HD2{
hJDi
lli&x,.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
MULTIPRODUCT QUEUEING NETWORKS WITH
DETERMINISTIC ROUTING:
DECOMPOSITION APPROACH
AND THE NOTION OF INTERFERENCE
by
Gabriel
R.
*
Bitran
and
*
Devanath Tirupati
WP # 1764-86
March 1986
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
MULTIPRODUCT QUEUEING NETWORKS WITH
DECOMPOSITION APPROACH
DETERMINISTIC ROUTING:
AND THE NOTION OF INTERFERENCE
by
R. Bitran
and
*
Devanath Tirupati
Gabriel
WP # 1764-86
*
Massachusetts Institute of Technology
March 1986
I
Multiproduct Queueinq Networks with Deterministic Routing:
Decomposition Approach and the Notion of Interference
Gabriel
R.
Bitran
and Devanath
Tirupati
Massachusetts Institute of Technology, Cambridge, Mass.
ABSTRACT
Queueing networks have been used to model the performance of a variety of complex
systems, however, exact results exist for a limited class of networks. The
used extensively to analyse networks of queues
is
methodology that has been
the decomposition approach.
In this
paper,
we
consider open queueing networks with multiple product classes, deterministic routings and general
We
examine the decomposition method
arrival
and
that
provides estimates of key parameters with an accuracy that
it
settings.
service distributions.
Recognizing
previously ignored.
this
weakness,
we
is
for such systems
not acceptable
in
enriched the approach by modelling a
We consider interference among
products and describe
its
effect
and show
many
practical
phenomenon
on the variance
of the departure streams. The recognition of this effect has significantly improved the performance
of this methodology.
We
provide extensive experimental results
manufacturer of semiconductor devices.
based on the data of
a
1.0
Introduction;
Queueing networks have been used
to
model the performance of a variety of complex systems
such as production job shops, flexible manufacturing systems, computers, urban service systems,
communication networks
etc.
However, exact results exist
for the limited class of
Jackson type
For more general networks several approaches
networks (Jackson 1957,1963 and Kelly 1975).
leading to approximate solutions have been proposed in the literature. Reiser and Kobayashi (1974)
and Kuehn (1976, 1979) were among the
first
proponents of the parametric decomposition approach,
which was later used by Shantikumar and Buzacott (1981)
for single product
networks and by Whitt
This approach generalizes the notion of independence and product form solutions of
(1983a,b).
Jackson type networks
to
more general models
In this method, the squared
(see section 2).
coefficients of variation (scvs) of interarrival intervals at
each station are computed approximately.
The performance measures such as mean number of jobs, queue lengths
at each station are estimated
based on these scvs.
In this paper,
we consider an open queueing network with multiple product
classes,
We
deterministic routing for jobs in each product class and general arrival and service distributions.
such a class and show that the approximations do
examine the parametric decomposition approach
for
not perform well in certain instances with large
number
of products.
Our analysis shows
that,
under
the assumption that the interdeparture intervals are independent and identically distributed
(iid),
the squared coefficient of variation (scv) of interdeparture intervals for each product stream leaving a
station can be expressed as the
sum
of two terms.
The
first
term
reflects the influence of congestion
and the service at the station while the second term represents the
products in the network.
approach
fact that
to a
of a product stream
mentioned above.
all
presence of other
This result can be interpreted as a generalization of the decomposition
multiproduct network with deterministic routing. The generalization comes from the
we recognize
aggregate
effect of the
is
the interference
among products
ie.
at a given station, the scv of the departures
distorted by the presence of other products. This
In order to
compute
is
reflected in the second
term
this interference effect for a product at a given station,
other products into a single product. Therefore, the two product model
is
we
an important
building block in our methodology.
Since
it is
difficult to
determine the interference
efi"ect
of the other products exactly,
we obtain
two approximations that are based on the following assumptions about the behavior of the aggregate
product.
The
i)
arrivals of the aggregate product follow a Poisson process. This assumption
by the results of Franken (1963) and Cinlar (1972) and
number
of a large
of independent renewal processes
relies
is
inspired
on the notion that superposition
can be approximated by a Poisson
process.
ii)
The
interarrival intervals of the aggregate product have an Erlang distribution.
approximation
than
is
intended for cases with moderate number of products with an scv of less
interarrival intervals.
.5 for
Both approximations are easy
the
demand due
to
each product tends
to
compute and are asymptotically exact when the proportion of
to zero
Our computational
results demonstrate that these are
quite robust and provide reasonably accurate estimates for other distributions with the
The paper
organized as follows.
is
In section 2
briefly describe the parametric decomposition approach.
based on
(i)
above
to
compute the scvs
deterministic routing
for
we provide a review
In section 3
4.
these approximations in a network context.
The
of the literature and
we develop an approximation
to the
new
detailed study of the Erlang approximation and the
and 6 respectively. In section 7 we examine
The network chosen
the production facility of a semiconductor company. Finally,
is
based on real
we summarize
life
data and models
the results in section
8.
Review of the Literature:
The research on network of queues has focussed primarily on performance
the purpose of a brief review of the literature,
i)
we
evaluation.
For
classify the research as follows:
Exact Analysis.
ii)
Approximation Methods.
iii)
is
scv.
departure streams for networks with multiple products and
related computational results are discussed in sections 5
2.1
same
Computational experiments demonstrating the improvement due
approximation are described in section
2.0
This
Simulation and related techniques.
Exact Analysis
:
Exact results exist
for
Markovian systems.
A
seminal contribution in this area
the paper by Jackson (1963) which provides the results for equilibrium probability distribution of
the
number
of jobs
for a variety of
systems that are referred
to as
restricted class of networks can be described in the following framework.
Jacksonian networks.
This
1)
The system
consists of
N
stations or
Jobs which need processing at any
machine centers.
station form a queue at that stage.
2)
A first come first serve discipline is observed at each station.
3)
The
where
service time at each station follows
k; is the total
The system
4)
number of jobs
5)
The
number of jobs at
station
an exponential distribution with parameter pd.kj)
i
(in
queue and
in service).
can be described by an N-dimensional vector
state
at station
arrival process
i,
is
i=
(ki,k2,...,kN)
where
k, is the
1,2,...,N
K
Poisson with parameter A(K) where
the total
is
number
of jobs in the
system.
Job route
6)
is
modeled as a random walk
i.e.
the next step on the job route depends only on the
current station and not on the past processing history.
Thus, the routing behavior can be
described by a matrix R.
R=
where
{rij,i€[0,N].j€[l.N + l]}
ro,i is
the probability that an arriving job visits station
rjj is
the probability that a job visits station
'i,N +
1
and
is
is
equilibrium distribution,
if it exists, is
Probability that the system
=
being processed at station
i,
1^
ij
i.
quite general enough to model arbitrary networks. Jackson provides
sufficient conditions for the existence of
=
first.
the probabilty that a job leaves the system from station
The routing behavior
n(k)
j aifter
i
an equilibrium
of a product form
is in
distribution.
The main
result
is
that the
i.e.
state k in equilibrium
Br(k)ei(ki)e2(k2)...GN(kN)
where k =
state vector (ki,k2,...kN)
r(.)
9i(.)
=
=
a function of the number of jobs in the system.
a function that depends on the nature of station
probability distribution at station
i
i.
0; is
proportional to the equilibrium
assuming the arrivals at the station follow a Poisson
process.
andB =
a normalizing constant.
This result implies that
to
consider each station individually.
compute the equlibrium probability of a given
state k,
we can
Kelly (1975) extended Jackson's result to networks with more
general routings and priority behavior. Specifically he considered a multiple product network with
the following characteristics.
1)
2)
Job arrivals follow independent Poisson processes.
The
service time at each station
service time at a given station
is
is
homogeneous and has an exponential
independent of product type.
distribution.
The
The routing behavior
3)
each class can be described by a routing matrix as in a Jackson
for
network. This permits deterministic routing for some or
4) It is possible to define
on job
in position
Ofcourse, the
5) 5(i,nj) is
PjS
i
a variety of queue disciplines.
at station
together
j
The
nj jobs.
The
product classes.
Pj(i,nj) is
service rate for this job would then be
a function which defines another type of priority.
system
state of the
the proportion of effort expended
pj Pj(i,nj).
sum to one.
arrival at station j with nj jobs
6)
with
all
is
would occupy position
i
6(i,nj) is
the probability that a
in the queue. Again, the 5's
described by specifying the
number
sum
new
to one.
of jobs at each station
and the
product type in each position of the queue.
Kelly showed that even in this more general network, the equilibrium probability
distribution
=
n(k)
is
of the product form
B. r(k).Ai(ki). A2(k2)...
where k =
ki
=
Ai(.)
a vector describing the product type of jobs at station
= A function that depends on the
= a function that depends on
is
i,
state vector.
station
i,
a normalizing constant.
The product form
time
AN(kN)
state vector,
r(.)
andB =
i.e.
result holds in the case of random order service discipline.
not homogeneous, the result holds with a preemptive resume, last
In particular
it
does not hold for
first
come
first
come
When the service
first
serve discipline.
serve policy and non-homogeneous service times.
Similar results are provided for closed and mixed networks by Baskett et al (1975), Gordon and
Newell (1967). The reader
(1985) for
referred to survey papers by Lemoine (1977) and Disney and Konig
is
more references and
details of this approach.
While the product form results are interesting and useful, they are
practice due to the very large state space.
mean
Also, in
values of queue lengths, waiting times
etc.
many
and not
difficult to
implement
in
cases, the parameters of interest are the
n(k). Reiser
and Lavenberg (1980) consider
closed networks for which the product form results hold. For such networks, they have developed a
procedure (mean value analysis) to compute
The main
n(k).
with
result
is
values without evaluating the state probabilities
a recursive relationship describing the performance measures of the system
K jobs as a function of those with
In
mean
summary, the exact
K-1 jobs.
results encountered in the literature basically rely on the following
assumptions;
1)
Exponential distribution
2)
Homogeneous
for service
times
service requirement at all stations.
3) Priority discipline
independent of customer
class.
Poisson arrivals.
4)
However, these are overly restrictive
not extend to more general networks,
it
many
in
practical situations. Since the exact results do
has led to the development of approximation schemes
described in the subsequent sections.
2.2
Approximate Methods: The lack of success
in obtaining exact solutions for general
motivated research in developing approximations to evaluate performance measures
.
networks has
These may be
broadly described under the following four categories,
i)
Diffusion approximation,
ii)
Mean value analysis,
iii)
Operational analysis,
iv)
Decomposition methods.
The
first
mentioned here
three approaches are not directly related to the
for completeness.
We do
work
in this paper
and are
not describe these in any detail, but provide references for
the interested reader.
Diffusion Approximations are motivated by heavy traffic limit theorems and are based on
asymptotic method for approximating point processes.
The work by Iglehart and Whitt
(1970),
Newell (1971), Gaver and Shedler (1973), Kobayashi (1974), Gelenbe (1975), Harrison and Reiman
(1981)
reperesentative of this type of analysis.
is
Mean Value Anlysis
(1980) and
is
is
a heuristic approach similar to the work by Reiser and Lavenberg
intended for closed networks.
Schweitzer (1979) proposed an approximation that has
been extensively tested by Bard (1979). The resulting Schweitzer-Bard algorithm requires solution of
a non-linear system to estimate the performance of closed network.
Buzen (1976) was among the
first to
use operational analysis computer systems.
approach focuses on directly measurable quantities and testable assumptions.
distribution free
and
relies
(1978) provide a tutorial
This
The analysis
is
on flow balance and homogeneous service principles. Denning and Buzen
on the approach. This paper contains a detailed
list
of references on the
subject.
Decomposition Methods are essentially attempts
to generalize the notion of
independence and
product form results for Jackson type networks to more general systems. This approach relies on two
notions
-
(typically
first
a) the
nodes can be treated as being stochastically independent and b) two parameter
mean and
variance) approximations provide resonably accurate results.
The approach was
proposed by Reiser and Kobayashi (1974) and has been used by several researchers in developing
approximations.
Sevick et
al.
(1977),
Chandy and Sauer
(1978),
Kuehn
(1979),
Shantikumar and
Buzacott (1981), Buzacott and Shantikumar (1985) and Whitt (1983a) have all used a similar
approach. Essentially the method involves three steps-
Step
1:
Analysis of interaction between stations.
Step
2:
Decomposition of the network into subsystems of individual stations and their analysis.
Step
3:
Recomposition of the results
Step
1 is
critical to the
to obtain the
network performance.
decomposition approach and we describe
in detail using the
it
models of
Shantikumar and Buzacott (1981) and Whitt (1983a,b). These are representative of the method and
illustrate the approach.
Also,
we have used these as a bench mark
proposed in this paper. The interaction between the stations
a composite of three basic processes
compare the approximations
analysed by looking at the network as
-
a)
superposition or merging,
b)
flow through a queue or a station, and
c)
is
to
splitting or decomposition.
The
first
process typically represents the arrivals at a station while the third describes the
departures from a station. The splitting and the merging processes model the product routing in the
network.
processes.
We first describe the approximations used by Shantikumar and Buzacott for the three basic
They considered networks with a
single product, Poisson arrivals
and general service
times.
a)
and
cai
The merging process considers superposition
of arrivals at a station (say station
be the arrival rate and the scv of interarrival time of the flow from station
external arrivals).
The resulting mixture
is
i
to
approximately described with an arrival rate
j).
Let
\,j
j (i
=
for
\j
and scv
caj as follows:
A
1
=
AT
ca
^ y
i
caj
and
(1)
The flow process describes approximately
csj
are the scv of the interarrival and service time at station
is
=
J
c)
number
the scv of the departure stream from a station. If
j
with a utilization
pj,
the scv of
approximately given by
cd
p^
J
+ (1-p^)
cs
J
J
ca
(2)
J
In the decomposition process a product stream with arrival rate
\j
of substreams, each stream representing the flow from a station
network. The routing
i,
l\^ca
b)
the departures
a
(X
=
then the substream
is
i
assumed
is
to
be Markovian.
If p, is
= p
\.
j
to
is split
into
other stations in the
the probability that a job would follow path
characterized by the following parameters:
\
and scv of cdj
= p
cd
Expression
mean
1/A
(3) implicitly
+1-P
cd
(3)
assumes that interdeparture intervals of the product stream are
and scv of cd. Note that caJi =
iid
with
cdjj.
interactions between the stations are reflected by the scv of the arrivals at each station.
The
Shantikumar and Buzacott consider single product networks with Poisson arrivals and Markovian
Combining the approximations
routing and general service time distribution.
for the three basic
processes described above leads to the following linear systems.
N
= Ar„ +
a•.-'"0.."
y
ica-
y
a
(1-pV. ca. =Ar- +
Pj
into the
utilization at station j
caj
=
scv of arrivals at station j
csj
=
scv of service at station j
=
The
r
(p^r cs
^J J.' J
+
1
-
)],i=l,2,...N
r
J'
(5)
'
routing matrix
Note that
station.
a
system
=
{r^ j}
Y
(4)
J=l
net arrival rate at station j
R=
J.I
J J.'
=
Qj
J
=l
J=i
where \ = arrival rate
,i=l,2,...N
a r
-^—
J
(4) is
the standard flow balance equations to determine the net arrivals at each
internal flows Xij are given by airij. (5)
is
the approximation to compute the coefficient of
arrivals at each station.
In step 2, each station
is
analysed based on the partial information obtained in step
Shantikumar and Buzacott examine M/G/1 and G/G/1 approximations
performance.
interarrival
to
1.
evaluate the station
approximation requires only the mean and scv of the
In particular, the G/G/1
and service times.
Finally in step 3 these results are synthesized and performance measures for the network are
estimated. This procedure
is
mean queue
straight forward for
In developing the queueing network analyser
approach similar
to the
lengths,
and lead times.
(QNA) Whitt (1983a) has
essentially used an
one described above in analysing the interactions between the nodes. The
superposition or the merging process has been modified as follows:
= wca +
ch
J
where
chj is the
modified scv of the merged stream,
function that depends on the station utilization
follows.
l—w
(la)
J
pj
and
caj is
Xjj.
determined by
In the
QNA the
(1)
and
weight
w
w
is
is
a weighting
determined as
)
w
-1
=
l+4(l-p )^v-l)
J
wt
This modification
leads to
1)
is
^^
kj
J
motivated by the observation that neither the asymptotic method (which
nor the stationary interval approach (see Whitt 1982 for details) by itself perform well over
wide range of ca
for
approximating the superposition process, (la) above
both approaches and
is
is
a hybrid approximation of
based on the work of Whitt (1982) and Albin (1981,1982). Whitt also refines
the approximation of the queue process to provide for multiple servers and uses the
approximation
(3) for
the splitting process. Estimation of station performance measures in step 2 are
also modified to provide for multiple servers at each station.
use the
first
QNA
in the
deterministic routing for each product.
is briefly
is
is
formed by appropriately
done with the aggregate data.
The aggregation
described below.
m
= number of products
Afc
=
Hfc
= number of operations for product k
"k,i
arrival rate of product k
=
lH(x)
to
analyse networks with multiple products and
to
an aggregate product
First,
combining the product data and the analysis
Let
However, the approximations continue
two moments of the arrival and service distributions.
These approximations are used
procedure
same
station visited by product k at step
=
The external
1 if
xc
i
H and Ootherwise.
arrival rates
and the flow rates within the network are obtained as
m
k=i
m
k
t=U =
The routing matrix
r^j is
i
defined by
X
_
~
'i/
N
''ij
y
+ *—
0,1
(A.
X,
k,i
k=\
The mean and
the scv of the service time at each station
8
is
computed by
follows.
m
At
H\%n(k,[):-^,rJ^
X.
=
k=ll=l
m
*
II\\/K/+^^^«*'^^"*,/=^^
2.
^
xlcs
J
where
Xj
k
and
csj k
are the
+1)=
*=1/=1
J
mean and
n.
e
m
scv of the service time for product j at step k.
Both Shantikumar and Buzacott and Whitt report encouraging results based on experiments
However, the experiments are
approximations with simulation.
comparing their respective
The one exception
(except for one case) based on a single product.
is
all
the case with two products
presented in Whitt (1983b). However, in the manufacturing context, the problem with more than two
products
is
more common and relevant and provides the motivation
2.3 Simulation Methods:
for the analysis in this paper.
In the absence of (exact) analytic results except for very restricted cases,
discrete event monte-carlo simulation has been an obvious alternative to evaluate large queueing
networks. This approach permits use of more elaborate assumptions that are closer
main drawback
the computational requirement.
is
The process
is
time consuming and except in very
small examples only a limited number of alternatives can be examined.
Recent developments in
This
perturbation analysis suggest considerable promise in reducing the computational needs.
technique, developed by
Ho
et al. (1979) provides a
derivatives of performance
supporting the technique
method
measures with respect
is to
to estimate, in
to the decision
some extent experimental.
SCV of Departures from a Station
one single simulation run,
parameters.
The evidence
Recent work by Suri and Zazanis (1984)
and Zazanis and Suri (1984) examine some related theoretical
3.0
The
to reality.
issues.
with Multiple Products-
Characterization of Interference in the presence of multiple products
The basis
determination of
of the decomposition
mean and
approach described
scv of arrivals at each station.
It
in the
may
previous section
is
the
be noted that specification of
product arrival rates and routings determine the means exactly and hence the quality of the results
depend on the
to the
scvs. In the case of multiple product
estimation of scvs of each product stream.
Poisson distribution, the use of splitting process
perform well in some instances
(this conjecture is
networks with deterministic routing, this reduces
We
observe that
(3) to
if
the arrivals do not follow a
describe the scv of each product
may
not
supported by the computations described in sections
4 and
we
In this section,
6).
characterize, for each product, the scvs of departures from a single server
station processing multiple products.
In the process,
We
product due to the presence of other products.
discussion in this section will
aggregating
all
two product case
Our analysis is described below
The
iii)
refer to this distortion as interference effect.
clear that this interference effect
it
may
and
in detail
is
relies
be estimated by
on the following assumptions.
FCFS.
intervals from the station are
iid.
The product arrivals are independent. The interarrival times
for
each product are
These assumptions are consistent with the decomposition approach,
approximation,
(i)
(ii)
iid.
is
clearly an
together with the single server assumption implies that the sequence of job
arrivals (by product type)
In this section,
is
identical to that of departures.
we consider a
single server station processing multiple products
characterize, approximately, the scv of departures for each product.
generalizes the approximation for the splitting process given by
stations in a network are
now described by
The
(3).
result
is
and we
analogous
to
the approximations for the superposition
(1)
and queue
Notation:
p
=
number of products
station utilization
cs
= scvof service time
Xi
=
X
arrival rate of product
=
=
ca,
i
arrival rate at the station
= S
X,
scv of interarrival time of product
Xi
=
interarrival time of product
Pi
=
Xi /
d;
=
interdeparture interval for product
=
cdi
=
d
nlj
n;
(random variable)
i
(random variable)
scv of interdeparture intervals of product
inter departure interval
nli
+
i
from the station (random variable)
intervals from the station
= number ofjobs of the aggregate product
=
cni
i
X
= scvof interdeparture
cd
i
that arrive during an interarrival time of product
1
= scvof ni
E(.)
=
expected value of (.)
V(.)
=
variance of (.)
10
and
The interactions between
processes and this generalization.
m=
The
product of interest and the aggregation of the other products.
-
priority discipline at the station
The interdeparture
ii)
identify the distortion in the scv of a given
other products. Thus, in our methodology, the determination of interference reduces
to the analysis of the
i)
make
we
i
(2)
Note that the random variable
assumption
d, is
the
sum
of n,
random variables which are
iid
by
(ii).
Hence, E(di) = E(ni) E(d)
=
and
V(d,)
since,
cdi
we
=
=
get, cdi
=
E(d)/pi
E(ni) V(d)
+
l/(piA)
1/Xi
V(ni) (E(d))2
V(d,)/(E(di))2
=
V(d)/(E(ni)(E(d))2)
=
pi
The
=
cd
+
en,
+
V(n,)/(E(ni))2
(6)
scv of the departures of each product stream
is
characterized approximately by
interesting to note that (6) expresses the scv of the departure stream as the
term can be considered as reflecting the
first
effect of the
captures the interference due to other products. This
The expression
explicitly recognise the latter effect.
multiple products.
(3) for
It
reduces to
(3) if
Poisson (ca
b) the arrivals at the station are
is
It is
of two terms.
The
queue process while the second term
which does not
in contrast to that given by (3)
in (6)
may
be interpreted as a generalization of
either a) the routing
=
sum
(6).
is
Markovian with probability
p;
or
Also, note that, in the case of single product and
1).
deterministic routing (6) reduces to cd.
We observe that (6) is not easy to implement, since it is difficult to evaluate en; in the general
In the remainder of the paper,
case.
we propose and
test
approximations
to
determine
en,.
These
approximations are based on assumptions about the behavior of the aggregate product.
approximation
3.1 Poisson
for the
In this approximation
aggregate product
we assume
that the arrivals of the aggregate product follow a Poisson
process. This is motivated by the notion that superposition of a large
processes
may
be approximated by a Poisson process and
and Cinlar
(1972). In
subscript
from
i
what
we derive expressions
follows,
X,, the interarrival
is
time for product
number
of independent renewal
inspired by the results of
for cn^
and
cdi.
For simplicity, we omit the
i.
Let fx(x) be the probability density function of the interarrival interval for product
and
let
E(X)
=
First,
we derive
Pr[nli
=
n]
=
i,
1 / \i
The arrivals of products other than
i
follow a Poisson process with parameter A(l-pi)
the probability distribution of nli
probability that nli takes value n
Pr[nl^=
n]
=
Pdnl ^n\X = x]/'Jx)dx
^
,
'
(X(l-p)xf
since
=n\X = x]=
Pr[nl
'
exp{— X(l
n\
11
Franken (1963)
—p
)x)
'
(Ml
CO
we have
=
Pr[nl
-p)xf
=
n]
-p
•exp(-\(l
)x)f^x)dx
i!
It
can be shown that
£(nl
y
=
)
nPdnl =n] = a-p)/p
n=
and
Next,
we compute
£(n
=
)
1/p
E(nli2) as follows:
£(al^)=
y
T
nVKnl_ = n]=
n(n- l)PKnl, =
n]
^
+
•^-
I
n=
-
and
^
n(n-l)Pr[nl^ =
^
-2
n=
itcan be shown that
n(n—
/
n
l)Pr[rtl
=
+
=n] =
(
An approximation for cdj
is
)(1
— p T/p.
+ca
)
'
(£(nl
))
en
I
I
(1-P.
=
I
+
ca
'
r>
= (l-p) p +
=p^V(n)
'^i
I
(1
-p
)ca
I
given by
cd = p cd +
Note that approximation
+ca
(1-p)
= £(nl^)-
)= V(nl),
Since Vin
(1
[d-pP
(1
n
II
)
Vv-Wdx
^
'
=
I
and V(nl
-p)fc
(Ax(l
^"-2)'
(1-p)
hence E(nl^)
1
(A(l-p)xf~2
-
=
n]
nPAn\_^n\
n=
fi=0
(7)
reduces to
(3)
(I
—p
when
)
+ (1-p.)
p
'^i
^i
ca
(7)
I
the arrivals are Poisson (or ca;
=
1)
and thus
it
can be
considered to be a generalization of the approximations used by Whitt and Buzacott and
Shantikumar. However,
approaches zero,
(3)
it
differs
from
(3)
would suggest that
even qualitatively
cd;
As the following proposition demonstrates,
in the general case.
tends to one, where
(7) is
(7)
exact in the limit.
12
For example, as
indicates that cdi approaches
pi
ca;.
Proposition: Consider a single server station with multiple products, arrival rate A, utilization p
Assume
given service distribution.
interarrival times.
Then as
pi
that the arrivals to the station can be
-> 0, cdj -cai for all
Outline of Proof: Let Wj be the waiting (queue
WjS are identically distributed for all
modelled as a G/G/1 queue.
Oa
j.
+
w
Let
and a
approximated by
iid
i.
Note
service) time for job j.
that, in equilibrium, the
be the equlibrium waiting time. The station can be
An upper bound on
V(w), the variance of
w
is
given by Oa^ + 2 ot^, where
and Ob are the standard deviations of the interarrival and service times respectively. Let V{w)* be
the upper bound on V(w). (For details see Kleinrock, 1976).
Consider an interdeparture interval for product
wi and W2 be the corresponding waiting times
Then,
»|12
d.
=
x.
-
w,
+ w.
we have, V(x) - 4
for the
and
V(d.)
(
V(.w^-w^ ^ 4 V(w) ^ 4 Viw)' and
Since
\Cov{x.
^'^
Viw)*
[V(x.)
Note that V(w)* does not depend on
Let
i.
<.
]
x,
be an interarrival interval for product
and
two successive jobs of this product.
= V{x) + V{w,-wJ
12^ - 2Cou(x.,w,-wJ
il2^
i
,w^-w^\ ^[Vix) V(w^-w^f^ 5 2[V{xWiw)'
V{d)
<.
V{x)
+
4 V(u>)*
+
4 [V{x) V{w)
Also note that in equilibrium, E(di)
pi.
i
=
=
E(xi)
f'^
\'^
l/Xj
=
l/(Xpi).
Hence,
ca.-4X[ca.
V(u))*]
Since X and V(w)* are constant, as
As
(7) is in
the
number of products
p
'
pi
-*
-^ cd. -^
0, cdi
->
we can
p.
p, to
number
If the
number
be small and thus the approximation
of products.
The formula has other
of products processed at each station
ignore the interaction between stations and analyse each station independently.
mean and variance
assume that
]'
caj.
in the
implications as well for queueing networks.
large,
+ A\^V{w) p^ + ^\[ca.V{w)
we can expect
increases,
some sense asymptotically exact
ca
of each product stream would be preserved throughout the network
at every station visited by a product, the first two
moments would be
the
is
The
and we could
same as those
at
the time of the external arrivals into the system.
We make
two remarks about the quality of approximation
(7).
First,
we can expect that the
approximation will perform well as the arrivals are close to Poisson and the performance will
deteriorate with increasing deviations from the Poisson process.
approximation will be good
section
4.0
we describe
for
small values of
results of computations to
pi
(and large
number
Second, we can expect the
of products).
In the following
examine the goodness of this approximation.
Computational Results
In this section
we report the
results of computational experiments
comparing the various
approximations. In the absence of any exact results, we use simulation as a bench
the approximations.
This approach
is
common enough
13
in the analysis of
mark
to evaluate
queueing networks.
For
example see Fraker
Whitt (1983b, 1984)
Kuehn
(1971),
etc.
Shimshak
(1979),
The experiments were designed
We
from a single station processing multiple products.
Shantikumar and Buzacott (1981),
(1979),
examine the scv of the departure streams
to
considered the following four factors in the
design of the test cases.
1)
Number of products
(and the fraction
Four levels were considered
p):
Number of
for this factor.
products were set at 2,3,5 and 10. In each case the arrival rates and distributions of interarrival
times were identical for
from
2)
fraction of
demand due
each product
to
(p)
ranged
0.1 to 0.5.
The distribution of interarrival time: These were assumed
The Erlang parameter was
3)
The
the products.
all
Station Utilization:
set at 2,3
Two
and 4
be
to
with Erlang distribution.
iid
to yield 3 levels for this factor.
levels of station congestion, p
=
0.6
and
0.9
were
These were
tested.
considered representative of moderate and high utilizations in manufacturing settings.
4) Service
time distribution: The service times were assumed
The parameter of the
Thus, in
the fact that in
distribution
was
and
set at 2
to
be
iid
with Erlang distribution.
3.
48 problems were simulated. The choice of Erlang distribution was motivated by
all,
many manufacturing environments
much smaller than
that of a Poisson process
used in the reported literature
some other distributions
in the
to
It
may
is
The Erlang
scvs are typically smaller than one.
i.e.
family provides a range of scvs in this domain.
the variability in process times and job releases
be noted that the Erlang distribution has been
examine cases with scvs
than one.
less
experiments reported in section
we consider
In addition,
6.
To evaluate the performance of the approximations, we compare the results with the
estimates provided by the following alternatives.
1)
Approximation
(7)
together with
interference introduced in this paper
(1)
is
2)
Simulation.
3)
Application of approximations (la),
(1983a) and described earlier.
2.
and
This approach, which uses the notion of
(2).
referred to as
(2)
and
(3) to
The routing matrix
We refer to this as the aggregation approach
INTl
in
subsequent discussions.
the aggregate product defined by Whitt
for the
aggregate product was given in section
and the approximation
is
denoted by
AGPl
in the
rest of the paper.
4) Application of
AGPl. This
is
approximations
(1), (2)
and
(3) for
a generalization of the approach by
the aggregate product which
Shantikumar and Buzacott
multiple products and we denote this approximation by
Figures
1
to
is
defined as in
networks with
AGP2.
and 2 describe the behavior of the approximations as a function of the fraction p
two data sets specified by
compare the estimates
simulation as p tends
ca and cs and are representative of the results obtained.
p,
of cdi
and clearly bring out the
to zero.
fact that
In contrast, the approximations
14
INTl converges
The
for
figures
to that
given by the
AGPl and AGP2 which
are based on
random routing provide estimates that approach the value
one.
represents substantial improvement over both
AGPl and AGP2
INTl requires an estimate
departures
for scv of station
The same procedure
parameter.
is
used
for
AGP2
use
while (la) and
difTerences in the estimates of cd; arise from the use of (3)
and
be observed that INTl
in the estimates for cd,.
We
cd.
may
It
(7) for
(1)
(2)
and
Note that
estimate this
(2) to
AGPl.
are used for
The
the splitting process.
Tables 1-4 describe the results in detail and compare the estimates of cd; for the three
approximations with the simulation value. In each case, by symmetry, the cd, values are the same for
all
the products and the simulation value reported in the tables represent the
mean value
obtained in
the experiments. These results support the analysis of the previous section and permit the following
conclusions.
1)
The computational experiments suggest
AGP2 in the cases
is .39
the two product case
is
a distinct improvement over
is
0.81
and
.75
is
AGPl and
0.333 in table
.976 and .935 compared to the simulation value of .36.
within 10%. The results are less dramatic but
which
-
INTl
For example, consider the ten product case when ca
tested.
The AGPl and AGP2 estimates are
INTl estimate
that
by
AGPl and AGP2 and
.625 of
still
4.
The
significant for
INTl compared
to .545 given
by simulation.
2)
The approximations deteriorate as the
consider the ten product case in table
for
.5).
The error
2.
is
less
than
6% when
ca
is .5
The errors become progressively worse as the ca value decreases - error
.333 and
3)
arrivals diverge from the Poisson process. For example,
15% when ca takes
As expected the
(estimate of .545
of
10% when ca
is
the value .25.
quality of the approximation improves with the
number
of products (smaller
Pi).
4)
The approximations over estimate
the value of cd;.
The above conclusions should be tempered by the
fact that the
measures of interest are mean
queue lengths, number of jobs, and waiting times at the stations and not the
Hence,
it is
necessary to examine the impact of errors in cd on the queue length estimates at the
subsequent stations. Note that the
cd; of the
departure stream represents the arrivals at subsequent
stations and should be interpreted as ca at the next station.
approximations
Lq
=
to
p2/(2(l-p))(ca
observe that the two
moment
L = p + Lq
+ cs)g,
L = mean number of jobs
f(ca,cs,p)
We
evaluate the queue lengths are of the form
where Lq = the mean queue length
andg =
coefficient cd itself.
=
at the station
at the station
exp[-2(l-p)(l-ca)2/(3p(ca
This approximation
is
due
to
+ cs))]
Kraemer and Langenbach-Belz
Whitt and Shantikumar and Buzacott. The
latter, in fact,
(1976) and has been used by both
have used two other approximations. The
choice of the appropriate formula depends on the values of ca and
15
cs.
The function g
is
not very
sensitive to either ca or cs. Thus,
if
would result in an error of magnitude (x/2)% in the estimate of
in the estimate of ca
the estimate of L would be even lower.
occur
when
the value
is
illustrate this effect,
we consider a
departure streams from station
mean number
table 5
be the
1
at stations 2-6
of the
station
1
in table 6.
The
of jobs
Observe
that,
sets of
Note that the
We now examine
parameters are described in
by symmetry, the mean number of jobs
The table provides,
computed from an error
is to
All the
in figure 3.
in addition to
result.
AGPl, AGP2 and INTl
figures in the parenthesis in
We also provide an estimate
free estimate of the scv of the departures
illustrate the
performance of the approximation
to
lengths.
12%
error in the estimate of
and AGP2. For example,
note that
1,
mean number
the error in the
The corresponding
cd;.
of jobs
is
4.2%
AGP2 and AGPl
the arrivals are Poisson (cai=
(1984) indicate that
5.0
in case 2
AGPl
overestimate the parameter by 52.3% and 57.9%
1 for all
AGPl and AGP2 perform well.
in
figures for case 2 are 9.7% and
The M/M/1 approximation performs poorly with an error
when
from
estimate
18.2% respectively. These result provide additional evidence of the improvement by INTl over
respectively.
will
result reported in table 6 represents the average of the
results in the table are encouraging. For case
contrast to the
in
When
unlikely to be large.
at stations 2-6.
M/M/1 and M/G/1 approximations. The
by simulation. The intent
mean queue
the
The two
measure devations of the estimates from the simulation
mean number
is
network shown
and need a second operation
1
and the simulation
estimates, those resulting from
The error
note that large errors in the estimate of cd;
correspond to the experiments described earlier.
values observed at these stations.
the table
to
six station, five product
of jobs at stations 2-6 for two cases.
and the results are given
same
encouraging
Lq.
error
and would not affect the queue length estimates substantially.
products are first processed at station
the
It is
small and the impact on queue length estimates
the cdi are large, the errors are small
To
x%
ca and cs are of comparable magnitude, the impact of an
i),
of 124%.
The reader should
the computational results presented in Whitt
In that case INTl,
AGPl and AGP2
are the same.
Erlang Approximation
In this section
we
processing two products.
(whose en;
is to
As explained
earlier, the
two products represent the product of interest
be determined) and the aggregation of
additional approximations
the fact that
study, in detail, the scv of departure streams from a single server station
INT2 and INT3
INTl does not perform very
this section are based
Erlang distribution.
to
all
the other products.
estimate the interfernce
well for small values of ca,.
effect.
We
propose two
These are motivated by
The approximations proposed
in
on the assumption that the interarrival times of the two products have an
The choice
of the
distributions with scv smaller than one
Erlang
and
is
due
to the fact that it
this is of interest in
provides a family of
many manufacturing
settings.
Further, this assumption facilitates the analysis and leads to computationally viable approximations.
The computational
results of the following section demonstrate that the scvs are not sensitive to the
16
^
specific distributions but
depend on the
Arrivals at the station belong to two product classes
the other products.
ii)
which follows, we assume that
also. In the analysis
used with other distributions
i)
two moments only and hence the approximation can be
first
.
The arrivals in each group are independent of each other.
The interarrival times
each product are independent and have an Erlang distribution. The
for
parameters of the distribution are
\i, ki
and
\2.
the aggregate product). Note that ca, is l/k, for
iii)
As
before,
product of interest and the aggregation of
-
we assume
^^2
for products 1
=
1
i
and
and 2 respectively. (Product 2
is
2.
that interdeparture intervals from the station are iid with a scv of cd.
An estimate of cdj is given by (6)
,
cd,
^—
p =
let
=
pjcd
,
i=l,2
+ cnj
We describe below,the two approximations INT2 and INT3 to estimate en,.
5.1
Approximation INT2 In evaluating cni
product
1
(2),
parameters
we assume
(cn2),
the arrivals of jobs of product 2 (1) follow an independent Erlang ditribution with
A2, k2 (Ai, ki).
We
are, in effect ignoring the
between successive interarrival intervals of product
approximation
now
that during any interarrival interval of
for the first arrival of
(1)
However, note that,
an
1
(2).
product 2(1) only and
derive the probability distribution of nlj the
interarrival time of product
dependence in the arrivals of product 2
number
it is
this is
exact for subsequent arrivals.
We
of jobs of product 2 that arrive during an
1.
Let fxi(t) and fx2(t) be the probability density functions of interarrival intervals for products
1
and 2
respectively.
-1
*
^
*,
^^i^'^
t
= 'i'^rri]i^^^-v^''-«
*2-^
^X2W =
Property
1:
Let
X
^'^][tT^^^(-V)
-'"O
be a random variable with an Erlang distribution with parameters X and
cumulative distribution of X
is
k.
The
given by the following:
F^it)
=
1
-
*-Va<)'
i
e-^, t^O
=
Property 2 The probability density function for n arrivals (n-fold convolution) of product 2
with parameters X2 and nk2. Hence,
17
is
Erlang
—
The probability distribution of nli
is
obtained as follows
PKnlj-n]=
= n\Xl =x] =
Prinl
But,
Pr[nl^
Pr{n\
= n\Xl
=x]f^^(x)dx
^^n\Xl=x]- Pdnl ^ n + 1 |X1 =x]
j
J
"*2-Va2x)'
Pr[nlj^n|Xl=x] =
and
rj^it)dt=
1
y —p exp{-x^)
-
,
(n
Pr[il.
/lence,
=
'•
+ m^-l^^^^.
——
^
= H-^1 =^] =
exp(.
'.
— \^)
i = nk.
(n + l)*„-l.-
anrf
=n]=
Pr[nl
Jo
>
^^,
(n
+ l)*2-l
I
.
1
{k,-l)\
1
It
«^-l)!
—
=
= n]
is
nonnegative and
it
77
7;
exp{-{\^
(1-g)'
1
1
it
for given values of the
,
where q =
(8)
\ + ^2
is
a proper probability mass function.
can be shown that these probabilities sum
parameters by using
c^x
+ \^x} dx
rt*„
easy to obtain closed form expression for V(nli), the variance of nli.
compute
«cp(-^,^)
(^l-D!
X
can be verified that the distribution obtained for nli
Clearly, Pr[nli
^i
i\
y
=n] =
hence, Pr{nl
^
,
—r~ ^P^-^o^^
Jo
= n*2
-1
*
,j
(A^)
2
p
(8) directly.
^^^
it is
Since ni
is
to one.
It is
not
straight forward to
nli
+
1,
we have the
following.
E(ni)
cni
=
=
E(nli)
+
andV(ni) = V(nli)
1
V(nli)/[E(nli)+l]2
Hence cni can be computed from the parameters of the arrival
for the
5.2
distributions.
The
coefficient
second product can be computed in an analogous fashion.
Approximation INT3
:
In this approximation
we assume
instance of random incidence in the arrival stream of product
first arrival of product 2
that the arrival of product
2.
during an interarrival interval of product
18
The
1 is
distribution of the time
is
an
till
the
1
then given by the following
"2
i-l
J
fx2M^ =
7 I
=
'^2
The
^2
1
1
V
exp(-A^)
x^
,
'•
distribution for the subsequent arrivals
is
Erlang with parameters
from INT2 in the assumptions regarding the distribution of time
differs
distribution of nl 1 can be derived as in the previous case.
*2
It
j-\
till
A2, k2.
Thus INT3
the first arrival.
The
can be shown that
(i
+ k-\)\
=01=
P7{nl
1
(A.
-!)!*„
;=i
1
=
n*2+7-l
(l+k^-\)\
Pr{n\=n]—
1
and
(1-q)'
(^,-1)!*,
'
2
1
j
=
a
for
>
(9)
t={n-\.)k +j
\
Again, we do not have expressions in closed form for variance of nli and ni. For specified parameters,
cn^ can be
computed directly from
We make
a few remarks about INT2 and INT3.
similar and for low values of p,
Second, the approximation
is
time
is
INT2
is
likely to
we do not expect any
only in respect of the
during an interarrival interval
results.
(9).
is
large (or p
very small. This will happen most often
This hope
is
significant difference in the two estimates.
first arrival.
small),
two approximations are very
Hence,
if
the
number
of jobs arriving
we may expect INT2 and INT3
to give
good
perform poorly when the number of jobs arriving during an interarrival
applicability of the two approximations to be
when
the fraction p;
is
close to one.
more general than the case
for
We also expect the
which they were derived.
based on the fact that two moment approximations provide reasonable results in most
cases. Hence,
scv.
is
First, the
we expect INT2 and INT3
to
perform well
We empirically examine some of these issues
in the
for other arrival distributions
with the same
computational experiments described in the
next section.
6.0
Computational experiments
In this section
to test
we empirically
Erlang approximations:
test the
goodness of approximations INT2 and INT3 and report
the results of the computational experiments.
estimates of cdj by
INT2 and INT3 with
In the first set of experiments
we compare the
other approximations and simulation results for a two
product, one station system. In the design of these experiments the following factors were considered.
i)
The
fraction p;
section 4, p;
is
the proportion of demand at the station due to product
was the same
for all the products (1/m). In the present case,
i.
In the experiments of
we consider nine
Pi,
0.1 to 0.9 in steps of 0.1 to
analyse impactof changing the product proportion.
ii)
The arrival
Job arrivals
distribution:
in
each product class are independent of each other. The
scvs of the interarrival times for each product are set at .333.
with the same scv
-
a)
levels of
Erlang distribution with k = 3,
19
b)
We
consider three distributions
Uniform distribution over
[0,a],
and
c)
Beta distribution with parameters 2 and
approximations
to distributions
As
iii)
Station utilization:
iv)
Service distribution
In all 108
approximations
for
is
The
6.
intent
is to
examine the robustness of the
with the same scv.
in section 4,
assumed
we consider two levels of congestion,
to be
Erlang with parameters 2 and
problem sets were simulated.
p,
shown because they are very
in the cd estimates by
INT2 and INT3 give comparable
(p)
and service distribution
otherhand, for larger values of
The detailed
expectations.
i)
p,
close to this value.
INTl, INT2 and INT3 over
results
and appear
INT3 seems
The
10%
increasing
INT3 consistently
For low values of
and lead
This
is
the
consistent with our
to the following conclusions.
The estimates of cd are
in the test cases.
On
While the performance of INT2 deteriorates a
typically
little
with
gives good estimates of cd.
The scv of the departure streams
ii)
figures demonstrate the
perform better than the others.
results are furnished in tables 7-10
of the simulated values.
p;,
to
The
The
(cs).
results with other
AGPl and AGP2.
be the best method.
to
INT2 and INT3 are very good approximations
within
.9.
3.
simulation values in the figures correspond to the Erlang arrival case.
improvements
and
.6
Figures 4 and 5 describe the behavior of the
two data sets defined by station utilization
distributions are not
p=
is
cds for the three distributions tested
not sensitive to the arrival distributions.
is less
than
5%
in
most
The range
of the
cases. This supports our conjecture
that the approximations are robust and should give reasonable results for distributions with the
same
scvs.
We now examine the behavior of INT2 and INT3 for the multiproduct case. The test problems
are those described in section
The
The
4.
cdi
estimates given by INT2 and INT3 are displayed in table 11.
table also provides the simulation values for comparison.
results,
case.
it is
From
figures
and 2 and the tabulated
1
obvious that INT2 and INT3 perform better than the others even for the multiproduct
The value of p,
is
not greater than 0.5 and both
suggest that with increasing number of products,
explained by the fact that as the
number
INT2 and INT3
INT2 performs
give good results. These results
better than INT3.
of products increases, the aggregate product departs from an
Erlang process, and the corresponding scv of the interarrival time tends
situations, the
random incidence considered
We
modelling the interference phenomenon.
infinity,
within
INTl
5%
(the
will
in
error
is
to increase.
In such
INT3 becomes gradually counter productive
in
would expect that as the number of products tends
outperform both INT2 and INT3.
maximum
This might be
The errors
to
in the cd estimates are typically
12%) which would suggest that the errors in the queue length
estimates are likely to be very small.
Finally to see the impact of the Erlang approximation on the queue lengths,
five product, six station
example of section
and the simulation results are displayed
4.
The INT2 and INT3 estimates
in table 12.
20
The error
in the
for
we consider
mean number
INT2 estimate
for the
the
of jobs
two cases
are 1.51 and 2.13% compared to 27.7 and 52.3% given by
The corresponding figures for the INT3 estimates are
7.0
AGP2, and 30.67 and 57.9% given by AGPl.
1.51
and 1.12% respectively.
A Network Example
To
recapitulate, the intent of the approximations presented in this paper is to evaluate
performance measures in a queueing network and in this section we report the results of an
experiment in this context. The network
models the production
facility of a
for this set of
experiments
semiconductor company. The
is
based on real
facility is
station 2
more than
once.
We
data and
represented by 13 machine
stationsandprocessesjobsof 10 product families. The routing for jobs in each family
The network characteristics are described,
life
is
deterministic.
Note that product families
in detail, in the appendix.
visit
considered the following factors in designing the problems for this
experiment.
Arrival Distribution: The interarrival intervals for each product family were assumed to be
1)
iid.
For each product the arrival distribution was randomly assigned from
Erlang with parameters
The same
2)
2,3,4,
five alternatives
Station utilization was
utilization at each station
We
were considered
for the distribution of service time.
assumed
to be
uniformly distributed between 0.65 and 0.95. Again, the
was determined
in a
random manner.
Summary
presented in table 13. In the table we compare the estimates of
results for the ten problems are
mean number
by approximations INTl, INT2 and INT3 with those given by AGPl,
term.
Overall,
improvements that can be obtained by
INT2 seems
to
For each
time was randomly assigned.
generated ten problems for this experiment.
results demonstrate the
-
exponential and uniform distribution.
station, the distribution of service
3)
five alternatives
of jobs in the network
AGP2 and
simulation.
The
explicitly recognizing the interference
be the best procedure, while in most cases INT3 provides errors
between those of INTl and INT2. These results are consistent with those
in tables 1-4
and
11.
The
average error with INT2 and INTl are 1.78% and 5.95% respectively. In contrast, the average error
with
AGP2 and AGPl
are 23.67 and 27.39% respectively.
aggregation approach (AGPl and
is
AGP2)
The errors
in the
are typically abovel9.45% in the problems tested. Case 10
an exception with errors of 12.5 and 15.26%. This problem corresponds
arrivals of five of the ten products follow a Poisson process.
the two cases in which approximation
The estimates at the
AGPl and AGP2.
within
10%
INTl
It is
still
(86% of the observations).
is
21
(error 3.76%).
mean number
The corresponding figures
at nearly
60%
of jobs
was
for the other
AGP2 and AGPl
respectively.
may be
noted that, with
of the stations.
The maximum
station level statistics in table 14.
more than 20%
INT2
substantially better than those given by
approximations were 93 (71.5%), 14 (10.77%) and 9 (6.92%) for INTl,
the aggregation approach, the error
case in which the
also not surprising that this is one of
In the ten test problems, the error in the estimate of
We present the summary results for the
to the
(error 1.47%) performs better than
station level are not as good but are
at 112 stations
estimates by the
It
error
is
64 and 63%for
AGPl and AGP2
In contrast, the
respectively.
maximum
error with
INT2
is
18.8%.
We make
two remarks about the use of the Eriang approximations INT2 and INT3
estimates. First, note that for a given
en,
pi,
in these
can be estimated for discrete values of arrival scvs. Since
the scvs in the network are continous variables,
we obtain
en;
by linear interpolation in such cases.
Secondly, since the cniS need to be consistent with these scvs, an iterative procedure
determine them. In the test problems, typically three
to four iterations
is
required to
were required.
8.0 Conclusions;
In this paper
we have examined the use
of the parametric decomposition approach to the
analysis of multiproduct queueing networks with deterministic routings.
the approximation for the disaggregation or the splitting process based on
quite poor.
sum
as the
The analysis
in this
of two terms.
The
time while the second term
is
We have demonstrated that
Markovian routings can be
paper shows that the scv of product departures can be approximated
first
term represents the influence of station congestion and service
the interference term due to the presence of other products.
interference term explicitly recognizes the presence of multiple products
This
and can be considered as a
generalization of the one used by Shantikumar and Buzacott, Whitt and others in the decomposition
approach.
Since the estimation of the interference effect
some
tested three approximations that are, in
(INTl)
evaluate exactly, we proposed and
sense, asymptotically exact.
may
be approximated by a Poisson process.
The computational
performs reasonably well over the range of parameters tested.
INT2 and INT3 are based on an Eriang
for arrivals
with scvs
much smaller than one
However, INT2 seems
Our computational
increases.
substantially superior results
MMl
approximation
(for
approximations. This
in the estimate of the
results
compared
is
the aggregation approach.
0.5).
and are intended
While INT2 and
gives better estimates at high values of p
perform better than INT3 as the number of products
to
show that approximations INTl, INT2 and INT3 give
aggregation approach (AGPl and
to the
demonstrated by the
mean number
The other approximations,
example, scvs smaller than
INT3
show that INTl
of the approximation
distribution approximation for the arrivals
give comparable results for low values of p,
(greater than 0.5).
and
first
results
The quality
deteriorates a little as the scv of arrivals diverge substantially from one.
INT3
The
based on the notion that the aggregation of the arrival streams of a large number of
is
products
is difficult to
of jobs
is
The experiments
five product, six station
13% by INT2 compared
2
AGP2) and
the
MGl
example. The error
to errors in
excess of 50% with
also suggest that the results are robust with respect to
the arrival distributions and the critical factor
is
the scv. This
is
in the
same
spirit as the
two moment
approximations in the decomposition approach.
The
potential of the approximations proposed in this paper
is
demonstrated by the network
experiments with representative data from a semiconductor manufacturing company.
22
The
application of INT2,
INT3 and INTl
simulation figures. This
result in errors of over
In table 15
is
give estimates that are typically within 5 and
a substantial improvement over the approximations
10%
of the
AGPl and AGP2
that
20% in most cases.
we summarize our recommendations regarding the
application of INTl,
INT2 and
We
INT3. These are based on the computational results which support the preceeding analysis.
recommend the use of INTl when
greater than 0.5,
0.5),
INT2
INT3 may be used
or
we recommend use
the scv of arrivals
of INT3.
routings are representative of
arrivals and service times
is,
and scvs smaller than one are
Finally,
for
is
greater than
low values of
p.
.5.
When the
scv of arrivals
is
not
For larger values of (greater than
we observe that multiple products and deterministic
many manufacturing environments.
Also, the variability in job
according to our experience, usually less than that of a Poisson process
typical.
This enhances the jjotential for the approximations proposed in
this paper.
Acknowledgements:
The authors are
helpful
grateful to Professors Hirofumi
Matsuo and Stephen C. Graves
for their
comments on an earlier version of this paper.
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25
Utilization
=
scv of arrivals
scv of service
#
0.9
= 0.333
= 0.5
of products
Utilization
=
0.6
scv of arrivals = 0.5
scv of service = 0.333
#
of products
4
Figure
3:
The
5 Product. 6 Station
Example
>
*-
PRODUCT
1
PRODUCT 2
PRODUCTS
PRODUCT
PRODUCTS
Table
1:
Comparison of Approximations Estimation of cd
Station Utilization p = 0.9
scvof service = 0.333
#of
products
-
Table
3:
Comparison of Approximations Estimation of cd
Station Utilization, p = 0.6
= 0.3333
scv of service
#of
products
-
Table
5;
Parameters for the 6 Station. 5 Product Example
Table
7:
Comparison of Approximations Varying
Station Utilization, p = 0.6
scv of service = 0.333
Pi
•
pi
Table
9:
Comparison of Approximations Varying
Station Utilization, p
scv of service
Pi
=
0.5
=
0.9
p;
Table
11:
Performance of Approximations INT2 and INT3 with Multiple Products
Estimation of cd, the scv of departure streams
#of
producu
Table
13:
Comparison of the Approximations
for the
Network Experiment
L = Mean number ofjobs in the network.
e
=
% absolute error in the estimate of L relative to the simulation value,
c
is
the average of e for the ten problems.
Cas€#
APPENDIX: Data Base
Number of Stations =
13.
Number of products =
10
Product Routing Characteristics:
Product
for the
Network Example
ifiASEMtNT
Date
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