HD28
.M414
DEWEY
I*
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Heavy Traffic Analysis of a
Transportation Network Model
William P. Peterson
Lawrence M. Wein
#3673-94-MSA
April 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Heavy Traffic Analysis of a
Transportation Network Model
William P. Peterson
Lawrence M. Wein
#3673-94-MSA
April 1994
J
I
I
M.I.T.
MAY
LIBRARfES^
1 ? 1994
RECEIVED
HEAVY TRAFFIC ANALYSIS OF A
TRANSPORTATION NETWORK MODEL
William
P.
^
Peterson
Operations Research Center. M.I.T.
and
Lawrence M. Wein
^
Sloan School of Management, M.I.T.
Abstract
We
By
study a model of a stochastic transportation system introduced by Crane.
adapting constructions of multidimensional reflected Brownian motion
since been developed for feedforward queueing networks,
tional central limit
the case in which
theorem
all
we
(RBM)
that have
generalize Crane's original func-
results to a full \ector setting, giving
terminals in the model experience heavy
an
traffic
explicit
development
conditions.
gate product form conditions for the stationary distribution of our resulting
We
RBM
for
investi-
limit,
and
contrast our results for transportation networks with those for traditional queueing network
models.
Keywords. Transportation networks, queueing, heavy
AMS
traffic, reflected
Brownian motion.
1991 subject classification. Primary: 90B22, 60K25. Secondary: 60F17.
April 11, 1994
'Resecirch supported by Middlebury College under the Faculty Leave Program.
^Reseajch supported by National Science Foundation grant DDM-9057297.
Introduction and
1
In this paper,
we prove a heavy
network of
M
+
A'^
1
theorem
.
.
.
,
5/v- Vehicles in
i
—
1,
.
.
,
.
1
z,
=
i
0. 1, .... ^V.
group
S, are initially
—
to discharge
group. Terminals
1, ... ,0.
They
The network
The
fleet is
is
full
system model
mathematical
consists of a linear
served by a
fleet of
partitioned into service
and proceed
where they pick up waiting passengers. They make successive stops
and pick up passengers, then
i
=
N
1
join a single
is
queue at
all
repeated indefinitely by each vehicle
i.
i
require transportation to one of the terminals
and board arriving vehicles
served (FCFS) fashion subject to availability of seats. There
passengers that can be queued at any station, and
do not balk or renege. As Crane notes,
system, such as a municipal subway
where
I'eturn to terminal 0,
each have an exogenous arrival stream of prospective
passengers. Passengers arriving to terminal
t
given a
dispatched from terminal
remaining passengers disembark. This cyclic route
in the
1, is
To summarize, the system under consideration
2.
terminals, indexed
directly to terminal
for a transportation
Figure
in
vehicles, each having fixed passenger capacity V.
groups ^i,
at
traffic limit
The model, depicted
introduced by Crane (1974).
formulation in Section
Summary
this
line,
it
is
is
no bound on the number of
assumed that prospective passengers
model can be applied
in
in a first-come first-
to a public transportation
which there are no pubhshed schedules and
passengers arrive without reservations. His paper appears to be the only previous attempt
to analyze the performance of a
dynamic stochastic network model
of a
pubhc transportation
system with limited vehicle capacity.
Crane presents functional central
limit
theorem
results for his model, of the type ob-
tained in the seminal papers of Iglehart and Whitt (1970) on queueing systems.
the theory of what
at the time,
is
now
called multidimensional reflected
Crane was able
which a single terminal
in
Brownian motion was unknown
to explicitly describe his limit processes only for the case in
the network
is
critically loaded, in the sense that the
passenger arrival rate at the terminal equals the rate at which vehicle capacity
there.
Even
Because
in that case, the covariance
exogenous
is
provided
manipulations presented are quite cumbersome. Our
1
Group
queueing networks described
multiclciss
form of the
reflection
Harrison and Nguyen (1993). In particular,
in
matrix one can identity both customer routing effects
in
the
passenger
(i.e.,
destination information) and vehicle effects: the queueing models cited include only the
former.
will
We
example
discuss this in the context of an
be seen there that except
RBM
why
this
is
the case.
However,
Section
On
5.
the
RBM
limit
We
show
in
in artificial special cases,
conditions for a product form stationary distribution.
structure of the
in
a negative note,
can never
it
satisfy the
terms of the covariance
should be noted that even though
it
the stationary distribution cannot be found analytically, the numerical methods recently
developed by Dai and Harrison (1992)
for steady-state analysis of
be applied to analyze system performance
Our
limit results are
appropriate dimension
d,
formulated
where
D=
that are right continuous and have
convergence
will
be denoted by ^.
in
D[0,T]
.A.11
on an orthant can
our model.
in
terms of weak convergence
is
hand
left
RBM
in
product spaces D'^
the space of real-valued functions on
limits,
column
[0,7"]
Weak
under the Skorohod topology.
vectors are to be understood as
for
vectors, with
primes used to indicate matrix and vector transpose.
The Model
2
We
in
begin by summarizing the probabiHstic structure of the model, following the treatment
Three basic phenomena need
Crane.
to
be described: exogenous passenger arrivals to
the system, passengers' choice of destination terminals, and circulation of vehicles in the
network. For simplicity,
t
=
0,
and that
all
it
assumed that there are no passengers
is
in
the network at time
of the vehicles in the fleet begin their routes from terminal
i
—
at that
time. For present purposes, these assumptions have no significant effect on our limit results.
For
i
=
represent the
1,...,A'^, set
/"*
u,(0)
=
and
interarrival time in the
for
/
=
1,2,... let the
random
variable
ii,(/)
exogenous passenger arrival stream to terminal
(.
Define the counting processes
A,{i)
= max{/>0:u,(0) +
3
u,(l)
+
-
••
+
«,(/)
<
^}
(1)
for
>
t
over
that
0, so
.4,(<)
of
exogenous passenger arrivals to terminal
sume
described by an
is
that a passenger arriving to terminal
/'
N
x
A'^
matrix F
independent of the destination of any other passenger. As described
other words, T
strictly lower triangular.
is
passenger originating at
sequence of
i.i.d.
random
A'^-dimensional
/"*
passenger
in
otherwise. Observe that £'[\,(1)]
=
is
equal to
A'^-
if
1
the
sum
dimensional cumulative
j"*
component
among
passengers
the
Recall that vehicles k
a vehicle k G
j
=
0, 1,
.
.
.
,
z,
5,.
We
where
=
1,
...,
Vk,j{l)
assumed that when a
=
{Ui{r),r
= \Al) +
which we
as-
(.\i,i(Oi
---
+
shall
i
—
=
7,^
i
•
•
i*''
.
.
row of
.}
where
\t.iv(0)'-
1
has destination
1,2,
i\
j,
and
a
i
x,j{l)
is
zero
Next define the
F.
by
\.(r).
(2)
denote by U,j(r), gives the number of
who have
random
destination j.
variables
{t'fc,j(/).
represents the transit time for the vehicle's
to terminal j
>
for j
^^e probability that a
'^
7u
the last section,
in
M have been partitioned into service groups
hnk goes
We
(7,^).
introduce for each terminal
denotes the
F,
=
introduce sequences of
starting from terminal j; this
It is
where
F^,
arrivals to terminal
first r
Z]j=i
We
0.
vectors \,(/)
processes U,
of U,{r),
—
1
the arrival stream to
ihir)
Then the
.Also,
has destination terminal
i
=
has destination terminal j with probabihty
passengers always seek transportation to lower numbered terminals, so
in
;'
[0, i].
Passenger destination selection
7,_,,
number
represents the
—
I
vehicle arrives at a station,
if
j
>
1,
and
/"*
/
=
1,2,...} for
trip along the link
to terminal
any passengers
Consider
5,.
it
is
;
if
j
carrying
=
0.
who
are destined for that station disembark instantaneously, and any waiting passengers then
instantaneously board subject to
Then the sequence
availabilit\- of space.
u^fc(l)
=
i.'«:,o(l).
io,{l)
=
vUn
{wk{l),
I
=
With
this
understanding, we define
and
+ i2''^-J^-^)^
/
=
2,3,...
.
1,2,...} gi\es the times between successive visits of the
vehicle to terminal
^
.
Define the corresponding counting processes {C\-(t),
i.
[
Now
+
/ max{/:a',(l)
.
0,
wt,{l)
if
>
---
+
u',(/)
<0,
>
t
0} by
iiwk{l)<t,
t.
recalhng that each vehicle has fixed capacity V, we see that for
/
=
1,
.
.
.
,
A^,
= \-J2C,(t)
G,{t)
(4)
kes,
gives the total
number
provided at terminal
Two comments
i
of units of vehicle capacity
during
[0,^]
by vehicles
(i.e.,
available passenger spaces
group
in service
on
S,.
are in order here pertaining to the circulation of vehicles.
noted by Crane, the above framework can be used to accommodate a stochastic
at the terminals,
vehicles)
by simply envisioning that such a delay has been incorporated
First, as
idle
time
in the
most
recent link transit time. Observe, however, that this does not allow the idle time to depend
in
any way on passenger data. Second, the model
fails
to capture
the travel times for different vehicles. This would become an issue
not permitted, as would be the case with a single track
dissertation.
Crane (1971) outhnes an approach
any interaction between
if
passing of vehicles were
system, for example.
rail
In his
to incorporating such dependencies into the
model.
The counting
{U,{r), r
=
processes {Ai(<),
1,2,...} for
j
=
1,
.
.
.
,
t
A'^,
>
0} and {G',(i).
assumed
are
to
t
>
0},
and the vector sum processes
be mutually independent. These are
the basic building blocks for our system representation.
We
proceed somewhat differently
from Crane, developing a streamlined description that facihtates the proof
limit theorem.
t,
Let Z,(i) denote the
Ei{t) denote the
[0,<],
and
Yi{t)
number
number
of passengers
of our vector
of passengers waiting at terminal
who have boarded
vehicles at terminal
at time
;'
i
during
denote the cumulative number of units of capacity that go unoccupied on
vehicles departing from station
i
during
[0./].
The
following
is
a recursive construction of
representations for these processes. Let
AV(i)-.4v(^)-GV(0^
and
(5)
.V
'=.1+1
+
-Y]+,it]
where, ior
The
j
=
idea here
N,
.
.
j
= N-L...,U
- mf
YAt)
=
Z,{t)
= XAn +
E^t)
= AAt)-Z,{t).
Xj
{A-,(.)},
(7)
0<s<i
and
Yjit)
(8)
(9)
as a netput process; that
is,
between the total number of passengers arriving to station
number
(6)
,1,
.
to construct
is
<,(t),
Xj{t) represents the difference
during
j
and the cumulative
[0. t]
of units of vehicle capacity provided there over that time period.
If
one accepts
this
interpretation of (5)-(6), then (7) and (S) are standard representations from the queueing
and
literature,
is
(9)
simple bookkeeping.
To understand
ways that vehicle capacity can be provided
effectively three
(5)-(6),
at
;'.
observe that there are
The
This
of vehicles in service class 5j, as accounted for in the basic process Gj.
=
source of capacity at terminal j
-'V,
provide capacity at
when they
j
is
the only
so A'.v can be defined in (5) entirely in terms of basic
processes; this gets the recursion started.
sources of capacity represented in
through arrival
first is
(6).
.\t
terminals j
.\rri\ing vehicles
are carrying passengers
< N,
there are two additional
from service
classes 5, for
whose destination
is
j, since
i
>
these
passengers immediately disembark there, which frees up space for passengers queued at
Note that U,j{E,{t)) gives the number of passengers with destination
who board
vehicles at
from terminal j
+
vehicles reach j.
available capacity
j
by time
in transit
t.
1
during
during
The
is
i
£_,(<)
[0,i],
is
accounted
[0,^].
j
among
an error term, which
is
may
Observe, however, that a vehicle can contribute to
i
and j
for
the E,(t)
when those
required because the vehicles whose
by U,j(E,{f)) and Yj+i(t)
somewhere between terminals
j.
unused capacity on vehicles departing
In addition,
as represented by Yj+i{t), will provide capacity
for
j
some
/
>
not have reached terminal
ej(t)
j at time
only
t.
if
that vehicle
is
Thus, at any time.
the error
is
certainly
bounded by the
total capacity of the fleet:
0<tj(t)<MV
Using
bound, the error
this
for all
>
f
0.
j
=
l
iV
-
(10)
1.
be shown to be negligible under the heavy
will
traffic reseating
to be defined in the next section.
We
conclude this section with representations for the queue length and departure
processes at the terminals broken out by passenger destination.
be the number of customers waiting
Q,j{t)
for j
from
=
i
0,
.
.
.
,i
—
I.
Similarly
whose destination
is j.
let
at terminal
Then
for
z
=
.V
i
whose destination
i
number
D,j{t) be the
At each terminal
of passengers
is
let
z,
terminal
j,
who have departed
we have the following straightforward
representations:
The compound
Q„[t)
= U,j{Mt)) -
Q,o{t)
= Z.(i)-EQu(0,
D,j{t)
=
D.o(0
= EM)-'Y.D.,{t)-
processes
cussing equation
(6).
Z),j
U,,[EM)).
7
j
= h...,i-l,
(12)
= L....z-l,
and
queue length
phenomenon
3
The
in
(J.j will
in this
be treated
Z, at each terminal
heavy
(13)
(14)
way
in
will
i.
This
is
in dis-
be convenient for developing the
a corollary to that theorem, where
be shown that their limit processes are given by deterministic fractions of the
total
(11)
were already introduced, though not explicitly named,
Separating them out
main theorem. The processes
U,j{E,it))-
it
will
limit for the
a manifestation of the state space collapse
traffic results.
The Heavy
Traffic
Conditions
precise statement of our results involves a sequence of systems, of the type described in
Section
2,
approaching conditions of perfect balance between passenger arrivals and vehicle
capacity at each terminal.
Hereafter, a superscript n will be used to index processes and
,
parameters associated with the
and vehicle
interarrival
(Ai,...,A^) and G"
system
n"'
=
{Uii,
We
.
.
.
,
U,n) for
=
(G'j,
z
=
1,
will allow the
passenger
.
.
.
depend on
,6'JV)
.
.
.
,
is
fixed,
For simplicity, however, we assume
n.
which implies that the vector processes
do not vary with
A'
=
n.
require that there exist sequences of .V-dimensional vectors {A"} and {/i"} so that
the scaled processes A" and
G"
A" and the round
defined by
A"(0
=
n-'^-iA^int)- X^nt)
G"(i)
=
n-'^-(G^{nt)
satisfy functional central limit
processes
We
the sequence.
transit time distribui ions to vary, so that the basic processes .4"
that the destination probabihty matrix T
Ui
in
trip
theorems
(f.c.l.t.'s).
-
(15)
S'^nt)
(16)
For example,
if
counting processes C^ underlying the
the component arrival
G"
in (4) are
mutually
independent renewal processes, then the required results can be established under some
additional
moment assumptions. By
the classical Donsker theorem gives
0:^(1)
(2), f ,(/)
f.c.l.t.'s for
has a multinomial distribution for each
t
Therefore, recalHng the assumed independence of A",
as n
—
»•
oo,
.
.
.
where the processes on the
,
UJ^,)
^
G" and
[A', G',
right aie .V
+2
=
l,...,N.
the U,'s,
(17)
we have
Ul ...,Un)
(18)
independent, zero
drift,
vector Brownian motions, whose covariancc matrices will be denoted by
i
so
the processes C^" defined by
= n-'^HU,{[nt\)-T\nt),
(i", G", 6T,
r,
jV-dimensional
11,4,
Sg and
S^,
= h...,N.
The
heavy
definitions in (15)-(17) consist of a centering of key processes followed by the
traffic rescaling, in
which time
is
scaled by n and space by n~^^^.
in (18) gives the following interpretation of the centering parameters:
average arrival rate to station
z,
and
capacity are provided by vehicles
/3" is
in 5,.
The convergence
A"
is
the long run
the long run average rate at which units of vehicle
Ignoring the rescahng for the
moment, we introduce
the following centered versions of our basic processes. Define:
= G'^lO-^i"^
G'"(f)
=
Udt)
We
=
from Section
loading conditions,
in (15)-(17)
from these by rescahng;
we seek
For now, as an intermediate step,
n~^/^A"(n/).
state equations
(19)
L;{[t\)-r',t.
can later construct the tilde processes
A'^(t)
and
example,
to rewrite the
system
2 in terms of the centered processes. Anticipating the balanced
we expect the long run departure
rates to
rates, so the following defines a natural centering for the
£"(i)
Recall from (13) that Df^(t)
for
=
match the corresponding
arrival
departure processes:
= £"(0 -A"^
(20)
U,j{E^(t)), and observe that
=
u.AE^it))
r.,(£r(^))
+ 7.,^r(o
This suggests the centering
D:^{t)
where A"7,j can be interpreted
=
U,,{E^{t))-X:f,,t,
as the long run rate at
nal j board vehicles at terminal
;'.
By
=
A''^{t)-Cr^{t)
-y;;i(0 +
where, for j
=
1
,
.
.
.
,
for termi-
solving (19)-(21) for the uncentered processes and
substituting into (5)-(9), our system state equations
X;^(i)
which customers destined
(21)
+
become
{\l--3lr)t,
e;(0-
j
and
= .v-i,....i,
(22)
(23)
A'^
^i"(^)
=
-o^'JL^-^'"^^)^
(-^'
Z;(0
=
Xj{f)
E^it)
=
.4;(f)-z;(o.
+
and
y;{t)
(26)
Consideration of the rescaling to follow leads to our heavy
—
that as n
>
such a way that, for j
=
1,
^A
.
.
,
.
>
and
This requires that the quantity
in
parentheses
common
all
^r
rate of convergence for
^"
-
E
^
/3
Therefore,
we can
K'lu)
terminals
-^<0,<
Oj,
groups
5, for
t
>
j.
is
^.
converge to zero;
in (28)
j.
(28)
also specifies a
it
Following our earlier discussion,
who
originated at
i
can
A".7,j
disembark at
exogenous passenger arrival rate to each terminal
exactly balanced by the rate at which capacity
the rate at which capacity
(27)
see that (27)-(28) describe a sequence of systems approaching
In the limit, the
critical loading.
in
suppose
>
also be interpreted as the long run rate at which customers
j.
We
A'^,
-
V^(\" -
terminal
traffic conditions.
oo,
A"
in
(25)
made
is
provided at
j
by vehicles
in
is
j
group Sj plus
available there by passengers disembarking from vehicles
Note that by
letting 6
=
(^j,
.
.
.
,^7v)',
we can
restate (28) in vector
form as
v^((/-r')A"-,^") ^e.
It is
interesting to
derived by Crane.
He
compare our heavy
traffic
defines for each terminal
(29)
conditions with the stabihty conditions
an
effective service rate
fij,
which
in
our
notation would be given for the n"' system by
Here a
Ab =
imbalances
min{a,6} and
in loading.
bounded by the
The
[a]^
=
max{a.O}.
This set-up explicitly recognizes potential
(A" A^i") acknowledges that the departure rate from terminal
total rate at
which capacity
is
10
provided there.
The
[fi'J+i
-
/
is
^'}+iV accounts
anv excess vehicle capacity
for
+
at j
which then becomes axailable capacity
1.
at j
.
Crane's
stability conditions are
A;-//,;<0.
We
our heavy
will see that
traffic
relationship completely, write
=
//"
N x N matrix
the
first
superdiagonal; that
=
1
l...../V.
(31)
—
^"
—
0.
>
To explore
(/_r')A'' -,i",
5 having ones on
this
(32)
the main diagonal and negative ones on
is
5.,
Proposition
=
and define vectors ^" by
/.ly]'
{/.il
1.
if
J
I
-1.
if
;
I
0.
[
Then we have the
j
conditions imply that A"
r
Next, introduce an
for
=
=
= +
i
;
(33)
l
otherwise
following.
The
stability conditions (31) hold if
and only
if
5"^^" <
where the inequality
Proof.
is
First note that
to be interpreted
S~^ exists and
component-wise.
is
given by
'"'
I
If
we have A; A
(31) holds, then for each ;
and
(32),
we can rewrite
0.
i.i'J
otherwise
=
A; and [^;
-
A;]+
=
^^
-
A;.
Using
this
(30) as
a;-//;
In vector form, this says that
=
^;
5(A"
Thus, (31) implies that 5"*^"
<
+
(a;^i-/i;^i),
-^i'')
=
0"
;
= a'-i
and the stabihty condition
0.
11
(34)
i.
(31)
is
A" -^z"
<
0.
5
Conversely, the scalar form of
^^'^
<
is
f:^;<0,
.
=
1,...,.V.
(35)
J='
At
= N,
i
this gives 6"^
= A^ —
/?J^
j
=
k
j
=
k,
+
.
I,
.
,
.
.
.
,
0.
by (32) and
/Zyv
.
<
N
.
Suppose inductively that (31) holds
(30).
/^^
<
by
RBM
—
components
valid for
= E^;-
(36)
(35).
In the notation just introduced, our hi^avy traffic condition (29) says s/nO"^
will see later that
AJ^
gives, after cancellation,
A2-/'i:
-
for
Then, reasoning as above, we see that the equations (34) are
N. Adding these equations
This imphes A^
=
This implies component A^ of (31), because 6^
S~^6 <
emerges as
a
—
9.
We
necessary condition for positive recurrence of
this
means that the process must be
achievable as a limit from a sequence of stable systems.
Condition (29) also implies that
our
9^
—
>
0.
limit process.
In light of Proposition
1,
Following the steps leading to (36) above, this gives A" -/z''
—
*
in
Crane's set-up,
as asserted earher.
4
We
The Main Theorem
are
now ready
to state our
main theorem. Define rescaled versions
from our system state description
X"(0 =
Theorem
n-^I^X^{nt).
of the vector processes
as follows:
y'"(0
= ir"^Y"{nt).
Z^{t)
=
1 // the the basic functional cailral limit theorems (18)
conditions (27)- (28) hold, then
(X".Z".
>•'",£") =>
{X\Z',Y\E').
n'^/^Z^'int),
and
the heavy traffic
The
limit processes are specified
component- ir/.se by
X'^it)
= A%.(f)-Gy{t) +
x;it)
= A'^(t)-G;(t)- Y,
+
where, for j
=
iV,
.
.
,
.
y'*(-)
we can
1.
The key
(f'*(A,0
+ 7u^*(0)
(38)
1,
is
continuous and nondecreasing, and
¥'{) increases only when Z'{t)
Remark
(37)
;=iV-l....,l
-l]\,{t),
e^l
and
6Mt.
Z;{t)
=
X;(t)
E;{t)
=
A;{t)-z;(t).
limit process
is
Z',
+
characterize as a multidimensional
(40)
0,
{U)
and
Y;(t)
the
=
(39)
(42)
liirdt for
RBM
the vector queue length process, which
by the following steps. Substituting (38)
into (41) gives
= A'(t)-G;[t)- y:
z;{t)
+
{u:^{x^t)
l^JE:it))
t=J+l
+ 9,t-Y]\,(n +
To rewrite
Then F'
is
m)
y;{t).
this in vector form, introduce the vector process
a zero
drift vector
Recalling the matrix
S
Z'{t)
Substituting E'(t)
=
Brownian motion with covariance matrix
defined in (33).
=
A'{t)
A'{t)
—
F' defined by
-
G'it)
we can express
-
Z'(t). which
F-{t)
i-<
-
(43) as
T'E'it)
+
et
+
SY'(t).
the vector form of (42), and collecting terms
gives
{I
-r)Z'{t) =
{I
-r)A'{t] -G'{t)-F'{t) +
13
9t
+
SY'(t).
Because T
is
strictly
upper triangular,
—
(/
F')
guaranteed to be invertible, so we can solve
is
the preceding equation for Z'(t). obtaining
=
Z'it)
A'{t)
-(I-
r'r'[cr(t)
+
F'{t)) +rjt
+
RY-(t),
(45)
where
T]^{I-r')-'e
Now
R=
and
{I-r')-'S.
(46)
setting
at)
=
-{I-
A'it)
r')-i [G'it]
+
F'{t))
+
r,t,
(47)
equation (45) becomes
Z*{t)
Observe that ^
=
m+
RY'it).
(48)
a vector Brownian motion process with drift vector
is
//
and covariance
matrix
Q =
S.4
+
(/
- r')-\EG + Sf)(/ - T)-\
(49)
Therefore, equation (48), together with the properties (39)-(40) for
multidimensional
Remark
The
first
2.
RBM
The form
factor
is
with
drift vector
//.
covariance matrix
R =
of the reflection matrix
(I
—
Q and
Z*
V'*, identifies
reflection
as a
matrix R.
r')~^5 has a nice interpretation.
attributable to customer effects; the second to vehicle effects.
ditional queueing network, only the former would be present.
The
In a tra-
latter arise here because
unused vehicle capacity at one terminal translates directly into available capacity
at the next
terminal downstream. In Section 5 we present a concrete example of these calculations.
Proof of Theorem. Given
can be proved
in a straightforward
rescaling equation (22),
If
the recursive representations developed so
follows immediately
that X;^ =» X*^
=
manner by induction,
we obtain the following expression
from the basic
A'^ -G*,^
+
f.c.l.t.'s for
ejvc where t{t)
14
A,\
=
t.
far,
the theorem
starting at terminal j
for
,Y^
=
.V.
By
:
and Gat, and the heavy
traffic
condition,
This estabhshes (37). Observe that A'^
is
a one-dimensional Brownian motion process, and therefore continuous.
by rescaling
It is
easy to check
(7)-(9) that
0<s<t
By
Z^{t)
=
Xlr{t)
+
Y^{t)
E;,{t)
=
A-yit)
-
Zl.{t).
the continuous mapping theorem, we have that
with the result for
This expresses
Since X]^
Now
N,N —
{Yjl}
,
Z^
,
E'^) =^ {Yj!^,Z^,E^), jointly
and
(50)
X^, where
and Z^
in
continuous, the
is
above also
Yj!^
and
verifies (41)
Yi^it)
^
- mUX'^is)},
U<s<r
Z'nW
=
A';v(0
£.;.(0
= Ayit)-z'M-
+ r^(0
terms A'^ via the famihar one-dimensional reflection mapping.
first
and (42)
equation here implies properties (39)-(40) for j
.
.
,j
+
I,
and consider the situation
X^it)
.
at this terminal.
suppose by inductive hypothesis that the results of the theorem hold
1,.
= N The
Rescahng equation
at terminal j.
= A]{t)-G]{t)- Yl
for terminals
(23) gives
Dlit)
N
-hV^(A;-/?;- Y. ^r7u>->7+i(0 +
(si:
s"(^)'
where
A"(<)
First note that our basic
f.c.l.t.'s
=
n-'l' (U^jiE^int))
give (/i",G'")
the deterministic factors multiplying
for
UJj,
i
=
j
+
I,
.
.
.
,
t
^
-
X^f^.nt)
(AJ,G'J),
converge to
9j.
and by the heavy
For the
N, by inductive hypothesis. Combining
(52)
.
D"
this
traffic
condition
terms, note that £"" => E'
with the basic result for the
we can apply the standard Compound Process Functional Central Limit Theorem
(52) to obtain D^^ =» D'^,
where
15
to
is
a Brownian motion process. Also, the inductive hypothesis gives
limit
is
continuous. Finally, rescaling (10) gives
<
for
any
T >
(51) gives
X'
is
=> Vj'+,, where the
Vj'^j
0.
X^
itself
=
follows that e" => (, where ({t)
It
=^ X*, with
continuous.
X^
Now
conclude
0.
Putting
the one-dimensional reflection
Remark.
1
this section
,
,
Theorem
We
limit results for the
at
terminals
the state space collapse result mentioned in Section
is
as in (50)
define the usual rescalings
we have
1,
term implies that
(39)-(42).
by recording a corollary which gives
Jointly with the results of
This
these terms together in
mapping can be used
Y' Zj and E' thereby establishing
detailed queue length processes Q"^ defined in (ll)-(r2).
Corollary
all
as in (38). Furthermore, continuity in each
to get corresponding results for
We
<n-^'^MV
sup P{t)
0<t<T
i
2.
=
It
\,
.
.
.
,
N
that
shows that
in
the limit, the queue lengths broken out by destination can be expressed as fixed multiples
of the total
queue length
be described
in
Thus,
In fact, this result
is
queue length processes (11)
Qi(t)
in
£^"
=
Z", the
= t>(Ar(o) -
last
The key
Crane; what
in
to the corollary
is
is
missing there
to express the
terms of centered processes as
o^^iE^it))
After rescaling, the difference between the
—
the limit, each terminal can effectively
already contained
the development of the vector limit process Z'.
Since A"
in
one dimension.
Sketch of proof.
is
at each terminal.
term
first
+
7.;(^r(o
two terms
will
E?it))-
be seen to vanish
will give the desired result.
16
-
in the limit.
See Crane for details.
A
5
To
Simple Example
example
gives the simplest non-trivial
is
we consider a network which
illustrate the analysis presented so far in a concrete case,
of our results.
The network has
N=
2 terminals,
served by a single bus, which belongs to service group S2- Service group Si
we
set Gi{t)
=
0.
Observe that passenger destination information
by specifying the parameter
A2, Ai,
G2 and
f/2i-
721.
Assume
for
Thus, the building blocks
for this
alf^^
.
follows
completely determined
model are the processes
guaranteed by (18) are independent zero
f',"i
Brownian motions with variance parameters a\^. a\^ a^^ and
al,
empty, so
simphcity that the arrival processes are independent. Then
the basic hmit processes Aj, A\, G^ and
expression for
is
is
and
from the binomial distribution.
write out the components of the two-dimensional
RBM
=
what
In
Z' given
721(1
—
follows,
in (48) as
721);
we
drift
here the
explicitly
the limit for the
vector queue length process.
Following the development in Section
T.
Then easy computations
/
2,
we have
\
^
,
for this
M
example
-i
give
:/-r')- = (; '7
and
^/^-lc
R = {I-Vr'S=
The
basic
^
/
1
-(1
'^ -721,
"''
^
Brownian motion processes described above enter
1.
(53)
into our calculations in vector
form as
where F'
is
limit process
defined as in (44).
Following (47), the underlying Brownian motion ^ for our
is
^^^'[
O2
)'^[
A-it)-Gl{t)
17
)
1-Y 21
^
Figure
The
drift vector
t]
is
2:
Directions of reflection for example network.
Q
apparent here, and the covariance matrix
given by (49)
is
easily
calculated as
On
(t/,
=
the interior of the nonnegative orthant 5?^, the process Z'
Brownian motion.
f2)
reflection,"
At the boundary Z'
—
which corresponds to a displacement
0,
in
it
+ RY*
behaves
like
an
subjected to an "instantaneous
is
the direction given by the j"* column
This displacement, governed by the increase in Y',
of R.
if
is
of the
The
required to keep the process Z* from leaving the orthant.
minimum magnitude
direction vectors are
shown
in Figure 2.
Hitting the boundary Zj
terminal
is
normal
Z2
—
1.
This has no
effect
to the boundary.
corresponds to unused capacity on a vehicle departing
on the "upstream" terminal
corresponds to unused capacity on
As noted
in
Remark
2;
hence the direction of reflection
At the other boundary, the situation
additional capacity to terminal
effects,
=
1,
so
2 following the
a vehicle
is
more
departing terminal
interesting. Hitting
2.
This
will
provide
we anticipate a negative impact on the queue
main theorem, the
corresponding to the factorization
R=
{I
—
result
r')~^5.
The
is
there.
the combination of two
vehicle effect, described by
S, equates unused capacity at terminal 2 unit for unit with available capacity at terminal
18
This effect alone would suggest a 45-degree direction of reflection.
1.
passenger
recorded in (/
effect,
— ^')~^
note that every occupied unit of capacity at terminal
2 represents an expected 721 units of available capacity at terminal
passengers. This
is
foregone
when Z2 =
net decrease in the Z' direction
We now
(1
is
—
Combining these two
0.
whose density
is
721) per unit of increase in the
an
for
RBM
it
to disembarking
follows that the
Z2 direction.
RBM
limit.
=
0.
Then
covariance matrix
rj,
the condition R'^r]
For our transportation model,
it
follows
the condition S~^d
<
is
17
and
RBM
Z' on
reflection matrix R,
necessary for positive recurrence.
from (46) that R~^t]
<
Harrison and
process to have an equilibrium distribution
a product of independent exponential densities. Consider an
the nonnegative orthant having drift vector
starting at Z'{0)
due
1
effects,
turn our attention to the equihbrium behavior of the
Williams (1987) describe conditions
To understand the
=
S~^6.
Recall that at the
was seen to have a natural queueing theoretic
end of Section
2,
interpretation.
Harrison and Williams derive a skew symmetry condition which, together
with the preceding condition,
R
is
necessary and sufficient for the product form result.
has ones on the main diagonal, this skew symmetry condition
2n =
where
D =
diag(Q)
is
diagonal.
it
is
is
DR',
(55)
the diagonal matrix constructed from the main diagonal of U.
reflection matrix for the transportation
given by (46),
RD +
When
The
model always has ones on the diagonal because,
as
the product of two upper triangular matrices, each having ones on the
With the data from
(53)
and (54)
for
our two station example, condition (55)
reduces to the single off-diagonal equation
2721 <T^,
This equation can only be satisfied
two uninteresting solutions.
First,
=
if
-(1
-
721
)(< +
O-
(06)
both sides are equal to zero, which leads to one of
we could
set a\^
=
a^^
=0.
In other words, the basic
processes at terminal 2 would have to be completely deterministic.
721
=
1
and aQ^ =
0.
Second, we could
set
In this case, all passengers boarding at terminal 2 have destination
19
terminal
1,
and the
analogous problems
vehicle circulation
will
be encountered
N
condition relating terminals
and
N
an iV terminal model. The ofF-diagonal
in general for
—
1
have exactly the form
will
Given the lack of interesting solutions
to the
The problem
arises
results in the positive left
variance, note that,
(.56).
product form conditions,
ask what features of the transportation model make
network.
Furthermore, one can see that
deterministic.
is
it
differ
it
is
natural to
from a conventional queueing
from the positive covariance between terminals 2 and
hand
side in equation (56).
away from the boundary, each
results in 721 units of service capacity for terminal
To understand the
1,
which
sign of the co-
unit of service by vehicles in group S2
1.
Thus
this unit of service effectively
decreases the queue lengths at both terminals, which explains the positive covariance term.
By
contrast, in a traditional feedforward queueing network having the
service completions at station 2
customers routed to station
sponding covariance term
symmetry
is
would decrease the queue
same
there, but the portion of these
would increase the queue at that station.
1
negative in this case, and
it
basic topology,
becomes possible
Hence the
corre-
to satisfy the
skew
condition. Solutions for such models are covered by the results of Peterson.
References
Crane, M.A. (1971). Limit theorems for queues
tion.
Department
J.
Dai, J.G.
methods
&
Queues
in transportation
systems
II:
an independently dispatched
Appl. Probab. 11, 145-158.
and J.M. Harrison
(1992).
for steady-state analysis.
Harrison, J.M. (1985).
Wiley
transportation systems. Ph.D. disserta-
of Operations Research, Stanford University.
Crane, M.A. (1974).
system.
in
Reflected Brownian motion in an orthant: numerical
Ann. Appl. Probab.
2, 65-86.
Brownian Motion and Stochastic Flow Systems. New York:
Sons.
20
.John
Harrison, J.M. and R.J. Williams (1987).
.Multidimensional reflected Brownian motions
having exponential stationary distributions. Ann. Probab. 15, 115-137.
Harrison, J.M. and V.
Nguyen
(1993).
Brownian models
of multiclass queueing networks:
current status and open problems. Queueing Systems Theory Appl. 13, 5-40.
Iglehart, D.L.
in Appl.
and W. Whitt (1970). Multiple channel queues
in
heavy
traffic,
1
and
II.
Adv.
Probab. 2, 150-157 and 355-364.
Peterson, W.P. (1991).
A heavy
traffic limit
theorem
for
networks of queues with multiple
customer types. Math. Oper. Res. 16, 90-118.
Reiman, M.I. (1984).
Open queueing networks
441-458.
21
in
heavy
traffic.
Math.
Oper.
Res.
9,
3878
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