p;j.T.U3RARli:S-
DEWEY
Oewey
HD28
Heavy Traffic Analysis
Martin
I.
of Polling Systems in
Tandem
Reiman and Lawrence M. Wein
#3857-95-MSA
September 1995
M/^S£gHasc-rrs,,vsT,rL.)E
OCT 2 4
1995
L!3aAR;es
Heavy
Traffic Analysis of Polling
Martin
I.
Systems
in
Tandem
Reinian
AT.L-T Bell Laboratories
Murray
Hill, N.J
07974
Lawrence M. Wein
Sloan School of Management. M.I.T.
Cambridge,
MA
02139
ABSTRACT
We
The
analyze the performance of a tandem queueing network populated by two customer types.
interarrival times of each type
and the service times of each type
independent random variables with general distributions.
A
setup time
at each station are
is
incurred
when
a
and each server employ's an exhausti\e
polling scheme. We assume that the load on each station is identical, and employ heavy traffic
approximations to compute the sojourn time distribution for a customer that arrives to find the
server switches from one customer type to the other,
network
in
a particular state.
When
.setup times ai^e zero (except
and additional "product-form" type assumptions are imposed, we
time distribution
for
each customer type.
September 1995
perhaps
at the first station)
find the steady state sojourn
We
consider a
types, denoted by
Each customer
is
tandem queueing system with
.4
and B, arrive to station
1
/\
served once at each station, and customers exit after being served at station
own
general service time distribution at each station,
Although each customer type has
we
require the load on each station to be identical.
its
A
setup (or switchover) time
is
when
currently set up;
other customer type and serve
Our
goal
transient
is
there are no
all
of
its
all
more of these customers
cases. In the transient case,
customer who arrives to the network and finds
it
in
queue, switch to the
we attempt
customer type
in
example,
in contrast
performance analysis, where unconditioned or conditioned sojourn times can be
we compute
is
conditioned on the state, and
state-dependent.
The motivation
for
studying this problem comes from manufacturing systems.
queueing network models
for
the expense of large material, equipment and/or labor costs)
is
to reduce setup times at
when
a machine switches from
producing one type of product to another. To exploit these economies of
are forced to produce their products in large batches,
the most natural
way
Although
manufacturing typically do not include setup times, many factories
incur significant setup times and/or setup costs (the recent trend
is
both the
to find the sojourn time of a
in a particular state (described, for
considered, the transient sojourn time distribution
is
We
customers of the type
by the 2A'-dimensional queue length process and the location of each server). Hence,
hence
incurred
customers exhaustively.
to derive the sojourn time distribution for each
and steady state
to steady state
is
whenever a server switches from serving one customer type to the other.
assume that each server employs an exhaustive polling scheme: Serve
that
Customers of two
according to independent renewal processes.
A'.
at each station
single-server stations.
scale,
manufacturers
and the exhaustive policy considered here
to reduce setups without incurring unnecessary idleness.
In manufacturing settings, sojourn time (also called manufacturing cycle time, lead time
or throughput time) distributions help managers to release and schedule work, quote delivery
dates for customers, coordinate with downstream operations (such as distribution), price their
products and
set
performance metrics. Steady state sojotun times are helpful
strategic level decisions,
real
and are the best
for tactical
and
available estimates for factories that cannot generate
time queue length information. For systems with real time information capability, transient
sojourn times are preferable to steady state estimates for operational decisions, and can greatly
enhance performance
(see
Wein 1991
Our queueing network model
is
for a
simple example in due-date cjuotation).
essentially a set of polling
1
systems in tandem.
Although
the performance analysis of polling systems has generated an enormous literature, there are no
Karmarkar
studies that consider a network of interacting polling systems.
a fixed batch size queueing network approximation that
et al.
(1985) develop
useful for tactical level decisions,
is
but they do not capture the detailed system dynamics of the network. Since our queueing net-
work model appears
to defy exact analysis,
we employ heavy
the problem. Recently, Coffman, Puhalskii, and
to as
CPRI and CPRII,
Reiman
approximations to address
traffic
(1995a, 1995b)
under the traditional heavy
is
due to a time
normalizations:
traffic
On
decomposition that arises
scale
the time scale giving rise to a diffusion
process for the total workload, the individual workloads (for each class)
When
move (asymptotically)
viewed on the time scale that makes the rate of movement of the
vidual workloads positive and
finite,
be analyzed deterministically. The primary contribution of
workloads
for
this
individual workloads can
thc^
paper
a tandem queueing system.
is
the deterministic anal-
This deterministic analysis
yields a recursive procedure for the transient sojourn time distribution of each
for the
system described
at
the beginning of this paper.
To perform a steady
required.
We compute
customer type
state sojourn time
workload at each station
analysis, the stationary tlistribution of the vector describing the total
is
indi-
the total workload remains constant, and the individual
workloads move deterniinistically. Consequently, the dynamics of
ysis of the individual
henceforth referred
derived an averaging principle for single-server polling systems with
and without switchover times. This principle
infinitely fast.
—
the steady state .sojourn time distribution by making the additional
assumptions that setup times are zero and that the total workload vector has a product-form
distribution (product-form conditions for an approximating
restrictive
than the corresponding conditions
we derive a product-form solution
in
heavy
for
Brownian network are
slightly less
a traditional queueing network). In addition,
traffic for
the case where only the
first
station has
setup times.
The remainder
1
and the heavy
of this paper
traffic
is
The model
organized as follows.
normalizations are introduced
in
Section
2.
is
formulated
in Section
Section 3 contains the
deterministic analysis that forms the basis of this paper; to keep the analysis relatively easy
to follow,
we
restrict ourselves in this section to the case
where
all
service time distributions
have the same mean. The deterministic results are used in Section 4 to obtain the transient
and steady state sojourn time distributions
for the original
queueing network.
For several
simple examples, we also compare our steady state sojourn time estimates to those computed
via simulation.
In Section
5,
we
investigate several generalizations of the model, including
service rates that differ by customer type at each station, closed networks, networks with three
FCFS
customer types, the incorporation of
remarks are offered
Section
in
The queueing network
6.
ha.s
A' single-server stations
Customers of each type arrive to station
rates
1
equal
.
.
.
/.i.
,
type
We
completed at station
for
K. Each
j
A and
B.
according to independent renewal processes with
1
to station A',
Following traditional terminology,
A".
types,
we
and
exit
define a different
each stage of each type's route; these classes are denoted by Ai and Bi for
class has a different general service
= \j/p
Let pj
= Pa + PB-
for
is
customer
class of
=
1
and two customer
A^ and A^, proceed through the tandem network from station
after service
p
Conchiding
1.
The Model
1
i
nodes, and setup times at station
for j
=
.4,
B, and define the
time distribution, but
traffic intensity of tliis
all
service rates
balanced system by
Finally, let cJq denote the scjuared coefficient of variation of the interarrival times
and
let c~,
represent the squared coefficient of \-ariation of the service times for class
assume that a random setup time
is
incurred
when a
server switches from one class to
the other; however, due to the scaling of time and the rarity of switchovers in heavy
traffic,
our transient sojourn time estimates are independent of the setup time. Consequently, we do
not introduce any setup time notation. Finally, the server at each station serves each class to
exhaustion, and then switches to the other class.
2
Heavy
Traffic Preliminaries
In a typical heavy traffic analysis, one defines a sequence of queueing systems indexed by n
that approaches heavy traffic as n —> oc. Since a heavy traffic limit theorem will not be derived,
we avoid unnecessary notation by considering a
where
c
is
negative and of moderate
size.
great majority of the t'nnc to satisfy
single large integer n satisfying \/n{p
= A,B:) =
process and {M'j,(f),
remaining work
t
>
1
demand.
A',
In Section
let
4,
the waiting time experienced at station
;
i
=
c,
{Lj,{t),f
embodied
we
see that our sojourn time
n.
>
0} denote the unfinished workload
0} be the viytiiRl wniting time process.
for the server at station
1)
This condition requires each server to be busy the
estimates are independent of the system parameter
For class ji,j
—
in class ji
The quantity
L_,j(<)
customers at time
t,
denotes the
and Wj,{t)
is
time
/.
by a type j customer arriving to the network
at
Using the standard heavy
and
=
Zj,(i)
at station
CPRI
Wj,{nt)/,/n. Finally,
at
time
t.
and
CPR
II
i
employs exhaustive
and any
>
7"
define the normalized processes Vji{t)
let V';(0
perform a heavy
=
VA,{t)
\'i
is
well
approximated by
/
Jo
traffic
a single-server, multiclass queue that
any bounded continuous function /
for
f{uVi{t))du)dt,
(/
V,\i
and
The one-dimensional
\'b]-
traffic limit.
workload moves at a
the
movement
(2.1)
If
finite
time
is
total
workload
V'l
varies as
setup times are positive and a reflected Brownian motion
if
setup times are zero), whereas the two-dimensional process
heavy
j=A,B.
Jo
averaging principle reveals a time scale decomposition of the total workload
and the individual workloads
in the
Lj,{nt)/y/n
0,
a diffusion process (a Bessel process
if
=
V'e,(0 denote the normalized workload
traffic analysis of
Jo
This heavy
+
Their results imply that
polling.
f(Vji{t))dt
/
we
traffic scaling,
slowed
and positive
down by
(V'41
,
V/ji
)
moves
infinitely (}uickly
a factor of y/n. so that the two-dimensional
then the total workload \\ remains constant and
rate,
of the two-dimensional workload process
is
deterministic.
Moreover, the setup
times do not affect the deterministic movement of the normalized two-dimensional process,
CPRI
CPR II
and
also use (2,1) to derive an averaging principle for virtual waiting times in a
single-server polling system,
/
f{Zjiit))dt
In this paper,
station
1,
but
for
is
well
which implies that
approximated by
we a.ssume
/ (/ f
l^^^^\
du)dt,
= A,D.
j
(2.2)
that the workload averaging principle (2.1) holds not just for
each station
i
=
1,
.
.
.
,
A' in the
snapshot principle holds; this principle, which was
tandem network.
first
We
also
assume that the
observed by Foschini (1980) and proved
by Reiman (1984)
in the
context of a single class queueing network, asserts that in the limiting
heavy
.>cale,
the vector of total workloads
traffic
time
l'/^)
(\'i
does not change during a
customer's sojourn in the network. I'nder these assumptions, we undertake
in
the next section a
deterministic analysis of the 2A'-dimensional nornudized workload process. This analysis yields
the distribution of the normalized virtual waiting time Zj,.j
constant workload vector
(V'l,
denote the actual waiting time
after time
/,
\\). As
at station
(
is
- A.B.i —
typical in heavy traffic systems,
of the
first
type
7
A',
1
if
we
given the
let
\Vp{i)
customer to arrive to the system
then \Vjt{nt)ls/n converges together with the norniHlized virtual waiting time
process \Vj,{nt)/ y/n.
we need the steady
In order to carry out a steady state analysis of sojourn times,
distribution of
tlie
vector of total workloads
(V'l
behavior of the vector total workload process
There
1)^. ).
{{\'\{t)
no rigorous treatment of the
is
l'i-(O)-
state
'
>
*J}
i"
heavy trafhc under
the a-ssmnption that the service time distribution of a customer at a given station depends
on
type.
its
setup times are zero and the service time distributions de{)end
If
station, then the
and
in station
>
A-
is
the
Peterson (1991) covers our
in
apply to our case because the total workload
1
is
the
same under both
station.
disciplines.) It .seems clear
from
the vector total workload process
traffic limit of
same under exhaustive
depends only on the
The heavy
diffusion process
cases.
polling as in
Thus we use the
The
argument
traffic scaling
depends only on
FCFS
a.s
mean
long as the
though
results of Peterson even
arrival rates
roughly as follows.
is
fraction of each type served
and mean service times, so
is
time;
the
on
same
this scale,
as
m
based on results
FCFS,
drift of the limit
it
the
is
same
in
under which the stationary distribution
this condition
is
CPRI
in
and CPRII. the
so the variances should match. Although
the stationary distribution of the limit diffusion process
identified cases
The
variance of the limit diffusion process depends on variation in the centered and
normahzed processes over 0{n)
When
theorem
no rigorous proof that they apply (unless the service time distribution depends only on
the station).
both
traffic limit
arguments that the heavy
(with zero setup times)
is
Under the FCFS
(1984) can be applied.
under exhaustive polling may not be the same as under FCFS. (With no setup
1
heavy-traffic scaling
there
Reiman
insult in Peterson does not directly
times, the total workload in station
service time
in
on the
time distributions are allowed to depend on both station
in the sense that service
The
type.
theorem
traffic limit
no setup times), the heavy
discipline (with
assumption
heavy
onl\-
is
is
not
known
Peterson has
in general,
a product of exponential marginals.
not met, the numerical procedure of Dai and Harrison (1992) can be
used.
3
Deterministic Analysis
Throughout
is
fixed,
and
fashion. If
this section,
{Vj,(/),f
>
we denote the
we slow down time
0},j
= A,D:i =
arrival time of a
K
1
+
\'[u(0)
—
\',
and
Hence, the deterministic path of
.s,(/)
is
moves
at
= A.B.i =
1
A'
and
0,
then the
.s,(0),(
the customer type being served by .server
{\'j,{t).t
>
0}
is
dictated by
I'k)
(\'i
a finite rate in a deterministic
randomly arriving customer as time
conditions are random, and consist of \'j,{0).j
where Va,{0)
so that the total workload vector
\',4/(0), ,s;(0),/
=
1
=
i
initial
A'.
1
time
at
/;
as
f.
we
will see shortly,
V; for all
The
on
suffices to focus
VAi{t),
=
i
derivation of our
main
of time; that
result consists of three
^Bi(0
for
i,
steps:
is
we show
First,
equal to
The
?.
and
=
CAi
+
Cbi time
from the
same
cycle.
cycle consists of the service of class Ai
we
units;
time unitr.
for C/j,
refer to C, as the cycle length
we derive the normalized
refer to Cj, as the cycle length for class ji. Finally,
virtual waiting time
that, inde-
0} enters a unique limit cycle within a
C4, time units followed by the service of class Di customers
cycle repeats itself every C,
for station
main
the trajectory of the process keeps repeating the
is.
Second, we identify the limit cycle for station
customers
>
the process {V4,(<),f
initial conditions,
amount
and the
+
because V/t,(0
A',
1,
t.
pendent of
finite
it
limit cycle.
Before stating the main result, we illustrate our method on the single-server polling model
CPRI
and CPRII, and
workload vector
for a fixed total
.41
work arrives to station
work
increases at rate
pA
increases at rate
1
do not appear at
this
V'4i(f)
otherwi.se.
—
(I'l,
.
.
.
,
Since arrivals to station
\X-).
system
at rate p^. regardless of the
1
and hence
at rate one,
dynamics of VjAO^J
th(Mi informally describe 'he
decreases at rate
Because p
pg, rather than pa,
time scale
in the
=
1
in
when
heavy
1
— pa when
Bl
class .41
K
\
server depletes
being served, and
is
we assume that
>
{VAi{t),t
traffic limit.
The
\'.\i(0
Since setup times
being served.
is
=
A. B\i
are exogenous, class
1
state.
the heavy traffic limit,
class
=
in
0} follows the familiar
"saw-tooth" path that arises in the economic order quantity model with finite production rate:
see the top graph in Figure
process {VAi{t),t
and without
from
zero.
>
1.
initial
conditions (^41(0),
0} essentially enters a limit cycle at time zero.
we
loss of generality,
If fi
Hence, regardless of the
specify the start of a cycle as the
denotes the starting time of the
first
However,
moment
cycle at station
convenience
for
that V^i
{ti
1
=
the
.si(0)).
(<)
increases
10 in Figure
1),
then
VAi(t)
= {\-puXt-h)
for
fG[fi,f,
+
—-^]
1
(3.1)
Pb
and
,
,
VAiit)
=
\\
-
{I
-
_
PA)(t
~ ih +
1
Class
Bl customers
are served in (3.2).
(-/U
\',
r_
V^i
^-^))
-
for
^€
PB
1
-
pf)
\\
- PH
i
are served during the interval described in (3.1)
and
1
1
- Pa.
class .41
(3.2)
customers
follows that
It
=
\',
_
:;
^
- Pa
.
Cb\ =
I
- Pb
and
Ci
=
;
I
- Pa
h
.
:;
I
- Pb
(3.3)
Therefore, V'4i(0
=
for
t
j^ + j^
= h +
=
32 and the cycle in (3.1)-(3.2) repeats
itself
at this time.
VaoU
32 36
22
Figure
The workload
1:
Vai{0)
Now we
V'l
—
^1
+
(1
.41
—
=
=
t
(1 ""
\'i.
If
a type
A
=
16,
customer arrives
—
(^1
+
^_
'
)).
A
time units until
PB)(i
— h) time
type
.4
all class
VbiW =
time
at
U
and
=
As =
0.4,
=
si(0)
t
e
+
[f]
0.5,
=
S2(0)
customer arriving
Bl customers
at time
are served,
f
is,
in
llO
=
/x
120
1,
A.
Ji
i-ps'''
G
f^^V'i
[0, 1]
i-p.iJ'
i-PB
'
workload at time
[f 1, f
1
+
I-PB
Z ,\\
t,
or
must wait
and then wait an additional
time that
+
uniformly distributed on [tiJ\
denote a uniform random variable on
+
.y*
Z.41 ecjuals the class .41
'randomly' arriving customer will arrive at a point
let
98102
88
him
units for the class .41 customers in front of
throughout the limit cycle (that
we
0, V:42(0)
customers are being served, and
p.4)(f
"i
^'41(0
A
—
76
process dynamics for an example: A^
Vbi(0)
6,
66
use the limit cycle (3.1)-(3.2) to derive the normalized virtual waiting time
conditioned on
then class
=
54
44
is
—
^"
]!
and again assume that pA
p.4;
with probability
pg;
be served.
uniformly distributed
j^,
with probability
to
'
]) Hence,
+ Pb =
^-
if
then
(3.4)
'.41
UV\ +
^
~_
'
'
and hence
Zai
is
uniformly distributed on
(3.5)
0,
PAi
This result can be derived directly from the averaging principle of
waiting times by setting V\{t)
-
\\ for all
and
t
=
letting /(.r)
CPRI
and CPRII
for virtual
a similar analysis
/{x<.-} '" (2-2);
implies that
V,
Zbi
is
uniformly distributed on
(3.6)
0,
PBi
We know
p
<
1,
that E[Zji] should equal \\
=
if p,\
To maintain consistency with
pn.
this fact
when
us refine (3.5)-(3.6) to
let
Zj\
Wp
uniformly distributed on
is
for
0,
= A.B.
;
(3.7)
Pj
Stations 2
A' Ijehave in
a fundamentally different way than station
1,
and consequently
the averaging principle for virtual waiting times (2.2) does not hold for the downstream stations.
downstream
In particular, rather than receiving steady streams of both types of customers,
A and
stations receive aiternat/ng streams of type
actual timing of the arrival streams of cla^sses
.4,
by the dynamic location of the server at station
and
if
and
1
S2{t)
or S](t)
=
=
If .si(0
2.
= A
S2(0
,42\s
=
\'A2{t)
Moreover, V^2(0
are
"Bi
B. then
and
now ready
=
»A,
Fix
Ca,i+\
Za,i+\
=
(V'l,
=
.
+
.
-
(";
V',
.
+
and B.
;
i
is
f
which
0},
to station
I
S2{t)
if .si(^)
=
.4
1.
Hence,
— A and
is
1
the
dictated
if
.S2(/)
1
si{t)
= B
at rate one,
=
S2it)
= A
then \'A2i^)
then V'42(0 decrea.ses at rate one.
0.
and
n,
=
result.
tiA,
+
Let us define
n^t
for
=
;
1.
1,
(3.8)
Cb,
,
V'v)-
1)Ca,,
Then
for
Ci3,,+i
i
=
=
I.
{n,
- 1)Cb,
and
=
C,+i
with probability
^^-^-j-
for
j
with probability
^^^
for
j
=
=
+ jCax-
with probability
j^-^y
for j
=
0,
\-jCa,-
with probability
^^-^
for j
=
1,
V',+1
+
work from station
+ jCbi
- jCsi-,
1
i
.
influenced by the locations of
is
being depleted at rate
>
+
earliei
/.
and prove our main
1
customers. As mentioned
.42's queiie is receiving
= B and
for all
iv> +
1
remains constant;
.si(/)
^''
to state
C.4,
1.
if
+ Vb2(0 =
=
Theorem
work
then class
increases at rate one,
We
then class
••!
B
^
Let us consider the behavior of {l..i2(')-'
servers
type
0,
1,
.
.
.
.
.
(n,
-
n/j,
-
1:
,n.4,
-
1;
,?),4,
-
1:
niii
-
1-
,
.
1)C,;
(3.9)
(3.10)
and
V',+1
Zb.i+\
=
V,
+
•
(3.11)
The proof
Proof.
work)
by induction on
is
Our
at station 2.
step
first
i.
to
is
Let us
focus on deriving
first
show that the process
{V,.v2(0'
^
f
0} enters a hniit cycle
within a finite amount of time. Without loss of generality, we define the
to begin at time
—
(2
Bl
switches from class
52
class
rate
just after time
Since
conditions
initial
(
y^_
at rate
V'42(0
for
=
V'.i2(^2)
,si
If .si(0)
=
=
52(0)
Z
- Pa
for
1-
c A\
- PB
If .si(0)
=
and
.4
'
if
^^'^
is
^B2(0)
<
/'
'
Pa
^
A
)
.^2(0)
=
fhe four cases charac-
constant for the
alternates between decreasing
it
units,
and
c..41-
1
where
=
t^
(3.12
44.
Vo, .si(0
=
B, and
and constant phases
VbtW
C
until
l|Cs,+
B, then
The
grows
\'.42(0
length of the
^^^
.S2(0
'
=
^2^
where
\'>
1
\.
C Bl
—
1
V',42(0 rises
and
periods of length Cb]
•
^- then the process V'42(0
^2
c B\
c
1
C7i
Before reaching
initially.
first
time units,
^^'^
it
t-i,
growth period (which starts
(3.13)
must
at
rise
by
time zero)
and the length of any subsequent growth periods (there are none
V'/^2(0)^
is
1,
.41
Vb2{0) and then drop to zero.
for V'42
=
+
Cai
,
1
by the
fully specified
V;42(/) stays
c B\
Cbi
periods of length C41 and
^^^
Vb2{0)
Vbi(0)
is
v;42(o)
Cb\ +
B. then after remaining constant for
alternates between decreasing
:;
0}
^2 foi"
then
.4,
C41 time
V42(Q)
+
At this point V'42(0
respectively.
>
'
=
stays constant during
=
=
.S2(0)
readers are referred to the bottom graph in Figure
=
1
hence, V'42(0 increases at
^2 also);
We now compute
(0), .§2(0)).
S2(0)).
(.s-i(O),
1
If 5i(0)
^hc server at station
^2.
the s(>rver at station 2 will be serving
0,
fixed, the deterministic process {V/i2(0.^
Cb\ time units and staying constant
f2
r
t2
=0}. At time
time units. Thereafter, until V'42(0 reaches zero,
-
1
Since
V'l,
(and perhaps just before time
to
V'.\i(0), V'42(0),
terized by the values of
first
=
cycle at station 2
first
f2-
and V2 are
V'l
r4i(0
:
to class .41.
starting at time
1
>
'n\i{t
wortcload (unfinished
tlie
Once
C'ai-
\'.42(0 starts decreasing,
it
proceeds as
previous case.
in the
Hence,
V:4i(0)
^;
1
where
-Pa
/|jj
is
Vb2(0)
+
-
t^
Cm
V?
^'^{Vb.(0)>^} + C fli
the indicator function for the event
Va2{0)
12
=
:;
1
- PB
H
Cbi
.r.
Sinularly,
Cbi
+
if .si(0)
Vi
C B\
= B and
1
C.4,
•S2(0)
=
,
(3.14)
.4.
then
Vbi(0)
1-PB
CJ.v
Mr I'bi(O)
,^<V^2(0)}-
(3.15)
Now we
^2.
look at the evolution of V^2(0 during the
Readers are referred to 1^2(0
-
alternates between tibx
(that
= U-
VA2{t)
[44,110] in Figure
The
1.
t
(;
process VA2{t) initially
Cai and constant periods
remains constant) of length C«i, as described by
\'A2{t)
+ - {h +
1)Cai
€
t
(possibly zero) growth periods of length
1
time intervals where
is,
for
which starts at time
full cycle,
first
-
l)Ci)
for
e [h
t
+U-
l)Ci
,
h+
-
(;
l)C,
+ Cai]
(3.16)
and
VA2{t)=]CAx
=
for j
1,
.
.
—
,nfli
.
The
1.
for
truncated when
is
G
/
=
The
=
V2
+
{riBi
for
t
- l)C.u +
- l)Cut2 +
(hb,
\'A2{t) reaches its
constant until server
^42(0
[^2
+ {j-\)Cy-^CAxJ2 + ]Cx]
growth period, which
last
VA2{i)
te[i2
for
f
(n«i
maximum
is
(3.17)
given by
- ih +
(nsi
-
1)C,)
-
+
-
(ngi
l)Ci
Vo
-
(3.18)
l)C4i],
value of V2; at this point, the process remains
exhausts the class Al customers in queue:
1
e [h
+
-
(nei
1)C,
+
process then alternates between hai
V2
-
-
1
(ubi
- 1)CaiJ2 +
-
("bi
1)C,
+
Cai]-
(3.19)
depletion intervals of length Cb\ and constant
Cai. given by
intervals of length
=
VA2{t)
for
t
- U - l)Ce, -[t-
V2
e [h
+
for
t
(nai
+j -
G [h
+
2)Ci
{I,
+
{nBi
+ CA1J2 +
+J -
2)C,
("31
+j -
+
C..U
)]
l)Ci]
(3.20)
and
VA2{t)
for j
for
is
=
=
1
V2
- jCbx
-
1.
Once
VA2{t)
=
V2
TiAi
-
{hbi
when
(n.-u
- DCfl, -[t- ih +
VA2(t)
=
for
V'^v2(0 reaches zero,
Bl customers
serving the class
/
e
[f2
{'iBi
+j-l)Ci+
Cai]
(3.21)
again, the last depletion cycle
te[i2 + {nBi+nAi-2)Ci+CAi,i2
truncated
+j-l)Ci,i2 +
in
{iiAi
+
n/ii
- 2)Ci + Cax)]
+ {nAi+nBi-2)Ci+CAi + V2-{nAi-'^)CBi]
and the
linut cycle
concludes
when
server
1
(3.22)
completes
queue:
+ (".u+"«i-2)C,+Cu+V'2-(n,u-l)CBi,r2 + (7!..u+»/?i-l)C,].
(3.23)
Notice that at time
the
same
as at time
Class ,42
and
+
the system state iVy\i{t),VA2{t),s\{t),S2{t))
l)C'i,
Hence, the system repeats the same cycle of length
(3.9) holds for
i
=
B2
is
when
time blocks of length
—
the 7)01
growth periods
1
we view
and
(3.18)
—
(3.19) together as
a randomly arriving class
.42
=
—
Vo
one
customer
is
interval,
rhen
all
-
ni
which point
at
during
t
(3.16)
—
l)C'ai,
.4
customers only arrive
intervals correspond to
its
subsequent
=
1)~' for j
must wait
until
time
class .42 begins exhaustive service,
+
intervals are of length Ca\\
in (3.21) finds the server serving
given
is
/741
1
t>
1
If
dining each of these intervals.
ecjually likely to arrive
—
jC'bi with probability (nj
arriving at time
{iii
constant intervals dining V',v2(0's descent in (3.21).
1
customer arriving during the constant periods
class .42
)Ci thereafter.
- l)C.u and
These time
type .42 customers, and hence the virtual waiting time there
Z.42
{riy
truncated growth interval (3.18) and
in (3.16), the
constant period (3.19), and the n^}
1
1.
customers are being served.
class .41
—
(/'i
is
served in (3.16)-(3.18) and (3.23). Therefore,
calculate the virtual waiting time for class .42, notice that type
to station 2
A
~
"fii
B'2 are served in contiguous
and
respectively,
To
(".-ii
served in (3.19)-(3.22) ^uld class
is
classes .42
to-
+
^2
—
(n bi
—
b)'
A
1.
l)C'i
\'A2{t)-
+
Therefore,
customer
class .42
lo
—
—
(''si
l)C.ii-
plus an additional V'.i^O time units for
the class .42 customers ahead of him to be served. Therefore, the virtual waiting time equals
+
V'42(0
Z.42
=
+
^2
^2
+
^2
+
["01
("SI
—
-
l)(^Bi
j](^'bi-
- ' Substituting 1^2(0 from
which occurs with probability
(3.16) into this expression gives
-
(iii
1)"^ for j
—
l....,;)si
—
1-
Similarly, class .42 customers arriving during (3.18) have virtual waiting time V2. Finally, class
.42
customers arriving during (3.19) find their class
waiting time
A
Pa
1
4
>
is
V'4,(0
for
directly affected only by its
=
= Pb =
1.
Thus. (3.10) holds
/
=
\',,
in
\'4 ,_,_i(/)
=
and hence
also have virtual
1.
=
1.
Hence,
if
similar analysis of c hiss B'2 customers yields (3.11) for
station
:
V'o.
in service,
upstream
station.
;
0} then our entire analysis holds for
i
Finally, notice that each
we
>
define f,+i
1,
=
inf{f
>
except that we take
equations (3.r2)-(3.15), becau.se Vji{t) decreases at rate one, not
1
-
pj, for
I
Performance Analysis
In this section,
we analyze the performance*
of the oiiginal
tandem queueing system. Recall
that two types of sojourn tiiue analyses were identified in the Introduction: Transient and steady
state.
The
goal of the transient analysis
is
to e"<timate a customer's sojourn time conditioned
11
on the system state
time of
at the
In our heavy traffic analysis,
arrival.
normalized virtual waiting time Z^, given the vector workload {V\,
service times are not
known
actually observable.
However,
system at time
then
t,
at tlie
if
heavy
in
.
Since a customer's
V,).
,
.
time of arrival to the system, the workload process
we
let
Qj,(t) denote the
traffic Vi{t) is well
n
number
V; is
not
of class ji customers in the
approximated by
y^—
Jn J
+
j=r—
ft
in fact, the latter
.
we estimate the
for
/
>
(4.1)
0;
process converges weakly to the former process in the heavy traffic limit. Let
us define
QM{t)+QB^{t)
for
L,it)
t
>
(4.2)
0,
/'
which
L,{t)
an observable process. Supf)0.se a customer arrives to the system at time
is
=
L, for
and
tions (3.3)
=
i
K
1
If
.
-^
we make the substitutions
(3.7)-(3.11). then the
heavy
traffic
and
for Zj,
and, for
;
=
A'
1
WA,i+l
-
for
0,
J
= A.B
1.
L,+\
+ jCb,
with probability
—^
for
j
=
0,
Lt+i
-
with probabilitv
—^-
for
j
=
I,
+ jCai,
with probability
—
for
—
with probability
—
jCfji,
and
L, +
=
^'B.i + l
]
L,-^.\
jC,\,.
^-r
111
)
i
-^^—^
for j
=
=
0,
1,
where
Cai
—
1
and, for
;
=
1,
.
.
.
,
A'
-
- PA
1,
no,
=
CB^ =
1
- PB
finds
equa-
parameter n cancels out of these expressions
uniformly distributed on
is
and
-y^ for V, in
and we get
Wji
/
and
Ci
=
Lx
+
-^Bi
Now we
estimate the steady state sojourn time Sj, for class
ji
under some additional
as-
sumptions. Most importantly, we assume that setuj) times are zero. Peterson has shown that
(Vi,
.
.
.
.
product-form stationary density
hail
\X)
1
=1
where
as long as the network data satisfy a certain
skew-symmetry condition. The skew-symmetry
condition, given by equation (62) of Peterson, reduces in our case to
-^.4^0
for all
;
=
.
.
.
,
A
A'.
special case of (4.11)
slightly less restrictive
is
Of course
0,
1,
.
.
.
,
A'
-
symmetry
+ ^BCbo =
is cj,
Xac^j,,
—
+ \bCb,
c~ for j
=
B
.4.
and
1,
where
r~^
^
Cq
is
=
;
than the standard product form conditions
(4.11) also allows for other interesting solutions, such as
A'
for .Jackson
c^,
= Cy
J
-
1.
which
networks.
= A.B.
/
=
Harrison and Williams (1986) show that the skew-
po.ssihle.
condition, together with the stability condition
sufficient for the
exponential product-form solution.
Brownian motion on the orthant can
l)e
<
/;,
1
for all
skew-synunetry
If
numerical procedure developed by Dai and Harrison (1992)
reflected
(4.11)
employed
is
i.
are necessary and
not satisfied then rhe
for the stationary distribution of
conjunction with our deterministic
in
analysis to estimate steady state sojourn times. For the remainder of this section,
we iissume
that the skew-symmetry condition holds.
Since the
random
with parameter
variable
heavy
u,
Y = nX
is
exponential with parameter ^^«"'
traffic analysis preui':ts that
(Lj
.
,
.
.
Z,/^)
,
if
A'
is
exponential
possesses an exponential
product-form density with parameters
9,
=
9,
-?=
/"
=
2(1
^
Ej = A.B^jf^
-p)
^
_.,,
>>
-('^jO
-,
,
+
(4.12)
.
Cj,)
Hence, equations (4.3)-(4.8) can be combined with (4.12) to characterize our estimate of the
stationary virtual waiting time U^,, which
before. 5^,
time
for
=
type
VI j,
j is
-|-
Tj, gives
^,^i
is
independent of the heavy
traffic
parameter
/;.
As
the stationary .sojourn time for class ji and the stationary sojourn
Sj,.
Unfortunately, the recursive nature of (4.3)-(4.8) prevents us from explicitly writing the
probability distribution for the virtual waiting time
1.3
U j,. However, performance measures
of
moments, can be computed by integrating with
interest for Wji, such as tail probabilities or
respect to the stationary distribution of
For example, equations (4.4)-(4.5) imply
L,).
(Z,i
that
/rru\r
£;U,4,,+i Li
r
L,+i
=
]
CB,(nBi(nB, -
+
r
L,+i
,
-
1)
-
n.4,(n^,
1))
(4.1J)
-
2(n,i,
+
2(n4,
+nB, -1)
ne,
1)
and
for
J
=
1,
.
.
^
^
,
.
A'
-
^
Hence, the steady state expected virtual waiting time for class .42
1.
Jo
Jo
2(\^^^^^f^]
+
\^^^^^^^]
is
-l)
91626-"'''' e-'''^'dxidx2.
Our
results reveal several insights
how the
exactly
for all
by
;
lengths,
same
(4.7),
follows
it
and hence batch
for all
i.
about system behavior. Expressions (4.7)-(4.8) quantify
service cycles at a station are forced to synchronize with the service cycles at
upstream neighbor, and how
its
by
sizes,
this effect ripples
(4.8) that Cj,
through the tandem network. Since
nondecv^asing
is
Zj\
is
;
for j
= A.B;
so that the cycle lengths of each customer type grow in the
uniformly distributed, but Zj, takes on
Finally, notice that since Cj, increases with
greater frequency as
;,
;),
very different for
is
-
1
>
1
distinct values for
the quantities
move towards
n.4,
/
and ns,
than
1
>
Similarly, by (4.4)
(4.5),
fiie
we conjectm-e
is,
^^
>
2
cycle
is
the
as
for station
1.
will
equal one with
both
(but not necessarily converge to) the value 6 as
we conjecture that the imhalmivc between
moves downstream.
that
n,
same proportions
increases. Hence, equations (4.13)-(4.14) suggest that
i
£^[^Bi] should slowly
consequently,
in
tend to be larger at downstream stations. Moreover,
one moves downstream. Also, the virtual workload
1:
(4.15)
£'[H''.4i]
i
and
increases;
customer types dissipates as one
that the variabiJitv of the sfationarv
sojourn times at each station should be smaller at downstream stations. Although we have not
been able to prove these two conjectures, they have been borne out
(that
is,
by generating random samples
Let us
for all
for
i,
now
for L,) of the recursive
focus on the symmetric case where A 4
and -fP-
is
an odd positive
any given station
(
integer.
and ff^j,] =
the heavy traffic analysis of a
0,"',
=
A/j.
is
Wb,
iiai
=
"Bi and C.\,
=
Cb,
are identically distribut(>(l
equal to the corresponding quantity in
tandem system under the FCFS
14
Monte Carlo simulations
equations (4.7)-(4.8).
Then
Hence, W,\, and
whicii
In-
discipline; of course, the latter
Class
Station
not necessarily equal
knowledge of
class.
The
Vl,(fi)
'•!
\'/,(^).
Define
i,
The
VA.,(f,).
=
inf{/
>
:
terms of
/,.
Given
V/,(^).
\'ii(t)
virtual waiting times Z^,
=
/
dynamics cannot be analyzefl without
limit cvrle
^,(0)} to
.s,(/) y^
analysis proceeds by recuisively (leri\iui;
equations for 1^,(0 and
and
-
l',(0)
A'
1
—
1,
Z/,
first
time server
t
switches
and then developing an expression
t,.
the next step
corresponding to (3.16)
and
he the
to calculate the
is
The
(3.23).
can then be obtained. Define
n/,
=
cycle
liniil
cycle lengths C\,
and
''''"','"'"
.
for
C/,.
There
are two different structures to the limit cycle here, depending on whether
1)
V',,,
+ i(^", + i)
-
-
{n,,
l)rk,Ck, e (O.Cfc,]
or
In case
while
1) class
s,(t)
=
that does not
When
.s,(f)
=
Note that case
2)
exhausted while
is
li
.s-,+i(0
=
/.
show up
k\ s,^i(t)
=
and
I,
cannot occur
in Cfise 2) class
if r^.,
=
1,
so this
is
exhausted
li
is
a
phenomenon
in the ecjual service rate setting.
viewed on the time scale wh.ere the total workload process changes (and individual
workloads are moving
infinitely (juickly). this process
Because the
service rate case.
workload at stations 2
total
cycle, the constraint that the total
w'orkload achieved over a cycle.
has a feature not present
A'
workload can never go negative
As
may change
is
equal
in the
over the limit
imposed on the minimum
a consf»()uence. the analysis of the vector total workload
process must take into account the "oscillations"
each station's total workload.
in
Multi-Type Networks
5.2
Another natural generalization
The
simplest example
is
(
and
is
the case of
more than two customer
a perfectly synnnetric three-type, two-station network; that
type has arrival rate A and
length at station
to investigate
\',
all six
classes have service rate
be the normalized workload
denote the normalized virtual waiting time
for
any
V2.
After analyzing numerous problem instances,
two
cases,
//.
As
before,
at station
i,
for
let
i
=
is.
each
C, ecjual the cycle
1,2.
Also,
class at station 2, conditioned
we have the following
types.
conjecture.
on
let
I'l
Zi
and
There are
depending upon wliether
C,
-f > C
1
3
(case 1)
(5.1)
"
or
Vo
1
-^G[0,-)
C,
In case
Ci
(case 2)
(5.2)
o
we have
1,
C2=
3
C,
2
(5.3)
and
-1
=
Z2
w.p.
^J
3
.3
J
=
0,
J
=
1.
4J
(5.4)
-1
^
J
In case
for
(;)[,iifl-2)
for
3
2,
Ci
C,
1
(5.5)
C,
and
2Ci
3
,2Ci
Zo
=
3
./
W.p.
(6[i^l-l)
w.p.
(6[A.l-l)
for
J=0
2[ol
for
;=0
2|A^
for
;
-1-
<
3
['.-(fSl-l)^-.^
>v-p.
(6f^l-l)-'
=
1;
1,...
(5.6)
In case
units.
1,
the three customer classes at station 2 are served hi contiguous blocks of -^ time
However,
in
described by the
case
first
two equations
two equations of
(.5.6).
between
-
c^ K'l
Vj,t+\{f)' the
each class
2,
i^
There are
<iiid
and once
first
in case 2
would presumably make the analysis of a
station.
The presence
when setup
,
and ^u(x) —
first .station
0,
2
<
k
times are
of setup times alters the vector station-level workload
present,
it
is
sliown in
There
traffic.
in
the
< K, where
18
c
=
is
first
of the infinitesimal drift vector, along the lines presented in
"P^Pg"
length
1
tandem network with setup times
for the
in
Since the server location s,{t) dictates the dynamics of
process converges to a Bessel process in heavy
=
which alternate
last
more complex.
process; indeed, with only the
/i2(x)
subcycle given by the
third generalization allows us to perform a steady state analysis
incurred at the
theorem
in the longer
six blocks of service in a cycle,
U^ CV
Setup Times at Station
Our
in (5.6)
complicated service process
third station considerably
5.3
served twice in a cycle, once in the shorter subcycle
is
II
that the workload
no analogous heavy
station.
CPR
\/n{l
CPR
-
II
p),
A
heinistic derivation
yields //i(x)
.s
is
traffic limit
the
=
mean
^-^^^
,setup
-
r,
time
over a cycle (switch from
goal
satisfies the
and
afj
Yamada
B
and from
skew-symmetry condition described
— — a~^l{|,_j|=i},
1
(1986) that for A'
=
and x
to .4),
<
'
<
j
7^
Section
in
=
where a
A',
remarked there that the extension
theorem
traffic limit
some backing
CPRII
to A'
>
2
is
4,
so that
ii'/[\,\cj^^
unique
is
+
^Bcjij-
an
earlier result of
is
Yamada
covered by
ary distribution has a product form, with the
exponential. In particular,
=
<
A',
(It
was also
(1986) also proves a
1984).
Yamada
(1986), there are no results
We
conjecture that the station-
marginal a ganmia distribution (obtained
first
and the other marginals
'
ril3+l)'
"'
^
0'
1
^
'
^
^''
'
= Xp^pBS.
current theory of multidimensional diffusion processes does not provide a
verifying the above conjecture.
"Basic Adjoint Relationship'
Brownian motion
proposed solution
in
One way
(BAR)
to lend
jilausibility
it
for this process,
satisfies the
for
taking this lengthy detour, we merely point
BAR. Rather than
other stations the state dependent
drift
is
it
works.
first station.
\\'ith
extremely complicated because the
of the drift depends on the cycle time at station
-
to derive (without proof) a
analogous to the one proven to hold
have restricted our attention to setup times at the
k relative to station A
is
means of
an orthant by Hiurison and Williams (1987). and show that our
out that this program has been carried out, and
5.4
;
we conjecture that
i-k)
7r(.ri
reflected
<
was shown by
in distribution.
as the stationary distribution of a one-dimensional Bessel process),
We
It
1
state dependent service rate that gives
that character'ze the stationary distribution of the process.
The
2q~',
to our Jissumption here of convergence (especially in light of the discussion in
linking their result to
f3
=
af,
Yamada
straightforward.)
tandem networks with
for
Although the existence of our process
where
Since onr
2 a process fitting an appropriately completed version of the above
description (including boundary behavior) exists, and
heavy
the workload vector.
is
performance measures, we assume that the covariauce matrix
to obtain steady state
is
B
to
.4
A',
setup times at
A'*'^
component
which depends on the workload at station
1.
FCFS Queues
Most manufacturing systems have some workstations that incur
workstations where setups are negligible. At the latter stations,
19
FCFS
sizeable .setups
is
and other
a commonly employed
queueing discipline. In
this subsection,
we show
FCFS
network that consists of any combination of
paper without altering our
this
queues and polling queues.
tandem FCFS queues can be placed
a set of product-form
results; the
tandem
that our analysis can be extended to a
network analyzed
in front of tlie
Poisson output from the
note that
First,
FCFS queue
last
in
preserves
the product-form stationary distribution.
Now
suppose that
FCFS
in
queues are appended to
customer overtaking cannot occur, the sojourn
given by the workload.
Mme
of
all
tlie
Because
end of the network.
customers
in
FCFS
the
stations
is
Using the snapshot principle, the sojourn time distribution can be
calculated from the vector of station level workloads.
Now suppose
{^
n?
-I-
-I-
is
1
a polling queue.
Then
j.K+m+i{t)J > 0} only through the server location process
details,
Theorem
(3.11) hold for
:
station A'
\'4 /v+„,(/)
;
=
A
=
I
Sf(+rn{t)-
can be applied directly to analyze station A' -I-
1
-I-
A+,n,
in.
\'.j
To prove
/^•_^,„^i(/)
0}
reached within a
is
m
-I- 1
;
A'
1
finite
affect
Although we omit the
that
is,
equations (3.8)-
amount of
=
inf{^
>
time.
Closed networks
5.5
A
Suppose that customers exiting station
erating a closed cyclic network.
immediately return to station
Although customers never enter or leave
manufacturing context where each customer represents a "job",
replacing a completed job that
are referred to Dai
network models
Two
(>xits
station
and Harrison (1993)
for
for
A
type
it
is
it
thereby gen-
this network, in a
useful to think in terms of
manufacturing systems.
returns to station
1
If
upon the nature
If
the customer type of each reinjected job
the latter case, the total population size
otitput
is
is
exogenously
is
obtained;
fixed l)ut the nunil)er of ctistomers in the
fluctuate over time. Manufacturing system
managers need
mix of job types equals the exogenous demand mix. The most
to guarantee this
which
a job departing station A' does not change
prespecified (in a deterministic or probabilistic fa.shion) then a single-chain network
may
in
then we have a multichain network, where the population size of
each customer type remains constant.
of each type
Readers
1.
an extensive discussion of multidass closed queueing
types of closed networks can be formulated, depending
when
1,
with a new job that enters station
customers or jobs are injected back into the system.
in
in
-I-
one only needs to verify that f^'+m+i
this fact
=
stations
match between the desired mix and the
realized
mix
to ensure that the
<iirect
is
network
and
reliable
way
to use a single-chain
network where the mix of entering jobs equals the exogenous product mix; see Harrison and
20
Weill (1990) for further discussion on this point.
Consecjuently,
Section 2
tomers.
J
we focus on
tlie single-cliain
Let us alter the ba.sic uiodel in
torniulafion.
exogenous arrivals and maintaining a fixed population
i)y (iisaliovving
size of A' cus-
\\e assume that the long run pr()])ortiou of exiting jobs that are reinjected as type
customers
is
where
c/j,
(7.4
+ ga =
Also assume that the service time distributions are
1.
associated with each station rather than each class, so that the .stiuared coefficients of variation
satisfy r^,
=
cj^^
for
;
=
A'.
1
Recall that in the deterministic analysis of the open queueing network in Section
J
work arrives
to station
deterministic analysis of this closed network, station
of Qj for J
= A,B:
the open and
eciuations (3.3).
replaced
l)y
r/,
for j
and
(3.6)
(3.-5),
= A.B.
=
1
(3.11).
(3.8)
—
A'
1
all
1.
Observe that
.4,/?.
work
receives type j
1
furthermore, the interactions l)etween stations
losed networks for
i
=
at the constant rate pj. for j
1
i
and
Hence, the results
i
+
in
+ p^ =
in the
1
a similar
at a steady rate
are identical in
1
in Section 3.
may be more
namely
robust (with
respect to the heavy traffic assumptions) tiian the open network results because q^
p:^
type
hold for the closed cyclic network with pj
Moreover, the closed network results
whereas we needed to assume that
3.
open network
The
analysis.
+
qij
=
results
1.
on
unequal service rates do not carry over to the closed network case because the imequal service
rates lead to a fluctuating arri\al rate of
type
is
If
being serv'ed at station
A)
work
at station
1
(depending upon which customer
of the closed network.
we further assume that setup times are zero and
all
A
service time distributions have
identical sciuared coefficients of variation, then the steady state
remarkably simple
results, as
rate of our network
is
shown
traffic analysis
f/j
throughput rate of the
{.r
Moreover, this result holds
€ /?^
for A'
The
throughptit
network considered by
given by eciuation (11.2) of their paper. Their approximating
and the precise nature
:
(i.e..
=
Ylk=\-'^k
=
/'"'-"^
} This
L^-)
2 stations even
uniformly
result holds regardless of the
if
the s(iuar(>d coefficients of variation differ
(whose analysis reduces to reflected Brownian
interval [0.//"'], the virtual waiting time (listril)\ition at station
f
is
deterministic or random) of the reinjection mechanism.
at the various stations. For the two-station case
motion on the
FCFS
implies that the stationary workload process [L\
distributed on the simplex
values of
is
traffic analysis yields
Section 11 of Dai and Harri.son (1993V
identical to the
Dai and Harri.son (1993). and
heavy
in
heavy
^
[l
+ h.f-^)]
for
1
21
0</<i^
for
^
> -^
.
1
is
given by
We
also obtain
E[Wji]
=
= ^-^ and
£'[Vl''ji]
E[Wj2]
=
g
and by
V'ar[iyji]
= y^^^-
(4.16),
Var\\Vj,
3
U
[f-l
/
Jo
If in
addition
q,\
=
r/B
=
k-
th™
6
Concluding Remarks
Fi^oiii
tli(^
viewpoint of manufacturing systems aj^plications, the biggest shortcoming in
queueing network theory
per we
make a modest
is
umiouhtediy the faihuc
att(Mni)t at addressing this
a set of i)oning systems in tandem.
heavy
traffic analysis of polling
and steady state
cases.
By
to incorporate setu[) times.
In
tliis
pa-
shortcoming by analyzing the j)erformance of
exploiting the time scale deconii)osition inherent in the
systems, we derive sojotun time estimates for both the transient
The simple form
of
Theorem
1
allows us to gain
some understanding
of
the l)ehavior of these com])lex systtnns. In i)articular, as one moves from upstream stations to
downstream
stations, the batch sizes tend to get larger, the imbalance t^etween
becomes
pronounced and the
less
in Sections 5.1
network
and
will not
varialjility of
sojourn times
is
customer types
reduced. However, our analysis
5.2 suggests that results for the general multi-type, multi-station
come
tandem
easily.
Acknowledgment
The second author was
at
su[)])orted
l)y
MIT, NSF grant DDM-9057297, and
Laboratory
carried out.
at
a grant
EPSRC
the L'niversit>' of Cambridge for
from the Leaders
for
Manufacturing Program
grant GR/.I71786; he thanks the Statistical
its
hospitality while
some
of this
work was
W'c are grateful to Chalee Asavathiratham and Beril Toktay for performing the
numerical comjiutations reported
in
Section
4.
23
Appendix
wo jnovide some more
In this appendix
details of the analysis of the
case, considered in Section 5.1. For the sake of brevity
we only
unequal service rates
state results, omitting derivations
and proofs.
The
step in the analysis
first
can be obtained. In particular,
for \]i{it)
—
t]
for
*.+l
(
=
=
A'
1
V't
—
+
,+ i(0)
^T—
-
1
and
a recursive determination of
is
ii,
.
.
i/^-i are defined
.
{0)^j
s
It
.
for
j
from which an expression
i,,
by
= A.B.
(A.l)
2,
max
+
r^,f,
I
-
1,
1
ik.Ck,
:a.2)
=
.s,(0)
if
=
/,+i
.s,+,(0)
Vi.,,+ i(0)
=
A-:
+
vi.,,+i(0)-f,
+
(a.,
-i)c,,
^|Vi.,+,(0)<(",)
{Vt,.+i(0)>/,}
'•i,Ci,
if
,s,(0)
=
/
and
=
.s,+i(0)
+
M
''k'C'ki
k;
(A.3)
i,+i
=
V,,,+ i(0)
+
+ l)n,0,
(m^.,
if
=
.s,(0)
A-,.s,+,(0)
=
/
(A.4)
and
\;.,+ i(0)
-
/,
+
(//,
-
1)C/,
-
"u.,a.,Q, € (-oo/{„,^_^_i},
O.,],
where
'"/c!
=
+
=
f,
,
V;,+i(0)-f,
+
(n,
-1)0,
^{i,,,+,(0)>(,-(n,-i)O,}
<,
+
(777J..
+
1)C,
+
(r;,
-
1)-'[^"U.
+
l)a-,a.,
+
/',
-
^(r,,, + i(0)<f",-(r,,-i)C,.}-
\',,,
(A-a)
+ ,(0)]
(A.6)
if
=
^,(0)-A-,s,+ ,(0)
f,
+
i
and
/
=
Vi.,
\';,,+
+ i(0)
+
i(0)
-
f,
/•„/,
+
mi.n.Ci,
+
(r,,
-
1)C,,
if
-
.s-,(0)
"U-.'l-.a.,
€
(Q.,, a-.a,]:
=
=
/
.s,
+ i(0)
(A.7)
and
\i,,
+ i(0)
+
(/•,,
-
1)/,
-
/n,,a,,CV-,
€
(O.C/t,],
where
771,,
=
rk.Ck,
^{11..
+ (0)>(l-r,,)f,}
and
24
!
^{Vl,.
+ ,(0)<(l-r,,)(,]
(A.8)
=
h+i
mi,C,
+
+
C\,
-
(r,,
+
l)-'[r»;,a.,a.,
-
C\.,
V,,,+,(0)]
s,(0)
if
=
=
,s,+,(0)
/
(A.9)
+
and
V;,,+i(0)
The
quantities
-
(r/,
-
l)i,
V;2(f-2)
=
+
v;+i(0)
= _,|,
oci/{,„,^
a.,Cfc,].
given by
^rt'
^'//\'(^/\')
i/,,+i(^+,)
e (O, -
m/,.,r;t,Q-,.
-
in,
iji,
+
-
(n,
i)Ci,
^,(0)
if
=
(A. 10)
A-
and
=
\i,+i(i,+i)
for
/
=
A'
1
-
1.
v;+i(o)
+
(n,
-
i)f,
=
^,(0)
if
=
Notice that, as expected, Vi,+i{f,+i)
(A.u)
/
V',+i(0)
if rt,
=
=
n,
With
1.
the vector of \]i(t,)'s in hand, we can calculate the limit cycle ecjuations for Vk,(t) and
corresponding to (3.16)-(3.23) (they
not be written out here), and then derive the cycle
will
lengths O-, and Ci, and virtual waiting times
^"/,i+i(^-n)
=
«;,
Vi,(f)
and
Z^.,
and
Z/,
Section
in
a.s
3.
Define
=
Ilk,
(A. 12)
ri,Ci,
There are two possible
+
class 1,1
1
[Cki, rkiC'ki]
random
case
1,
is
exhausted while
and
class
/,
/
+
.s,(?)
=
variable on the interval
[«, 6],
A-,.s,+i(^)
and
let
Ci,,+\
=
{rk,
- l)U[OX'k,] +
(n/,
=
-
+
1)C,
=
.s,(/)
/.
+ i(f,+i)
In case
=
s,
+ i{t)
—
—
(n;,
2, V;,
=
I.
+ i(r,+i)
—
(n;,
1
are as follows: For
ni,,ri;,Ci„
nk,Cr
+
{ni,
-
Ci+i
=
(lu-i
+ nu -
-
\)ri,Ch
ir>t,
-
-
+ i{U+i)+ jri,Ci,
—
l)i\;Cki €
=
i
—
A'
I
(A. 13)
l)r,,C/,,
(A.14)
(A.15)
1)C',,
in,
Under
1.
>n,rkjCkt.
w.p.
and
(0, C\.,]
Let U[a,b] denote a uniform
1
\'tj
6
l)7'A:,Cfc,
"w.p." be shorthand for "witli probability".
our results corresponding to Theorem
=
l';,
1,
exhausted while
is
1
Cfc,,+i
Zk^t+l
In ca^se
limit cycles:
+
n,,
-
for
^'
=
n,,
1
-
1,
1
(A.16)
Zi<.,+\
=
{i-k,
-
l)U[i).Ci„]
+
r,,,,i(/,4.i)
-
1
jri,C,,
w.p.
Ilk,
+
"h -
for
J
-
nk,
\
-
1.
1
(A.17)
Zk.,+i
=
{rk,-l)U[O.Vi,, + i{t, + i)-(n,,-l)rk,Ck,] + Vu^i{J,^i) w.p
V',,,+i(f,+i)
(nt,
+
-
{ni,
n/,
-
-
l)rk,Ck,
l)Ck,
(A.18)
25
Zk.r+i
=
-
i'-k,
-
l)C-[0, [ni,
+
l)rk,Ck,
w.p.
+
{nk,
-
ni,
-
Q.,
V',,,+, (/,
+
,)]
+
a.,V',,,+,(<,+i)
+
rk.ini,
-
l){r,,
-
l)Ckt
(A.19)
Zl.i +
i
=
(n, -l)t'[0, C;,]
+ V',,,+i(f,+i)-Cfc,-7rfc,0.,
1
w.p.
+
'U-t
=0
for J
nu -
nu -
2,
1
(A.20)
Zi,,+i
=
-
(/•/,
+
1)L'[0,C/,]
V',,,+i(^,+i)
1
-
w.p.
Ck, +jrk,Ck,
rik,
+
for
-
"i,
J
=
2
Ill;,,
1
(A.21)
=
Z,,,+,
-
in,
Cu +
w.p.
+
l)U[O.Ci,
-
\i,+iii,+i)
-
V/.,+i(^+i)
nk,ri,Cu]
+
+
V/,,+, (/,+,)
(rt,
-
1)Q..
(A.22)
nk,ri,Ci,
and
Zi,,+i
=
-
{ri,
'U-,r/,C/,-\
Notice that
For case
(A. 13)
2.
we
-
1)C'[0, nt,n,C/,
V',,,+
=
?
i
=
(r;.,
—
A'
1
=
(n,
-
/
=
!,..., A'
—
1.
1,
+
ni,)rk,Ckt,
(A.24)
Q,i+i
=
('U-,
+
ni,)ruCu,
(A.25)
(n,
=
-
+
(ilk,
(A.26)
nt,)C,,
-
l)-'[n,,r,,a.,
\',,,
-
+ ,(^ + i)]
jr,,C,,
(A. 27)
+
+
for
r/j
{rik,
for
i)c/[o, C/,]
=
if /-t,
1
=
1
w.p.
Zi,,+i
- \)nk,ruCu
(r,,
Ck,,+i
- l)r[0,a,] +
'U-.
-
(A-23)
Thoorcm
C.-i-i
Zk,+
r/,V;,,+i(/,+i)
-,,,+, (/",+,)
(A. 23) ri'ducc to
have, for
+
i(/,+i)]
=
J
in,
+
-
III,
1,
"/,
I
/,,+!
(^+i)
-
Ck,
-
1
ji-k.Ck,
w.p.
for
Ilk,
+
J
=0
II,,
-
2,
III,
(A.28)
Z,,,+,
=
in,
-
l)f'[0.O,]
+
^,+^{t, + ^)
1
w.p.
for
>ik,
Zi,r+i
=
(r,,
-
l)L'[0,n,,C,
ni.C,
w.p.
-
(n.
+
Ck,
+
{rik,
-
j)rk.Ck,
(A.29)
=0
Ilk,
"/,
-
(r,,
-
-
ir'[ii,,n,Ck,
(ilk,
J
-
1)
+
'[n/,r/,C/,.,
-
-
V,.,+i(f, + i)]]
V;..^i(f,+i)]
ni,)Ci,
26
+
{iik,
+
n,,)rfc,C/,,
(^-30)
1)C,,
Zi, +1
-("/,-
l)'-;-,G,
(A.31)
-a-,
- l)~'[n/,n,0-, -
('7,
V',,,+
w.p.
("A-,
z,l.l+\
=
-
(;•„
+
V',,,
l)f
(a,
'[0,
+ i(r, + i)
r,,
- DC,, -
-
[ni,
-
\)C\
Ck,
"(l)^!
+
(n,,
-
/^/,
-
D'-;,0,
+
('7,
- D^'["hr,,C,, -
V,,,+ i(<,+i)]]
+ (r,,-DC,,
DC, -
-
w.p
+
i(^+i)]
(ru-,
+
(A.32)
- D'7,0, + {vki ("fc, + riu)Cu
77,,
D"M"7;'V,C(t,
-
V^,,,+i(<,+ i)]
and
Zi., +
\
=
(//,
-
Df'[0,
+ 'l,[{in; +
(>H;
(7(,,
+
ni,)ri,Ci,
ni,)ri,Ch
+
ni,)ri,Ci,
-
-
(7-/,
-
l)'^[ni,ri,Ck,
-
\'i_,
+ i{f,^i)]]
(A.33)
'H-X\,]
{'1,
-
l)~'^[n,,ri,Ck,
W.[).
('It,,
+
>ii,)Ci,
-
Vi,,+i(t, + i)]
References
[1]
Boxma, O.
and H. Takagi. 1992. Editors, Special Issue on Polling Systems, Queuoing
J.
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