^"i%i
HD28
.M414
no.
35/9-
^3
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Fluid and Diffusion Approximations of a
Two-Station Mixed Queueing Network
Vien Nguyen
WP#
3519-93-MSA
January 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Fluid and Diffusion Approximations of a
Two-Station Mixed Queueing Network
Vien Nguyen
WP#
3519-93-MSA
January 1993
^^-^^4
1993
Fluid and Diffusion Approximations
of
A
Two-Station Mixed Queueing Network
Vien Nguyen
Sloan School of Management, M.I.T., Cambridge,
MA
02139
Abstract
The
subject of this paper
is
a two-station mixed queueing network with two customer types:
"Open" customers enter the network
at station
1
and depart the system
after receiving service.
Meanwhile, a fixed number of "closed" customers circulate between stations
1
and
2 indefinitely.
Such a mixed queueing network model can represent a single-stage production system that
services both make-to-order
for this
network under the
first-in-first-out service
of the workload process at station
This result
is
We present fluid and diffusion limits
discipline. We find that the heavy traffic limit
Brownian motion (RBM) on a finite interval.
and make-to-stock customers.
1
is
a reflected
surprising in light of the behavior of the original mixed network model, in which
the workload at station
KEYWORDS: mixed
1
need not be bounded.
queueing networks, make-to-order production, make-to-stock production,
fusion approximation, reflected
Brownian motion, performance
analysis.
Contents:
2.
A
A
3.
The System Processes
4.
The Limit Theorems
5.
Proof of the Fluid Approximation
6.
Proof of the Diffusion Approximation
7.
Refining the Brownian Approximation
8.
Numerical Examples
9.
Appendix
1.
Two-Station Mixed Queueing Network
Make-to-order/Make-to-stock Production System
-
References
January. 1993
dif-
'Open" customers
c'
X
la
m,.
C,
'Closed" customers
N
Figure
is
A Two-Station Mixed Queueing Network
A Two- Station Mixed Queueing Network
1
This paper
1:
devoted to the analysis of the network pictured
stations serving both "open" and "closed" customers.
1
and depart the network
stations
1
and 2
departures, the
by
.V the
number
Open customers
after being served. Closed customers,
[3],
We
mi and m2
will
on the other hand, circulate between
to be the
mean
and
1
let
One
traffic intensities at
stations
1
pi
=
Amo +
P2
=
I.
expects that the
is,
approximations of
We
and
/.
denote
be the
Pi
=
Amo +
P2
=
Qr^-
traffic intensity at station 2
Oy
—
1
(1.3)-( 1.4)
as A^
when
In addition, let ('o(0
mean
service time of
1
and
is
l/m2. Consequently, we
repsectively, to be
2,
mi
m2
:i
is
given by a'^/m2. where
traffic intensities are
1)
oy
is
given by
—^mi
(13)
m2
(1.-1)
approaches
1
as the
number of
closed customers
—
oo and accordingly, equations (1.1)-(1.2) can
N
is
ht^
taken as
large.
are interested in the process 1^1(0. defined to be the total
at time
We
:i.2)
than one. Moreover, the actual
strictly less
increases; that
time
mj < m2- From Chen and Mandelbaum
For each finite N, the actual throughput rate of closed customers
1
mo
one can verify that the relative throughput rate of closed customers
number
in
service times of closed customers at stations
aissume throughout this paper that
can define the "relative"
a
enter the network at station
of closed customers in the system.
these customers. Set
respectively.
which consists of two
of closed customers in the network remains constant
Let A be the arrival rate of open customers at station
2,
1.
Because there are no external arrivals of closed customers nor are there
for service.
number
Figure
in
amount
of work found at station
and Ui(t) be that part of the workload corresponding
1
to
open and
closed customers, respectively.
We
show
will
in this
paper that when station
is
1
nearly saturated,
namely
Pj
= Amo
when the
the following approximation holds
is
a reflected
m2
»s 1,
closed customer population
«
Here, l^i'()
H
is
A'^
large:
iW^{-),Uoi-)M:{-)).
Brownian motion on the
interval
[0, 1712]
9-
=
N(\mo +
— -l)
CT-
=
Xml(cl +
ct)
with
(1.5)
drift
and variance parameters
(1.6)
2
+ '^ic\ +
c^^.
(1.7)
Moreover, the partial workloads corresponding to open and closed customers
live in fixed
propor-
tions according to
t^oC)
These results seem counterintuitive
=
(^ - 1)
in light
where the workload process at station
1
Our
if
goal in this paper
one
is
= AmotVrCO-
of the behavior of the original mixed queueing network,
On
the other hand, neither (1.5) nor (1.8) should be
familiar with certain properties of queueing networks in heavy traffic.
to prove a heavy traffic limit
is
(1.8)
need not be bounded above and the partial workloads are
not subject to deterministic relationships.
completely surprising
^i'(*)
theorem that
justifies the
approximation
stated in (1-5).
For the reader to better understand the contributions of this paper,
results within the context of diffusion
Queueing networks are said
it
is
helpful to cast our
approximations of open and closed queueing network models.
to be "multiclass"
if
the service time distribution as well as the routing
of customers at each station can depend on the class designation of the customer.
class" networks,
"single-
customers at each station are indistinguishable, meaning their service times are
identically distributed
and
workload processes
is
all
customers at each station follow the same routing mechanism. For
Markovian routing, Reiman
single-class networks with
a reflected
Brownian motion
in
[19]
proved that the diffusion limit of the
the positive orthant.
a similar result for multiclass networks in which the routing
the
In
same work, Peterson
also
showed that the
is
class specific
by fixed proportions of the overall workload at that station.
essentially requires that
all
Peterson
[18]
proved
deterministic and feedforward.
In
workloads at each station are given
The feedforward
structure, which
customers travel from lower numbered stations to higher numbered
ones, turns out to be an important restriction.
The
generalization of Petersons work to include
routing with feedback has proved to be quite difficult and the source of the difficulties contains
deep and subtle theoretical
issues. In the case of a multiclass single-queue
network with feedback.
Reiman
[20]
was able to prove a theorem
approximation of the workload process
to justify the
by a one-dimensional reflected Brownian motion, and the proof due to Reiman was subsequently
simplified by Dai
(12, 13]
insights
drawn from these
results.
Harrison and Nguyen
The Brownian system model proposed by Harrison and Nguyen
Brownian motion on the nonnegative orthant, and
reflected
if
With
[7].
proposed a Brownian system model to approximate a general multiclciss queueing network
with feedback.
RBM
and Kurtz
was
any queueing network.
well defined for
it
is.
essence, a
in
was generally thought that such an
Nguyen
Indeed, Dai and
[8]
have shown that
the vector of wokload processes were to converge to any continuous limit, then that limit must
be the Brownian system model described
in
[12,
Wang
However, an example by Dai and
13].
[9]
conclusively verified that there exist queueing networks for which Harrison and .Nguyen's Brownian
approximation do not
irregularities
[21],
much
Whitt
exist.
[23]
provided another example that further illuminated the
and nonconvergence of the workload process. Due
progress has been
made toward
to the
work of Taylor and Williams
identifying sufficient and necessary conditions for the
RBM's. However, the convergence of open multiclass queueing networks
existence and uniqueness of
with feedback remains a wide open question.
Similar progress has been
network models.
the area of diffusion approximations for closed queueing
workload process
in
4]
have proved
fluid
such a network
is
an
RBM
and diffusion
[6]
all
theorems
limit
particular, the diffusion
Extension of Chen
difficulties as in the
for a closed
[21],
open
manufacturing
job classes which are served at a station share a
From Taylor and Williams
RBM
same
propose a diffusion approximation
system, but with the restriction that
service time distribution.
In
on a simplex.
to the multiclass networks involves the
Dai and Harrison
weak sense) a unique
[3,
queueing networks with .Markovian routing.
and Mandelbaum's work
counterpart.
in
Chen and Mandelbaum
for single class closed
limit of the
made
common
one can verify that there exists
(in
a
corresponding to the proposed approximation. However, there are no
proofs to verify that the workload processes
in fact
converge to the said
RBM.
one may suspect that mixed queueing network models, as a combination
In light of these results,
of open and closed queueing networks, will exhibit similar properties under the diffusion limits.
That
is,
the diffusion limit of the workload process can be cast in the form of a reflected Brownian
motion, and
on an
in
interval.
particular, the workload
due to closed customers can be expressed as an
Moreover, one can conclude from the theory of multiclass queueing networks that
the class-specific workload proceses at each station live
processes due to open and closed customers at station
in fixed
I
proportions, so that the workload
is
Because station
substantially
networks.
more
One may
1
is
in
then
1
must be an
RBM
on a
finite
essence a multiclass station, however, the proof of this network
intricate than the corresponding proof of the single-class
open and closed
therefore view mixed queueing networks as an intermediate stepping stone
between the well understood single
models.
It
are deterministically related
follows from these observations that the workload process at station
interval.
RBM
class
networks and the more challenginR multiclass queueing
The remainder
of the paper
and proof, we discuss
is
Before we turn to problem formulation
organized as follows.
is
next section a make-to-order/make-to-stock production system that
in the
naturally modelled by the mixed queueing network under study. Section 3 defines the processes
that
we use
in
our analysis. Our main results are stated
theorems are then given
in Sections 5
mappings, which we discuss
These proofs
6.
rely
4,
and the proof of the
we discuss
in Section 7.
limit
on the properties of a certain pair of
Appendix. Our approximation scheme
in the
of the Brownian limit, which
and
Section
in
is
based on a refinement
Finally, Section 8 contains the results of
several numerical experiments.
We
end
this section with
product space of functions /
some
;
[0,oo)
on (0,oo). The space D''[0,oo)
—
*
D''[0,oo)
£
is
the r-dimensional
^^ that are right continuous on [0,oo) and have
endowed with the Skorohod topology
is
of processes in D'"[0,oo) and .Y
The space
technical preliminaries.
X"=^X
we write
D''[0,oo),
[2].
mean
to
left limits
For A"" a sequence
A'" converges to A' in
distribution.
All vectors will be envisioned as
vector whose components are
On
we
occasion,
column
The dimension
ones.
all
will also write e{t) to
We
vectors.
mean
of
the identity
e
to denote a
(column)
should always be clear from context.
e
map
confusion as to the appropriate interpretation of the letter
For/: [0,oo)^$R,
use the letter
e{t)
=
t.
Again, there should be no
e.
set
=
11/11,
sup
|/(s)|,
0<s<(
and
for a vector- valued function
=
f
{fi, f2,
lt
A
=
,
fr)'
'
[0,
>2)
—
3?'",
we
let
(ll/illt,...,IIMI.)'.
sequence of functions {/"} converges to a function / uniformly on compact sets
each
t
>
||/"
0,
—
that /(x")
D''[0,oo),
>
—
f{x)
f\\t
u.o.c.
we write
2
—
A
X^
—*
as n
—
oo.
>
Finally, for a
X
u.o.c
if
We
say that /
is
continuous at x
if
x"
^
A'
if
for
x u.o.c. implies
sequence of functions {-A"} on D'"[0,oo) and
almost surely, X"' converges to
(u.o.c.)
A' a process in
uniformly on compact
sets.
Make-to-order/Make-to-stock Production system
Production systems are typically categorized as "make-to-order" or "make-to-stock," corresponding
to the
two scenarios
for finished
in
which new jobs are triggered by customer orders or by replenishment orders
goods inventory, respectively.
In a make-to-order system, a
system each time a customer places an order.
A
either in the
over time.
in
form of orders waiting
is
released into the
make-to-stock system, on the other hand, maintains
a finished goods inventory from which customer
inventory triggers a job release
new job
demands
are filled.
Each order
fulfillment
from
the system; hence, the total
number
to be processed or as finished
goods inventory, does not change
of "jobs" in the system,
^^
o
Make-to-order
requests
o
Make-to-stock
requests
Inventory
Replenishment
orders
Figure
more often the
It is
2:
A
Make-to-order/Make-tostock Production System
case, however, that production
systems employ a combination of both make-
to-order and make-to-stock operations. Figure 2 shows such a system with a single processing stage.
We make
the cissumption that orders for make-to-stock products that cannot be
of inventory are simply lost (no backlog).
to
model
this system,
One can employ
the mixed queueing network
we use station
are serviced.
A
1.
Figure
1
1
naturally represents the production
model the finished goods inventory from which make-to-stork orders
2 to
service at station 2 signals that a make-to-stock request has been
turn triggers a replenishment order for station
to station
in
to lack
where make-to-order products are represented by open customers and rlosed
customers take the place of make- to-stock products. Station
center and
due
filled
In this case, a closed
1:
that
customer at
is,
.station
filled,
whirh
iii
a departure from station 2 then proceeds
1
represents an order for make-to-stork
products, whereas a closed customer at station 2 takes the form of a finished good.
We
note that station 2 only approximates the
In particular, consider a
During
The
order to be
it is
times.
[<i,<2]
process for make-to-stock pro<luris
during which the finished goods inventory
this time period, make-to-stock requests continue arriving with
first
order,
time interval
demand
—
clear that t'
That
"excess
filled is
life"
is,
the
first
<2
the
first
one to arrive
does not,
in
Denoting by
same
subject of this paper
d.
mterarrival
is
There
is
enipiv
iiiii>'s
the arrival time if ilus
distribution as other interarri\.ii
not significant
in
traffic limit (see Iglehart
are, of course, several
an "Order-up-to" policy
in
the sense that both systems
and Whitt
[16]).
other policies that should be considered
wiiirh a batcii of .V
—
ii
in
\<\
the behavior of such a system as the workstation operates umi-r
example, make-to-order products (or similary, make-to-stock products)
One can employ
t'
i
service time of a busy period at station 2 should be characterized
distribution. However, this difference
first-in-first-out policy.
t2-
general, have the
can be shown to converge to the same heavy
The
after
i
is
may
i
hf
I't
receive higher pn-'niv
make-to-stock
recin-'-is
-ir-'
sent to the workstation whenever the inventory level n falls below
interesting option
is
to process the
In this example, the
requested.
are the
rate for make-to-stock products
P) be a probability space on which
and identically distributed
(i.i.d.)
random
variables {u{i),i
squared coefficients of variation (SCV) of u(i) and
designate open customers as customers of class
and closed customers
at station 2 as class
2.
sequence of open customers to be {X~^u{i),i
of the
and
open customer
i
at station
2 are defined as {mit;i(j),f
one interprets A as the
definitions,
of open customers at station
and
2 are
=
mi and m2,
Setting u(0)
=
0,
1.
time
let
is
server
divided
1
1,2,...}, {vk{i),i
0,
Vk{i), respectively.
closed customers at station
With
=
this categorization,
1,2,
.
.
.}
among
=
0, 1,2,
amount
we
=
and
c^.
arrival rate of
—
the
1
customers,
set the interarrival time
1,2,...}, respectively.
open customers and mo as the mean
mean
—
convenient to
and we denote by moi'o(0 the
1,2,...} and {m2V2{i),i
Similarly, the
time
servici^
1
With these
time
servict^
service time of closed customers at stations
u{0)
:
+
+
u(i)
<
1
1
as
Xt}.
(3 1)
t.
At station
the two customer types, and we denote by To(t) and 2i(<) the
=
amount
To{i)
+
1
.
this
nf time
T\{l)
)
be the counting process associated with class k service times.
= ma\{i >
1
and
Ai{t)
ib
c^
as class
1
of time that server j has spent working up to time
arrival processes for class
I4(t),
1,2,...}, k
respectively.
Skit)
Denote by
=
We denote by
We will find it
has devoted to open and closed customers, respectively. (Clearly, Bi{t)
Let Sk{t), k
The
=
define the arrival process for o'pen customers at station
Bj{t) be the
mi < m2-
Service time sequences for closed customers at stations
1.
AqU) = max{j >
Next,
given by l/m2.
are defined four independent sequences of independent
where these random variables are positive and have unit mean.
0, 1,2,
is
The System Processes
3
!F,
Another
which make-to-order products are
at
natural, in the context of this example, to consider only cases in which
Let (Q,
.
processing times for make-to-order and make-to-
demand
stock products, respectively. Finally, the
It is
mean
critical level n'
cyclical basis.
parameter A corresponds to the rate
mo and mi
Similarly,
two product types on
some
0, 1,2,
=
2
:
mfci>fc(0)
+
+
rrikVkii)
<
t).
(3 2)
customers, respectively, are then given by
S2{B2{t)),
the partial
A2it)
sums process
Vkit}
=
ir.i)
for class k service times,
= ^7nkVkii}.
:=1
Si{T,{t)).
'31)
Setting
Mk(t)
it
=
= mkVk{l) +
VkiAkit))
follows from (3-l)-(3.5) that Mif(t)
is
amount
the
have arrived to the associated station by time
Li{t)
then L,{t)
We
with
all
for station
those customers at station
i
at time
=
(3.6)
customer classes at station
ail
=
M2(t),
=
Q\{0)
i.
= N;
and (32(0)
that
W,{0)
=
I.
is,
the
Letting
defined to be the remaining service time associated
i,
t.
who
define
L2(t)
\fi(i),
process for
iii|)ut
(35)
closed customers at station 2 and no open customers in station
denote the workload process
all
we now
assume throughout the paper that Qo(0)
system starts with
W^,(<)
+
Mo(i)
the immediate workload
is
will
=
mkVk(Ak(t)),
of immediate work from riass k customers
If
t.
+
either
0,
queued or receiving
service,
we have
Wm = V2{Nl
(3.7)
+
(3.8)
and
W,(t)
Defining
I,(t)
=
/
—
=
W,(0)
L,(t)-B,(t).
B,{t) to be the cumulative idleness process at station
X,(t)
=
-
L,(t)
and
i
(3.9)
t
to be the workload netflow process, write (3.8) as
W,{t)
We
=
W,{0)
+
+
X,{t)
(3.10)
I,(t).
require the idleness processes satisfy the following properties:
The
first
statement
is
continuous and nondecreasing with
/,
is
/,
increases only at times
t
when
is
=
=
(3-11)
(3-12)
U.
a simple consequence of the properties of an idleness process, and the second
statement holds for any work-conserving system. That
when there
IV',(/)
/,(0)
is, it
states that the server remains idle only
no work to be processed. The vector processes
A', IV, I,
U.Q
are then dpfined in the
obvious manner.
It
remains to characterize the "allocation" processes T{t)
arrival
time of the customer currently
With FIFO
service discipline,
in service at station
1
if
=
(To{t),T\(t))'
W\{t)
>
and
.
set
Let
denote the
r](t)
ri(t)
=
t
otherwise.
we must have
Tk(t)
=
Mk(riU))
+
eik(f),
(313)
where
fifc(f ) is
eik(t)
=
amount
the
otherwise.
of service the current customer has received
The amount
are denoted by Uq and Ui
of work at station
is
of class k and
associated with open and closed customers
1
and are given by
respectively,
,
that task
if
Uo{t)
=
Mo{t)
-
Toit)
=
Mo{t)
-
-
eio(0
(3.14)
Uiit)
=
Miit)
-
Tr{t)
=
M,(t)
- Mi(q{t)) -
en(t)
(3.15)
Moiriit))
Next, define Qk(t) to be the queue length process associated with class k customers (including any
customer who may be
in service).
It
follows from the previous definitions that
Qo{t)
=
Aoit)
-
Ao{r]{t))
(3.16)
Qi(t)
=
Ai{t)
-
Arirjit))
(3.17)
Q2(<)
=
N + A2{t)
As we expect, Qi(f) + Q2(0 = Qii0) + Q2{0)
does not fluctuate
=
A2irj{t)).
(3.18)
^, so the number of closed customers
Furthermore, Q2(0
in time.
-
in the
network
completely determined by Qi(t). Finally, observe
is
that
r^(t)^t-\\\{ii{t))
where 62(0
'f
'^
W'lO =
and otherwise
currently occupying station
The
limit
is
(3.19)
equal to the remaining service time of the customer
theorems proved here apply to systems that
we
satisfy conditions of
and service times
terms of the basic sequences of unitized random variables {u{i)
=
To construct a sequence
0, 1, 2.
>
{A("',n
1},
{mj^ ,n >
1},
and service times are taken
the n
system, A'"'
is
fc
=
of networks
nii^
to be u'"'(f)
sequence
is
and m^.
set to be n, that
The convention
superscript "(")"
The
•
here
is
is,
traffic,"
and
:
;
>
network are defined
1},
{i'A:(')
:
'
>
1},
of positive constants
0,1,2. In the n"' system of the sequence, the interarrival times
=
u(!)/A'"' and v]^ (i)
the arrival rate of open customers and m)^
in place of A
for the
we further require sequences
= m]^
is
various customer designations. Define the relative traffic intensities pj
and
"heavy
require the construction of a "sequence of systems"
to be indexed by n. Recall that the interarrival times
fc
e2{t),
1.
in order to rigorously state these conditions,
in
+
Finally, the closed
we
define AT*"'
to denote a
For example, Aq
=
the
i'/ic(')'
mean
respectively.
For
service time of the
as in (1.1)-(1.2) using A*"'
customer population of each network
in
the
n.
parameter or a process associated with the n'" system by the
refers to the external arrival process of
open customers
in
the
results in this paper apply to processes that have been "scaled."
Let A'*"' denote
a "generic" process associated with the n"" system. The "fluid scaled version" and
"diff"usion scaled
n
system.
version" of the process A"*"', denoted as
A"(<)
=
-A(")(n')
n
A"
and A", respectively, are defined via
and
X"(/)
= -A<")("^0n
We
also define
= \x^"Hri^t) =
.V"(0
--V"(<)
n^
assumed that the following conditions hold
It IS
the arrival rates and
i
=
mean
— oo <
<
9
—
pj
.
= Amo + mi/m2
Theorem
Theorem
Theorem and
3.1 .4s n
=
where .4^(0
Xt,
mfc,
(3.20)
1
number of
closed
must become approximately
— oo,
Vo{i)
=
1.
"sufficiently fast."
is
following two theorems are direct consequences of the functional strong law of large
bers, Donsker's
num-
the functional central limit theorem for renewal processes:
W''2'(0)
mot. V;
—
=
mo almost
m,t.
=
l'/
and
surely
m^t.
= ^t.
S'o
S',
=
^t
.
S'^
= ^t.
3.2 iA^,V^,V{',V.p)=>{A^,Vo ,Vi' ,V{) where A^ Js(0,Ac^) Brownian motion and V^
(0,m^c^) Brownian motion, k —
The
—
assumed that there
requires that as the the
It
large, the relative traffic intensity at station
Moreover, the rate of convergence
is
is
it
and m^"
—.^asn^oo.
the heavy traffic condttton.
(3. 20) is called
customers becomes
The
Furthermore,
network. First,
oo such that
n(p',"'-l)
Condition
for the input processes of the
service times converge to finite constants, A*"' -^ A
This implies that p,"
0, 1,2.
exists
n
0.l.'2.
following result, which establishes that "remainder" terms converge to zero under scalmg, will
be needed
in
our proofs. For j
=
1,2, let
fu(0=-4:;'(n'0,
e-^(i)
=
-e[';\nt),
n
and
^2(0
Lemma
3.3 For
lkl()l|.
-0
Proof.
It
j
=
1,2
= -4"'(n'0.
r,{t)
=
-e["\nt).
n
and each
n
t
>
0,
\\('^j()\\t
—
0,
\\^ji-)\\t
~
0,
asn-^oo.
follows from the definitions of fij(() an<l f^(t) ihat
<
<
~
fi,(0
e2(/)
<
~
<
max
max
max
»noio(')+
miii(i)
I<i<52(<)
l<i<.4o(()
rM|ii(;)
1<I<.^-J(()
An
application of
Lemma
3.3 from Iglehart ami \\
liiti
[16]
proves the lemma.
Ikji)!!'
-
^'
""(^
The Limit Theorems
4
Theorem
4.1
(The Fluid Approximation)
heavy
If the
condition (3-20) holds, then
traffic
(W^",/",Q",r')^(iy*,r,Q',f')
u.o.c.
where
=
W,'{t)
(4.1)
W;{t)^m2
Qoii)
/,•(<)
f*it)
Theorem
4.2
=
0,
(4.2)
=
Q'lit)
=
i;(t)
=
=
Xmot,
(4.3)
1
0,
(4.4)
T^{t)
(The Diffusion Approximation)
= —t.
Ff the
=^
(IV",r,C/",Q")
Qm =
0,
(4.5)
heavy
traffic condition (3.20) holds,
then
{W'J',U',Q'}
where
W:it)=Cx{t} + r{t)-^i;(t)
^j
is
a {6,a'^)
py;(0
= m2 -
Browntan motion
(4.7)
^^^(O
(4.8)
= XmoW^it),
C/o-(0
(4.6)
U'{t)
=
^P^i'(i),
C/2(0
=
W2(t)
(4.9)
771.2
QUO =
/*
is
—
'
k
f^fc'(0,
=
0.1/2
(4.10)
continuous and nondecreasing with /*(0)
/' increases only at times
t
with
W*{t)
=
=
(4.11)
(4.12)
0.
Equations (4.6)-(4.8),(4.11), and (4.12) characterize IV{ as a one-dimensional reflected Brovvnian
motion on the interval [0,m2] with
and
(1.7), respectively.
drift 9
and variance a^
,
where 9 and a^ are given by
Properties (4.9) and (4.10) e.^press the deterministic relationships between
queue lengths, partial workloads, and overall workloads that are characteristic of Brownian
of queueing networks (for example, see [12,
In Section
1,
equation (4.9), states that
in
of the (limiting)
traffic nii.xing principle."
the heavy
limits
13]).
we noted that the boundedness
be viewed as a consequence of the "heavy
in
(3.20)
traffic limit,
workload process at station
This principle, which
can
born out
the class specific workloads at each station
are deterministically proportional to the overall workload at that station.
We
rationale for explaining the boundedness of the workload process at station
10
is
1
1,
can develop another
which perhap.s may
Open customers
be more intuitive, by considering the arrival process to this station.
station
While station
at rate A
1
"2
not empty, the arrivals of closed customers to station
is
When
resembles a renewal process with rate l/mj.
all
closed customers are at station
the arrival process for closed customers are temporarily "turned
the rate at which work arrives
of time
Let
which station 2
in
falls
below the
empty, station
is
heavy
critical
Theorem
[0.6].
and
—
Tr{dz)
_
our
=
b
m2.
=
=
^2
'""
1'
during the period
If 9
=
then 3i
0,
=
32
=
and
(T^/2b
tt
is
29/(7^,
^-—E,
_ g-«6'
(4.13)
1
p{:)d: where
(''')
= -pfzi
pi^)
The
is,
t
Otherwise, setting k
p«6
That
period of time,
[10].
4.3 (Proposition 5.5.5. [10]) Set
uniform distribution on
the
traffic level.
for that
lim \i;{t)
t—'OO
have the following result from Harrison
and
however,
and set
,
=
/?,
We
off,"
1.
1
displays "non-heavy traffic behavior."
1
be the steady state distribution of \V^
IT
arrive to
following deterministic time change theorem, due to Whilt
[22], will
be helpful
in
proving
results.
Theorem
sequences
4.4 (Deterministic
inn
where Cn
is
Time Change Theorem)
nondecreastng with Cn{0)
=
0-
Let {/n,n
If {fn,Cn)
>
1}
ind {c„,n >
converges u.o.c.
to a
1}
he
continuous
pair (f,c), then /n(cn(0) converges u.o.c. to f{c{t)).
Proof of
5
Lemma
Proof.
5.1
for each
The lemma
t
>
is
0,
||tVi"(-)||(
tlic
—
Fluid Ai)proximation
a.s
n
—
proved via a bounding argument
sequence of open queues with two customer types
Aq
and S2
and wp
let
from
type
1
(3.1)
•x,.
and
(3.2).
We
denote by
and
.4o
in
1.
which W^
bounded above by
a
Recall the definitions of the processes
the arrival process of type
customers arrive according to the renewal process ^j
11
is
The sequence
customers
of service
.
times for class k customers
sums
associated partial
given by {m)^ Vk(i),i
is
X^-)it)
=
l/o'"'(4"*(0)
Zi^)(t)
=
^(")(i)
y(")(<)
=
queue provides an upper bound
<
Applying the
fluid
0},
and as
in
(34), we denote by
the
V^.
process. Defining
+
+ vf>(55"'(0)-<
(5.1)
y(")(<)
(5.2)
sup {x^"n«)}~
*
(5.3)
-'
0<s<t
this
>
for station 1 in the sense that
Wl''\t)
<
Z(")(<)
for all
t
>
0.
(5.4)
scaUng to x^"^ equations (5.1)-(5.3) become
nt) = Ki<it)) + KiS^{t))-t
z"(<)
=
y"(t)
=
x^{t)
sup
0<i<t
It
follows
from Theorem
almost surely for each
ll^"(-)ll(
-^
0.
and
y"{t)
{xT-
>
as n
—
oo.
(5.7)
—
4.4 that ||x"(.)||f
Because the mappings (5.6)-(5.7) are continuous,
follows from (5.4) that for each
it
(5.6)
and the Deterministic Time Change Theorem
3.1
<
+
(5.5)
t
>
0, ||W^r()i|(
^
almost surely as n
— oo.
I
Proofof Theorem
4.1 Because |B,"(0-flr(s)l
<
l*-s| and |77(i)-T;"(s)|
we can conclude from the Arzela-Ascoli theorem that there
which
is
Tq''
,
<
|<-s| almost surely,
subsequence nt,
exists a
fc
=
1, 2,
.
.
.
on
T"*, B"*, and Sj* converges almost surely to continuous limits and the convergence
uniform on compact
Observe that I^{t)
Applying the
From equation
=
sets.
-
t
Denote the corresponding
,
=
B^{t), hence /f* -^ /• u.o.c. where /*(<)
fluid scaling to
(5.12),
limits by Tq
T*, 5', and
t
-
flji respectively.
B'{t).
equations (3.8) and (3.16)-(3.19), we have
W^(t)
=
W^{0) + V^^{SnB^it)))-B^{t)
(5.8)
QS(f)
=
i"(0-i"(r)"(0)
(5.9)
Q'iit)
=
S^iB^{t))-S^{B^iri''{t)))
(5.10)
Q^{t)
=
l-Q'^it)
(5.11)
;j-(t)
=
t-Wi'{fj^it))
+
^{t).
(5.12)
we have
limsup||.-rj"(.)||,
<
limsup||P^i"(77"(-))||(
<
limsup||^^i"(.)||i
n—'oo
<
+
limsup||e-^(.)||,
n—-oo
n—*oo
n—oo
+
limsup||6-^(.)||i
n— oo
(5.13)
0,
12
where the
result of
inequality follows from the observation that
first
Lfmma
3 3
Lemma
and
51. Scaling (313)
=
T?it)
Applying Theorem
follows that
T^
—
and the
t
the fluid convention,
l/,"(5?(fl2"(7j"(0)))
(3.1), the Deterministic
r*
in
<
Tj(t)
+
last inequality is a
we obtain
f7i(0-
Time Change Theorem
Lemma
4.4,
33, and (5.13),
it
where
u.o.c.
= \mot
TZit)
and
f,-(f)
= ^flj'CO.
1712
Again applying Theorem (31) and Theorem
we
4.4 to (5.8)-(5.1 1),
get (W2''.Q'"')
—
(iVj'Q*)
where
u.o.c.
W^{i) = ^2 +
—f^{t) -
fljC)
mi
= ^2
(5.14)
and
Q5(O =
QI(O =
0,
Finally, (5.14) implies that there exists a finite n'
all
<
>
and
n/t
>
= for all >
T7(0 = ^<, Si'(0 =
that /2"*(0
that
Because
n*.
may
I^
t,
and
and positive
increase only at times
and n^ > n*.
<
Q\{t)=\.
0,
/,'(0
=
Hence. /^(O
l^it)
=
=
0,
for all
i
such that IV'^''(0
e
for
t
(5.15)
which tV2'(0
=
^
^
0, it
converges to the same limit
Theorem
4.1,
>
Because each subsequence
0.
t,
from which
we may conclude that (IV, /",Q",
it
Txit)
-
/ij,"'(0
=
.4i"'(0
»7<")(0
=
r?(")(<)_<.
(3. 13)
and (3.19)
=
in this
V;'"*(0
A(")<
=
i;<"'(/)-m<.'''f
way, we obtain
V,(S2{B2{T](t))))
+
m,S2{B2(n(t))) -
—hinii)) +
77)2
rri
=
follows
in
turn
T"") itself
I
.
begin with the definition of "centered" processes
Centering
follows
=
Proof of the Diffusion Approximation
6
We
in
for
S.J(0
of (PV", /",Q", T") contains a convergent subsequence and each of these subsequences
converges to the limit described
^s-
Vi(S2{B2{rj{t)m+ miS2{B2{r](t))) -
-—Wi(n(t)) + —(2{t) +
mj
mj
(u(t)
13
1
—hinV))
U}2
—m +
'"2
^u(f)
=
+
V,iS2{B2{n{t))))
-
m,52(52(r,(<)))
—
X,(r,(0)
(712
-^/2(r?(0) +
m2
—
m2
e2(0
+
- —hivit))
m2
fn(<)-
(6.1)
Next, set
6(0 =
VoiMt)) + moAo{t) +
+ miS2iB2{t))+(xino +
ViiS2{B2it)))
V
6(0 =
t^2(0)
+
V'2(5i(Ti(O))
+m2S2{B2{ri{t)))
One can
+
—
+—
mi
+
m25,(Ti(<))
- Unit)) +
£2(0
—
-\)t
m2
(6.2)
J
V'i(52(fl2(r?(0)))
m.\
eii(0-
(6.3)
invoke (6.1)-(6.3) together with the observation that /i(f)
=
/i(rj(0) to express the netput
processes (3.9) as
Xi{t)
=
6(0-—
^2(0
m2
(6.4)
^2(0
=
)/2('7(s)).
6(0-A(0-(l-—
m2/
(6.5)
V
Applying the diffusion scale to (3.10)-(3.12), we have the following expressions
for the scaled
work-
load process:
(n)
w'r(o
=
-er(o
- ^/2"(o + /r(o
(6.6)
7712
wm = ait) /"
is
+
hit)
f
1
-
-J^j
-
i^i^it)
/2"(r?"(0))
continuous and nondecreasing with /"(O)
/" increases only at times
when Wp{t) =
t
+
-j;^i2(t)
=
(6.7)
(6.8)
0.
(6.9)
where
6"(0
=
Ki^oit)) +
r"|,"'/iS(0
^^ + AHm<"' =
6"(0
=
W^2"(0)
+
+
+
Vi"(52"(fl2"(0))
ml"'52"(SJ(0)
+
nf
1
j
(6.10)
(")=
=
=
V'2"(5r(Ti"(<)))
=
=
+ m<"'5r(7r(0) + ^Vr(5J(B2('?"(O)))
TTlj
=
=
,,""(<)
=
t
+ -W^i^-it)) +
n
By
in
-
^r('?"(0)
+
f5(0
-e^(t).
n
the Skorohod representation theorem, we
Theorem
(")
=
+m^2^S^iB^{ri'^(t)))
+ ^^11(0,
(6.11)
(6.12)
may and
3.2 holds u.o.c.
14
will
assume henceforth that the convergence
—
Lemma
6.1
Proof.
Let Z'"' be the process defined
is
For each
>
t
-
0,||r7"()
||(
—
as n
in
Lemma
oo.
51, and note that Z" -~ Z' uo.c. where Z'
a one-dimensional reflected Brownian motion with
hmsup
n— CO
hmsup
<
-n''{-)\\t
•
II
n—"oo
<
=
Proof.
The convergence
as n
u.o.c.
—
>
.
Because ir|"
(<)
<
H
n—'OO
n
— oo
-||Z,"()||,
Ti
0.
= '"j— m"(')
oc where ^' is{9.(t^) Broitnian moftoi} nnrf^^iO
of ^" follows from from (3.20),
Theorem
Theorem
3.2,
4.4,
and
Li
nima
Define
6.1.
=
(,7(0
V2"(5T(:^(0)) +
(")=
=
C2"(0
An
cr^
+ Hmsup -||f?()||,
1
Hmsup
n
^ ^'
and variance
limsup-!-||PVr(-)lk
<
6.2 ^"
/i
-UPV^H'T"!))!!.
n—'oo
Lemma
drift
application of
Theorem
3.2,
Theorem
Theorem
4.1,
(6.13)-(6.14) that ||C()II(
^2"(o
=
((113)
=
=
^Vr(52"(fl?(r/"(0))) + m^"'Sr(77(/)).
Iglehart
in
1
"'2'''^2"(B?(r7""(/)))
and Whitt
[1.5],
shows that S"
(6 11)
— —m,~^^^V'
*•
u.o
c
Fmm
and the Deterministic Time Change Theorem, we can conclude from
—
as n
1^7(0)
—
:c for each
+ cr(n +
C2"(o
-
/
>
and
s^r(rr(o)
=
i
1.2.
+ f^o +
Writing
,n
2
-jto
'"i^'^'
ml
'1
the theorem follows as a result of
Lemma 62
proof,
that
the
that
Theorem 31, Lemma 33, and Lemma 61.
implies that e'^"(<)
> m2/2
we may therefore assume that ^" £
for all n sufficiently large.
D^
,2
'^o'"
^" " -
^
From
(6
I
For the purpose
.5).
we may
also
<,{ .>iir
ciiiicln.l.-
4J(0- /"(O - (1 - "«*r'/"i2"V2'( '?"{«)) has no jumps downward, from which it follow. 1I1..1
assumption of Lemma 9.7 is satisfied because /" is a nondecreasing process. Finallv. iictmn
17"
£
A,
we conclude from Theorem 91 that there
satisfying (6.6)-(6.9),
and that
(WJ")
is
given by the
15
exists a unique pair of processes (II
mapping (IT",/") =
(*. *)(4"-
»?")
"
/''
I
Proof of Theorem 4.2 From Lemmas
Brownian motion,
continuous,
W)
(/*,
^2J{t)
= m2 —
Theorem
follows from
it
=
where {I',W')
u.o.c.
^i{t),
I'
($,
6.1-6.2, (^".^J.rj") -^ (^t.^^rj*) u.o.c.
and q'{t)
9.1 that
*)(r,
=
Wi
is
and variance
is
continuous at
v')- Specifically,
{^',t]'),
we have from
nondecreasing and continuous with /*(0)
is
/* increases only at times
implying
Because Brownian motion
t.
($,^)
with W*{t)
t
where ^j
=
is
is
{9,cr^)
almost surely
hence {I^,W^) —*
Lemma
9.2,
=
0,
a one-dimensional reflected Brownian motion on the interval [0,m2] with drift 9
cr^.
Next, observe from
(3. 19)
and
(5.
13) that
;)"
—
t)'
»
where
u.o.c.
77*(f)
= -W'(i).
Centering
we obtain
(3.14)-(3.15),
USit)
=
Mo"(t)-Mo"('?'"(0)-A(")m("'^"«)
U^{t)
=
M,"(0-Mr(/?"(f))--J;^'?"(0,
7712
with
=
Mo(<)
Kr(^o(0) +
=
Because Brownian motion
and Theorem
and
(/"
—
[/j*
is
4.4 that ||A/,"(-)
u.o.c.
=
continuous,
-
m<,"'.4S(0
(")
=
it
follows
MP(jfi-))\\t -^
from
Lemma
for
a.s.
each
t
61, Theorem
>
and
i
=
3.1,
1,2.
Theorem
Thus, U^
where
and
Uoit)^\moWi{t)
= —Wi{t).
u;{t)
1712
Similarly, noting that
Qlit)
=
iS(f)-.4S(r,"(0)-A(")77"(0
QUO
=
.4^(<)-.4!,'(^"(<))-^^"(<)
Qlit)
=
A^it)
1
-
A^rj-it))
-
-^nt)
where
A-,{t)
= S^{B^{t))--L^I^{t)
and
16
.45(0
= 5r(T,"(0)--|;^rr(0,
—
3 2.
TJ
it
follows that
Q"
— Q'
with
u.o.c.
Qo(0
=
g-(0
=
—u:(t)
xw;(t) =
"'0
—
-Ltv7(o =
c/,-(o
r7i2
and the proof of Theorem 4.2
is
complete.
Refining the Brownian Approximation
7
We now
turn to the following important question: Given a mixed network with parameters A. c^,
m,, and
c^,
=
i
0, 1,2,
and a
finite
number
customers,
;V of closed
estimates for the network using the theory developed
in
how do we obtain performance
the previous sections?
the following approximation for the workload process at station
Theorem
4.2 suggests
1:
j^(Wr{N'')^\V'{)
where
(1.7),
is
IV'j*
an
RBM
and variance a^ are given by
(1.6)
=
.V
(xmo +
V
a^
=
Xml{cl +
—
m2
1
m?
cl)
+ ^{cl +
m2
cl).
reversing the scaling in equation (7.1), one obtains the approximation
W{)^\V{)
where
W
is
an
RBM
on the interval [0,m2N], whose
H
and whose variance
of
and
namely,
9'
By
drift 6'
on the interval [0,m2] whose
(7.1)
W
is
again
cr^.
We
= Xmo +
(7.2)
drift
—
^
-
/i
is
given by
(''S)
1.
in 2
then apply
and other performance measures of
Theorem
interest.
In
4.3 to obtain the steady-state distribution
particular,
writing
W''('^5o)
to
random
variable associated with the stationary distribution of the process {n'(f)./
setting 6
=
rriiN, k
=
2^/ct^
,
mean
>
0}
the
and
we have
EW{oo) =
=1,
{
,' ,L
**
17
--,.
,
-
otherwise.
(71)
Morover, the throughput rate obtained from the Brownian approximation
a =
I
Equations
12, 13]),
-
^ \T^
much
ajm^ where
(7.5)
otherw.se.
form one possible approximation
traffic
to developing our refinement
is
for a two-station
approximations
however, one typically needs to "refine" the Brownian limit
that, as
given by
^
previous works on heavy
in
performance estimates. The approach
method
1
(7.4)-(7.5) in particular,
As was noted
mixed network.
[6,
and
(7.2),
V2M
"^;
J
is
in
(for
example,
order to obtain good
to arrive at an approximation
as possible, yields estimates that agree with the exact solutions in those
known. Henceforth, our
special czises for which the exact solutions are
results will be
benchmarked
against the following special case.
From
the theory of quasi-reversible queues
product form solutions
[17] (see also [1]),
the mixed network in Figure
1
has
if
mo = mi = m,
Ca
=
Co
=
c?
=
c^
and
(7-6)
=
(7.7)
1-
In this case, observe that
= 2m(l
(?
Denoting by
at station
1,
-K/j).
P(fc,
/)
the steady-state probability oik open customers at station
and
N
—
/
closed customers at station
P(fc,/)
2,
= Gf ^1' NoSi
and
G
is
- Am,
closed customers
we have
where
rjo
1, /
/yi
=
(7.8)
m
,
the normalizing constant
G=(i-^)f:(-^)
-'70/
TT'c.y^
/c=0
and Let us write Qo(oo)
{Qo(0'^ ^
0}i
^nd
customers at station
to
mean
the steady-state
similarly, let us use
1.
From
(7.8),
random
variable associated with the process
Qi(oo) to mean the steady-state headcount of closed
we obtain the following
statistics:
.
and
=
a'
l-^P(n,.V)
.V
(Recall that
a'/mj
is
the throughput rate of closed customers.) In particular,
if
p
=
r;o
+
^i
=
1.
equations (7.9)-(7.11) simplify to
Eg,
Finally,
when p <
1,
=
we have the following
(7.13)
J
limit as the
number of
closed customers in the system
mcreases:
Refinement
We
1:
Replace
b by
lim
NTco
EQo
lim
Ntoc
EQ,
=
1
-
'/O
I
-
^0
=
(7.15)
-
^1
-
71
'^-^
(7.16)
m2N/a.
justify this modification via the following argument.
increases whenever that station
"functional" Little's
interpreting
l/m2
Law which
is
The
empty, or equivalently, at times
follows from equations (4.9)-(4.10),
idleness process at station 2
t
when Q\(t) =
we have Q\[t) =
By the
.V.
(l/m2)\'V{t),
as the throughput rate of closed customers. This refinement essentially replaces
the naive throughput rate by the approximated throughput rate
be verified from (7.5) and (7.14) that a^
= 2m and
In the case
p =
U.
it
can
the refinement gives
\-2bj
in
=
a/m^.
(7.17)
1-f
from which we obtain
a
Note that
in this case,
Refinement
2: Set
.18)
V +
1
the Brownian appro.ximation agrees with the exact solution (7.14)
Q,(/)
=
[& / m2)W (t) / p
19
This modification of equations (4.9) and (4.10) consists of two parts.
rate used in the so called "functional" Little's
a weighting factor of l/p
processes.
As noted
in
is
taken to be
d/m2
the throughput
rather than l/m2. Second,
then applied to the relationship between queue length and workload
is
Section 6 of
may be obtained with such
Law
First,
[13],
empirical experience suggests that a better approximation
a refinement. Writing Qi(oo) to
>
with the stationary distribution of the process {Qx{i)J
we have
0},
random
the
variable associated
=
0,
the approximation
is
the special case of
in
/i
^EPV(oo)
=
EQi(oo)
mean
m2p
a
{
7712
p V
m2N
2Qf
N_
'
2
which agrees with
(7.13).
asymptotically exact as A^
Moreover,
it
follows from (7.4) that
=
Qo{t)
This refinement
is
-
%-
=
J?!
lim EQi(oo).
l^- oc
XWit)/p.
essentially identical to the previous one, in that a factor of l/p
can also be shown that with this modification, the approximation
N
—> oo whenever
<
is
is
introduced.
asymptotically exact as
0,
lim EQo(oo)
=
-\-fIp
p
iV-»oo
\
no
1
-
-
'70
=
=
EQo(oo)
lim EQo(co).
'^^<:
^l
However, note that we do not obtain the exact solution
for finite
-ElV(:c.)
P
A
/m2,V
p
\
'2a
'to
1
where the
0,
rn2p
It
p.
<
1
=
1
3: Set
//
— oo:
lim EQi(oo)
N—'oo
Refinement
when
last equality follows
because
rjo
+
'/i
-
=
20
I
'to
/ .V
+
V
2
1
N
,
even when p
=
0,
since
To summarize, our approximation procedure
RBM
on the interval [O^m^N/a] with drift
The queue
/i
and variance a^ given by (73) and
Qo{t)
=
-rV(/)
P
(7.19)
Qi(0
=
-^W{t).
(7.20)
m2p
< d <
=
Theorem
Proof.
1
that satisfies equation (7.5) for o// ranges of the parameter set A,
7.1
/i
Let b
=
0,
= m2N/a. Then
it
is
easily verified
there exists a unique
from
(7. .5)
a =
/i
c^,
m,,
0,1,2.
For
Consider
must be
itself
approximation to be consistent, we must now show that there
In order for the
exists a unique
i
an
length processes are then approximated via the mappings
approximated.
c^,
.
(1.7), respectively.
Note that the formulation of the approximation includes the quantity d, which
and
W
consists of replacing the workload process by
<
0.
We
< d <
1
that satisifes (7.5).
that the unique solution
is
given by
1 )
to the transcendental
N + o-2/2mi
need to show that there exists a unique solution r S
(0.
equation
/(x)= (l-e-"/")(l-r)--^/i =
where
First,
(7.21),
a.?^
observe that /(I)
= — (m2/mi)/i >
[
dx
Because a <
solution X
G
0, it
follows that df(x)/dx
(0, 1) to (7.21) (see
Figure
8
7.
0.
(7.22)
>d as j
Next, f(x)
],
0.
Finally, differentiating
we obtain
df{i)
The
(7.21)
subject of this section
is
+
>
3).
e-"/^ [[
-
ax-'^{[
<
J
<
for
The proof
for
1
-
x))
(7.23)
.
and consequently there
^ >
exists a unique
I
proceeds similarly.
Numerical Examples
the performance of the refined approximation described
in
Section
Using product-form networks (namely, those whose parameters satisfy conditions (7.6)-(7.7),
we compare estimates obtained from the Brownian approximation against exact
theory predicts that the approximations are asymptotically exact, that
21
is,
as the
solutions.
number
Our
of closed
""2 „-
.
Figure
In
6.
cases, the
all
approximation
1%
within
is
N
of the exact solution for
expect, the quality of the approximation increases as p\ becomes closer to
>
System
One can
2.
The accuracy
is
see that the approximations are
of these estimates are clarified
the performance measure of interest
10%
with
N
of exact solutions for
>
As Figure
10.
of closed customers. Figure 11 bears out our expectation that
N
of
but
are required for good estimates.
>
in all cases, ^V
Some
X
£
>
0.
(iii)
=
So
or a{t)
X
=
<
=
si
a,
(xi,j;2)
<
<
for all
t
•
•
for
•
t
G Dj,
>
t
=
0;
<
s,v
£
[s,_i,s,).
a
<
D
€
1,
larger values
solutions.
in
the proofs of Sections 5-6.
Chen and Mandelbaum
[5].
that satisfy the following conditions:
>
e'i{s)
>
j(0)
£
e for all s
(i) j:(0)
>
0;
Define C^ to be the continuous
[0, t].
0,
>
e'r(s)
(
for ail
/
each
finite
<
and constants
t,
<
oq
there
ai
<
a finite
is
<
a2
particular, observe that e(<)
In
<
c
> o}
that have the following properties:
for
(iii)
A, and
£
:
D
Denote by A the set of functions a €
a{t)
closer to
namely,
,
C^ = {j e C^
<
is
queue length
Appendix
Let D^[0,<] be the set of x
D^
length of open customers
10 shows, the estimates are
pi
devoted to characterizing some mappings that are used
no downward jumps; and
hcis
functions in
(ii)
10% from exact
15 gives estimates that are
of the results here are adapted from the work of
Fix
(ii)
is
N
Here, the estimates perform poorly for small values of A^,
9
This section
when
respectively,
1,
Finally, Figures 11-12 displays estimates for
5.
also
intprmediate values of
for
The queue
Figures 9-12.
in
Figures 9 and
in
good even
As we
1
Figures 7 and 8 display the queue lengths of open and closed customers at station
for
15.
<
1,
let
u'
=
mapping (w,y) = {^.^)(x,a) defined by the following
wi{t)
=
<l>i{x, a){t)
=
xi(t)
+
W2{t)
=
^2{i.a)(t)
=
X2{t)
- yi(0 +
yiit)
yi
are nondecreasing with y,(0)
y,
increases only at limes
t
-
(u'i,[r2),
y
•
=
—
(i)
a
is
number of
• such that
f
is
nondecreasing;
subintervals
either a(1)
an element of A
i
For
{y\.y2) be a solution of the
properties:
cy2{t)
(
—
1
-
(9.1)
c){y2(i)
-
y2{a(t)))
+
cy^it)
—
(9.2)
(9.3)
where w,(t)
=
(9.4)
Observe from (9.1)-(9.2) that
e'w{t)
due to the monotonicity of
2/2.
=
e'x{f)
+
The main
(1
-
r){y2{t)
-
.V2(n(0))
result of this section
23
is
>
e'x{t)
the proof of the following:
(9.5)
Theorem
9.1 For each x
satisifes (9-l)-(9.4)-
x € Cj and a{t)
process
The
if
—
X2
following
Lemma
{I
is
£ D^ and
is
A, there exists a unique pair of processes {uj,y) that
In other words, the mapping ($,^)
well defined on
is
— t, then ($, 'J') is continuous at {x,a).
— c)(y2 ° a) has no jumps downward.
a special case of Theorem 2.5 of Chen and
9.2 Suppose that x G
and {w,y)
G
a
D^ and
=
a{t)
Madelbaum
Then {w,y)
t.
—
Finally, y
D^
x A.
'^(i,a)
is
Moreover,
a
if
continuous
[5].
are uniquely defined by (9.1)-(9.4),
uniquely given by
wi{t)
=
xiit)
+
yi{t)
-
cy2it)
W2it)
=
X2{t)
-
yi{t)
+
cy2{t)
yi{t)
=
sup {xi{s)
-
cy2{s)}~
0<3</
- sup {x2(s) - yi(s)}~
=
y2it)
C
Moreover, fixing a(t)
Lemma
=
9.3 Given x G
t,
"^
D
is
0<s<(
continuous on C^.
and a G A, define
fi{z){t)
=
-
sup {x{s)
{l
-
c)z{a(s)}}-
(9.6)
0<s<t
f2{z){t)
=
- sup {x(s) + (l-c)(z(s)-z(a(s)))}-.
C
There exists
Proof.
a
unique solution
The key
to the proof
to (9.6)
is
and
is
a contraction
the solution uniquely satisfies (9.7).
the observation that a{t)
\\h{z){-)-f,{z'){-)\\i
Hence, /i
mapping
(9.7)
0<s<t
in
<
t,
from which we obtain
<
{l-c)\\z{a{-))-z'{a{-))\\t
<
(l-c)||z(.)---'(.)||t.
x and there exists a unique solution to
(9.6).
the fix point solution of (9.6) and observe that
z{t)
=
sup {x{s)
—
(1
—
c)z{a{s))}~
+
(1
-
c)(z(5)
-
z{ais)))
-
(1
- c)z(s)r
+
(1
-
c)izis)
-
z{a{s)))}-
+
(1
0<s<t
=
sup {x{s)
Q<a<t
<
sup {x{s)
=
C/2('Z)
-
c)
sup {---(s)}"
0<s<t
0<s<t
+ (1-C)Z(0,
24
Denote by
where the
first
inequality follows because
monotonicity of
r.
<
Therefore, z{t)
f2{-){t)
<
<
s
<.
On
last equality
is
a result of the
the other hand, noting that
+ xis)-{l-c)z{a(s))>0
z(t)
for all
nonnegative and the
z is
we have
=
f2(~)(t)
+ z(s)-
- sup {x(s)-ll- c)z{a{s))
C
>
cz(s))-
0<j<(
- sup
{
— c;(s)}~
c 0<s<.t
=
and
Now
a solution of (9.7).
2 is
=
fi(z'){t)
^(0.
let z'
be another solution of
(9 7)
We
have
{x(s)-(l-c)z'(a(s))}-
sup
0<3<t
=
sup {x{s)
+ {l-c){z'is)-z'{a{s)))-{l-c)z'{s)}-
0<s<t
<
sup {x{s)
+ {l-c){z'{s}-z'{a{s))) +
cz'{s)}-
+
sup {---'(s)}"
0<s<t
0<3<t
= At)
and
=
Mz'){t)
sup {xis)
+ {I-
c)(z'(s)-z'{ais)))-i\-c)z'(s)}
0<s<(
>
sup {-;'(«)}"
0<3<(
=
because i(s)
to /i
,
but
Remark:
/i
+
(1
h;i-s
-
r)(z'{s)
-''(0
-
+ cz'(s) >
z'{a(s)))
a unique solution so
Suppose that
a{t)
=
for all s
>
we can conclude that
for all
t
>
z'
and x{0) >
z{t)= sup {j(a)}- = - sup {x(s) +
C
0<a<t
Lemma
9.4 Given x 6
the interval [0,r).
Thm
D? and
a
e
there exists
—
have shown that
:' is
a >;ohiiion
I
z.
It
then follows from
(l
-c)r(s)}"
Lemma
'J
:}
lii.ii
.
o<j<(
A, suppose that (ir.y)
a
We
0.
is
the
unique extension of(u'.y)
25
unique solution of (9.1)-('J
to the interval [O.r].
i )
""
Proof.
To extend
"^('")iy(^)
-
the definition of {w, y) to the endpoint
r,
observe from (9.2)-(9.4) that (u(r)
—
y('"~)) satisfy
wi{t)
=
(wi{t-)
+
xi{t)
-
xi(r-))
+
(y^ir)
u;2(r)
=
{w2{r~)
+
X2(r)
-
12(1""))
-
(1
y2(a(r)))
-
+ (1 -
c)
-
(y2(r)
-
-
yi(r-))
c){y2{T~)
-
(yi(r)
-
-
c(y2(r)
y2ia{T~
- y2(r-))
(9.8)
)))
yi(r-))
(9.9)
+c(y2(r)-y2(r-))
(9.10)
y.(r)-y.(^")>0
(9.11)
Mr)'(y(r)-y(r-))=0.
(912)
Because
e'iwir-)
x{t)
-
i(r-))
-
=
e'.c(r)
+
(y2(r)-y2(a(r)))
>
e'i(r)
>
0,
where the
[5]
+
first
inequality
is
(1
a result of
-
-
(y2(r-)
c)
y2(afr-)))
—
that (9.8)-(9.12) produces a unique solution for y{r)
For X
£
T)
Theorem
follows from
(9. 5), it
+
(1
-
c)
4.3 of
-
(y2(r)
y2(a(r)))
Chen and Mandelbaiim
y{T~).
I
and a G A, define the mappings
u(0
=
yi(x,a)(t)
=
sup {x(s)
+
{I
-
c){u{s)
-
u(a{s)}})-
('J
13)
0<j<t
v{t)
For a sequence Tk, k
=
=
g2{x,a)(t)
x{t)
+ il-c){u{t)-uia(t)) +
cuit).
(9 11)
1,2,..., let us define the "shifted" processes
(uo
It is
=
x'^it)
=
v{n) +
u^'it)
=
u{t
«)*=(<)
=
u{a{t
v''{t)
=
v(t
x{t
+
Tk)
- x{n)
+ Tk)-u(Tk)
+
+
Tk))-u{aiTk))
Tk).
straightforward to verify from (9.13)-(9.14) that
u''(t)
= yf(x^a)(0
=
- sup {x^ is) + il-c){u'' is)
C
v^{t)
=
g^^{x\a){i)
=
0<s<t
x^{t)
-{no
a)' (s))}~
(D
!.->)
'
^
+ {\-c){u\t)-{uoa)^(t)) +
26
cuHt).
i9l(3>
Remark:
If
we define the mappings
=
/i,(z,a)(0
- sup {x(s)}-
(9.17)
C 0<j<«
=
/.2(x,a)(0
it
is
clear that (/ii./jj)
+
x(t)
cfn{x,a)(t),
a special case of (31,92) with a equal to
is
(918)
where
e
e(t)
=
<
is
the identity
map.
Lcmnia
9.5 For each / G Dj
and
(here exists a pair of functions
G A,
a
(w,y) that
sattstfes
(9.1)-(9.4).
=
we
Proof.
Noting that
We may
cissume without loss of generality that
(9.5) that ifi(O)
>
it
suffices to set
f
It is
=
(<I>,
prove the
first
>
it'2(0)
lemma
for i
1/2 (otherwise,
it
G C\.
<
Fi.x
follows from i £
i5
<
1/2.
C\ and
1/2 and we proceed similarly).
For an increasing sequence of times
{iv,y)
1.
vl')(x, a), it will
k
Tjt,
be necessary to
=
and a pair of functions (u\y) satisfying
1,2
refer to the following "shifted" processes:
x*(0
=
w,{Tk)
y*(<)
=
y,{t
{y2oa)'(t)
=
y2(a(t
u;*(0
=
wit
+
x,{t
+ Tk)-x,{Tk)
(9.19)
+ n)-y,(Tk)
(9.20)
+ n))-y2{a{Tk))
(9.21)
+ n).
(9.22)
straightforward to verify that the mappings (9.I)-(9.2) yield
u{-(0
=
xt(0 + y{''(0-cy2'(0
w^it)
=
j:^(0-yf(0 + (l-c)(y^(0-(y2oa)'--(0) +
(9-23)
(9.24)
cy^'(<);
moreover,
e'w''{t)
=
e'x^t)
+
(1
- c)(y^(0 -
(y2 o a)'(t))
(9.2.5)
and
e'l^O
+
(l-c)(yl^(t)-(y2oa)''(t))
(1
=
-
e'x{t
c)(y2{t
+
+ n) +
Tk)
Set To
0, y^(t)
=
0,
e'w(Tk)
- y2{n)) -
(l-c)(y2(t
where the equality follows directly from equation
=
=
(1
-
+
e'x(t
c)(y2(a(<
+ n)-y2(a{t +
+ n)-e'x(n)
+
=
/ii(r?,a),
- y2{a{n)))
(9.26)
Tk))),
(9.5)
and ob.serve that by monotonicity, y2('i(0)
by the mapping (9.17), namely, y?
T,))
and
27
let
=
for all
/
>
U.
Define y^
w° be given by equations (9.23)-(9.24) with
fc
fc
=
=
Observe that
0.
and hence w^ are uniquely defined by Theorem
j/j
=
If f 1
=
>
inf{t
=
hand,
a
if
the identity
is
=
<
oo.
-
ti
+
.
.
+
.
tk
where the
-
(1
c)(y2(0
=
+
e'x°{s)
{w'^,y^)
—
for
<
<
=
(y2 ° a)'^(0
on
(2/2
is
nondecreasing. Hence, we
-
c){y^{s)
-
o a)''(<))
y^ia{s)))
-
for k
=
Wiitk+i)
>
e'x''itk+i)
=
e'x{Tk+i)
>
1-6 >
=
monotonicity of
shift
time
to 7\ via the
y2 be defined by (9.15), namely, y2
>
=
on the other hand,
=
a{t)
t,
[0, s;
definition of
(u;,
Using the special structure of
tk+x
(1
=
inf{t
the definition of Ti,
—
c)(y2(0)
that e'x*^(<)
conclude
^2
Define for
ifc
[0,<i],
£
t
—
+
>
=
and we
Define
[0,<i].
0,
+
+
- c){y^(tk+i)- {y2oaf{tk+i)) -
(1
-
(1
-
c)(y2(Tfc+i)
-
y2(a(Tk+,)))
6
6
(9.28)
8,
mappings
=
(9.19)-(9.2I) setting
3j(x2,a).
—
then y2(0
—
(y2 °
Ti) whether a{t)
y) can then be
The
=
I:
Set y\(t)
I.
2/2(^(0)))
(1
0.
-
—
and
functions w\, yl are then defined using
a)HO = for
= t, in which
<
<
s/
case
—
Ti
=
for
Thus
.
y2
where a
a,
is
>
we have from
~
c)(y2(f)
"^2(0)
—
—
(9.28) that w\(0)
=
"^{(O)
=
For
-
+
(y2 o a)*^(0)
In addition, w\{s)
-
>0.
c)(y2'(0
e'x^it)
:
S(
—
either a constant or the identity
all t
1
(
> ^- ^^^
— ^t(0
"^^.n
A:
=
1
has no
-
wi{Ti) >
w^^it)
6
is
Ti using
map
si
—
Ti.
uniquely
2.2.3 of
Lemma 9.3
Lemma 9.4.
over intervals of
<
(9.29)
^}-
and consquently e'x^(O)
same argument
as before to
+
show
jumps downward, from which we may
+ {l-c)(y](s)-{y2oa)^{s))-
e'x^{s)
<
let
y'2ia{t)))
use the
t
we invoke Theorem
extended to the endpoint
time, one can thus uniquely define {y^,w^) for
By
x €
or a{t) takes on a constant value, in which case the remark following
[10],
The
conclude that
Let [s;_i,s/) denote the interval corresponding to the function a that
I.
defined over the interval
applies.
In either
may
6 for
contains Ti. If a{t) takes a constant value over this interval, then (y2 o a)^{t)
Harrison
[0,si).
has no negative jumps on
u;f (<)
w^{s)
the other
y2.
(9.23)-(9.24) with k
If,
—
On
si.
a solution of (9.1)-(9.4) over the time period
is
and observe that
-
(1
=
a)''{t)
(1
associated with the function
first interval
~
y^
+
equality follows from (9.26) and the last inequality follows from (9.5) and the
first
We now
+
(9.27)
6}.
note from (9.25) that e'x°(0)
First,
Let [0,si) be the
8.
- w^t) <
(yj o a)'=(t))
over this interval, then y2{t)
e'x''{t)
=
Moreover, w^{s)
0.
wi{Tk+i)
let
>
c){y^{t)
<i
no downward jumps and
have shown that {w,y)
Tk
-
(1
us cissume
let
w^{0)
map
we may conclude that
Si) because x has
h >
+
e'x*(0
a takes a constant value over this interval, then (y2 o
If
case,
:
oo then we are done so
c)(y2(0)-y2(a(0)))-t«i(0)
[0,
For
[10].
0, set
tk+i
a.
of Harrison
2.2. 3
wl{s)
>
6 for s
G
[O.f^]-
1
'
y(0
=
<i
0<i<Tk
y{t)
^, _-^~
y{n) + y^{t-n) t>Tk.
'_,
.
,,,
28
(9-30)
.
The
pair {w,y) thus constructed
observe that
for
=
it
and
In particular,
=
y*
let
a solution of (9 l)-(9.4) over the time period
if
>
e'i*(<fc+,
=
e'xin+i) +
>
1-6 >
+
)
(
-
1
)
-
(y2 o a)*((t+,
=
In either case, w''
is
process (w,y) on the interval
(9. 22).
Our
[0,T)t] is
some
finite
(9.5)
and the
t
for
which Tn
construction for i
<
>
for all "
t
))
Finally,
-
6
6
pair {w,y) that satisfies (9.1)-(9.4) on the interval
—
^"'^ ^^^ y\
''i(^f.'j)- If
''^
odd, we set y^(t)
's
>
G Cj
To do
t.
If
1
=
defined according to (9.23)-(9.24). Similarly, we use
is
thus complete
is
if
map
given
we can show that
in (9.30)
for
each
us suppose to the contrary that there
so. let
t
The
even or odd, respectively.
is
then constructed via the concatenation
there exists finite n' with Tn«
t,
r2]
(9.31)
either (9. 27) or (9.29) to define tk+\ depending on whether k
fixed
[0,
6.
we can construct a
is
c)(y2*(/fc+,
- c)(y2(n+i) - y2(a(n+i))) -
{I
even, set j/2(0
fc
g'l{x2,a).
and property
g
t
1,
Iterating in this way,
[O.Tfc].
is
is
even, we have the following inequality due to
definition of Tk
a-,(7fc+,)
-
>
1*^1(7;)
\-6 + e'x{Tk+x)-
e' x(Tk)
+
{\
-
c){y2{Tk+,)
-
y2(a(Tt+,
(1
-
c)(y2fT;.+
-
y2{a(Tk+i)))
)))
-(\ - c)(y2{n) - y2(a(Tk))y,
odd, we have
for k
u'2(7it+,)
-
>
u;2(7;.)
Because a G A, there are a
identity
there
map
^ -
6
-(I
-
finite
+ e'x(n + i)- r'r{n) +
-
c)(y2(Tk)
number
of intervals partitioning [0,<] sucli that a
interval [s/_i,S() such that Tk
(1
-
C)
[(y2(T^+,)
)
xj2(a(Tk))).
From the
or a constant value over each subinterval
must be an
,
G
- y2(«(n +
,
-
U2{Tk)
-
either the
finiteness of these subintervals.
[^/-i.s/) for all k
)))
is
>
k'
.
For such k. either
y2(a(Tk))]
=
or
(1
-
c)[(y2(71. + ,)
- y2(a(n+,))) - i'MT,) -
depending on whether a
In either case,
is
the indentity
we may conclude that
max
sup
J
Tk<t<t<Tk+i
{wj{t)
map
for all k
'/i(.i(r,))]
=
(I
-
c)(y2(r,+,
)
-
y2(Tk))
>
0,
or a constant value on this subinterval, respectively.
>
- wj{s))> - -
k'
f'
->i\\Ax
2
J
29
sup
r»<3<<<Tt+i
\xj(f)
-
Xj(s)\.
(932)
It is
Mandelbaum
straightforward to extend identity 2.8.G. of Ciien and
[5]
to this setting, which
states that
-
(w(t)
sup
"S'^'^"
Substituting (9.33)
max
E
for all s,t
>
for all k
It
[0,t]
with
—
|s
<
<|
t).
>
k* imply that T^
tj
-
\xj{t)
The
so there exists
[0,t],
Xj{s)\
<
f
3
of
jumps over each
>
for all k
k'
x(s)|.
(9.33)
77
2
>
~
(9,34)
such that
V
<
t
However, this contradicts the finiteness oft.
.
Lemma 94
interval [0,<].
unique extension at each jump point of
Lemma
-
inequality (9. 34) together with the assumption that T^
remains to extend the construction to r G Dj. This
number
|j;(<)
\xj{t)-Xjis)\>l(l--8] >0.
sup
uniformly continuous on
is
sup
u<s<t<v
we have
in (9.32),
max
However, x
<
w{s))
is
is
done by noting that x has only a
then applied to show that
finite
tliere exists
a
I
x.
9.6 Lei x and a satisfy (he conditions Theorem
Then
9.1.
(9.1)-(9.Ji)
have
unique
a
solution.
The proof proceeds
Proof.
{w,y) be the process constructed
satisfying (9.1)-(9.4).
1.
y and
2.
if
y(r)
=
3.
if
y{t)
=
y'(7') at
y'{t)
in
the proof of
Lemma
and
9.5,
let
[.5].
Let
(w',y') be another process
Suppose we can show that
coincide on
y'
Chen and Mandelbaum
as in the proof of Proposition 2.4 of
on
[0, S]
some
t
e
<
[0,
for
>
some
0,
5
>
0;
then the also two coincide on
r) then y{T)
=
[r,
r
+
6]
some
for
positive
6;
y'(r).
Defining
it
follows
from
(1) that r
T
=
>
6.
sup{<
>
Suppose
y(s)
:
<
r
=
y'{s) for all
Then
oc.
<
(3) holds
This contradicts the definition so we can conclude that r
=
00.
s
<
/},
hence y and
y'
The proof now
coincide beyond
rests
r.
on establishing
(l)-(3).
The proof
of (1) follows from the construction y"
then follows from (1) by applying a time
application of
Lemma
shift as
111
in
Lemma 9.5. The proof of (2)
Lemma 95. The proof of (3) is an
the proof of
the proof of
I
9.4
30
Lemma
9.7
then y
continuous.
is
We
Proof.
(9.8)-(9. 12) of
If
X2{) —
- c)(y2oa)() has no downward jumps where
(1
=
only need to show that y(t)
Lemma
94. As argued
in
y{t~) for
all
the proof of this
>
t
Lemma,
(u),y) satisfies (91)-(94),
Consider the problem posed
there exists a unique solution for
y(t) — y(t~). Because X has no downward jumps, u;i(r~) + ii(r) — ri(r~) > iii(r~)>0.
because we assume that X2{-) — (1 — c)(j/2 o a)() has no negative jumps,
-
-
-
W2it-)
+
>
w2{t-)
+ il-c)(y2(t)-y2(t-))
>
U!2{t~)>0.
Hence, y{t)
—
y{t~
)
J-2(0
=
X2ii'
)
( 1
c)
- y2(a(r ))) +
(y2(a(0)
(
1
-
c)
(yjCO
Moreover,
- y2(r
))
I
the unique solution.
is
in
Let us define the modulus of continuity
u„Ax)=
|x(r)-r(s)|.
sup
(9.35)
0<r.3<(.|r-l|<r)
Lemma
9.8 Let
£
x. x'
D
and
a, e
-
{u,v)
£ A where
e{t)
{9i,g2){x,a),
—
Define
t.
(i/, v')
=
(yi, yj)!-^', e),
and
{u'^v')
>
and
=
Fix
t
We
have the following inequalities:
=
(u'.v''')
(yf,y.t)(x*.a),
{glg'^){x'' ,e).
set
ll"()-"(a())||,
<
-u;,,,(x),
(9.36)
c
||u*(.)-(uoa)*(-)||,
<
(9.37)
-a;,,+r.(-r),
c
Proof.
||u*(.)-»''()lk
<
l\\x''()-x''()\\,
+ 2(^^y,,^TAr).
(9.38)
l^^(-)-v'^)||,
<
2\\x''{)-x'''{)\U
+ 4(^^^^r,.,+n(x).
(9.39)
From Lemma
9.3,
we have the equivalent representation
u{t)=gi{x,a}{t)= sup
{j:(s)
0<3<t
31
-
(
1
-
c)u(a(s))}-
.
(9.40)
With
we have
(9.40),
<
-
u{t)
u{ait))
=
sup
-
{ix{s)
x{a{t)))
-
-
{I
c)(u{a{s))
-
u(a(a{t))))}-
a(t)<s<.t
<
sup
\x{s)
-
x{a(t))\
+
{I
-
sup
c)
a(t)<s<t
\u{a{s))- u{a{a{t)))\.
a{t)<s<t
Thus
||u(.)-ti(a(.))||t
and equation (9.36)
\\u^{-)
-
is
a)'=(.)||t
-
u,r,A^)
+
{I
<
u;,,,(x)
+
(l-c)|lu(.)-u(a(.))||t
proved. Equation (9.37)
(u o
-
<
+
c)\\u{a{-))
u{a{a{-))}\\t
proved similarly by observing that
is
+
<
||"(-
<
2||u(.)-u(a(.))||,+ r,
<
-w^,i+Tfc(^)
Tk) -u{a{-
n))\\t
+
|u(Tfc)
-
u(a(T^))|
2
where the
Next,
last inequality
we obtain from
is
an application of
(9.15)
(9. 36).
and (9.37)
<
ll"'(-)-"''(-)ll^
-\\x>'i-)-x''(-)\\t
+ ^\u'(-}-{uoa)\-)\\t
c
<
c
-J\x'{-)-x''i)\\t
+ 2(^'^y,,+TA^).
Finally, observe that
li'^'(-)
-
v''{-)\\t
Proof.
+
Wx'i-)
<
2\\x'{-)-x''i-)\\,
and the proof of the lemma
Lemma
-
<
is
x''{-)\\,
t
>
0.
We
will
construct the processes {w,y)
-
+
u''{-)\U
+(1 - c)\W'{-) - (u oa)'{-)\
+ 4(^^y,,+T,{^)
finished.
9.9 Suppose that x E C^ and a(t)
Fix
c\\u'(-)
make
—
=
t.
Then
^
is
continuous
at
(x,a)
use of the procedure described in the proof of
($, ^)(a;,a) in the interval
[0,<].
Lemma
Let 8 be the postive constant
used in the procedure, Tk the sequence of (increasing) times obtained from the construction
Tk
—
t.
We may
T)t-i),
and
let
n* the (finite)
assume that Tn« =
t.
iteration (starting with iteration 0).
number of
(tk
—
iterations required to construct {w,y) up to time
Denote by {w ,y
We
9.5 to
)
write (u*^,u^) to
32
the shifted processes defined on the k
mean
the processes obtained by applying
)
same mappings used
to (z',a') the
procodure
+
and w(t
Fix
\\z'
-
=
Tk)
w^(t) for
We
want
to
n and
[W -
a||,
0.
<
x\\t
nhtaining
(u'*, y*).
Wp
then define {v,u) by the concatenation
in (9.30):
>
(
in
"0
=
<
<
<
<
n{t)
i
~
q,
<Tk
~
9-41)
/*+i.
show that there
<
t
,
{
exists
>
rj
such that for any
£ C^ and
x'
a'
g A with
then
<f.
||vl',(i',a')-*,(x,a)||,
Note that
-
||*,(i',a')
From Lemma
<
*.(x,a)||,
||*.(x'.a')
we can conclude that there
9.2,
-
*.(x'.«)||,
exists
>
rji
any
\\x'
—
j;||(
<
r/i
If
.
we can show thai there
exists
-
||*.(x',a)
(9.42)
vJ/.fi-, a)||,.
such that
<'-
||v]/.(x',a)-*.(i,a)||,
for
+
r/2
>
such that
\\a'
-
a\\t
<
772
implies
\\^,{x',ci')-^.{r\a)\U<'-,
lemma
then the
is
proved by setting
rj
=
min(r/i
.
r;2).
Because
c^
\ f
i' is
continuous, there exists
>
»72
such that
f
/
A-l
and
"...(-')<*
where
/I
We
=
+
1
first
2/c.
We
[8
(i^)(i^)..(if^)]"',
assume henceforth that
.show that (iv,rj) and
(I'.ii),
\\a'
—
a\\t
<
(M4,
r?2-
constructed as described previously, satisfy the following
condition:
max||y,(.)-ti,(.)||T,
From
(9.45)
and
max||y,(-)-u,()||,
lemma
is
proved
First,
we prove
(9. 45).
|'
^^
(
/
(9.45)
we can conclude
(9.43),
so the
<2(i-^) i!_l L^^
if
<2(^—^j
(
-^—^j- j a-^
,,
(
we can establish tho processes we constructed
=
w^(k)(n) +
0\t)
=
v,(k){n)^A(t^Tk)-x\[n)
33
)<
^,
satisfy {v. u)
Define
a*(0
x'
r\(t+Tk)-x\{n)
=
(<I>,
*)(j'
.
a
).
where
=
j(ib)
^m{Tk)-
1 if
ifc
is
odd
=
and_7(ib)
2
even or zero. In either case, observe that a
is
A:
(t)
— l3
(t)
=
i>j(k)iTk) so
l|a'(-)-/?*(-)llt
We
if
+
<
l|yi(-)-"i(-)llT.
<
2max||y,(.)-u,(-)||T,.
||yi(-)-"i(-)llT.
(9.46)
proceed by induction and the observation that
sup
From
(9.38)
max
and
(9.46),
\y,{t)
-
|y,(Tfc)
-
+
Uj{Tk)\
||y,^(-)
-
u'^{-)\\t.
we can conclude
\yj(t)-Uj{t)\< (l
sup
<
u,{t)\
+ -)ma.x\\yji-)-Uj{-)\\T,+-2(^-^]u>r,^j{x').
(9.47)
straightforward to verify that
It is
max
sup
\yj{t)
-
Uj{t)\
-
=
\\yji-)
<
2(—2-) w^2,ri(-r')-
Uj{-)\\t,
Equation (9.45) thus follows with an inductive argument.
It
remains to show that {v,u) as constructed satisfy {v,u)
enough to show that
odd.
It suffices
Because
e'x'{t)
\\e'x'{-
Here, the
v'-,f,Jt)
>
<
for
to do so for k even, for the
=
e'w{t),
<
^fc+i,
where j{k)
argument
=
2
(<t>,
if
for the case of k
t
^)(j:', a').
is
To do
even and j{k)
=
.so,
I
if
it
is
A:
is
being odd proceeds similarly.
we have
+ n)-vU-)-uj'2i-)\W,,
first
<
=
=
l|u^f(-)-t'i'(-)lk,.
=
\\92i<^'.e)C)-g',{p',a')(-)\\t,^^
<
2\\a'i-)
<
4max||y,(-)-Uj(-)||T,
<
6.
-
l3'{-)\\tk+,
+
4
e'x'it
+
Tk)
(9.44), respectively. Hence,
-
vi{t
+ n) >
34
W2{t
^,„T., (^')
+4(^^)u;,„r,^(x')
inequality follows from (9.39); the second inequality
two inequalities follow from (9.45) and
(^i^)
is
a result of (9.46); and the last
by the definition of
+ Tk)-6>0
7^.,
for
<
<
< ^+1. But
V2{t
+
Tk)
=
e'z'{t
>
e'r'{t+n)-vx(t + n)
+
Tk)
+
{l-c){u:i{t
+ n)-U2(a{t + n)))-Vi{t +
Tk)
>
and the theorem
is
proved.
Acknowledgements:
I
am
I
grateful to
during the course of this research.
discussed
in
Section
I
Mike Harrison and Larry Wein
would
like to
for their helpful
comments
thank Marty Reiman for sharing the interpretation
4.
35
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