HD28
.M414
DEWEY
ALFRED
On
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
the Convergence of Multiclass
Networks
in
J.G.
Heavy
Queueing
Traffic
Dai
and
Vien Nguyen
WP# 3489-92-MSA
October, 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
On
the Convergence of Multiclass
Networks
in
J.G.
Heavy
Queueing
Traffic
Dai
and
Vien Nguyen
WP# 3489-92-MSA
October, 1992
MASSACHUStnS
0^^
INSTITUTE
rr-HMrjiOGY
NOV
1
4 2000
LiSRARIES
On
the Convergence of Multiclass Queueing Networks
in
J.
Heavy
Traffic
G. Dai^ and Vien Nguyen
Georgia Institute of Technology and Massachusetts Institute of Technology
Abstract
The
subject of this paper
is
the heavy traffic behavior of a general class of queueing networks
with first-in-first-out (FIFO) service discipline.
For special cases which require various as-
sumptions on the network structure, several authors have proved heavy traffic limit theorems
to justify the approximation of queueing networks by reflected
Brownian motions (RBM's).
Based on these theorems, some have conjectured that the Brownian approximation may
in fact
be valid for a more general class of queueing networks.
In this paper,
class of networks.
networks
we prove that the Brownian approximation does not hold
Our
findings suggest that studying Brownian models of
may perhaps be more
for such a general
non-FIFO queueing
fruitful.
Contents
1.
Introduction
2.
Preliminaries
3.
Conjecture and the Main Theorem
4.
A
5.
Proof of Theorem
6.
Concluding Remarks and Open Problems
Pseudo Heavy
Keywords:
Traffic Limit
Theorem
3.1
Multiclass queueing network, heavy
traffic,
diffusion approximation, reflecting
Brownian motion, performance analysis.
AMS
1991 subject classifications: Primary 60K25, 60F17, 60G17; Secondary 60J15, 90B22,
68M20.
October, 1992
*
Research supported
92-09586
in
part by two grants from Texas Instruments Corporation and by
NSF
grant
DMS
Introduction
1
The past few years have witnessed
a surge in research activities dealing with Brovvnian ap-
proximations of queueing networks
[3,
5,
10,
7,
15,
This
16].
line of research suggests that
certain processes associated with a queueing network can be approximated by reflected Brow-
nian motions (RBM's), and these approximations are justified by so called "heavy traffic limit
theorems." Although such limit theorems have been proved only for special cases which require
various assumptions on the network structure,
approximations
may be
in fact
some authors have proposed that
Brownian
the
used for a more general class of networks that operate under
the first-in-first-out service discipline [15, 16].
we prove that the Brownian approximation
In this paper,
networks.
class of
that
We
do so by proving a "pseudo" heavy
is
not valid for such a general
theorem, which states
traffic limit
the process associated with the queueing network converges to a continuous limit, then
if
that limit
RBM
must be the
queueing network example developed by Dai and
approximation
not well defined.
is
Our
We
Brownian approximation.
specified by the
Wang
[11] for
findings suggest that
then present a
which the specified Brownian
it
may be
fruitful to consider
a more general class of approximating processes. Furthermore, our findings signal that other
may
service disciplines
We
yield
tractable structures.
consider a network composed of d single server stations, which we index by j
The network
populated by
is
renewal arrival process E^
k customers
is
more
who
assumed that
=
c classes of
>
{Ek{t),t
customers, and each class k has
0} (possibly null), where Ek(t)
arrive at the network by time
=
£'^(0)
coefficient of variation
its
.
.
.,
.
.
number of
For each customer class k
t.
c^ ^.
mean.)
(The
We
SCV
of a
denote by
random
E the
.
,
tf.
=
class
l,...,c,
it
Ec are independent and a^ >
for at least
variable
is
defined as
its
variance
c-dimensional process with components
Ei,...,Ec- (All vectors are envisioned as column vectors.)
El,
the
1,
and customer inter-arrival times have mean l/a^ with squared
(SCV)
divided by the square of
is
=
own exogenous
its
one
We
assume that
For each
k.
ak
k,
is
arrival processes
interpreted as the
long-run average arrival rate for class k customers. These customers require service at station
s{k),
and
and their service times are independent and identically distributed
SCV
c^
1^.
The
station s{k), a class k customer
the network with probability
P =
(Pjt/) is
leave the network.
with
mean
m/t
service time sequences for the various customer classes are independent of
one another and are also independent of the arrival processes.
matrix
(i.i.d.)
1
becomes a customer of
— ^;
Pki,
class
independent of
all
/
Upon completion
previous history. The transition
taken to be transient, which simply means that
Hence the networks we
assume that the waiting buffer
of service at
with probability P^i and exits
are considering are
all
customers eventually
open queueing networks.
We
at each station has infinite capacity, and that customers are
served at each station on a first-in-first-out (FIFO) basis.
Hereafter,
we
will refer to
such a
network as a multiclass open queueing network.
Such a description of a multiclass network
Nguyen
[15, 16].
(The
class of
case of the setup in [15, 16].)
is
now
quite standard, as in Harrison and
queueing networks described here
Figure
1
is in
fact an
important special
shows an example of such a network, which Dai and
Wang [11] have studied. Customers arrive at station 1 according to a Poisson process with rate
ai = 1. Each customer follows a deterministic route whose sequence of station visitations is
1,
1, 2, 2,
1,
after
which the customer departs from the network. Hence, each customer makes
1
and we designate those customers
5 stops before exiting the network,
in their
Ic'-^
stop as class
k customers.
In his pioneering
paper on queueing networks, Jackson
[22]
assumed that customers
or occupying any given station are essentially indistinguishable
customer completing service at station
independent of
P,j,
customer
i
previous history.
all
was extended by Baskett
move next to station j with some fixed probability
Thus in Jackson's networks, each station serves a single
will
et al.
and Kelly
[1]
[24] to
networks populated by multiple types of
customers, each type following a deterministic route.
paper subsumes those considered
=
\,
.
.
.
,d and each
time
at
t,
Readers are referred to Harrison
>
<
Wj{t) denote the
0, let
customers who are queued at station j at time
any customer who may be
for
_;'
all
in [1, 24].
The routing mechanism described
that customer
in service
there at time
must wait Wj{t) time
t.
If
/,
the d-vector with components W\{t),
a
...,
sum
Intuitively,
when
the system
is
new customer
for station j.
Wd{t). Define the process
Set W{t) to be
Wj — {W]{t),t >
in
pj
pictured
in
t
will
1
1.11
P:
>
0,
is
a special
is
set to be
=
let
1,
us for the
J
form of the heavy
we are interested
how
in
1,
Section
fast
=
is
2).
busy.
the traffic intensities for the network
= m\ + m2 + m^ and
are given by p\
explanation of the underlying concepts,
Condition (1.1)
in
be large for
can also be interpreted as the long-run fraction of time that server j
Figure
0} and
the natural way.
heavily loaded, the workload \V(t) at time
As an example, because the arrival rate
f
arrives to station
Let pj be the traffic intensity at station j (this term will be be defined
t.
The quantity
fixed
of the impending service
units before gaining access to the server, so
W be the corresponding c/-dimensional workload process defined
large
and Harrison
plus the remaining service time
one can also describe Wj{t) as the virtual waiting time process
let
[14]
in this
[15, 16] for further discussion.
For each j
times for
from one another, and that a
hence these networks have been called single-class networks. Jackson's model
class,
and Nguyen
visiting
p2
=
'^3
moment assume
+
To
"^4.
facilitate
our
that
1.
traffic
W{nt) goes
condition as described in Section
to infinity as n
— 00.
It
2.
For
has been widely
believed that
W-(t)
W'^it)
(1.2)
= ^W(nt)
-^W{nt) =^
W'{t)
as n
—'CO,
v/n
where
W=
(SRBM) and
will
be
{W'{t),t > 0}
the
is
symbol "=>"
clarified in Section 2).
(1.2) does not hold in general.
a d-dimensional semimartingale reflecting
Brownian motion
denotes weak convergence (the notion of weak convergence
Our main
contribution
in this
paper
is
the proof that conjecture
QNET
Method
Performance Analysis
for
Exact analysis of a multiclass network models
arrival times have exponential distributions
same exponential distributions
station have the
commonly
limited essentially to networks whose mter-
is
and whose service times
BCMP
referred to as
(cf.
Baskett
et
for all
al.
customers at each
Such networks are
[1]).
For networks with other distributional
or Kelly networks.
forms, no exact formulae are available to facilitate the analysis.
In recent years, a promising
been proposed. The
new approximation scheme, known
step in a
first
QNET analysis
is
as the
QNET
queueing model by
to replace one's "exact"
an approximating Brownian model, see Harrison and Nguyen
The second
[15, 16].
method, has
step involves
steady-state analysis of the approximating Brownian model; see Harrison and Williams
Dai and Harrison
[7]
and Dai and Kurtz
[18],
For a queueing network with d stations, this
[8].
analysis requires that one determines the stationary distribution of a d-dimensional reflecting
Brownian motion.
summary
Finally,
from that stationary distribution are
statistics derived
used to obtain approximate steady-state performance measures for the original systems. Unlike
previous approximations, the Brownian approximations culminate
distributions; readers can find
Dai and Harrison
16],
QNET
method,
accuracy and
as well as its
complete sojourn time
for
Readers are referred to Harrison and Nguyen
[16].
and Dai, Nguyen and Reiman
7]
[6,
examples of Brownian estimates
Nguyen
distributions in Harrison and
estimates of complete
in
[10] for
Those interested
efficiency.
[15,
more discussion on the
in the
QNET
software
package can send an email message to dai@isye.gatech.edu.
Heavy
Traffic Limit
In cases
where
Theorems
corresponding theorem
(1.2) hold, the
There now
exists a variety of
structures.
The
Whitt
[20, 21],
first
fic
is
traffic limit
theorem
theorems
for
for
traffic limit
theorem."
networks with certain special
networks of queues
is
due to Iglehart and
smgle-class queues in series. For single-class networks whose
similar to that of Jackson's networks, but
may have
"heavy
general distributions,
Reiman
[26]
whose
inter-arrival times
proved that under the heavy
and
traf-
condition, the normalized queue length process converges weakly to a reflecting Brownian
motion (RBM). The
Reiman's proof was
[25]
later simplified
[5]
to
was
first
by Johnson
networks
proved an analogous heavy
The term
RBM
definition of an
Chen and Shanthikumar
son
heavy
traffic limit
who worked with
routing structure
service times
heavy
called a
is
in
presented
Reiman 's
[23].
in
Harrison and Reiman
result has
[17].
been extended by
which stations may have multiple servers. Peter-
theorem
traffic limit
for multiclass
feedforward networks.
"feedforward" denotes a routing structure in which stations can be numbered so that
customers always travel from lower numbered stations to higher numbered ones. Reiman
proved a heavy
Kurtz
[9]
traffic
limit
theorem
for a multiclass
have greatly simplified Reiman's proof.
A
one station feedback queue.
heavy
traffic limit
theorem
networks were discussed
Until recently,
it
Chen and Mandelbaum [3]. Strong approximations
in Glynn and Whitt [13] and Chen and Mandelbaum
was believed that a heavy
queueing networks of the type described
theorems, Harrison and Nguyen
[15,
limit
theorem should hold
in this section.
16]
Dai and
for single-class
closed networks was proved by
clciss
[27]
for single
[4].
for multiclass
Based on existing heavy
open
traffic limit
proposed Brownian models to approximate these
networks. Unfortunately, Dai and
Wang
have found two-station and three-station networks
[11]
which Harrison and Nguyen's Brownian models
for
(A more
to exist.
fail
example showing no convergence was given by Whitt
which he discovered chaotic
in
[30],
explicit interesting
Wang
behavior for certain multiclass open queueing networks.) Building on Dai and
we prove
in
paper the following general
this
There
result:
exist
A
by-product of our result
that
is
example,
queueing
multiclass open
networks for whicli the scaled workload process \V"(t) does not converge
limits.
s
any continuous
to
the normalized workload process converges to a
if
continuous limiting process, that process must be the reflected Brownian motion identified by
Harrison and Nguyen.
Organization of the Paper
We
introduce some additional notation and definitions
heavy
traffic
We
conjecture and the main theorem of this paper.
a "pseudo heavy traffic limit theorem," which
our main theorem
Section
in
is
we
Finally,
5.
we
2.
In Section 3,
state the
prove our theorem by way of
state and justify in Section
some open problems
discuss
we
in
4.
The proof
Section
of
6.
Preliminaries
2
We now
define several processes that will be used later in later sections. Let
be a sequence of
(i)
equals
if
1
routing vectors for class
i.i.d.
the
i'
(2.1)
cumulative
<j'=(r)
Finally, set C{j) to be the set of
CO) =
{k
:
s(k)
require that C{j) be
Next, set
C
=
all
This set
j).
nonempty
for
=
sum
0*^(1) -H...
=
=
^
from class k by time
from internal
t.
it
by time
t
.
.}
all
other components are
+
/(r).
_;
in
at station
Harrison
_;',
[14].
that
We
components
'ftGCO),
otherwise.
and by Dk(t) the
Letting Fk{t) denote the
transitions,
.
d.
1
i
o'^(2),
processes
the external arrival process for class
of customer visits to class
and
{<?l)^"(l),
component of the vector
called the constituency of server
is
each j
I
is
/,
Z'*^
customer classes k that receive service
C,,
Recall that Ek{t)
The
customers.
to be the d x c incidence matrix with
(2.2)
result
it
customer next goes to class
class k
zero. Also, define the c-dimensional
is,
Section
in
k.
Denote by Ak{t) the
total
number
number
total
number
of customer departures
of visits to class
it
by time
t
that
one has as a matter of definition
C
Ak{t)^ Ek{t) + Fk(t)^ Ek(t) +
(2-3)
J2'^'k(D,(t)).
1=1
Let {vk{l),Vk('2),
and
let Vfc(r)
.
.
.}
be a sequence of
be the cumulative
sum
i.i.d.
service times associated with class k customers,
process defined by
We
denote by V{A{t)) the c-dimensional process whose t'^ component
given by rt(.4i.(0).
is
and we set
Note that L
work
= CV(A(t)).
L(t)
(2.4)
=
>
{L{t),t
for server j
0}
is
brought by
a cf-dimensionai process; one interprets Lj{t) as the
who
those customers
ail
amount of
have arrived at station j by time
t.
The
process Lj was referred to as the immediate workload input process for station j by Harrison
and Nguyen
[16].
Let Yj{t) be the
let
We
Y{t)
=
(Vi(0,
•
•
be the corresponding vector process. (Prime denotes transpose.)
W=
W{t)^Lit)-te +
t
Yj{t)
finish
=
such that Wj(t)
(2.6)
To
{lV{t),t
>
j
Y{t),
= - mf {Lj{s)-s),
Q=
may
increase
hence
0,
our description of the network model,
(2.7)
t
0} as follows:
the d-dimensional vector of ones. Clearly, the idleness process Vj()
only at times
and
-^'(i(<))'
(2.5)
e is
to time
of cumulative idleness experienced by server
can express the d-dimensional workload process
where
up
amount
let
l,...,d.
us define the fundamental matrix
=(/+P 4-
(/-P')-i
=
j
P^
+
...)'.
The {k, /)'^ element of Q represents the expected number of visits
who starts in class /. Thus, defining A = {\\,
A^)' via
.
.
made by
to class k
a customer
,
.
A
(2.8)
one recognizes that \k
is
= Qa,
number of customer
the long-run average
visits to class k per unit
The
time (resulting from external arrivals as well as internal transitions).
at station j
is
total traffic intensity
then defined by
p,= Y.
(2.9)
^krrik.
kecu)
Let p be the vector of traffic intensities at stations I,.
traffic intensities in
To
which
have
is
the c x c diagonal matrix with diagonal elements mj,
rigorously state our convergence result,
is
the space
all
finite left limits
functions /
on (0,oo).
topology, see Billingsley
[2].
X
A:"(-)
e
One can
D''[0,oo),
we write
For a function
/
express the vector of
= CMX,
p
M
,d.
.
matrix form via
(2.10)
where
.
:
[0,
oo)
:
[0,oo)
—
=> A'()
— R and >
if
t
,
=
.
,
m^.
which
R'^
{^"}
D'^[0.oo)
of D'^fO,
put
sup
0<3<«
|/(s)|,
D'^[0,oo),
are right continuous on [0,oo)
JV" converges to
0,
.
we need to introduce the path space
The path space
For a sequence
.
is
endowed with the Skorohod
cxd)- valued
X
and
stochastic processes and
in distribution.
and
for a vector of functions
=
/
(/i
,
A)'
,
[0,
:
oo) -~ R*^
l<-(ll/iil<
A
sequence {/"} of functions /"
sets to /
(u.o.c.)
{X"}
:
[0,'3c)
— H^
[0,oo)
:
if
each
for
t
—
we write X"()
.V() u.o.c.
if
>
t
>
put
0.
IIAII*)'.
—R
of D'^[0. oo)- valued stochastic processes
and
is
-
||/"
0,
and
X
G
uniformly on compact
said to converge
f\\t
D'^[0,
^
—
as n
For a sequence
x;.
oo) defined on a probability space,
almost surely, A'" converges to
A'
uniformly on compact
sets.
Conjecture and the Main Theorem
3
In order to rigorously state a
of networks" indexed by n.
m"
Let a" and
the n'
heavy
traffic limit
Our setup
theorem, we need to consider a "sequence
here follows closely that of Harrison and Nguyen
[16].
be vectors of inter-arrival rates and service times, respectively, associated with
network
in the
We may
sequence.
assume without
loss of generality,
however, that the
routing matrix and the squared coefficients of variation for inter-arrival times and service times
remain fixed across the sequence of networks. Define p" to be the vector of
for the
matrix similarly to
n'-^
We
(2.9).
are interested in the sequence of networks such that
a"^a,
(3.1)
traffic intensities
m"
— m>0,
and
^{p^ -e)~
(3.2)
where
rate,
e,
as before,
and
is
known
is
— oo,
asn
(3,
the vector of ones. Condition (32) requires that p"
as the heavy traffic condition.
the normalized workload process
W^
rV"(<)
(3.3)
Before we state the conjecture,
we
As n
^
oo,
we
—
1
at
an appropriate
are interested in the limit of
defined by
= -^Wint),
define
t
>
0.
more scaled processes. For each
t
>
and
n
>
1.
set
V"
where
It
[x]
is
K"(^)
=
^(V'"(M)-m"n<),
1.^(0
=
-^{n{[nt])-P:knt),
Donsker theorem that as n
£"
(3.5)
(3.6)
,c,
the integer part of x. (Again, note that the processes
follows from the classical
(3.4)
1=1,
(!>••"
=>^',
,-=1,
— oo
k
=
<!>'
l,...,c,
do not change with
n
)
where
and
^^
^'',
4'
=
('
1,
.
,c) are (c
•
+
motions with covariance matrices F", F' and
=
that F"
diag(aic^j,
.
.
accl^), F'
,
.
^
.
independent c-dimensional zero-drift Brovvnian
2)
F'
=
(;
= diag(m^c^
Because Brownian motions are continuous and £",
can and
.
.
,c), respectively.
.
and
"^fcC^,,-)
j
i{k
P.fc(l-P,fc)
r
1,
V"*,
=
It is
easily verified
is
a matrix defined by
c)
are independent, we
F'
l
<!>''"
(i
=
1,
.
.
.
,
assume by the Skorohod representation theorem that the convergence
will
in (3.4)-
(3.6) holds u.o.c.
Conjecture
1
the heavy traffic conditions {3.l)-{'i.2), the sequence of
Under
load processes W"-
uniformly on compact
sets as
n
—
•
That
cxd.
3.1
There
{W'{t),t
u.o.c, as n
open queueing networks for vohich the scaled workload
exist multiclass
and proved
in the
Theorem
of
conjecture
If the conclusion
is still
traffic result
stated in
in [11].
We
Theorem
4.1
leave the proof
m
conjecture
changed
1 is
to
{lV'",n
>
1} being D-tighi. the
false.
definition of tightness
A
"pseudo" heavy
next section together with the Dai- Wang example
proof of the corollary
4
3.1 is the
false.
is
1
3.1 to Section 5.
Corollary 3.1
The
Theorem
to the proof of
0}
— oo.
process W"' does not converge to any continuous limit. In particular, Conjecture
The key
>
is
-=W''{nt) ~W'{t),
(3.7)
Theorem
W
defined in (33) converges to a continuous process
normalized work-
—
is
is
given
in, for
example, Section 3.2 of Ethier and Kurtz
given at the end of Section
Pseudo Heavy
Traffic Limit
[12].
The
5.
Theorem
Set
G = CMQP'AC,
where
M
=
diag(mi,
diagonal elements Ai,
t
for the n'^
Theorem
.
.
.
,
.
.
,
.
m,.)
A^.
=
diag(A), and diag(A)
Recall that Yp{t)
system, and Y"{t)
4.1
and A
is
is
1 is true,
the sequence of normalized idleness process
the diagonal matrix with
the cumulative idleness of server j by time
the d-dimensional vector with
Assume Conjecture
is
components V",
namely, that the convergence
Y^
converges
to
Y'
u.o.c.
and
.
.
.
,
VJ*. Set
in (3.7) holds.
Then
the limiting processes
.
{W',Y') must
satisfy
+ G)w'(t) = cr(Ao + c.\[Q Icit) +
(4.1)
(/
(4.2)
W'{t) >
0,
(4.3)
y "(0)
=
0,
(4.4)
yj'() increases only at times
V
Theorem 4.1 states
Harrison and Nguyen [15, 16] is
The remainder
some important
that
Section 5 of
in
is
1,.
.
.
,ci
and
Let r"(<) be the d-dimensional vector defined
is
slightly different
a concise proof of
>
t
4.2 below.
With
customers to have departed from station j
—
n^(i\
(A ^\
~
"'^!t(''7('))
J
*:n -
^
if
^'
4n(^n(,)j^
0,
will
in
r"(<),
=
is
t
>
and n >
1,
referred
complete proof
in this
3.1 in Section 5.
^'^"(O
Peterson
We
4.1.
>
0,
begin by introducing
and to be
page
[25,
i
if
lV"(f)
103],
=
0.
r"-{t),
enables us to give
one can verify that the number of class k
/
given by
is
currently serving a class k customer.
use /l"(r"(<)) to denote the c-dimensional process whose
For each
Nguyen summarily
define T^{t) to be the arrival time to
if
s{k) by time
server j
d.
Otherwise.
I
We
y'it).
the obvious manner. This definition of
in
from what was given
Lemma
+
I
offer a
devoted to proving Theorem
=
For j
We
[16].
theorem to prove Theorem
this
station j of the customer currently being serviced there
which
-^^
Brownian model proposed by
true, then the
is
=
0. j
the correct model. Harrison and
of this section
notation.
=
such thai W'{t)
t
(3.7)
if
pseudo proof)
paper and consequently use
+
^^('^^•oj
continuous and nondecreasing,
!s
Remark.
to this result (namely, the
E
/fc'*'
component
is .-ifc(T-^%.)(0)-
define
f"«)
=
-T^'int),
f"(t)
= -^{nt -
r"(ni)),
and
A^it)
= -.4"(nO,
.4"(0
n
The
first
Lemma
two lemmas below hold
in
=
{Sk(t),t
>
-
A"t)
general without the assumption of convergence in (3.7).
Wki-)\\t<^,
5j^
(.4"(nO
,/n
4.1 For each sample path and each
Proof. Let
= 4-
t
>0,
there exists k independent of n such that
t=l,...,c,
n>l.
0} be the renewal process associated with class k service times.
Let T^{t) be the cumulative time that server s{k) has devoted to class k customers in
of time. Then, the
is
D]^{t)
=
number
5?(T^"(t))
<
=
5n0
E^^it)
+ Y.
c
<j>'(5'n7r(0))
< E^t) + Y.
follows from the functional strong law of large
renewal processes.
t
Therefore, from (2.3), we have
k=i
The lemma then
units
of class k customers to have departed from station s{k) by time
c
A'^it)
t
^\sm))-
fc=i
numbers
for
random walks and
From the
definition of T^{t).
(4.6)
it
f
where e"(0
is
if
=
\Vj^(t)
0,
follows that
= r;(o + iv7(r;(/))-.;(o.
and otherwise
customer currently occupying server
We
are close for large n.
under the heavy
Lemma
=
I,
.
.
.
follows
from the
—=i^{nt) -~
^n
cation of
traffic scaling,
the processes
f^"(<)
and W^{t)
is
negligible
(2),
Lemma
.
.
is
.}
—
as n
u.o.c.
0,
:o.
'
definition of e"(f ) that
<
~
vjf
and note that
r^(t).
,d,
lim
where {vj(l),
—
traffic scaling.
4.2 For j
It
t
begin with the following lemma, which proves that £"(/)
n—oo
Proof.
=
= r;(o<ir;(r;(o).
show that under the heavy
results
equal to the remaining service time of the
Define r"(t)
j.
PV';(r;(<))-€;(o
The next three
is
f"(<)
max
< max
1^(1),
k^C{:) \<,<A2(t)
•'
the sequence of
service times for class k customers.
i.i.d.
An
appli-
3.3 from Iglehart and Whitt [20] yields
0.
v^'"<"'
D.
Lemma
4.3 Suppose the convergence
tn (3.1) holds.
f"(f)
where
e is the
— e<
Then
u.o.c,
d-dimensional vector of ones.
Proof. Let iy,"(t)
=
iPV"(nt). Then,
Kif"{t)) - -^]{nt) = -T^int) <
Because t"{s)
<
s for s
>
rv;(f;(t)),
0,
1
With the assumption of
Lemma
(3.7)
and
Lemma
4.4 Suppose the convergence
f"(0
—
4.2,
the
lemma
in (3-7) holds.
W'*(<)
proved.
Then,
u.o.c. as
9
is
n
— 00.
Proof. Because
-
w^ir^^it))
€^(t)
=
t
-
r;(f
)
< n-;(r;(o),
we have
rv^;(f;(0)
The lemma
Lemma
- ^s"("*) = ^;(0 < K(f^{t)).
4.5 Suppose the convergence
It
follows
— A<
4.2
and 43.
Then
tn (3.7) holds.
i"(<)
Proof.
Lemmas
follows immediately from assumption (37) and
u.o.c.
from (23) that
C
k=i
where E"(0
= ^F"(nf),
£)"(/)
A''{t)-Xt
(4.7)
=
=
^D^'int) and
$''"•"(0
=
]^4>^{[nt])
hv
=
/t
1
c.
Therefore.
E'^(0-af + Xl(<l>''"(^,"(/))-P(D^(0)
fc=i
+
P'(D"(i)
-
AC'f"(<))
- P'XC'ite -
f"(<)),
where we have used the fact that
A
and Pk denotes the
k^^
row of P.
= a+P'A,
Using equation
(4.5),
expression on the right hand side of (4.7) by ^"(f"(t))
we can replace D"(<)
when n
is
large.
Hence, by
the third
in
Lemmas
4.1
and 4.3 and functional strong law of large numbers, we have
hmsupp"()-
A-
lit
n—^oo
Because (/ - P')-i >
< limsupP'||/i"(r"())n— oo
Af"(.)||(
<
P' limsup ||.4"()
n— '>i
-
A
||(.
0,
hmsup||^"()- A||( <0,
n—»oo
and hence
lim
n—oo
|L4"()- A-
||(
=
0.
D
Lemma
4.6 Define for each
lit)
Suppose the convergence
t
>
0,
= Q Icit) + ^^*(AfcO - P'XC'W'it)]
in (3.7) holds.
.4"(<)
—
Then
T]{t)
u.o.c, as
10
n
— oo.
.
Proof.
First, note that
.4"(f)
*^''"
= £"(0 + Yl
+
(-^fc(^.sU-)(0))
P'.4"(f"(0)
- P'A"C"f"(0
and
c
Thus,
c
*:
+
P' (i"(f"(t))
-
P'
=!
-
+
r,(f"(0))
(A"C'f"(0 - kC'W'{i))
P' (^(r"(0)
-
nm
.
Hence
(4.8)
<
iii"(-)-'?(-)ii<
Because f"(s) <
s for all s
ii-^"(-)-r(-)iit
+
Eii<^''"(-^it(^r(fc)(-)))-^'(^/.-)ib
+
P'||i"(r"(.))
-
r?(r"(-))||t
+
P'||A"C'f"(-)-AC'^V'*(-)ll(-
>
and
j
=
1,
.
.
.
d,
,
+
P'lh(r"(-))
-
r?(-)lh
we have
IK(r"())-r?(r"(-))||f<IM"(-)-^(-)lkTherefore,
(4.9) (/
-
it
follows from (4.8) that
P')l|i"(-)
-
n{-)\\i
<
ll^"(-)
-
C(-)\\t
+
E
ll*^''"
(-^^(q^c)!-)))
-
^'( A.«)||<
fc=l
+ P'^r^'i)) -
r7(-)||(
+
P'||A"C'f"(-)
- AC'ir()||t
= at).
Again, note that
Q>
0.
Premultiplying both sides of (4.9) by Q, we have
\\A"{)-rii)\\t<QC{t).
Because C"(^) ~^ 0. ^^ have proved Lemma 4.6.
Proof of Theorem 4.1. To prove Theorem 4.1, observe that from
W'"(t)
By Lemmas
4.5
and
CV"
4.6
= CK"
(.4"(f ))
+
CM'^A'^it)
+ V^(p" -
e)t
(2.4)
+
and
(2.5),
V'"(t).
and assumptions (35) and (32), we have
(i"(0)
+
CA/'*.4"(0
- Cr (AO + CA/Q
+ ,A(p" -
(r(0 +
E
11
e)t
^''(^'tO
- P'AC'Py (o) +
/?t
—
n
u.o.c. as
>
Because the mapping defined
oo.
in
(2.6)
continuous.
is
Theorem
4.1 follows
immediately from the continuous mapping theorem.
Remark.
Because a Brownian motion
the matrix I
+ G must
be nonsingular
Multiplying both sides of (4.1) by
W'{t}
For each
>
f
is
order for a solution of the system (4.1)-(4.4) to exist.
+ G)~'
(/
RCMQ
= RCVi^i) +
one has
,
icit) + Y. ^^AfcO
)
+ R3t +
RY'(t).
0, set
X'{t)
Then X'
=
/?
almost surely not a process with bounded variation,
is
in
D
a
Brownian motion with
r-
RCMQ U" (0 +
= RCei^t) +
= R0
drift vector 6
MQ
= Rc ^^\ +
(
r''
^
+
^''(^fcO)
Y.
and
Atr'-'
+
R3t.
covariarice matrix
q'm' c'r'
]
W
Without worrying about the adaptness of these processes, one recognizes that
martingale reflecting Brownian motion (SRBM) starting from zero defined by
(4.10)
=
W'it)
,
and (4.2)-(4.4) with covariance matrix
and Williams
SRBM
[28] for the definition of
exists, the
matrix
R must
+
X'{t)
RY'{t),
T*, drift vector 6
<
and
>
reflection matrix R: see
SRBM
and
exists
Proof of Theorem
5
is
present the example
rem
does not exist for certain networks.
4.1
unique
Wang
Dai and
according to a Poisson process with rate
their k
1,
1,
2, 2,
1.
exponentially distributed with
m=
The
mean mk
show the
Figure
in
1.
(The index n
Section
in
1,
—
1,
(k
=
<
e for
1,
.
.
.
,
5),
each n and
lim
Customers
indicates the
v^(p"-e) =
12
(-1,-1)'.
IV
in
Theo-
arrive at station
n'*^
system.)
n.
1
Each
visited are in
we designate those customers
independent of
(1/10, 1/10, 22/27, 5/27, 8/10)' and
p"
limiting process
service times for class k customers are
"^^(-Tt)
Then a"
R
a completely-5 matrbc,
from the network, and the stations
As explained
stop as class k customers.
Choose
q^*.
5 stops before departing
the following order:
if ft is
an
in law.
[11] to
Consider the two-station network pictured
customer makes
if
3.1
Now we
in
[28]
Reiman
proved that
be completeiy-5. In particular, the diagonal elements of
are positive. Conversely, Taylor and Williams [29] proved that
then the corresponding
a semi-
0,
SRBM. Reiman and Williams
an
is
assumed
in
to be
=
For the specific data above, one can check that det(/ + G)
^
there exists a vector u
such that u'{I
Now, assume that Conjecture (37)
process and
+ G) =
thus
I
+G
is
singular
Tlierefore.
0.
By Theorem
true.
is
0.
the normalized workload
4.1,
idleness process (W'^,Y") converges to the limiting processes
[W.Y')
u.o.c,
where
+ G)W'{t) =
(/
(5.1)
.^(0
+
Jt
+
Y'{t),
and
m = ccm
is
a
Brownian motion with zero
/
+
CMQ
\
c
lc{t) + Y, etht)]
and covariance matrix
drift
C
r
=
cr AC + CMQ{r^ + Yi Ajtr*)Q'MC'.
Multiplying both sides of (5.1) by
It is
= -u'0t-
u'^(t)
(5.2)
easy to check that F
Remark.
If
Note that the
0.
Brownian motion
fore the conjecture
u'Y{t)^
for
alU >
0.
a positive definite matrLx and hence u'^
uTu >
motion with variance
variation, while a
is
we get
u',
left
hand
side of (5.2)
is
is
a zero drift
Brownian
bounded
a process of
almost surely not a process of bounded variation. There-
is
cannot possibly hold, and Theorem 3.1
is
Q
proved.
one takes
m=
R=
the reflection matrLx
{I
(1/10,1/10,23/27.4/27,8/10)',
+ G)~^ becomes
~
Because the diagonal elements of
Reiman and Williams
[28]
there
is
R
/
-310/27
16 \
\
20
-27 y
R
are negative,
SRBM
no
W
not a completely-5 matrix. Hence by
associated with the corresponding reflection
SRBM
matrix R. Therefore,
W^
Proof of Corollary
3.1 For x G D|^d[0,oo), define
can not converge to an
is
n
in this case.
/•oo
J{x)=
e-"[J(j,ii)Al] du,
/
Jo
where
d
J{x,u)= sup ^|j:,(0 -
c,(<-)
0<Ku;_j
It
follows
from
Lemma
4.2 that
j(py")_0.
almost surely as n
W' =
{W'it),t
topology)
is
>
—
oo.
Therefore, by
Theorem
3.10.2 of Ethier
0} of a convergent subsequence of {W''{-),n
continuous.
not exist. Therefore,
From
the proof of
{W^{},n >
1}
Theorem
3.1,
cannot be D-tight.
13
>
[12],
any limit
1} (under the
Skorohod
and Kurtz
we know that such
a process
W
does
D
Concluding Remarks and Open Problems
6
In this paper,
we have proved that conventional heavy
general multiclass open queueing networks.
such that the corresponding heavy
for the
moment.
We
In his
example, VVhitt
may
load process
traffic limit
conjecture that
with each server, the convergence
[30]
To
when
identify a
traffic limit
theorems do not hold
theorems prevail seems to be a formidable task
there
is
a single service time distribution associated
in (3.7) holds.
demonstrated that the non-convergence of the normalized work-
be caused by large fluctuations of the workload. In
[30],
tions occur because batches of customers with short service times build
way
to avoid such fluctuation
of-the-line processor-sharing)
is
to
up
employ some kind of processor sharing
among
these large fluctua-
is
just beginning, see the
One
discipline (like head-
the customer classes at each station.
ing disciplines. Research in this direction
the queues.
in
It is
investigate the heavy traffic behavior for multiclass queueing networks under
Williams
for
maximal subset of multiclass networks
worthwhile to
non-FIFO queue-
appendix
to Harrison
and
[19].
Acknowledgment. We
are indebted to
Ruth Williams
for
commenting on an
earlier version
of this paper.
References
[1]
Baskett,
F.,
Chandy, K. M., Muntz, R.
R.,
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[3]
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School of Mathematics and
Sloan School of
Management
Massachusetts Institute of Technology
Industrial/Systems Engineering
Georgia Institute of Technology
Cambridge, Massachusetts 02139
Atlanta, Georgia 30332-0205
USA
USA
email: vien@mit.edu
email: dai@isye.gatech.edu
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