HD28
.M414
\NST
IBAUG 18
4*«
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
CAPACITY ALLOCATION IN
GENERALIZED JACKSON NETWORKS
Lawrence M. Wein
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
^018-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
>
966
CAPACITY ALLOCATION IN
GENERALIZED JACKSON NETWORKS
Lawrence M. Wein
Sloan School of Management
Massachusetts Institute of Technology
Cambridge.
n018-88
MA
02139
CAPACITY ALLOCATION IN GENERALIZED JACKSON NETWORKS
Lawrence M. Wein
Sloan School of Management, Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
We
consider a capacity allocation problem for a generalized Jackson network (a Jack-
son network with general interaxrival time and service time distributions). The problem
to determine the service rate (or capacity) that minimizes the expected equilibrium
is
customer delay subject to a linear budget constraint on the capacities. The problem
analyzed using a Brownian approximation of the generalized Jackson network.
ing capacity allocation
for
is
A
Jackson networks.
a generalization of the
numerical example
is
clcissical
The
is
result-
squaxe root capacity allocation
provided that demonstrates the allocation's
effectiveness.
Introduction
1.
Consider a queueing network with
nite capacity waiting
and the
K single-server stations, each of which has an infi-
room. Customers arrive at station k according to a renewal process,
interaxrival times
have
finite
mean
A^^^
(variance divided by the squaxe of the meaxi)
at station k, are next
history.
It is
=
1,
...,
K,
c^/^.
finite
so
all
squared coefficient of variation
Customers, upon completion of service
routed to station j with probability Pkj, independent of previous
assumed that the Markov routing matrix
than one, so that
k
and
P=
(Pkj) has spectral radius less
customers eventually exit the network.
no immediate feedback of customers
is
allowed.
We
also
The
assume Pkk
=
for
service times at station
BECE»VEO
k
=
1,
K axe independent and identicedly distributed random variables with finite mean
...,
and
fj,^^
and routing
service times,
cire
FIFO
served
arrival rates
rates.
squared coefficient of variation
finite
Then
decisions are
is
it
=
A(/
eissumed that the
so that the system
is
assumed
—
of interarrival times,
be mutuedly independent. Customers
P)~^, where A
traffic intensity
This model
stable.
to
The sequences
each station. Let the A'— vector 7
(first-in first-out) at
be defined by 7
c^j^.
reduces to a standard Jackson network
is
pk
=
(Ajt) is
=
fkl fJ-k
the
<
1
=
(7^) of effective
K— vector
for
each
fc
of arrival
=
1,
...,K,
called a generalized Jackson network, since
when
it
the interarrival times and service times at
each station are exponentially distributed.
Queueing networks are very useful models of computer, communication, and manufacturing systems.
tion.
The
of the basic design issues for such a network
classical result
Kleinrock
in a
One
[3].
on
this subject
The problem he
that of capacity alloca-
the square root capacity assignment derived by
is
considers
is
is
to choose the vector of service rates
/x
Jackson network to minimize the expected equilibrium sojourn time per customer
equivaiently, the expected equilibrium customer delay or the expected equilibrium
of customers in the system) subject to the budget constraint X]it=i
the unit cost of capacity at station
this
problem
is
k,
and
D
is
^tA^it
=
the total available budget.
(^jt)
(or,
number
D, where d^
The
is
solution to
to choose
''^'
Y.UV^^
allocates just
)for* =
S^
In the case where the unit cost of capacity
first
=
is
among
(1)
the same at each station, this assignment
enough capacity to each station to
then allocates the excess capacity
l,...,^.
satisfy its effective arrival rate,
and
the stations in proportion to the square roots of
their effective arrival rates.
Notice that Kleinrock assumes capacity
capacity
is
tion); see
is
continuous, whereas in most applications
actually discrete (for example, parallel machines at a manufacturing work sta-
Shanthikumax and
Yax)
[5],
[6]
for recent
2
work on server allocation
in
Jackson
many
networks. However, in
allocated in large enough
thermore, even
when
applications, such as
communication networks, the capacity
numbers that the continuity
the continuity assumption
is
cissiimption
is
is
quite reasonable. Fur-
not realistic, this assignment
still
offers
a back-of-the-envelope calculation that can add some insight to the allocation problem.
Kleinrock also assumes that
Thus,
distributed.
it
is
all
interaxrival times
assumed that each
of variability. However, in
and service times are exponentially
station, in effect, exhibits the
most queueing systems, some stations exhibit more
than other stations. In a manufacturing environment, for exzmiple,
stem from the occurance of machine breaJcdowns or operator
Bitran and Tirupati
[1]
same cmaount
variability
this variability
unavailability.
may
Recently,
have analyzed capacity allocation in multiclass queueing networks
with general interarrivai time and service time distributions. However, they do not obtain
closed form results,
and so
making extensive numerical
In this paper,
we
very difficult to gain any insight from their model without
it is
calculations.
derive a simple capacity assignment for the generalized Jackson net-
The problem we analyze
work.
interarrivai
and
is
the same as Kleinrock's problem, except that general
service time distributions are allowed.
squared coefficients of vaxiation for
and independent of the value of the
stant,
is
all service
As
in Kleinrock's problem, the
time distributions are assumed to be con-
service rates chosen.
Our optimal assignment
a square root allocation that reduces to Kleinrock's assignment
for k
=
heavy
I,
...,K.
traffic
The
result
is
heavy
theory developed by Reiman
traffic
c^^
=
c^^
=
1
obtained by using an approximation based on results from
our results axe rooted in heavy
to the
when
[4]
traffic theory,
and Harrison and Williams
[2].
Although
they appear to be quite robust with regard
assumptions, as can be seen in the numerical example of Section
3
4.
2.
The Brownian Approximation
Brownian model proposed by Harrison and WiUiams
In this section, the
briefly
summaxized. This model
is
a refinement of the heavy
generalized Jackson network derived by
Reiman
that the total load imposed on each station
precisely,
we assume there
is
-v/nll
approximately equal to
—
which case n
=
process of interest
is
100
satisfies
<
pk\
where Qk{i)
n
is
is
the
number
=
^^^,
[2]
t>0,
forfc
=
1.0 for
Q*
—
each station
The primary
(Ql) defined by
l,...,K,
(3)
^,
and
Under the balanced heavy loading condition
(2),
show that the scaled queue length process Q*
distribution.
stationary distribution has a product-form density function
symmetry condition holds anaong the network
[3]
and
0.9
of customers queued and in service at station k at time
mated by a process Z that has a unique stationary
Williams
More
(2)
the balanced heavy loading condition (2).
the large integer specified in (2).
Harrison and Williams
capacity.
1.
the scaled vector queue length process
Ql{t)
its
requires
n such that
As a canonical example, one may think of pk being between
k, in
data.
that the stationeiry distribution of
It
if
They
is
also
and only
if
well approxi-
show that
is
this
a certain skew-
has been suggested in Harrison and
Z may
be approximated by the product
form distribution even when the skew-symmetry condition does not hold exactly, and
suggestion
be
approximation of a
The Brownian approximation
[4].
exists a large integer
majc
traflac
will
[2]
this
incorporated into our approximation scheme. Under this assumption, Harri-
son and Williams show that the expected number of customers at station k in equilibrium
is
2
'^'^
for
2(//it-A)
4
it
=
1,.., A',
(4)
where, from equation (27) of Reiman
[4]
and equation
(2.23) of Harrison
and Williams
[2],
The Capacity Allocation
3.
Using the Brownian approximation of the previous section, the capacity allocation
problem
is
to choose the vector
//
=
(^jt) so as to
minimize
y —^—
K
(6)
subject to
K
=
Y,dkfik
D.
(7)
fc=i
To
solve this problem,
we
define the Lagrangian
K
i=
L by
^
2
w
E2of^ + -(E<''«-^).
fc=i
where
tt is
the Lagrange multiplier, and set
/x;
Substituting these values of
value of
TT
Notice that
when
c^^
—
7fc
+
^==
for
for k
fc
=
=
1,
~
K. This
yields
l,...,K
(9)
tt,
and substituting the
(9) yields the square root capacity assignment:
^M^(EZ^fl^)(or
^Ik
...,
into the constraint (7), solving for
//jt
back into equation
,l^,, +
=
j^ =
^
f°^
A;
=
1,
...,
A', then,
k
=
by equation
l
A-.
(10)
(5),
J
afc=Afc
+ 7fc+^7;P;it
5
=27fc,
(11)
and thus our cissignment
(10) reduces to Kleinrock's assignment (1) under the exponential
assumptions.
The capacity assignment
(10) allocates the excess capacity to station k in proportion
The quantity
to the square root of the variability paxajneter a^.
three terms.
The
first
term measures the amount of
fc,
eind the third
term measures the
The main
k from the other stations in the system.
following:
in equation (5) has
from
variability arriving to station k
outside of the queueing system, the second term measures the
the server at station
a],
amount
of variability from
totaJ variability arriving to station
insight gained
from
paper
this
is
the
the optimal capacity assignment compensates for the highly variable stations
by adding more capacity to them.
4.
An Example
Consider a three station network with A
=
(6,2,0), c^
=
(.54, 4,0), c^
=
(.54,2,-08),
and
/O
1/3
2/3\
P=
\0
Assume
all
(12)
.
/
1/2
capacity costs axe the same, so that di
=
=
d2
=
di
that the skew-symmetry condition in Harrison axid Williams
problem data. The vector of
effective arrival rates for
will consider four different cases of
case.
the
Let us define the
sum
values of
all
is
our problem
traffic intensity
can easily be shown
not satisfied by this
7
=
D
varying in each
is
(6,6,4).
We
p of the problem to equal ^^._i fk/D, which
sum
were chosen to correspond to values of p=.6,
four cases,
[2]
It
our problem, with the total budget
of the effective arrival rates divided by the
D
1.
we performed simulation experiments on
the Brownian approximation (denoted by
B
assignment (denoted by K) from equation
(1),
6
in
Table
1)
The
four
respectively.
For
of the service rates.
.7,
.8,
and
.9,
is
three capacity assignment rules:
from equation (10), Kleinrock's
and the proportional capacity cissignment
(denoted by P), where capacity
rates; that
is
allocated in direct proportion to the effective arrival
is,
fil
=
D
2''
for
(INSERT TABLE
The
results are displayed in Table
traffic intensity
(13)
HERE)
1
Each combination of capacity assignment and
the average customer sojourn time (and
average server utilization
all
1,...,K
was tested by 200 independent runs of 10,000 customers each. For each
we computed
case,
1.
=
A:
levels,
values of the traffic intensity
95%
and the average queue lengths
/?,
confidence interval), the
at the three stations. For
the Brownian approximation outperformed the pro-
portional assignment rule, which in turn outperformed Kleinrock's assignment rule. Since
exponential distributions were not assumed in this example,
proportional rule outperformed Kleinrock's rule.
results
were derived under heavy
traffic conditions,
at all levels of the traffic intensity. For
of the
It is
p
=
it is
not surprising that the
interesting to note that, although our
the Brownian allocation performed well
.6, .7, .8,
ajid
.9,
the percentage improvements
Brownian assignment over the proportional assignment were 5.0%, 6.5%, 8.0%, and
6.4%, respectively. Notice that, of the three rules, the Brownian assignment produced the
most imbalanced average
utilization levels, but the
most balanced average queue lengths.
References
[1]
G. R. Bitran and D. Tirupati, "Trade-off Curves, Targeting and Balancing in Queueing
Networks," submitted to Oper. Res., 1977.
[2] J.
M. Harrison and R. Williams, "Brownian Models
of Queueing Networks with
Homo-
geneous Customer Populations," to appear in Stochastics, 1987.
[3]
L. Kleinrock,
Communication Nets: Stochastic Message Flow and Delay, Dover Publi-
cations, Inc.,
[4]
M.
I.
New
York, 1964.
Reiman, "Open Queueing Networks
in
Heavy
Traffic,"
Math, of Oper. Res.
9,
441-458, 1984.
[5]
J.
G. Shanthikumar cind D. D. Yao, "Optimal Server Allocation in Systems of Multi-
Server Stations,"
[6]
J.
Management
Science 33, 1173-1180, 1987.
G. Shanthikumar and D. D. Yao,
"On
Server Allocation in Multiple Center
facturing Systems", to appear in Oper. Res., 1987.
8
Manu-
CAPACITY
ALLOCATION
61
/
067
Date Due
Mar.
2
/yj
4
IFlit
Lib-26-67
MIT L1BRAHIF5
3
TDflD
DD5
3Sfi
flMfl
,./W^^^