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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
The Trouble with Diversity: Fork-Join Networks with
Heterogeneous Customer Population
Vien Nguyen
#3482-92-MSA
October 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
.
The Trouble with Diversity: Fork-Join Networks with
Heterogeneous Customer Population
Vien Nguyen
#3482-92-MSA
October 1992
M.I.T.
NOV
LIBRARIES
5 1992
iRkUuvicU
The Trouble with
Diversity:
Fork-Join Networks with Heterogeneous Customer Population
Vien Nguyen
Sloan School of Management. M.I.T., Cambridge,
MA
02139
Abstract
Consider a feedforward network of single-server stations populated by multiple job types.
Each job requires the connpletion of a number of tasks whose order of execution
is
determined
by a set of deterministic precedence constraints. The precedence requirements allow some tasks
to
be done
in parallel (in
which case tasks would "fork") and require that others be processed
may
sequentially (where tasks
straints, interarrival time distributions,
may
Jobs of a given type share
"join").
vary across different job types.
tiie
same precedence con-
and service time distributions, but these
We show
ciiaracteristics
that the heavy traffic limit of certain processes
associated with heterogeneous fork-join networks can be expressed as a semimartingale reflected
Brownian motion with polyhedral state space. The polyhedral region
faces than its dimension,
this setting.
networks as
effects that
One can
(i)
many more
and the description of the state space becomes quite complicated
in
interpret the proliferation of additional faces in heterogeneous fork-join
articulations of the fork and join constraints,
occur when jobs fork and join
KEYWORDS: ioxV-]o\n
typically has
in their
and
(ii)
results of the disordering
sojourns through the network.
networks, heterogeneous customer populations, reflected Brownian motion.
non-simple polyhedral state space, diffusion approximations, heavy
Contents:
Summary
1.
Introduction and
2.
Model Description
3.
Representations for Processes of Interest
4.
A
5.
Additional Notation and Preliminaries
6.
The Main Results
7.
Proofs
8.
Concluding Remarks
Sequence of Systems
in
Heavy
Traffic
October, 1992
traffic analysis.
Introduction and
1
We
consider in this paper the class of feedforward fork-join networks with heterogeneous customer
The network, which
populations.
consists of d single-server stations,
Each job requires the completion of
types.
a
to be
performed
in parallel (in
may
sequentially (where tasks
We
stations are
numbered
numbered
higher
We
in
populated by multiple job
The precedence requirements
is
allow
determined
some
tasks
which case tasks would fork) and require that others be processed
join).
Jobs of a given type share the same precedence constraints,
and service time distributions, but these characteristics may vary
interarrival time distributions,
across different job types.
is
number of tasks whose order of execution
by a set of deterministic precedence constraints.
is,
Summary
restrict attention to the case
where the network
is
feedforward: that
such a way that jobs always flow from lower numbered stations to
ones.
present a heavy traffic analysis for processing networks of the type described above.
was shown
Nguyen
in
that
[18]
when
the customer population
is
— that
homogeneous
It
when
is,
all
customers share the same precedence requirements, interarrival time distributions, and service time
distributions
— the heavy
traffic
behavior of the network can be approximated by a d-dimensional
semimartingale reflected Brownian motion (.SRBM) whose state space
hedral cone
in
the polyhedral region
strictly sequential processing) [11.
is
greater than
One can
d.
articulations of synchronization constraints
In this paper,
we show that the heavy
12,
13,
embodied
in
processes associated with hetero-
many more
SRBM's
faces than
with polyhedral state
its
homogeneous coun-
and the description of the state space becomes vastly more complicated
This result
is
surprising
when compared
of faces
the fork and join constructs.
traffic limit of certain
space. However, the polyhedral region typically has
number
the
19, 20],
interpret the presence of additional faces as
geneous fork-join networks can also be expressed as c/-dimensional
terpart,
a non-simple convex poly-
the nonnegative orthant. Unlike the corresponding results for conventional queueing
networks (networks with
in
is
to those a.ssociated with conventional
in
this setting.
queueing networks,
where the form of the limiting process does not change with the presence of multiple customer
types (this
is
a result of the "state-space collapse"
phenomenon)
[19, 21].
One can
interpret the
proliferation of additional faces in heterogeneous fork-join networks as results of the disordering
effects that
occur when jobs fork and join
in their
sojourns through the network.
Processing systems that are characterized by parallel as well as sequential processing exist
many
industrial settings.
Nguyen
[18] for
Readers
may
refer to Baccelli
and Makowski
a survey of several interesting applications.
The
[5],
Avi-Itzhak
generalization of
[18]
[16],
in
and
to allow
multiple job types constitutes an important extension from the practical point of view. Most current
treatises of fork-join networks
Liu
[4]
consider a fork-join network
more than one
sizes.
assume that
Baccelli
in
all
customers are statistically similar
[5].
Baccelli
and
which a job may send batches of tasks (that may include
task) to processing stations, and jobs of different types send batches of different
and Liu
still
assume, however, that
all
jobs share the same feedforward deterministic
Such a model can represent,
routing structure.
stations are capable of performing
example, systems
for
more than one kind
in
which some processing
and Liu are motivated by
of task; Baccelli
multiprocessor systems running parallel programs.
The
Mandelbaum, Nguyen, and Schwerer
recent works by Adler,
network model
studying new product development. The model they describe, which they simply
for
a "processing network model,"
call
The
studied here.
visit
key restriction
workstations
paper, which
in this
a feedforward
in
more encompassing than the
is
not assumed
is
in
is
We
organized as follows.
must
refer to a
Section
We
Finally,
7.
end
summarized
are then
:
on (0,oo). The space C^fO.oc)
in
W
[0,oc) -~
D'"[0,oo) and A'
is
model
in
Section 2 and
these
traffic limit results for
Section
in
The proof of
are given in Section
with some technical preliminaries.
product space of functions /
of processes
an
is
Before
we introduce some additional notation and preliminary
some concluding remarks
this section
yet an
is
Section 4 defines such a sequence.
the heavy traffic limit theorem for several special cases.
in
heavy
In order to state the
"sequence of systems."
The main theorems
5.
the network
the customer population
in
give a formal description of the
3.
stating the heavy traffic limit theorems,
results in Section
in
that jobs must
must be captured.
define the processes of interest Section
processes, one
in [1, 2, 3]. is
future work. However, as the work by Adler,
Mandelbaum, Nguyen, and Schwerer demonstrates, heterogeneity
essential characteristic that
networks
class of fork-join
manner. The possibility of feedback
important generalization that must be considered
The paper
propose a processing
[1, 2. 3]
6,
illustrate
these theorems are then given
8.
The space
D'^[0,cc)
that are right continuous on
[0,
endowed with the Skorohod topology
G D''[0.oo), we write A'"^=>A'
where we also
to
mean
>c)
[6].
is
the r-dimensional
and have
left
limits
For ,V" a sequence
A"" converges to
X
in
distribution.
For /
:
[0,.Do)
^
>R,
set
=
WfWt
sup
|/(.)|,
0<J!<(
and
for a vector- valued function
/
=
(/i, /2,
|t
A
sequence of functions {/"} converges
each
t
>
0,
||/"
-
f\\t
^
/r
)'
[0,
:
oc)
—
9?'",
we
let
(ll/l||<,..-,||/r|lt)'.
to a function
— oc. For a
A'" — A' u.o.c
as n
process in D'"[0,oo), we write
compact
=
.
/ uniformly on compact
sequence of functions {A'"} on
if
.sets (u.o.c.)
D'"[0, >:
almost surely, .V" converges to
A'
)
and
if
for
A
a
uniformly on
sets.
In our
heavy
traffic limit
theorems, the weak limit obtained
nian motion whose state space
is
a polyhedral cone
in
is
a .semimartingale reflected Brow-
the nonnegative orthant.
process having drift vector d and covariance matrix P will be denoted as
martingale reflected Brownian motion with these
R, and state space
is
S
is
denoted as (5,
^, P,
drift
(0,
A Brownian motion
r)BM;
and covariance parameters,
/?)SRBM.
likewise, a semireflection
matrix
TYPE
TYPEl
Figure
1:
Tasks and Precedence Constraints of
consideration consists of d single-server stations and iiosts p types of jobs. Jobs
of type q arrive to the network at rate a,.
activities.
we
Hereafter,
Aq denote
Each type
_;'
=
and we denote by
s{k),
the set of tasks (or classes) in job type
A = AiU.. .UAp Our convention
tasks by
fc,/
=
The order
straints,
.
.
mean
write q{k) to
be to index workstations by
will
1,
in
job requires completion of a number of
q
refer to a job-type/activity pair as a task or a class interchangeably.
k receives service from station
Letting
Job Types
Model Description
2
The network under
Two
2
.
,
A'.
which tasks are executed
is
which are articulated by way of a
j
1
if
task k
is
mean
service time for task k.
set
A'}.
{1
=
the
I.
.
.
.
job types by q.r
,d.
=
For notational convenience, we define A^
the job type q for which k £
_
i.j
q,
r^
Task
a, for
—
tasks k
all
1.
.
..
and
,p,
£ Aq and we
Aq
determined by a
A' x A'
set of deterministic
precedence matrix P
=
(Pyt/)
an immediate predecessor for task
precedence con-
defined as follows:
/
(2.1)
'it;
otherwise.
(Because
all
elements of the precedence matrix P are zeros and ones, routing of tasks
We
deterministic within each job type.)
of P such that the resulting matrix
that each
tcisk is
we can now
is
strictly
begin.
exists a
upper triangular.
performed exactlly once and
is
In
never repeated.
clearly
column and row permutation
terms of the model, this means
From
the precedence matrix P,
define the set of immediate predecessors as
=
V(i)
From
assume that there
is
the modelling point of view, V{1)
is
{keA
Pki
=
1}.
the set of tasks that must be completed before task
(2.2)
/
can
Figure
2:
Tasks and Precedence Constraints of
For the two job types depicted
the precedence matrix P
is
in
Figure
Two
Job Types
A\ = {1,2,3,4}, ^2 = {5.6.7},
1,
'P(4)
=
{1.3}. and
given by
^0001000
10
10
0000000
P=
1
1
We
allow tasks to
may be
We
map
capable of performing several types of tasks, each task
define the constitutency of station
/
=
C(i)
We
write c(i) to
mean
a{j)
is,
is
1
=
.
.
.
,
is
performed at exactly one station.
as the set of tasks tiiat are served at station
r/
{keA
-.sik)
=
for station
_;
=
1
.
.
.
,
J as
i:
(2.3)
/}.
the cardinality of the constituency set Cii).
and successor station mappings
That
many-one-to fashion, implying that while each station
to stations in a
Next, define the predecessor
follows:
Tr(j)
=
{i
= s{k):h<£V{l),l€C{j)}^
(2.4)
a(j)
=
{i
=
(2.5)
s{k)
iev(k).tec(j)}.
k{j) denotes the set of stations whose output feeds directly into station
the set of stations that receive input from station j
In
j,
analogously,
conjuction with the predecessor and
successor station mappings, we
now
define a d x d 'routing'" matrix IP
=
whose elements are
IP,j
given by
'
if/GT(j)
1
=
IPu
We
assume that there
is
network
(2.6)
,
otiierwise.
column and row permutation
a
upper triangular;
strictly
i
terms of the model,
in
of IP such that the resulting matrix
means that we assume jobs
this
such a way that jobs never return to a station
in
previously visited.
it
among
condition implies that there are no precedence constraints
constitutes the feedforward routing assumption stated in Section
It will
0.
be useful to think of new arrivals to the network
With that interpretation
in
mind
let
=
iiV{k)
accordingly. Finally,
=
{i-G A,
Al
=
{keA,:Pki =
;
P/t
(or equivalently
that cr(0)
^
= Ofor
in (t(0) receive
consisting of four workstations.
7r(4)
=
{1,3}, and a{0)
A node
j
is
than one task
/
=
predecessors
We
if
G
t
node
is
=
if it
join
G Vik) have completed
involve a joining of tasks, this simply
which
its
last
At join nodes, the
{/:^(i)
(2,7)
and ends with the tasks k £ A^.
.4°
>l°(<:)). let
(2.8)
s(0)
=
0,
and redefine
(2.4)-(2.5)
=
{0}}.
customers, we argued
in
Figure
7r( 1
)
=
1
are processed in a network
7r(2)
=
7r(3)
{0},
=
{0,2},
contains a task k G C{j) such that k G 'P(0 for more
node
if
'P{k) contains
in [18] that
is
more than one element
for
said to be complete, or a unit,
some
if all
of
their service.
at each station in a
FIFO manner. At nodes
means that the tasks
are served in the order of their arrival
arrival time of a (complete) task
predecessor completes service.
local, or station-level, information.
in
G
For this example, we have
assume that tasks are served
to the station.
station
only external arrivals and the feedforward assumption guarantees
Similarly, station j
/
"dummy"
OkrMleA,}.
constituent task k G C{j). At a join node, a type k task
its
a
{1,2}.
said to be a fork
.
from
/e A,}
all
Figure 2 shows how the two job types depicted
0.
1.
we define
<T(0)
Thus, the stations
In addition, this
us define for each job type q
A°
=
traverse the
the tasks at each station. This
as originating
Clearly, processing of a type q job begins with those tasks k
Next, set V{k)
is
is
that do not
defined to be the time at
Note that such a service discipline considers only
For the special case of fork-join networks with homogeneous
such a policy
is
equivalent to the global scheme of serving tasks
the order of the arrival of their associated jobs. In this setting, where different customer types
traverse different routes through the network, one oiiserves a fundamentally different
In particular, a task corresponding to a later arrival
may
enter a
downstream
tasks associated with earlier jobs. For example, consider the scenario depicted
shows the status of
five
jobs
in their
intermediate stages of processing.
phenomenon.
station before those
in
Figure
The network
3,
which
contains three
(SS)
Order of
Figure
type
1
3:
A
Fork-Join Network with Jobs
on the buffer between stations 3 and
2
Figure
in
2.
in
We
job (the third job to arrive to the network) has overtaken the
the
his,
first
a downstream station
station
is
if
server
job to be processed by server 4
the order of their arrival to a station
may
may
the order
1,
Each job
1,2, 1,2.
ask the reader to focus attention
completes
1
will
his
first
two arrivals and
is
first
type
the
first
next task before server 3 can finish
be the type 2 job. The policy of serving tasks
in
therefore result in serving jobs out of order. Moreover,
incur delays due to the need for resequencing tasks (for example,
if
the
a join node) that were overtaken by other tasks at upstream stations (for example, due to
forks). In Figure 3, for instance, server 4
is
©
hereafter referred to as buffer (3,4). Note that the
4,
job to reach buffer (3,4). Moreover,
O
(i
Intermediate Stages of Processing
in
jobs and two type 2 jobs, which arrived to the network
type follows the routing requirements described
DA
arrivals:
nonempty.
In this
must remain
paper we investigate how such
idle
even though each of his incident buffers
a disordering
is
manifested
in
the heavy
traffic
limit.
3
Representations for Processes of Interest
To construct the
basic stochastic processes associated with the fork-join network,
probability space
(fi, J",
1}
and {vk(i),i >
unit
mean. As
will
1}, q
P) on which
—
are defined .sequences of unitized
l,...,p, k
be explained
in
=
1
A',
where
the next section,
u,(i)
and
random
i;jt(i)
where each
us
assume a
variables {»,('),
'
>
are strictly positive with
we require very weak assumptions regarding
the joint distribution of these sequences of unitized variables. However, readers
to think in terms of the concrete case
let
is
a sequence of
i.i.d.
may
random
find
it
iielpful
variables and
From
the sequences are mutually independent.
these sequences, the interarrival times and service
times for the network are constructed by setting the interarrival time for the
be a~^Ug(i). and the service time for task k of
arrival rate for
for each k
new type
jobs and
q
mean
the
is
r^.
this
job to be
r;^-
(
)
1 7,/
job of type
/
Recall that
.
q\j
q to
the average
is
service time for task k. Also recall that
=
A;-
a,
E Ag-
To construct the external
=
arrival process for type q jobs, set Uq(0)
Nq{t)
= max{m
:
2J
a^ '"^(0
<
and define
i}
1=0
For k
=
1, ...
be the partial sums process associated with the service times for tasks
A', let \]:(t)
k,
[t]
Next, define for each k £ C{j) and j
=
L,Ut)
The process
all
Ljk(t)
is
Vk(Ng^k)(t))
=
=
1,
.
.
.
,
d,
u-n-( 1)
+
+
k
nv^,{^\^,^(f)),
e
C(j).
called the class k total workload input process for station
_;;
task k service times associated with those jobs that enter the network during
Ljk{t) includes service times corresponding to tasks that
time
t.
it
is
[0,/].
the
sum
of
Note that
not arrive to station j until after
Set
\keCu)
because
may
(3.1)
t
difference
is
amount
the potential
j
of work that can be processed in
t
units of time, ij{t)
between the workload input and the potential workload output, and
for this
is
reason
the
it
is
called the total workload netflow process at station j.
Let us choose an "external" station j G
set Ajk{t)
=
Next
^q{k){^)-
external arrivals, Ajk(t)
Similarly Mjk(t)
is
is
let
the
amount
the
Mjk(t)
number
=
further notice.
cr(0), fixing j until
Ljk{t) and A'j(/)
=
<,j(t).
For each k G C{j),
Because station
j hosts only
of task k that has actually arrived to station j by time
of task k work that has arrived to station j in
[0,/]
and Xj[t)
is
t.
the
corresponding immediate workload netflow process at this station.
From
Section 2.2 of
[10],
we can
verify that the processes
Wj and
Ij
are uniquely defined by
the following three statements:
Wj(t)
Ij{-)
is
=
Xj(t)
+ !j{t)>0
for all/
>
(3.3)
0;
continuous and nondecreasing with lj{0)
Ij(-) increases
only at times
t
when Wj(t) =
0;
=
0;
(3.4)
(3-5)
moreover,
Ij
mapping
given by the continuous
is
/,(/)
One
interprets
= -
as the cumulative idleness process for server
Ij
process at station
That
j.
in service.
anywhere
in
we
If
=
=
Wj{t) as a consequence of j G
process for server
Next,
set r]j{t)
k in
[0,t],
it
=
amount
^j(t)
+
Hereafter
o'(O).
be the arrival time of the customer
otherwise, letting Yjkit) be the
t
follows
€ik(t)
eijfe(0 's
=
present
is
I,(t),
(3.7)
we
refer to Zj{t) as the total
from the FIFO service
in service at station j at
amount of time
workload
the
amount
As
otherwise.
server
_;'
time
t
if
Wj(t)
>
0.
has spent serving tasks
discipline at each station that
YMt) = MMnAi)) +
where
of work for server j that
_;'.
let r]j[t)
and
of the impending service times for
then
t,
Zj(t)
and Zj{t)
and Wj as the immediate workload
_;'
plus the remaining service time of any task that
/,
define Zj[t) to be the total
the system at time
(3.6)
sum
Wj{t) corresponds to the
is,
(complete) tasks waiting at station j at time
may be
'nf{-V,(s)}-.
0<s<t
en{t),
(3.8)
of service the current task has received
a matter of definition, qj(t)
is
if
that task
is
of class k and
bounded by the immediate workload process
as follows:
Wjirjjit))
where e2j(0
=
if
occupying server
j.
Wj{T]j(t))
=
t,
denoted as Djk(t),
is
t
-
n,{t)
=
.
.
.},
t,
then for each
_;'
/
For each task k
=
follows
it
=
the service time of the customer currently
k to have departed from station j by time
all
(3.10)
of partial work associated with tasks k that
=
II;.p(^'(
)
(3.11)
Ujk{t)-
C(j) the processes A',(<), W,(t). l,{t). Z,{t), D,i{t),
A,k{t)=
its
and
U,t(t)
is, if
j
€
^(j)
have been defined.
arrival process to be;
''^'
'''^'
{
I
Con-
to all stations in the network.
immediate predeces.sor stations have been "treated", that
C(j) one defines
is
from the previous definitions that
manner, these definitions can be extended
such that
£
(3.9)
Ljk(t)-yjk{t).
a trivial consequence that Zj{t)
e
e2j{t).
Sk{Yjk(t)).
amount
Ujk(t)
sider a station
+
then given by
present anywhere in the system at time
In an inductive
's
number of tasks
the
Finally, defining Ujk(t) to be the the total
it is
\V,{r^J(t))
{Sk(i)J > 0} be the renewal process associated with the task k
Djkit)
Moreover,
<
and otherwise e2j(0
Letting Sk
service times {T^Vkil), TkVk(2),
<
m\n,^p^^.^D,^l),(t)
otherwise.
(3 12)
One can
(We
interpret Aj^it) as the
number
take the convention that work
is
of complete tasks k that have arrived to station j by time
t.
associated with complete tasks, so incomplete tasks present
no work to the server.) The immediate workload input process and immediate netflow process
for
station j are defined, respectively, via
Mjkit)
=
=
VkiAjkit))
+
rt [(^.(1)
.
.
.
+
r;.(.4j;..(0)]
(3.13)
and
\
-v,(o
(
=
\keC{j)
The workload
process Wj, the idleness process
Ij,
J
the total workload process Zj. the departure
process Djk, and the partial workload process Ujk are then defined exactly as
vector processes
A'^,
L,
V^.
The throughput time
.4,
W
.U, A',
of a job
is
,
Z, D, and
/,
U
in (3.3)-(3.11).
are then defined in the obvious manner.
the length of time between the job's arrival and
departure from the system. Let Tg(t) be the throughput time of the next type
network after time
t.
The intermediate
time through task t," which
of definition,
is
the
process
amount
The
k
Tfjf.(t),
6 Aq,
is
<]
its
subsequent
job to enter the
defined to be the "throughput
of elapsed time until task k
is
completed. As a matter
we have the relationship
Tq{t)= max{T,i(0}.
To
define the intermediate processes Tgk(t),
%{t) = a;'
u{l)
we
first
define for each job type q the process
+ a-'u(\{t)) + a;'u{Nq(t) +
+
I),
interpreted as the arrival epoch of the next type q job to enter the network after time
task k
G A°,
Because a type
=
and
4>,(0
job begins immediately with tasks k G
q
to station s{k).
Furthermore, because tasks are served
of time this task
must spend
immediately after
is
its arrival
at station s{k)
is
For other stations
in
k
^qk(i)
's
the arrival time of this task
in a first-in-first-out
amount
of
manner, the amount
work found at station s(k)
(which includes the service time associated with the new
the network, the
Suppose that
>^°,
precisely the
the total sojourn time of the job through task
as follows.
For each
let
$,fc(f)
Tqk{t)
t.
is
random
=
<!',(<)+
Thus
k.
processes 4>,i.(0 and T,a.(0 are inductively defined
a task such that T,/(/) has been defined for each
$,/,(«)
arrival).
max
7^^,(0
/
6 V{k), and
.set
(3.15)
l€V[k)
Tqkit)
=
^nmxTqi{t) + W,(k^{^qk{t))-
(3-16)
Recall that the arrival time of a task
task
Thus, maXi^-p,i^.Tgi{t)
tasks.)
s(k), ^qk{t)
is
precisely
its
G
the
is
amount
precedessor task
C{j), let pjk
—
is
Sequence of Systems
theorems stated here apply
limit
completed.
the throughput time through task
in
Heavy
k.
Traffic
systems that satisfy conditions of "heavy
to
(If
multiple predecessor
its
be the workload factor at station j associated with tasks
AfcTt
is
of time that elapses until task k •arrives" at station
time of arrival, and Tgk{t)
A
4
it
its last
requires a join, there could be a gap between completion times of
it
The
the time at which
is
k,
For
traffic."
and define the
total traffic intensity at station j to be
keC(j]
keC(j)
The system
is
said to be stable
"approximately"
if
<
for
1
j'
=
1,
.
.
.
,
and
J,
it
is
said to be in heavy traffic
The precise formulation of our heavy
for each j.
1
pj
traffic limit
if
pj
is
theorem requires
the construction of a "sequence of systems," indexed by n, whose corresponding traffic intensities
converge to
pj
1
for all j.
Recall that the interarrival times and service times for the network are defined
the basic sequences of unitized
k
=
1,
.
.
.
constants
,
A'.
{oiq
random
To construct a sequence
,
"
^
variables {uq(i)
of fork-join networks
,n>l},q =
l}i {'"t
:
\,...,p,k
time
X]^
for task k.
and
^
in
n*^^
Setting
AJ."
=
place of Xk and
The convention
here
the superscript "(")"•
the n
system, a,
is
a,
>
1},
{('/;(/)
In the n
=
ii,(!)/ag
(i)
the arrival rate of type q jobs and
is
for k
£ Aq,
:
'
>
1},
=
9
we further require sequences
— l,..,K.
the interarrival times and service times are taken to be Uq
respectively. For the
/
in
terms of
1.
.
,p,
of positive
system of the sequence,
and
r^."
= ^
^fc(')'
mean
service
i^" (')
is
define the traffic intensities p'"
the
as in (4.1) using
Tj.
to denote a
parameter or a process a.ssociated with the n"* system by
For example, Nq
refers to the external arrival process for type 7 jobs in
system. Define the centered processes
ir\t)
assumed that the following conditions hold
It is
mean
the arrival rates and
for the uiput processes of the
service times converge to finite constants,
—
-
Aj.
k
=
1,
.
.
.
,
A'.
This implies that p
exists a d- vector 9
=
{dx
...
,
—
-^ p^
Condition (4.2)
but also that the rate of convergence
it
assumed that there
is
limit
-Oj
is
^
1,
It
To explore
jV*,
-
0j
it
<
Q
is
r^.
and
>c
oc.
and
"'sufficiently fast"
—
and r^.
n"
assumed that there
is
(4.2)
requires not only that p^
uniform for
=
all
1
at each station,
stations.
Finally,
such that the following functional central
cxd:
(iV",l/",L")=>(,VM'*,L*),where L'
and
— cc' <
.... d,
as n
a d x d covariance matrix
is
theorem holds as n
—
—
called the heavy traffic condition.
is
)
,64) such that for each j
ni/2(p(")_l)
Furtiiermore,
'^*'"''*--
Z];.pC(
Xf.
^/c
•
fc
In)
network. First,
V are
is
a
(0,^) Brownian motion
Brownian motions with zero
also
drift.
(4.3)
the implications and restrictions of assumption (4.3), write the scaled netflow process
(3.2) as
E
^;(<)-
+
i,\.«)
n'/'(p;"'-i)/.
fceC'(0
It
follows from (3.1)-(3.2) and a.ssumptions (4.2), (43) that
i;;-(0
=
Cj{t)
=
V:{\ki)
Yl
+ TkN;^^p),
^Jfc(0
+
and
(4.4)
(4.5)
^j<.
keCij)
Defining the d x
c
constituency matrix
C
C,k
with elements
=
/
S
(_
1
ifkec(i)
,,
IJ
otherwise
(4-6)
,
and setting
r
= CQC,
(4.7)
one can conclude that
(Ar",K",L",,e")^(^*>'*,L',r). vvherer
Next, recall that Sk
Theorem
1
of
is
[14], (4.3)
is
(^.r)BM.
the counting process associated with the partial
i^-^)
sums process
From
implies that
S]:^-T-^^h'^.
Finally,
I4.
consider the special case in which {uq{i),i
independent sequences of
i.i.d.
random
(4.9)
>
1}
and {vk{i},i >
1},
are mutually
variables such that Ug(i) and Vk{i) have squared coefficients
11
of variation c^. and
defined to be
rate A,
,
its
c^^^
respectively (the squared coefficient of variation of a
variance divided by the square of
its
mean). Then
and a simple application of the functional central
proves that Ng
converges to
=^N^,
(0,n)BM
where
by
A^^'
is {0,
Theorem
Xc^g)BM.. Because L ^
is
theorem
a
is
+ <(t))
variable
is
a renewal process with
for
renewal processes
compound renewal
2.1 of [23]. In particular, the covariance matrix
^fc^'(d-
Qu
limit
A',"
random
is
[6]
process, L"
of the form
where
/i
=
1,
.
.
.
,
c(/);
=
convention that c(0)
—
I2
0.
.1
c(s(x'(
1
For a station
i
\\.
));
.;
.
-1
=
;.
.
.
./././'-2
c[s[x^~^
1
//,_i
^
_
)),
and we take the
with depth greater than or equal to h and x
^'T
(/),
we
define the set of indices
=
C\x)
{/
=
=
lk)-h
(/,
c(0;/2 =
l
c(s(r,\))
l
:
...;
In addition, let
r^'-tx)
Taking the network
=
{/
(/i,
w^
=
(1,5)
w'i
x^
(1,6)
2-?
y'
=
=
(3,5)
zi
=
(3,6)
T
(4), for
lemma can
Lemma
5.1 For each station
=
(0,0)
u^^
(2,0)
x]
y?
=
=
(0,0)
y|
z?
=
(2,0)
-I
{(1,1), (1,2), (2,1), (2,2)}
Cjix)
=
{(1,1)}.
£T
T
=
=
=
=
(4)
= (w.x.y,:) where
(0,0)
(0,0)
(5.6)
(2.0)
(2,0)
Moreover, Cj(x) identifies the particular "branch(es)"
)'.
Using the notation established above, the
_;'.
be verified directly.
i
=,.
.
.
,d and
max
6/^;
a
sequence of numbers associated with tasks kj,
^
= max
bki
The
b^.
Ma,in Results
6.1 Suppose thai assumptions (^.2) and (^.3) hold.
(C ir
=
(5.5)
j}.
as describing a "path" of tasks traversed by the various
''*(/')
6
i
=
and
=
V
where for each
2
)
=
the path i that would include a visit to station
following
i^
£2(1.)
think of each element x
Theorem
s(xf,
:
which we have
job types on their sojourns towards station
in
e £''(x)
...,/,)
Figure 2 as an example, we have d(4)
in
For the moment, consider x £
One may
=
I,
.
^''
.
.
,d and k
is a
{d,r)
,
G
z" M/"
,
C{i):
,
r )=>[C
U^ =
Browman
,
v
,
z'
,
Then
w
,
r
),
0,
motion;
13
(61)
.
'-
s{l)l
(6.3)
(6.4)
U'
=
U'k
max
+ max
Z'
P,k
teV(k)
^n,
Ps{l)i
/*
is
/'
increases only at times
continuous and non decreasing with /'(O)
W'(t) =
with
t
(6.5)
\neP[m)
.
—
P,[n)n
0,"
(6.6)
0.
(6.7)
Set
if
=
7fc/
=
i-
or
l-Ps(k)k-
'fi--/.
-Ps{t)h
if
=
/
0,
otherwise
(6.8)
I
For notational convenience, we henceforth write
of a station
i,
define for each x
£
T
and
(i)
I
1^
1
.
mean
to
s'"
.
.
,
six]"-
Denoting bv h the depth
).
,
d the following factors:
E
+
We
=
ji'
i-
then define the convex polyhedral cone
(6.9)
S
to be
d
n
^ =
lev
It is eeisily verified
that
>
2
=
-<;
(
'1
,
,
^d)'
-
-'.
E
^u(-^-)-i
(i)
if r
€ 5. For each
/
=
1,
.
.
.
,
cf,
we
also define the
d
F'=
6.2 For each
^'
i
is
—
1
a (5, F)
.
J
I
=
boundary
set
1
ce5:r, -^/i„(x)r, =0
(J
d(.).
Theorem
(6.10)
^
(6 II)
i
d,
Brownian motion;
(6.12)
(6.13)
W: = Z:/* is
/'
max
\'^l3„(x)Z'];
(6.14)
continuous and nondecreasmg with
increases only at times
t
14
with Z'{t)
/,'(0)
G F'
=
0.'
(6.15)
(6.16)
That
a
the process
IS,
Z'
SRBM
convex polyhedral cone S- The
R=
where I
I
Remark:
in
is
The
Theorem
as defined in
6
Z' has
1
a
is
drift 6,
d-diinensional
SRBM
whose state space
is
covariance matrix F, and reflection matrix
the d-dimensional identity matrix.
Nguyen
results of
may be
[17]
SRBM
applied to show that such an
is
well defined
the strong sense.
To
interpret
Theorem
note that statements (6.13) and
6.2,
(6. 15)
are reiterations of the charac-
terizations given in (37) and (3.4), respectively. Moreover, the approximation (6 12) of the netflow
process by a Brownian motion was justified
recall that
stituent jobs on their
station
i
is
T
each element x £
way
to station
of
Section 4 under assumptions (4.2) and (4.3). Next,
describes a ''path" of tasks traversed by the various con-
'''(/)
minimum amount
the
in
/.
Equation
states that the
(6. 15)
work found among
the "paths" leading up to station
all
In other words, (6.15) articulate the constraint that a task at station
all
of
its
precedessor tasks have been completed (this
interpretation in mind, statement (6.16)
is
is
immediate workload
i
cannot be processed
the definition of a join node).
is
then equivalent to (35).
argued that the additional faces correspond to the fork and join constraints
demonstrate
in
until
With
this
Nguyen
In
in
it
[18],
the network.
was
As we
an example, the polyhedral state space associated with heterogeneous fork-join
networks typically has many more faces than
may be
i.
Thus, each idleness process
associated with potentially multiple boundaries on the state space 5.
will
at
homogeneous counterpart. These additional
its
interpreted as results of the disordering effects that occur
when jobs
fork
and join
faces
in their
sojourns through the network.
Example
The heavy
1:
The Sample Fork- Join Network
traffic limit of the
network pictured
Figure 2
in
4'
is
a (^,r) Brownian motion;
/*
is
continuous and nondecreasing with /*(0)
is
given by
6.17)
=
/* increases only at times
t
with Z'{t)
=
0,
I^ increases only at times
t
with Z^{t)
—
p33Z2{t)
I^ increases only at times
t
such that one of the following conditions hold:
Z;it)
-
Zj (0
Z;{t)
-
(p^,p:ie
Zlit)
-
P44Z:(t)
=
0,
-
/
=
0;
1,2;
—
OR
P47P33)Z2it)
=
-
Zlit)
-
P47Z^(t)
0,
OR
=
0,
^4(0 - P47Z'(i) - P44P36Z2{t) - P44^3(0 =
0.
+
0;
P47P33ZUt)
15
OR
Hence, setting
1
.4
=
Example
2:
Common
Multiple Customer Types with
Consider a fork-join network
which
in
customers share the same routing constraints, but the
all
and service times may be
interarrival times
Routing Structure
different across job types.
correspond to a particularly simple case of the networks considered
j
G
cr(0), it
follows from (6.5) that
pikZ*
By induction on the depth
-
For stations
of stations, one can
becomes
verify that equation (6 5)
-
U'k
U'f.
=
This class of networks
this paper.
in
+
PikZ'
Ptk
niax
-
2",,,
/3,fc
51
^""
"iP'
(6-26)
^'m-
meC(i)
Because
customer types share the same the same routing structure, the sets {s[n)
all
m
are identical for each task
G
C{i)-
Hence the
last
term on the right side of (6.26)
is
:
n
G V(rn)}
equal to
/
-Pik
p,m
max Z* =
U:k
=
-p,k
max Z'
\meC{i)j
and equation (6.26) reduces to
Hence, for
z
=
1,
.
.
.
,
P.kZ:
d and k G C{i),
^*
a
is
W' =
U'k
/'
=
is
(0,
F) Brownian motion;
Z' — max Z';
PikZ'\
continuous and nondecreasing with /'(O)
/' increases only at times
t
with [V'(t)
Fork-join networks with one customer type (that
is,
above agree with those given
Example
3:
in
Nguyen
It is
0;
0.
homogeneous
a special subset of the networks discussed in this section.
results
=
=
fork-join networks) are clearly
straightforward to verify that the
Feedforward Multi-Class Queueing Networks
Consider now the feedforward multi-class queueing network studied by Peterson
described in Peterson
exception:
that
is,
I
[18].
[19]
The networks
[19].
The networks
are essentially similar to those considered here with one important
in [19]
there are no join nodes.
require that Vlk) contains at most one element for each task
(Peterson's work does not explicitly consider the case
17
in
k\
which
tasks
may
=
analysis.) Recall that Z'{t)
^%)i
W' + max
(6.27)
leV(k)
=
z'
we
p(/t)
"^^-^
Y.
=
p(p''-i(fc)).
Ps(l)l
and
k,
=
Letting Uqq
(6.28)
p'^
i^Vik)
the one predecessor task of task
recursively define p^{k)
Ps(i)i
f'Tt/,/
\v:{t)+
k^C[x)
denote by
to his
YlkeCii) '-''fcO- hence equations (6.2)-(6.5) imply
Pik
We
would not pose much hardship
fork, but the inclusion of the forking structure
efine p(0)
IV'J
=
0,
=
Setting p^(k)
0.
=
p(t).
equations (6.27)-(6.28) thus
reduce to
u:.
Pxk
W' +
^^'<P'"^))P'(^)
Ps(p'{k)]p'(k)
^'
z:
=
Similarly to the previous example,
w:{t)+ Y.
s(pHk))p^
ik)
^-
we can use induction
show that
to
for a station
;'
of depth
/i,
h
9= 1
z:
Readers can
Theorem
=
w:{t)
+ Yl
E
M-in'(p«(.))-
verify that this agrees with the result obtained by Peterson [19].
6.3 Under assumptions (4.2) and (4.3), {T^,
r;(<)
=
T;,{t)
=
.
.
.
,T^)=^(T{
,
.
.
.
I
.Tp) where
maxr;,(<),
keAl
max
T,'(0
+
tV-,),
T;o(/)
=
0.
ieV{k)
If
we denote by
/,
the
PERT/CPM
"longest path operator'" for type q jobs.
Theorem
6 3 implies
the representation
^;(0 = ',(w;,,),teA)As discussed
That
is,
in
in
Nguyen
the heavy
[18],
expression
traffic scaling,
to the length of time that a job
(6. 29) is
an example of Reiman's "snapshot" principle
the fluctuation in workload levels
spends
in
(6-29)
is
insignificantly small relative
the system, hence a "snapshot" of the system at the time
of a job's arrival remains representative throughout the job's sojourn
in
the network.
Equation
(6.29) expresses the remarkable result that sojourn time analysis of a fork-join network
phrjised in terms of the familiar longest path analysis of
task times are
now
[20].
PERT/CPM
methods, where traditional
replaced by waiting times at stations corresponding to the tasks.
18
may be
Proofs
/
By Skorohod's
representation theorem and the continuity of Brownian motions, we can and will
assume that the convergence
in (4.3) is
almost surely uniform on compact
sets: that
we henceforth
is,
assume
{N',V',L')
We
begin the proof of
Theorem
consequence of assumptions
^ ^'
Lemma
7.1 ^"
Lemma
7.2 For
Then
—
f"^
Proof.
and
«•
€2,
(4.2)
and
1,
.
.
,
.
~^
A', j
=
(N',V',r)
few preliminary
u.o.c.
results.
(7.1)
The
first
lemma
is
an immediate
(7.1).
u.o.c. where ^'
=
fc
6.1 with a
-
is a (^,
d,
1
= CClC
F) Brownian motion where F
let €^i^{t)
=
n-^^^e['l\nt) and €^j{t)
=
n-^/^€^^]\nt).
u.o.c.
Note that
max
0<fll.'(O<
r<"'ufc(/)
and
0<
The lemma
Let
r7j(f)
4'^*(*)
< max
max
follows directly from assumption (7.1) and
= ^-^'"'("0
Lemma
7.3 If
Proof.
Because
Wf
and
— W*
r7"(f)
<
t,
r]'^(t)
=
u.o.c, then
it
7.2
Lemma
7.4 If
Proof.
It
—
rj"
—
W'
e
u.o.c where e(i)
n-'/'||tV;(r?-(-))l|t
<
n-i/2||VV7(.)||,
u.o.c, then
r]^
l-V'"
—
W'
- w;n\\t <
\\w;{-)
-
and Whitt
I
[15].
=
/.
+
+
/
>
0,
"-'/'lk5(')llt
n-'/'||£5(.)||t.
u.o.c,
it
follows that ||e()
-
»7"()||(
^
0.
I
-- Wj' u.o.c.
follows from equation (3.9) that for each
\\ri;(')
3.3 of Iglehart
rjj"'(nf)).
<
and the assumption that
W^
-
Lemma
follows from equation (3.9) that for each
l|e(-)-'7;(-)||*
From Lemma
n'^l'^int
r|."'(.;i(;)
py;(-)||«
19
t
>
0,
+ W^(n^{-)) -
K(-)\\t
+
Mi-)\\t-
(7.2)
As
Lemma
a result of
and the continuity of
7.3
converges to zero. Invoking
The proof
Theorem
of
Lemma
G
''(0), it follows
=
theorem that {Xf, W]", q, Z^)
W;{t) = X;{t)
+
/;(0, and
Lemma
follows from
U\{t) =
pjk^'{t) =
two terms on the right side of
rj^
-~ IVJ u.o.c.
7.2,
I
pjkZ'At).
\Ve begin with the
namely, stations j £
cr{0).
Because
cr(0).
A'^"
=
^;(t)
+
-
7.4,
I-{t).
=
where XJ
= -
^;, i;{t)
info<,<( A';(s),
Because
L;,(»7;«))
+
-
P%^r,j(t)
and the continuity of
e^i)^
that
L'f.
('J^.
—
Joint convergence of the processes of interest
U').
is
u.o.c.
I
6.1: Define
Y^l(t)
= n-i/2(r;;'(n0-p;:'0
and note that as a consequence of
lished for
it
G
C(j),
all
it
7.5
and y;,(0
=
r,-iy;';*(n<),
(3.11),
y,",(0
With Lemma
=
- (7,(0.
L,",(0
we may assume inductively that the convergence
stations with depth h or
follows
less.
(7.3)
in
=
=
h
-\-
\.
if^(t) =
{
A, for all
^"/eT'tfc) \s:(,),iy:(,),it))
/
G Vik).) Setting
^"^.(t)
20
+ ^;on(/)/(o|
= n-M^2*("0.
0,
otherwise.
equation
(3. 13)
estab-
For each task
from (3.12) that
AU^) =
A*;
Theorem 61 has been
Consider a station j with d(j)
Mfc)(t)
(Note that
where
a natural conse-
quence of their continuity.
Proof of Theorem
^^ for
equations (3.3), (3.6), (3.7), and the continuous mapping
L^,{t)
Lemma
0.
each task k £ C(j) when j 6
— (X', WJ,I^,ZJ) u.o.c.
=
(7.2)
0.
7.1,
Z;(0 =
U^,[t)
it
for
Lemma
from
first
one concludes that
6.1 holds for all stations of depth
Note that V(k)
Proof.
j
Theorem
7.5
the
,
proceeds by induction on the depth of stations.
6.1
following result for stations of depth
Lemma
7 2,
W'
gives
^
Because d(s(/)) < h
^s(l)i
e{t)
-^
=
t.
for all tasks
"oc- where
K{l)l
Consequently,
M;,(t)
=
if
G V(k),
/
=
V;,,),!*)
V{lc)
^
0,
-
Ll^yit)
—
M^f.
mm
K;(A,.0+r,
follows from (73) and the induction hypothesis that
it
and
t7,,);(i).
-
YJ^^^^
e
p,,,,,
where that
u.o.c.
MJf. u.o.c. with
{-i",-'^'
V','
(P.(0/0
+
- rr^f'^OiiO}
^r' i:(/)/(0
;e7'(Jt)
n*(^fcO+
=
where the
V:{Xkt)
+
nim {-r-H7(A,/) + rf^
^fc
^fciVV)(f)
last equality follows
we have M"^
—
-
because q(k)
—
-
rjAT*;)!/))
^r' 's'/He)}
^^n^,
nnax
Tfc
+
(r,'(A,/)
q(l) for
(7.4)
G Vik). On the other hand,
/
if
=
P(A-)
0.
M*;. u.o.c. with
*
M;; = V';(A,/)+rtA-,)(f).
Applying the continuous mapping theorem
^'(i)
=
E
=
Cj{^)
+ r,A^;,)(o)-
(^fc'(^/cO
E
-
^j*^
=
and we use the convention that max
to equation (3.14), A'"
E
—
A'* u.o.c.
%a-^ 7f';m/(^)
u.o.c.
and
t)"
is
0.
to prove the convergence of
—
W*
u.o.c.
Consequently,
=
YJ'it)
from which we can conclude V"
—
That (WJ'.P.Z")
=
u;,{t)
L'Jf.{t)
-
YJl{t),
it
it
u.o.c.
Y;(t)
Because U^^i^)
U^
(\V' I' Z') u.o.c.
L',(t)-T;,[t)
=
L]f,(t)
=
PjkZ
Pjk
(3.6),
7.4,
and
we have
17"
All
-~
e
follows from (3.8) that
-
P%''v]{t)
+
e-,,{t),
where
rkN'^,^{t))
max
leV{k)
again a
(3.7).
= M;,(t)-pjkW;{t).
- (v;r(A,0 +
+
is
.
,
From Lemma 73 and Lemma
(7.5)
follows from (7.4) and (7.5) that
=
(t)
.
M;,(;7;(0)
— Y*
+ ^/
"^-'*^
consequence of the continuous mapping theorem by virtue of equations (33),
that remains
where
+
pjk
Ps(l)l
21
Tk
[/J\.
—
and
U'^. u.o.c.
max -^^^^^ + PjkWJit)
Pjm
>
p,
.
max
neV(m)
.
Ps{n)n
Proof of Theorenj
and rewrite equation
6.2: Define fki as in (6.8),
U'k
=
rym
Z,
P,k
V^
2^
+
,
[
.
a],
I
=
1.
.
.
.
enumerate the elements of
.c(i).
7.6)
\yieP(m) Ps{n)n
^p,
met(i)
Recall that
s{n)n
max
I
1km
(6.5) as
Applying
C{i).
Lemma
5.1 to (76).
we
obtain
=
U:,
Z:+ max
p^k
Similarly, equation (6.3)
is
max
r€7(.)
Substituting (6.2) and (6.3)
in (6.4)
W' =
W'
in
=
Z' —
=
z-
Ps{x,)T,
Lemma
5.1,
(7.8)
l^P^a'.\^,=
Ps{iu)i-i
'
l
and applying
(7.8),
Z' - max
^
J
we have
s(xt)n
p,^.
the above expression, we obtain
max
>
Z',
p,„i
i
,
+
max
c(.)
7
>
c(.)
„ ^1
u:,.
"^(^(^Ii
E
max
r=(xl,x2)eT
^.a;7
(7.9)
.u;. ;'^"'-'^'"M
(
For notational convenience, we henceforth write
(7.9) recursively,
(7.7)
.
'
\
equivalent to the following expression as a result of
A, =^, -
Substituting (7.7)
-ii£iifi|
1^^
;^j
X6T(0
s"*
to
mean
s{x"^
^
).
Substituting (7.7)
in
one can verify that for a station of depth h
c(0 c(.')
c(.)
w-
=
Z--
z;,
+
•
•
•
x€7
(»•)
c(.)
cis"-')
£• E
It is
P< T.
straightforward to verify that (7.10)
is
,.
••7,h-,
equivalent to equation (615) and the theorem
proved.
7.10)
^-1 Z'/,
is
thus
I
22
Remark:
Substituting (6.2)-(6.4)
equation (6.5), we obtain
in
U'k
=
'-s{l]l
W' + max
Pxk
leV(k)
=
Because Z'{t)
YikeCu)
^^'fcC)'
'^
Psii)t
follows that
'-
max
depth
in
the
same manner
as in the proof of
Theorem
6.2,
we can show that
for a station
;'
of
(7.11) can be written as
/i,
z:
= w: +
max
J2
Readers may recognize that equation (7.12)
in (6.15).
amount
7.11)
/'.(O'
ceCo'^'^*''
Proceeding
s{l)l
p,k-
It
states that the
amount
is
E
of total work in the system for station
6.3: Define <I>^(0
(7.12)
fv-;.
Pxai
an "inverse" formulation of the relationship described
of immediate work destined for station
Proof of Theorem
+
K^ +
Pxai
i
i
is
the
maximum
of the
found along each path to that station.
-i^(")/
= n'^^
'{nt)
and
<I>^;.(<)
=
-i^(")/
n-'$y^'(nt).
In addition, note
that
t
hence by
Lemma
3.
<^lp\i)
<t+
of Iglehart and Whitt
.3
max
[15],
—
$^
We
follows
from
The theorem
is
(7.13)
and Theorem
6.1 that
presented
in this
neous customers.
here apply for
=
.1^(0
^^^
—
and
e u.o.c.
and
TJ'^.
—
T'^ where T^^
=
then proved by applying induction on (3.15)-(3.16).
paper a heavy
We made
W*^^^{t]
I
Concluding Remark,s
8
We
7,13)
e u.o.c.
begin with tasks k G ^°, for which
^^,{t)
It
Uk{i)/\[^\
traffic analysis of
feedforward fork-join networks with heteroge-
several assumptions to simplify the exposition, but the results proved
more general networks
as well. For
example, we assumed that each station
23
is
staffed a
single server. Using the
results to fork-join
are reliable,
[13, 8].
it
is
machinery developed by Chen and Shanthikumar
possible to analyze networks in which stations
total
2).
Acknowledgements:
for the
many
is
I
servers
experience server breakdown
[20].
(The
an example of such networks would thus become a special
In this case, the issue
workload input process
may
all
accommodated within the framework presented here
Lastly, batch arrivals can be
Example
one can extend these
networks of multi-server queues. Secondly, whereas we assumed that
model discussed by Baccelli and Liu [4]
case of
[7],
reduces to calculating Q, the covariance matrix of the
[20].
would
like to
thank Professors
J.
Michael Harrison and Avi
Mandelbaum
helpful conversations throughout the course of this research.
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25
for
Priority Queues:
Preemptive-Resume
Date Due
JUN.06
1994
Lib-26-67
MIT
KPRARIFS
nilPI
.-.
^
3
TOfiO
007nm3
M