
(see below).
q-
If a
2
is finite then
00
= r:
r=l
rq
r
=
00
00
I:
I:
qk <
00
,
r=l k=r
implying that the generating function
1 - Q(z)
1 -z
converges at z = 1.
However,
[
1 - Q(z) ]
l-z
.
(5.9)
Thus
is finite and has the value
A further deduction from (5.9) is the formu:J,.a
mii 1
+
=
valid when Ilal < 1.
a
31
1
-j.la l
i-l
I:
r=o
q,
r
(5.11)
:;6.
The probability distribution {q } defined by (5.8) will be
I'
recognised as the equilibrium distribution of queue length of the queue
M/G/l (Kendall, 1951, equation (15) ) in which the service distribution
function is A(x) and arrival provess is poisson with parameter~.
This queue is the dual of the system GI/M/l with which we are concerned,
the roles of interarrival and service distributions being interchanged.
That the results of the lemma are not surprising can be seen by the following
considerations.
In random walk terminology the T +l are first passage
ii
times from i to i+l with paths that are step functions on the integers
0,1, ••• ,i+l, the barrier at 0 being reflecting and the barrier at i+l absorbing.
'Upward' jumps occur according to a renewal process with distribution
function A(~)and 'downward' jumps according to a Poisson process with
parameter f.L •
If we reverse the roles of the distribution functions
A(x) and l_e~x , place an absorbing barrier at 0 and a reflecting one at
i+l, and commence the walk in state 1, then T + becomes the first passage
l1 1
from state 1 to the origin of the dual process.
This latter formulation
shows that T + is also the length of a busy period of the finite capacity
ii l
M/G/l queue in which the number of waiting c~stomers is limited to i .
limit al(l~al)-l is well known as the expected length of a busy period
of the unlimited M/G/l queue.
The
s6.
The queue GIIMll (continued)
Let R(t) = EM(t) be the expectation of maximum queue length in
(0, t] •
~eorem
If ~al < 1, A(x) is not e lattice distribution function, and a 2
is finite, then
6:
(6.1)
11m { Ret)
t+oo
Proof:
Lemma 4 implies that
Fii+l(x)~ F(x)
as i+
be shown thatF(x) is non-lattice with mean ~l
00.
Provided it can
= al(l~al)-l
and finite
second moment, Lemma 5 ensures the satisfaction of condition C.
First note
that for 11(e)/ ~ 1 and Res ~ 0
J
00
A*(ei1!~1(e»
=;:
e-x( Si1!~1( s» dA(x)
o
=
~
j=o
J
00
1 j (s)
e-x (Si1!)
o
j
.u:&.
dA(x)
jl .
00
=
j
E 1 (S) aj(s).
j=o
Taking Laplace transforms of (5.1) we have
i-l
00
!!i+l(S) = a (e) + E ares) F!-i-r i+l(S) + Fti+l(S) E a (s).
r=l
r=i r
~s a trial solution of
(6.3) take
(6.2)
(6.3)
38
Then
1 i (8)
= ao(s)
i-l
r
+ t ares) 1 i (S)
r=l
.
i
1 (S)
i
+
00
(6.4)
t a (a).
r=i r
~ 1 1 (S) = 1(e) exists for Re s ~ 0 and
Since (Fii+l(x») has a limit,
is the Laplace-Stieltjes transtorm:of F(x).
Furthermore the series
00
t a r (s) is convergent.
r=o
from (6.2),
Taking the limit as i
00
in
(6.4) shows that,
r
00
1(s)
+
= ao(s) + t ares) 1(S)
r=l
= A*( s-f1.L ;.q( s) ) •
The functional equation (6; 5) has been extendvely studied (see, for example,
~
Takacs, 1961, Lemma 1) and it is known to have a unique root 1
within the unit circle.
= F*( s)
The inverse transform, F(x), of F*(s) 1s the
distribution function of the duration of a bUSy period of the queue M/G/l
which is d~l to the GI/M/l system under consideration.
F(oo)
=1
If lJ.al .< 1 then
and
00
1J. 1 =
J
xdF(x)
=
(6.6) .
o
If in addition, a 2 is finite, then
00
1J. 2 =
J
2
x dF(x)
(6.7)
o
We have only to show that F(x)is non-lattice under the hypothesis that A(x)
is non-lattice.
But this is easily proved if~we rewrite (6.5) in terms of
Fourier transfOrJms and appeal to the criterion for lattice distributions
(Lukacs,
1960, page 25).
The limit
(6.1) folloW's on substituting the
above values of t.L 1 and t.L 2 in (2.3) and using (5 .10) •
Further deductions are, from (3.2),
(6.8)
a
for
2
2
2al (l;.Lal )
t>
vex) -
~
and from (3.5),
'
Pr[ let) ~ x ] - (1;.a1) ~ [1 - F(U)]
al
du
(6.9)
0
~(t) is the time one must wait, att, before the queue attains its
next max:J,mum.
J
00
ljf*( s) =
e -sx dljf(x) 1s
o
(l-j.!alH 1.-1' (8) J
ljf*(.s)
:II
-~-----
where I'(s) is the smallest root of
00
J
o
x dljf(x)
=
(6.5).
(6.10)
40.
J
2
00
~a2 + a (1-j.J.a1 ) ,
3
2
x d\lT(x)
J
3a (l-j.J.a ) 4
l
l
=
o
'
00
provided
a
3
=
x3dA( x) is finite.
o
The distribution function A(x) has so far been assumed to be
non-lattice.
'Queueing systems of interest are often ones in which A(x)
is in fact lattice, a case of particular1m:,portance being that of
deterministic input;
A(x) = {
o
1
Suppose now that A(x) is lattice and without loss of generality suppose
its possible points of increase occur only at the positive integers.
Successive queue maxima can then only be attained at times 1,2, ••• ,
The renewal sequence (T i i+l) is
and each F i i+l(X) is lattice.
discrete and it is clear that theorems corresponding to the above can
be developed in'this case.
The explicit formulae will differ in one
important respect due to the fact that if K(t) is now the renewal function
of a discrete process (Yi ), its second renewal theorem takes the form
(Feller, 1949, page 103)
1J.
1
2
~2
(6.11).
- 1
1
The time parameter t now takes only the values 0)1,2, •••.
The final
result (6.1) must be modified by the subtraction of (~l)-l from the
right hand side, otherwise the argument is essentially unchanged.
have
We
41.
If ~a1 < 1, A(x) is a lattice distribution function
Theorem 7.
with unit period, and a 2 is finite, then
;;: { H(t)
_ t( 1-~a1)j =:
(6.12)
a1
In the special case n/M/1, where
=
A(x)
, x <1 ,
, x
~
(6.13)
1 ,
(6.12) reduces to
- (1"1J.)'
H( t) - t(i~).. l~
(6.14)
By way of comparison, for the queue M/M/1 in which
A(x)
1 - e- x ,
=
(6.1) yields
H( t) -
t(l~) ..
( 6.15)
r"ii .
(6.16)
Intermediate between these two cases is the Ek/M/1 queue, k=l,2""J with
x
kk u k - 1 du ,
(6.17)
A(x) =:
e- ku
(k-1) I
o
J
a
2
=:
1 +. k-1
(6.17) reduces to (6.15) when k
=:
For this choice of A(x)
is
(6.1)
1 and approaches (6.13) as k
+
00
•
42.
-....IL
-ljl -
(ljl)
2
~
( 6.18)
+~
The limit on the right side of (6.18) is equal to the limit in (6.16)
when k =/1 and decreases as k increases.
However the former does not
approach the corresponding limit in (6,14) as k
for this is, of course, that the sequence
+ ~
The reason
(Xi) to which (6.14)
relates is lattice whereas that corresponding to (6.18) is non lattice
for each finite k.
An important problem still unsolved concerns the
asymptotic behaviour of R(t) when ~al = 1.
original GIIMll queue and its
MIGll
In this case both the
dual are null recurrent and a
stationary distribution of queue length does not exist for either
system.
The first passage distribution functions Fi i+l(x) still
have the limit F(x) defined in.(6.5), anu. although F(x) is honest it
has infinite expectation.
McGregor, 1961)
The only known result (Karlin and
when ~al = 1 is for the Poisson queue
H(t) "
t~
MIMll ,
References
Feller, W. (1949). 'Fluctuation theory of recurrent events'.
Trans. Amer. Math. Soc., §I, 98-119.
Karlin, S., and McGregor, J. (+961). 'Occupation time laws for birth and
death processes' • Fourth Berkeley Symposium, g, 249-272.
Kendall, D.G. (1951). 'Some problems in the theory of q~eues'. J.Roy.
Statist.Sbe. (Bl, !i, 151-185.
Lukacs, E. (1960).
'Characteristic functions'.
Hafner, New York.
Smith, W. L. (1954). 'Asymptotic renewal theorems'.
Edinburgh, (A), §!t, 9..48.
Proe. Roy. Soc.
Smith, W.L. (1958). 'Renewal theory and its ramifications'.
(B), gQ, 243-302.
J.Roy.Statist.Soc.
Smith, W.L. (1961). 'On some general renewal theorems for nonidentically
distributed variables' • Fourth Berkeley Symposium, g, 467-514.
Smith, W.L. (1963). 'On the elementary renewal theorem for nonidentically
distributed variables' • Institute of Statistics, University of
North Carolina Mimeo Series No. 352.
Ta~cs, L.
(1961). 'The transient behaviour of a single server queueing
process -with a Poisson input-';.
Fourth Berkeley Symposium, g,
535-567.
Titchnarsh.. lll. c. (1960).
Oxford" London.·
.,
,,
'The theory of :fUnctions I .
2nd edition.