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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ORDERING TANDEM QUEUES
IN
HEAVY TRAFFIC
Lawrence M. Wein
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
*2017-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
t7^
ORDERING TANDEM QUEUES
IN
HEAVY TRAFFIC
Lawrence M. Wein
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
*2017-88
MA
02139
^? IT.
UBRAHltS
RECEIVED
ORDERING TANDEM QUEUES
IN
HEAVY TRAFFIC
Lawrence M. Wein
Sloan School of Management, A'lassachusetts Institute of Technology
Cambridge, Massachusetts 02139
We
more
consider the queueing system design problem of finding the best order of two or
single-server stations in tandem.
Customers arrive to
this
system according to a
renewal process, and each station has a different general service time distribution. The
objective
is
to
approximation
An
minimize expected equilibrium customer delay.
is
existing heavy traffic
used to reduce the design problem to a directed travelling salesman
problem. Using an interpolation scheme, we combine the heavy
light traffic results to
traffic results
with existing
analyze the design problem for a two-station tandem system with
Poisson arrivals.
1.
Introduction.
Consider a tandem queueing system with A' single-server stations, each of which has an
infinite capacity
customer
is
waiting room. Customers arrive according to a renewal process and each
served once at each station, with the order of the stations being the same for
each customer. The interarrival times have
finite
mean A~^ and
of variation (variance divided by the square of the
k
=
fif.
1,
...,
and
K axe independent
finite
and
mean)
identically distributed
squared coefficient of variation
c|.
Cq.
finite
The
random
squared coefficient
service times at station
variables with finite
The sequence
mean
of intercirrival times and
the sequence of service times at each station are assumed to be mutually independent.
Customers are served FIFO
the traffic intensity pk
=
(first-in first-out) at
A/^jt
<
I
for k
=
each station.
K
it
is
the system
1,...,A', so that
consider the design problem of finding the order of the
Also,
assumed that
is
stable.
We
stations that minimizes the
expected equilibrium sojourn time per customer (or equivalently, the expected equilibrium
customer delay or the expected equilibrium ntunber of customers
The primary motivation
In
for
in
many
for this
problem stems from the design of a production
cases, the set of tasks (either fabrication or assembly) that
each job can be done in any order. Probably the most
Greenberg and Wolff (1987),
If
in the system).
is
common
line.
need to be performed
example, as mentioned
the insertion of components on a circuit board.
each station has a deterministic service time distribution or each has an exponen-
tial service
time distribution, then
it
is
known
well
respectively) that the sojourn time distribution
is
(Friedmaji (1965) and
Weber
(1979),
unaffected by the order of the stations.
In the general case, however, the departure process from queues are not renewal processes,
and the problem becomes much more
and Pinedo (19S2a,b) obtained
difficult to analyze.
Tembe and
results for systems with deterministic
Wolff (1974)
and non-overlapping
ordered with probability one) service time distributions. More recently, Greenberg
(i.e.,
and Wolff (1987) consider two-station systems with Poisson
limit,
i.e.,
as the arrival rate A goes to zero.
arrivals in the light traffic
They examine the
interesting case
where the
service times of a particular customer at the different stations are not independent.
The only paper
in the literature that suggests
Whitt (1985). Applying the approximation methods
Whitt (1983), he describes the solution
heuristic design principles.
Whitt suggests that
it
is
what
for
to
do
in the general case
is
networks of queues developed in
in several specifd cases
and obtains four simple
Although the design principles perform well
in
some
cases,
desirable to calculate the approximate value of the expected
sojourn time for each of the K\ permutations, and to choose the order with the lowest
value.
Our results
are
most
closely related to that of Whitt.
2
Using an approximation based on
results
from heavy
(1987),
we reduce
traffic
theory developed by Reiman (1984) and Harrison and Williams
the problem of ordering queues in series to a directed traveling salesman
problem (TSP). The distance from
city
i
delay incurred at station j by placing station
is c'j/{fj.j
—
A),
which has a clear
TSP
to city j in the
intuitive
i
corresponds to the customer
directly in front of station j
.
This value
meaning: in designing a tandem queueing
sys-
tem, one should attempt to precede the bottleneck stations with stations that have a low
squared coefficient of vziriation of
traffic,
its
service time distribution. This malces sense in heavy
since the squared coefficient of variation of the interarrival times to a station will
be very close
in value to the
squared coefficient of variation of the service times
preceding station, and the lower that this value
station. Since
many
efficient
is,
at the
the less congestion will occur at the
algorithms exist for solving TSP's (see, for example, Held and
Keirp (1970,1971)) our procedure offers a computationally less expensive alternative to calculating
K\ permutations using simulation or the Queueing Network Analyzer (QNA)
all
package (Whitt (1983)).
Furthermore, our results are consistent with
all
four heuristic
design principles proposed in Whitt (1985), and, for the two-station case, our results and
Whitt's are similar, but not identical.
Using the interpolation scheme proposed by Reiman and Simon (1987), we combine
the heavy
traffic results
with Greenberg and Wolff's
light traffic results to
obtain the
expected customer sojourn time for a two-station tandem system with Poisson
This interpolation approximation
the system.
server has
From
this
is
appropriate for
all
arrivals.
values of the traffic intensity of
approximation, the optimal order of servers
is
foimd when one
an exponential service time distribution and the other has a general service
time distribution.
In manufacturing settings, jobs are often sent back to a station to be reworked
the operation performed on the job at the station was not successful.
this situation, the
heavy
feedback of customers
is
traffic
approximation
is
To accommodate
generalized to the case where immediate
allowed at each station; that
3
if
is,
a customer completing service at
station k will go to the next station with probability pk,
queue
is
at station k with probability
and
will return to the
end of the
l—pk- The problem of ordering queues where feedback
allowed also reduces to a TSP.
The remainder
of this paper
is
organized as follows. In Section
proposed by Harrison and Williams (1987)
number
is
K
These
stations in series.
used to reduce the design problem to a directed TSP. In Section
results are
we compare our
3,
results
In Section 4, the interpolation approximation
presented for a two-station system with Poisson arrivals. In Section
approximation
the Brownian model
used to calculate the approximate expected
of customers present in equilibrium for
with those obtained by Whitt (1985).
2,
the heavy
5,
is
traffic
generalized to include probabilistic feedback of customers.
is
2.
In this section the
The Brownian Approximation
Brownian model proposed by Harrison and Williams (1987)
will
be
used to calculate the approximate expected number of customers present in equilibrium
for
K
stations in series. This
model
is
a refinement of the heavy
traffic
approximation of
a generahzed Jackson network (a Jackson network with general, rather than exponential,
interarrival
and
service time distributions) derived by
Reiman
(1984).
approximation requires that the total load imposed on each station
to its capacity.
More
precisely,
we assume that there
max
i<k<K
As a canonical example, one may think
k, in
which case n
In the
=
100
satisfies the
\/n\l
—
pu\
<
is
The Brownian
approximately equal
exists a large integer
n such that
1.
of pk being between 0.9
(1)
and
1.0 for
balanced heavy loading condition
each station
(1).
Brownian approximation of the iv— station tandem queueing system described
in Section 1, the
primary process of interest
4
is
the scaled vector queue length process
g*
=
(Ql) defined by
Qlit)
where
Qki'f) is the
and n
is
(1),
number
=
Qk(nt)
^^,
0, for
A:
=
and
of customers queued
1,...,A',
(2)
in service at station k at time
Harrison and Williams (1987) show that the scaled queue length process Q*
of that paper,
function
if
it
Z
that has a unique stationary distribution.
can be shown that
and only
t,
Under the balanced heavy loading condition
the large integer specified in (1).
approximated by a process
this stationary distribution has a
is
well
From equation
(6)
product-form density
if
<^o
It
>
<
=
^1
=
•••
=
(3)
^K-i-
has been suggested in Harrison and Williams (1987) that the stationary distribution of
Z may
be approximated by this product form distribution even when (3) does not hold
exactly,
and
this suggestion
is
Under
incorporated into our approximation scheme.
this
assumption, Harrison and Williams show that the expected number of customers at station
k in equilibrium
is
2
^k
2ifik
where
it
follows
iovk
=
l,..,K,
(4)
A)
from equation (27) of Reiman (1984) and equation (2.23) of Harrison and
Williams (1987) that, for queues in
al
It
-
=
series,
Xcl_.^+Xcl,
for k
=
can be seen from equations (4)-(5) that our problem
l,...,K.
is
to find the
(5)
permutation
7r(l,
...,
A')
that minimizes
K
k=l
Since X^jt=i c\l{iik
~
A)
is
2
,2
^^
unaffected by the ordering of the queues, this problem
alent to a travelling salesman
problem with A'
-|-
1 cities
indexed hy k
=
0,
...,
is
equiv-
A',
where
queueing system and
city zero represents the outside of the
k
=
1,
...,
A'.
The asymmetric
c^/{fj.j
(^
As mentioned
in Section
—
{d,j) for the
i
represents station k for
TSP
is
defined by
to city j corresponds to the expected
by having station
j
A;
otherwise.
A)
the distance from city
1,
customer delay incurred at station
directly precede station
i
_;'.
Comparison to Whitt's Results
3.
In this section,
=
distance matrix d
city
we compare our
results to
some of those obtained by Whitt
(1985). For
the special ca^e of two stations in series, Whitt shows that the stations should be ordered
so that 6i
<
(^2,
where the quantity
S^
is
defined by
6k
=
(l
-
Pk)(cl
-
(8)
cl).
Going through the analogous calculations using equations
tions should
be ordered so that
/xi<5i
<
Thus
/i2<52-
if
(4)-(5),
the
same ordering
is
quite
common
in practice, the
Example
1
in performeuice
is
of the queues; this occurs in
To examine how much improvement
the expected customer delay for the two-station
station j.
From equations
r(2, 1)
-
(4)-(5),
r(l,2)
r(2,l)
it
follows that the sta-
the two queues have equal means,
the two approximations schemes suggest the same ordering.
nearly equal means, which
it
When
the two stations have
two schemes
of
will often suggest
Whitt (1985).
possible, let us define T(i, j) as
tandem system, where
station
i
precedes
follows that
_ (4 (c2 +
We now consider three cases where it is
ci){^i,
-
A)
cl)(/ii-A)
+
+
{cl
(c2
+
cg)(/i2
-
A)
c2)(;.2-A)-
desirable to have station 1 before station 2, in which
case the quantity in (9) represents the relative improvement in performance by switching
to the better ordering. If
jjli
=
^2 ^^d c\
<
2
C2
c\,
_
then the relative improvement
is
^2
Cj
cl+c\+ 24
(10)
This quantity decreases as Cq increases, and approaches an upper bound of 0.5 as
and
c\
=
Cg -* 0; this
c\
=
(?
and
upper bound
<
/zi
is
consistent with equation (11) in
then the relative improvement
[I2,
Whitt (1985).
cj
—
>
If Cq
<
is
+ c2(;.l+2^2-3A)-
c2(;il-A)
This quajitity also decreases as Cq increases and approaches an upper bound of 0.5 when
Cq
=
—
and A
maximal.
>
Thus, in both these cases, a
/Z].
When cl>
=
c\
=
c\
c^
and ^1 >
/i2,
(eg -c2)(a^i
Pi),
which increases as
c^
—
>
Cq.
p\ increases
As
in
performance
then the relative improvement
-
is
is
^l2)
^^.
+ c2(^i+2a.2-3A)'
c2(;zi-A)
This quantity decreases as
50% improvement
(/ii
— /X2)//ii(l —
as the service rates of the stations
become more
c^ —> 0, this
and
quantity approaches
imbalanced.
Whitt suggests the following heuristic design principles
=
K
stations in series:
=
^^2
If cl
<
c]
=
cl
=
...
=
c]^,
then
fii
< /i2<
•••
<
A'A' is
desirable.
(P3)
If Cq
>
Cj
=
C2
=
...
=
c\-,
then
fi\
>
>
••
>
Mk'
is
desirable.
(P4)
If cl
<
c]
<
cl
<
...
<
c]^,
and
(PI)
If
(P2)
^1
=
for
•••
A^K'?
then c\
<
<
c\
//i
<
< c\
...
IJ.2
/X2
<
•••
<
is
desirable.
f^K,
then the order
is
desirable.
W^hitt proves that (P1)-(P3) holds under his approximation scheme, but principle
(P4), which includes (PI) and (P2) as special cases, remains a conjecture.
that (P1)-(P3) are consistent with our approximation procedure,
is
and that
We now
prove
principle (P4)
true under our procedure.
Proposition
1.
Proof.
=
tional to,
If /ii
Principles (P1)-(P3) axe consistent with our approximation scheme.
...
by equations
=
/iA',
then the expected customer delay in equilibrium
is
propor-
(4)-(5),
cl
+ 2j^cl +
k
=l
7
cl.
(13)
This
clearly
is
minimized by choosing
ci-
which
is
= max
Cl,
(14)
consistent with (PI).
=
If Cj
...
=
=
c^'
then the average customer delay in equilibrium
c-^,
is
proportional
to
^2^ + 2c^5:-^.
If Cq
<
c^
,
(15)
one would choose
\<k<h
to
minimize customer delay, and
>
if Cq
/zi
c
,
one would choose
= max
(17)
/ifc.
1<«:<K
Equations (16) and (17) are consistent with principles (P2) and (P3), respectively.
Proposition
and
=
let bk
of the stations
in position
which
is
—
f^k
is
7r(/:),
desirable for A"
K stations be indexed by
^ ior k
=
Let gq
l,...,/\.
denoted by a permutation
for k
defined by
The proof
Principle (P4) holds under our approximation scheme.
2.
Proof. Let the
is
=
=
7r(A;)
=
fc
for
tt
A:
=
<
...
=
(7r(l), ...,7r(A')),
1,
...,
K, and
< ax and
a^
let
b^
<
=
...
<
c\ for k
bx-
=
An
where station k
0,
...,
K
ordering
is
placed
Thus, we need to prove that the identity permutation,
I,...,/!.
A;
by induction on
2
|
=
1,
A', offers
...,
A', the
number
the least expected delay.
of stations.
By
(6), this
permutation
is
if
^ + fi<^ + ^.
h\
62
^2
(18)
^1
This holds, since
ii(ai
Now
stations.
-
ao)
+
62(00
-
02)
<
(ai
suppose (P4) holds for A' stations, and
It
follows by (6)
-
ao)(6i
let
-
62)
<
us consider a system with A'
and our inductive hypothesis that any permutation
8
(19)
0.
tti
+
1
of the
1
<
...
<
that has 7r2(l)
=
7ri(l)
A' + l stations that does not have 7ri(2)
than the permutation
tt2
7ri(A'+l) achieves no smaller expected delay
and
7r2(2)
<
<
...
the inductive hypothesis imphes that the identity permutation of A'
larger expected delay than
any permutation
Therefore, the only two permutations
TT
=
(A'
(AT, 1,
+
...,
AT
—
1,
A'
+ 1). The
left
7r(A')
=
A'
stations offers no
1
and
7r(
AT
= (K +
to consider are n
Similarly,
+ 1) =
A'
1,1,..., A')
+ l.
and
1,1,..., A') if
+
Ol
is
-^<-^
+ J^.
Ok+1
OK+l
20
Oi
true, since
bK-\-i{ao
The
such that
+
1).
identity permutation offers a smaller expected delay than
-^
This
tt
+
tt2{K
-
ciK+i)
+
bi{ak
-
oq)
<
(oq
-
a;^-)(^Ar+i
- h) <
identity permutation offers a smaller expected delay than (A',
dn
TOi
This inequality holds
Oft" —
dK
Qo
Qjt
...,
AT
(21)
—
1,
AT
+
1)
if
Qft" —
+ -T^ + r-^ < r^ + T^ + T^-^b^+i
Ok
oi
ok'+i
&A'
(22)
if
bK^K+iiao - ax)
Reexpressing {uq
1
1,
0.
— ax)
by (oq
+
bibK+i{aK-i - Qq)
— a^-i +
hK+\{ciQ-aK-i){hK
4.
An
Qa'-i
—
+
Q/c),
bibxiax - clk-i) <
it
0.
(23)
follows that (23) holds, since
-W) + hK{aK-i-aK){hK+\-hi)<Q.
|
(24)
Interpolation Approximation
Using the interpolation scheme proposed by Reiman and Simon (1987), we combine
the heavy traffic results of Section 2 with Greenberg and Wolff's hght traffic results to
obtain the expected customer sojourn time for a two-station tandem system with Poisson
9
This interpolation procedure yields an approximation of the expected customer
arrivals.
sojourn time for
we
aJi
From
values of the traific intensity of the system.
derive the optimal order of servers
when one
this approximation,
server has exponentially distributed service
times and the other has a general service time distribution.
Consider a two-station tandem queueing system with Poisson arrivals of rate
Sk be the service time for a random customer at station
Si
and 52 are independent, and
of variation
cj..
Let
G
let
Sk have
mean
finite
be the distribution of S2 and
let
^
k, for
and
/ij^^
=
finite
random
the
1,2.
We
A.
Let
assiune that
squared
coefficient
variable S2e have the
distribution
P{S2e <t}
=
I
^i2
1
-
G{s)ds.
(25)
Jo
The expected customer sojourn time
at the first station
is
easily obtained
from the
Pollaczek-Khintchine formula. Let /(A) denote the expected customer sojourn time at the
second station when the arrival rate to the system
that /(O)
=
/i^
is A.
Greenberg and Wolff (1987) show
and
=
/'(O)
fi:'E[{S2
-
5i)+]
+
^^2'E[{S2e
Consider a normalized version of /(A) defined by F{X)
-
(26)
5i)+].
—
=
{fj.2
-
5i)+]
A)/(A).
Then F(0) —
1
and
F'(0)
By
=
the results in Section
fi2fi:'E[{S2
2, if
-
5i)+]
interpolation
=
F
/(A)
(27)
fi^'.
i±^.
=
(/Z2
(2S)
— A)~^F(A). By Reiman and Simon
unique second order polynomial F{X) that
(28) (with
-
method approximates F{X) by a second degree polynomial F{\), and
then approximates /(A) by /(A)
exists a
E[{S2e
condition (1) holds then
lim F(A)
The
+
satisfies
F(0)
=
1
(1987), there
and equations
replaced by F). This leads to the approximation of /(A) by
=
(/.2
-
^r'iC-^ - f^T'E[{S2 - S0+] - ^i2'E[{S2e - 50+])A^
10
(27)-
+
{^i2^iT'E[{S2
- S,r\ +
£[(52e
-
-
5i) + ]
/i2-')A
+
1
(29)
.
j
Equation (29) and the Pollaczek-Khintchine formula can be used to compare the
expected customer sojourn time for the two different orderings of two-station tandem
We now
queueing systems.
derive the optimal order of servers in the special case where
one server has exponential service times with mean
and the other server has a general
yi~^
service time distribution G. Let G*{ii) be the Laplace transform of the general distribution
and
let
similar
the general service times have
theorem
in light traffic
Proposition
mean 7"^ and squared
and only
server in front of an exponentiai server if
Proof.
As
in
Greenberg and Wolff,
Poisson arrivals, where the
has general service times.
first
minimized by putting a general
if
system
let
is
A
be a tandem queueing system with
server has exponential service times
Let system
B
and the second
Then the expected sojourn time
at the
X
and the general
second server in system
85 the expected sojourn time at the first server in system B, since
By equation
Poisson arrivals.
than system
B
if
and only
(29),
if (/i
—
system
A)~^
is
server
be a queueing system with the servers in the
opposite order. Let the exponential service times be denoted by
times by S.
A
A
is
service
the same
both servers receive
has a smaller expected total sojourn time
than
less
(A^-A)-^^^-(7-^+/i-^)i:[(-Y-5)+])A2 + ((^ + l)E[(X-5)+]-M-^)A + iy
Since
E[{X -
S)'^]
This result
then
c^
=
1
is
=
iJ,~'^G*{p), rearranging
terms gives our
consistent with three existing results.
and G*(fi)
=
-f/{^
A
was proved by Greenberg and Wolff (1987).
Expected customer sojourn time
3.
coefficient of variation c^.
+ ^), and
If
result. |
both servers are exponential,
the order of service does not matter, as was
11
(31)
shown
in
Weber
(1979). If one server has constant service times
=
distributed service times, then c^
it
follows
result
and G'{n)
3 reduces to a light traffic
5.
Finally,
model described
A
end of the queue
previous history), and will
pk
=
move on
X/pkfJ'k for k
=
balanced heavy loading condition
equilibrium
is
there
equation
(5).
is
is
a special case of a
in equation (30), Proposition
Probabilistic Feedback
customer completing service at station k
at station
k with probability
1
—
=
\,...,K
p^ (independent of
to the next station (or possibly exit the system) with
1,...,A',
Now
1-
=
Since Y!>k=i ^Pkc\/2{p.k
—
1
for ^
A)
the
traffic intensity at
and we assume that these values
The expected number
(1).
pjt
=
will define po
= KPk-i{cl-i-l) + 2-pk) +
no feedback, then
generalized to allow for immediate
1 is
again given by equation (4), where al
(^l
which
first,
one takes A -+
in Section
probabiUty pk- For ease of notation, we
\s
Since
.
theorem of Greenberg and Wolff (1987).
probabilistic feedback of customers.
will return to the
if
Tandem Queues With
In this section, the
When
e~^l''
from (30) that the deterministic server should be
by Tembe and Wolff (1974).
station k
=
and the other has exponentially
is
=
now
is
of customers at station k in
defined by
k
Xpkcl, {OT
1,...,A',
satisfy the
=
l,...,K.
(33)
and equation (33) reduces
to
unaffected by the ordering of the stations,
the problem of ordering these stations again reduces to a travelling salesman problem,
where the asymmetric distance matrix d
'^"
_
~
=
{dij) is
now
defined by
r
I
if
P^{c]
-
1)
+
2
- p,)/(//, -
^^'^^
station
decreases as pj increases, and increases with p, only
c]
12
0,
otherwise.
A)
Thus the expected customer delay incurred by having
if
=
J
>
1.
i
directly precede station j
REFERENCES
[I]
Tandem Queueing Systems,"
Friedman, H. D., "Reduction Methods for
Oper. Res., 13
(1965), 121-131.
[2]
Greenberg, B.
S.
and R. W. Wolff, "Optimal Order of Servers
Light Traffic," to appear in
[3]
Haxrison,
M. and R.
J.
J.
Management
Sci.,
for
Tandem Queues
in
(1987).
WilHams, "Brownian Models of Open Queueing Networks
with Homogeneous Customer Populations," to appear in Stochastics, (1987).
[4]
Held,
M. and R. M. Karp, "The Travehng-Salesmem Problem and Minimizing Spanning
Trees," Oper. Res., 18 (1970), 1138-1162.
[5]
Held,
M. and R. M. Karp, "The
Trees: Part II,"
Pinedo, M.,
[6]
Traveling- Salesman Problem
Math. Programming,
"On
and Minimizing Spanning
(1971), 6-25.
1
the Optimal Order of Stations in
Probability-Computer Science: The Interface,
Tandem Queues,"
Vol.11, R. L.
Disney and T.
J.
in
Applied
Ott (Eds.),
Birkhauser, Boston, (1982a), 307-325.
[7]
Pinedo, M., "Minimizing the Expected Makespan in Stochastic Flow Shops," Oper.
Res., 30 (1982b), 148-162.
[8]
Reiman, M.
I.,
"Open Queueing Networks
in
Heavy
Traffic,"
Math. Oper. Res.
9 (1984),
441-458.
[9]
Reiman, M.
I.
and B. Simon, "An Interpolation Approximation
for
Queueing Systems
of Service in
Tandem Queues,"
with Poisson Input," (1987).
[10]
Tembe,
S.
V. and R.
W.
Wolff,
"The Optimal Order
Oper. Res., 24 (1974), 824-832.
[II]
Weber, R. R, "The Interchangeability of Tandem -/M/l Queues
Prob. 16 (1979), 690-695.
13
in Series," J. Appl.
[12]
Whitt, W., "The Queueing Network Analyzer," Bell System Tech.
J.,
62 (1983), 2779-
2815.
[13]
Whitt, W., "Best Order for Queues in Series," Management Sc. 31 (1985), 475-487.
14
6i 17
06b
\J.
/
Date Due
NOV 29
1991
Lib-26-67
MIT LIBRARIES
3
TOfiD
DOS
35fi
fll4