^ (UBRAWESJ .
CENTER FOR COMPUTATIONAL RESEARCH
IN ECONOMICS AND MANAGEMENT SCIENCE
Transient and Busy Period Analysis of the
GI/G/1 Queue: Part I, The Method of Stages
ty
Dimitris
J.
Bertsimas
and
Daisuke Nakazato
Sloan W.
P.
3098-89-MS
December, 1989
SLOAN SCHOOL OF MANAGEMENT
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS 02139
ALFRED
P.
Transient and Busy Period Analysis of the
GI/G/1 Queue: Part I, The Method of Stages
ty
Dimitris
J.
Bertsimas
and
Daisuke Nakazato
Sloan
VV. P.
3098-89-MS
December, 1989
Transient and busy period analysis of the
queue: Part
Dimitris
J.
I.
The method
Bertsimas
'
GI/G/l
of stages
Daisuke Xakazato
November
^
20. 1989
Abstract
In this
paper we study the transient behavior of the A/GEi/A/GE.vf /I
queueing system, where
MGE
is
the class of mixed generalized Erlang distri-
butions which can approximate an arbitrary distribution.
of stages
We
use the
method
combined with the separation of variables and root tinding techniques
together with linear and tensor algebra.
sions for the Laplace transforms of the
ing time distribution under
We
find
simple -iDsed form expres-
queue length distribution and the wait-
FCFS when
the system
is
initially
empty and
the
busy period distribution. \Ve report computational results by inverting these
expressions numerically
in the
time domain.
expressions derived our algorithm
Key
liords.
'Diraitris
J.
The research
program
at
is
Because of the simplicity of the
very fast and robust.
Transient analysis, busy period, transform methods, linear algebra.
Bertsimas, Sloan School of Management, MIT.
Rm
E53-359, Cambridge,
.\Ia
02139.
of the author was partially supported by grants from the Leaders for .Manufacturing
MIT and
from Draper Laboratory.
'Daisuke Nakazato. Operations Research Center, .MIT. Cambridge, M.\ 02139.
Introduction
1
queueing models have long been considered
Transient and busy period analysis
in
as very difficult problems.
many
Yet, in
situations
very important to study
is
it
the transient behavior of queueing systems. For example, systems often encounter
transient behavior due to exogenous changes, such as the opening or closing of a
queueing system or the application of a new control. Furthermore, even
with time homogeneous behavior the convergence to steady state
equilibrium behavior
is
demand
(for
so slow that the
not indicative of system behavior. Examples from practical
situations, in which transient
tems with frequent
is
systems
in
phenomena
start up periods
are important, include manufacturing sys-
and transportation systems with time varying
example airport runway operations
in
major airports).
Analytical investigations of the transient behavior of queueing systems are very
rare,
mainly because of the complexity involved. For the
queue length probabilities are known
for the
(see
in
Gross and Harris
[6]).
An
In
sums of modified Bessel functions
is
for the
the last decade, work on the transient behavior of queueing systems has
methods
in
emphasis was primarily mo-
are the randomization technique introduced by
integration
methods of the underlying Kolmogovov
Gross and Harris
this
[6],
The two
Grassmann
[5]
principal
and numer-
differential equations (see
section 7.3.2 and references therein).
paper we study various transient performance characteristics of the
MGEi,/MCE\f/l
In
[1]
queue.
tivated by the analytical complexity of the problems involved.
In
community
the recent work of Abate and Whitt
concentrated on numerical techniques. This change
ical
queue expressions
indication of the interest of the research
transient behavior of queues
M/M/1
as
M/M/\
system, which
a sequel paper (Bertsimas
et.
is
an important special case of the
al.
[3])
CI/G/l
queue.
we formulate the problem of finding
si-
multaneously the waiting time distribution and the busy period distribution of the
GI/G/l queue
with arbitrary distributions as a Hilbert problem.
The
MGE,
which
is
described
in
some
distributions, which
detail in Section 2,
dense
is
in
the class of mixed generalized Erlang
is
the space of
all
distributions and can approximate
arbitrarily closely every distribution. For a discussion of the properties of the
class see
Bertsimas
period distribution.
We
[2].
We
MGE
study the queue length, the waiting time and the busy
use the separation of variables technique together with root
finding techniques to establish closed form expressions for the Laplace transform
The
of the distributions under study (queue length, waiting time, busy period).
advantage of these closed form expressions
be used for numerically inverting them
we report computational
that they are relatively simple and can
is
in
the time domain.
In fact, in Section 6
results for inverting numerically the transform of the busy
period distribution.
These expressions
had over the years
time domain.
also explain the difficulty that the research
in establishing
community has
expressions for the distributions we study in the
Despite their simplicity
in
the transform domain, our expressions
involve roots of polynomial equations. In general, these roots can not be
analytically and even
if
they are
known they
analytic inversion extremely complicated
The paper
is
structured as follows.
if
are complicated
3
in
we
make
their
we describe the
MGE
distri-
we derive closed form expressions
the transform of the queue length distributions
In Section 4
to
not impossible.
In Section 2
bution and the notation we use. In Section
enough
computed
when
the system
is
initially
empty.
find an explicit expression for the busy period distribution while
Section 5 we analyze the waiting time distribution under
FCFS.
In Section 6
describe the algorithm to invert numerically the closed form expressions derived
the previous sections and also report
final section
for
contains
some preliminary computational
some concluding remarks.
results.
we
in
The
2
Notation
The
general Coxian class
Cn was introduced
Cox's
in
stage representation of the Coxian distribution
is
[4]
presented
pioneering paper.
in
Figure
be noted that this stage representation of the Coxian distribution
in
the sense that the branching probabilities
can be complex numbers.
distribution,
The mixed
As
a result, the
probabilistic interpretation, which
It
should
purely formal
can be negative and the rates
generalized Erlang distribution
where we assume that the probabilities
rates n, are reals.
^-l
q,
is
1.
7,
is
/x,
a Coxian
are non-negative and the
mixed generalized Erlang distribution has a
is
The
further exploited in this paper.
valid
Ra{t)
= The ATC
stage currently occupied by the arriving customer at time
R,[t)
= The STC
stage currently occupied by the customer
time
W{t)
is
being served
at
t.
— The
waiting time of a customer arriving at time
With the above
Markov chain with
definitions the
t.
system can be formulated as a continuous time
infinite state space:
{(A'(f),/2a(t).«5(O)..V(t)
=
fla(n=
0. 1
1.2
We now
introduce the following set of probabilities:
Pn,..,(f)
=
We
who
t.
Pr{.V(i)
=
n.R,{\)
= i.R,{i)^j\.
/?,(<)= 1.2
L.
=
P^Jt)
Pr{.\(t)
=
0.
A/}.
Ra{t)
=
i}
will also use the following notation:
•
T
=
the
mean
interarrival time.
-
=
the
mean
service time
and p
=
-
=
the
traffic intensity.
•
P^,{t)
=
is
a
column
whose elements
vector,
are the probabilities Pn.i.j[t)-
• ak{t).br{i)=the probability density function (pdf) of the remaining interarrival
(service) time
(r
=
if
1,.....\/).
the customer in the
Therefore, ai(t).6i(t)
ATC
is
(STC)
is
in
stage k
=
1
L
the pdf of the interarrival (service)
time.
•
a^(i).tf(t)= the probability to
stage j during the interval
t
move from stage
i
<
j of the
ATC
(STC)
to
without having any new arrival (service comple-
tion).
. a-,{t)
=
{a\{t),...,a^{t)Y,a-l{t)
=
(0
• h{t)
=
{b\{t),....blf{t)y.b1{t)
=
{0,....blit),...,bl'{t)y,
•
Tfn{s),ak{s),3k{s)
=
at(t),
.
.
.
,a^(t))'.
the Laplace transforms of P„(i), aX(t)i ^fc(0 respectively.
.
e-
=(o,...,i,...,oy
By introducing the following upper semidiagonal matrices Aq, Bq and the dyadic
matrices Ai
fii:
,
*"
.4o
•••
-(l-p,)>i
A,
=
-PiAi
.4,
=
•
-Az,
and similarly Bq, B\
time distribution we have
for the service
Q-;'(s)
=
e-;(75
+
.4o)-',
L
Qfc(s)
=
and similarly
Finally
-ei.{Is+ An)
for
(3^ (s),
(ll,...,In)'® (yi,..
.^,£7,^
sition rate matrices in
n::ai-p.)A.
'/
,
'
dk{s)-
we use the usual tensor notation
Finally, ii"'",£°,£~
L
= S^prKalis) - Y^PrAr
'.4,^1
.
,ym)'
(see also
=
Neuts
[10], p. 53)
(-Ciyi,-- ••Tnyi-
are transition rate matrices.
,Xr^ym)'
We
can express the tran-
terms of the matrices Aq, A\, Bq, B\ using tensor notation.
For example
£°
3
= -Aq®
The queue length
I
- I®
Bq.
distribution in the transient do-
main
The system MCEi/MGE\i/\
Markov chain with
a single
is
an instance of the homogeneous row-continuous
boundary (Keilson and Zachmann
6
[8]).
We
analyze
the
homogeneous part
of the
Markovian dynamics using the separation of variables
Then we analyze
technique combined with tensor algebra.
the boundary condition plus the
(i.e.
this
way we succeed
in
initial
the compensation part
condition) using linear algebra.
In
finding a closed form expression of the Laplace transform
of the queue length distribution.
We
Although our approach can be
principle applied even in the case of arbitrary
initial
in
conditions, the algebra required
The homogeneous
for
n
>
makes the
first
>
1)
empty.
initially
explicit derivation very hard.
Chapman-Kolmogorov forward
write the
1:
By taking the Laplace transform and using
n
is
part
Using the notation of section 2 we
equation
assume that the system
the assumption that Pn{0)
=
(for
we obtain:
s?;(s)
The compensation
Similarly, the
=
ir;_i(sk"^
+
^n(s)g°
+
^n+iis)g~
part
Chapman-Kolmogorov forward equation
J^Poi^)
for n
s^o{s)
5?i(s)
=
=
and n
=
1
is:
= Poit)^ + Pirn:
and therefore the Laplace transform of the the equations
Our
(1)
Po{0)
+
^'o{s)^
= ro(s)^ +
+
^xis)g°
>
1.
The
n
=
0, 1
is:
^[{s)g;
+
(2)
^2{s)g-
general strategy for analyzing these equations
find the general solution of (1) for n
for
is
the following.
(3)
We
first
solution of the Laplace transform
Tfn(s)
unknowns
M
are the
From
below).
we use
M
then written as a linear combination of
is
constants that depend on
we
(2)
(the coefficients
s
M
find Tfo{s) as a function of the
system of
(3) to find a linear
M
The only
geometric terms.
Dr,r =
coefficients D^.
M
[
Finally
equations with A/ unknowns. W'e exploit
the particular structure of the linear system to find a closed form solution for the
unknowns Dr,
=
r
M
1
we
a result,
.\s
find explicit closed
form expressions
Laplace transform of the queue length distribution.
tor the
The advantage
of our expressions
real time. In principle this
that they can be numerically inverted in
is
approach works
for arbitrary initial conditions.
able to find closed form expressions only in the case in which the system
empty,
i.e.
Theorem
nonzero
=
P„(0)
Under
1
n
for
the
>
1.
We
can now prove our
assumption that the system
initial probabilities are Po,fc(0),
and p <
1
is
the
is
We
were
initially
first result.
initially
empty,
i.e.
the only
transform of the queue length
distribution has the following form:
M
wn{s)
^DrPi{xr{s))®ai{s-Xr{s))iai{s-Xr{s)))^-^ {n>l}
=
M
L
Ms) =
^Po.fc(0)a";(5)
k=
+
^D,—-,Ji(x,(s))(Q-i(5-j:,(s))-ai(s)),
r=l
l
,
^'-^^)
where
p _
i:LxPo.kiO)<yk{s){-l)''3^'{0)
!-«,(.)
and X = Xr{s)
(for r
=
1, ..
.
,
MJ
are the
.\f
roofs of the equation:
ai(s-x)/?i(x)=
r
l3fJ(x)
;3'^'{xr{s))
it
x,{s)
""^^'^y xr{s)-x,{s)'
<
0(i.e.
||ai(s-
j-)||
<
1)
1
for3?(s)
>
Proof
We
first find
the general solution of (I). Equations (1) are partial difference equa-
tions with constant coefficients.
Following the separation of variables technique we
assume that a general solution Tn(s) of the difference equation
;r~(s) =
v..(s)t^,(s)(ti;(s))"-\
8
(1)
is:
which can be written
in
tensor notation as follows:
T„(S)
=
^{S)
®
n-1
11'{S){W{S))
With
Vi(*)
^{s)
=
Ais) =
\ rL{s) I
We
\
\
Vm[s) I
substitute the form of ?n(5) into (1) and using tensor notation we obtain:
s^(s)
®
^'{s){w{s)f-^
=
-/(5).4i
®
- ^{s) ®
^'(s)Bo(it<s)r-i
v^'(s)(u(s))"-2
- /(5)^o ®
- /(s) ®
t/i''(s)(a.(s)r-'
^~(s)5,(u.•(5))^
which by collecting terms can be written as
s(p{s)
®
^'{s)
+ ^{s){Ao + -)-.4i ®
)
t/r'(s)
+ ^(s) ®
.?(s)(flo
+
Using the standard separation of variables arguments (see Bertsimas
notation we
demand
that ^(s)
with an eigenvalue —y{s) and v'l*)
with an eigenvalue
— x^s). As
[2]) in
+
a row eigenvector of the matrix (.4o
is
is
=
u.'(s)fli)
a row eigenvector of the matrix [Bq
tensor
^^TTT
+
0.
-^i)
wi^s)B\)
a result,
(s-x(s)-y(s))[^(s)®T^'(s)]
=
0,
and therefore
s
In the following claim
we
=
i{s)
+
y{s).
(4)
establish the relation
among
Ui{s).
by computing the characteristic polynomials of the matrices
(Bo
+
u(s)fli)-
Claim
Proof
1
Qi{t/(s))
=
w{s) and w{s)(3\{x{s))
=
1.
(.4o
y(s)
-i-
and
^7777 -^i)
x{s).
and
Since x{s)
is
an eigenvalue of (flo
+ w{s)Bi),
it
satisfies the following
characteristic equation:
det[/r(s)
that
flo
+
ti'(s)Si]
=
0,
is
det[/r(s)
+
^
UrLiif^r+i{s))
Thus, since Bi has rank
+
1
= -
/?i(i(s))
+
w{s){Ix{s)
for
any rank
1
=
flo)"'5i]
rank, det(/ir(s) + flo)
full
Also
0.
+
Bo] det[/
Since the matrLV Bq has
But
+
0.
= Denominator [Ji(x(s))] =
matrix S, det(/+S)
=
+
trace(S).
ifi{s)
in closed
1
1
u(s)trace((/r(s)
trace((/r(s)
+
+
=
Bo)~'fii
0.
Bq)-^ Bi) and therefore
u(5)A(r(s))=l.
Using exactly the same methodology we establish that
ai(y(s))
We now
compute
in the
=
u-(s).a
following claim the eigenvectors
li'(s)
and
form.
Claim 2 The
and
vector cri(y(s))
the vector 3i{x(s))
we can choose
=
'fi{s}
is a
is
a
row eigenvector of the matnx
0*1(1/(5))
+
row eigenvector of the matrix [Aq
and
xp{s)
=
(
Bq
+ u(s)Si).
^^TTf
-"^i)
Therefore
/?i(x(s)).
Proof
We
Since
prove the claim
i*i
{x{s))
/fl'(i(s))(fio
+
=
for i3i{x{s)).
e*,(/j(s)
u'(5)5i)
+
The
case for 0*1(1/(5))
is
similar.
5o)~' we have that
=
e-i(/j(5)
+ BorHBo +
=
e-'i(/ir(5)
+ Bo)-\-Ix{s) +
=
-xis)i3iix{s))
10
+
e-'i
-
u(5)Si)
(/x(s)
+
^o)
uis)3,{xis))e'„
+
a'(5)5i)
= — i3i(s)e\. But from
since Jj (s)fli
claim
u,'(s)5i(i(s))
1,
=
and
1
thus
l3i{x{s)){Bo
As
and claim
a result of (4)
Qi(s
—
+
-x{s)3i\x{s)).n
and since we are looking
1,
S
3,{x{s))
where, because of (4) and claim
I.
3?(r)
<
3 For p
1
<
(i.e.
-
a,{s
x
Qi(s
In the following claim
=
ai(y(s))
=
stationary, the general
is
is
=
^n{s)
w{s)
for roots
r{s)) inside the unit circle, so that the solution
solution of equation (1)
Claim
=
w{s)Bi)
=
x{s)) {a,{s
- x{s))f-'
i[s) satisfy the equations
-
x)3i{x)
|iai(s
-
r)j|
<
=
1
for
1)
^{s)
we investigate the number of roots of
equation (6) has
(5)
,
M
roots Xr{s).
r
=
1
(6)
>
(6).
A/.
Proof
This can be easily established from an application of Rouche's theo-
rem
domain ^{x) <
in the
theorem see Bertsimas
trix
1,
.
[2].
For a very similar application of Rouche's
The same
result can
be established from ma-
geometric considerations by noticing that the roots ai(s— Xr(s)),
.
,
.
A/ are the A/ eigenvalues of the matrix R{s)
Assuming that the A/ roots of (6)
for
0.
T„(s),n
>
1
are distinct
in
Ramaswami
we can now write an
r
=
[12].
explicit expression
by taking linear combinations of the general solution form. Indeed,
there are coefficients
Dr.r —
\I
1
such that
\t
f„(s)
= Y. Dji{Xr{s)) ®
r
ai{s
-
Xr{s)) (qi(s
- Xris})^-' {n >
1}
(7)
=l
Remark: The distinctness assumption
This distinctness assumption,
sion for 7rn(s), n
>
1.
but
it
is
is
very convenient in order to find an explicit expres-
not critical however.
11
The
algebraic theory of rational
functions guarantees that
Xr{s)
—
some
ikis) for
'
if
r.k.
other words, we
In
that the roots are distinct and at the final stage
from
first
solve the
we show the
limit of (7) as
problem assuming
results are
independent
queue length and the
In fact, our final expressions for the
assumption.
this
we can take the
there are multiple roots,
busy period distributions are simple symmetric functions of these roots. So, finding
the limit
in
the case where there are multiple roots
The remaining unknowns
are the coefficients
we express
the following claim
is
D^
an
(r
eeisy task.
= 1,...,M) and
tto{s).
In
a linear combination of the Dr's.
Tro{s) as
Claim 4
M
L
?o(s)
.
= Y. PoA^Wkis) + YL Dr^^M^ris))
^'•V*^
(ai(s
-
x,(s))
-
ai(s))
(8)
.
r=l
fc=I
Proof
We
substitute (7) into the equation (2) and
we obtain
A/
SK'ais)
=
P^(0)
-
ro(s).4o
-
J]
a (Ji(x.(s))Si.-,) ai'(5 -
x.(s)).
r=l
Thus
M
^0(5)
=
PoWils + Ao)-' + J2
M
L
=
Y
/'O,)c(0)a-;(S)
+ J2
k=l
The next
Dr3,{xr{s))e',ilis
r
step in our approach
the following claim
we
is
-
xris))
+ Ao}-\ls +
Ao)-'
.
Dr—--f3i{Xris))
=l
(ori(s
-
Xr{s))
-
0.1(5)) .D
^^-y^'
to find the coeflRcients
establish the equations
Dr
(r
from which the
—
I,...,A/).
In
Dr
are
coefficients
computed.
Claim 5 For
all k
=
M:
I
M
y^^
1
l3^{xAs))
^
Proof
12
T.k=i Po.>c{0)c^k{s)
^g^
Using (7) we easily obtain that
Af
S7r[{s)
-
-
/i(s)£°
T2(s)^~
= Y^ Drth
(jr(s))-
r=:l
From
(3)
M
Y^ DrMxris)) =
r
=
Using (8) and since
-(/o(s)^ie-j)e-i.
(10)
l
q"!
(s)(
— .4ie*i) =
Q\{s) we obtain that
'^'
=
-(T'o(s).4ie-,)
Po(0)(/s
where we defined a'{s)
=
+
T D,-^/5i(x,(s))
1
Ao)-'Aiei
Po(0)(/s
+
+ ^o)~'- We substitute
this
{a,{s
equation
and obtain
in (10)
'r
STniT't
\-<
=
Since Ji'(s)
=
1
(^^
-
Qi (s)^i (Jr(5)) ^
,
x:aj,(x.(.)){/---^-^-«,(.)-^}.
e*i(/s+ So)"' and Ji(s)e*i
=
-;i*i'(s)5i, then
.v/
n
TD,—--J^'{xr{s)){Ixr{s)-{IXr{s)
+ Bo)-a^is)Bi}
=
c''is)e''i
^
.(s)
M
But
'
e"\(/s
\\
'
.
+
-.
1
=
-YDr-^/i'(xr{s)){Bo +
5o
As a
+
ir5i)-i
=
,/t'^4'^(,)
Yo'A'ix.is))
•rr(s)
system
and hence
€"\(5o
+
ai(s)Si )-^
=
result,
=
-^llL.^r(0)
=
-%iM5)^/,'(o),
i-ai(s)
;fr;
Therefore, for
ax{s)Bi).
all
k
—
l-ai(s)
(U)
1,...,A/ the coefficients Dr satisfy the linear
(9).
13
-
x.(s))
-
a,{s))
We now
Claim
solve the linear system (9) in closed form.
The linear system (9) has
6
the following solution:
i:LxP^,mc'k(s){-\)''3^'{^)
l-a,(.)
JF{xAs))
^
.
For
.^
k=l
all r
—
1,
.
.
x,{s)
Xr{s)
-
Xfc(s)
Proof
Let Cr
= ^L
we obtam that
'""'^'^
We
=
for all t
r
=
4fe^D,.
l
u = fc +
expand the above equation
J2
r
where
in
A/, J^^^.
1
(j^.n
Since for
Cr ow'^'^n
••
=
=
1.
1,...,A/
'•«
^^n
as a polynomial of rr{s)
E Cr<T,A-^r{s)r =
and obtain
1,
= l n=0
are the coefficients in the expansion.
"i.o
i
l
We express
matrix form and we obtain
1
all
(^i(*^))
w-i
{xm{s))
1
a-
A
s
Since by the definition of the
(7^.
Af-1
this equation
.
,
A/.-
By
letting x
We now
=
0.
we obtain that S
=
1
observe that the matrix A
e\i
a
is
and thus,
Vandermode matrix.
Using
Cramer's rule to solve the above linear system and exploiting the prop-
Vandermode matrLx generated from
erty that the determinant of a
(denoted by V'(ui,
_
.
.
,
u.\/)) is
given by ni<j("i
Ir-\{s).^.Ir + \{s)
Wjllis)
""
''
.
~
XSfjs))
T-w^cU
\-l r.i c\
.
.
.
u\f
,
obtain that
^'•^
";)•
U\,
_ -pj
~ li
Tfefs)
Therefore,
Having found
explicit solutions for the
expressions for the t„(5). n
>
remaining unknowns Dr we have explicit
The proof
0-
of
Theorem
As an additianal check of the algebra we compute
Summing up
(7)
and
(8)
sum
r
=
1
(-l)-^3;^(0)
we obtain that ^{s.
to one in the transform
1)
is
nontrivial and
=
-,
domain.
the generating function 'l'(s.r)
observation
it
is
is
1-;
which
is
ri,(5)
r-,.\f
the condition that the probabilities
Another interesting point
symmetric with respect
is
the fact that
to the roots Xr{s).
This
established by using the Lagrange interpolation
Theorem
assumption that the roots Xr{s) are distinct and therefore
If.
now complete.
the generating function
3,Jxr(s'\)-\
formula and the Chinese remainder theorem.
only in this case.
is
we obtain that
^M
For
1
I
all
was proved under the
the formulae are valid
however, there are multiple roots (say
j:,(s)
=:
Xj{s)). the
resulting formulae are simply the limit of the formulae given here as x,(s)
15
—
'
X]{s).
4
Busy period analysis
Ramaswami
[12]
has characterized the busy period of an
matrix geometric approach.
succeed
we show
Bp
Let
in
is
very suitable for numerical inversion in the time
in
we
call
in
the queue build up time.
the analysis.
the following properties.
A,,j
and
domain
Immediately
the system and the
STC
the system and the
STC
is
in
also define a
new random
This random variable plays
the time between two arrival epochs with
is
after the initial arrival
is
in state
epoch (there may be other arrivals
number
We
be the random variable of the busy period.
a critical role
arrival
his result considerably
the next section.
variable A,_,, which
customers
we simplify
using the
deriving a very simple formula for the Laplace transform of the busy
in
period distribution which
as
In this section
G/Ph/\ queue
state j.
in
t,
epoch there are n
while immediately after the final
+
between) there are n
In addition,
of customers never decreased below n.
1
customers
throughout the time
Notice that A,,j
is
A,,j
in
the
independent of
n.
Let R{t) be a matrix whose i,j element
=
the queue build up time. Let r,,j(s)
r(s) be the transform of R{t). Finally
eigenvalue u(s).
Theorem
2
We
where ir{s) are the
= E[e-'^n =
M
roots of the
Remark: (12)
is
assumption
no longer necessary.
is
the pdf R,.j{t)
= ^Pr[Aij <
E[e~^^'-i] be the transform of Ri,j{t)
let ^'(s)
t]
of
and
be a row eigenvector of r(s) with an
can now state and prove our basic result.
The Laplace transform
a(s)
is
a simple
1
cr[s)
-
(1
of the busy period
-
/?i(^))
Bp
J;^;^'^^^;^,,
polynomial equation
given by
ts
,
(12)
(6).
symmetric function of Xr(s) and therefore the distinctness
Proof
16
By
focusing on the last customer arrived in the busy period we write the busy period
dynamics.
LE
11=1
which the
and
in
*
6l""'V)
*
(
'*'
in
/
first
customers n and the state k of the
busy period
time to build n customers
a,{t)dt\
•^'
(13)
,
J
term corresponds
to the case
the busy period was the only one in the busy period
the second term (the double summation)
In this case, the
r
b\(t)Qr^l^]
indicates convolution, the
customer
last
E
\r = l
I
where the symbol
in
{Mt)
M".'(<) *
k=l
is
the
in the
STC
sum
this
we condition on the number
of
customer has found when he entered.
of two independent
random
variables:
The
queue (a convolution of n build up times Ajjt)
and the time to empty the system, since he
is
the last arriving customer
in
the busy
period.
Our
strategy
is first
the busy period.
We
and then using (13)
to find R{t)
are thus naturally led to the
times. Similarly to the
first
we write down dynamics
to find the
transform of
dynamics of the queue build up
passage time analysis as
in
(Keilson and
of the queue build up time in matrLx
Zachmann
[8]),
form by considering
the last arrival during A,,j
b\{t)
mt)
b-^'{t)
=
a,(n
b\'f{t)
r
+
/
\
...,^
bi{t)
£/?("'(<)*<
* b[''-'\t) *
(^
b\{t)
•••
b-l'{t)
)]ai(0
>
=
n=l
=
S(<)ai(t)
+
^
«(")(t)
*
[F„(Oai(0],
n=l
where S(t),F„(t) are the upper diagonal matrix appearing as the
sum and
Fn{t)
is
the matrix
composed of the three convolutions.
17
term
in
the
By taking
the
first
Laplace transform of the above matrix equation and multiply both sides from the
left
with the eigenvector of r(s) ^'(s), we obtain:
u{s)Cis)
= ^{s)C{[B{t) + f;
u"(s)Fn(t)]ai(t)}-
(14)
n=l
But,
3,{s)3, is)
+
{Is
+ uBO~' =
Bo
(/s
+
+
fio)-'
1-U^l(5)
Pm{s)I3i is)
since for every pair of matrices
C
^~^ ~
this in real
i
+ ir/c-'D
'
•
i
^y expressing
b\{t)
of
rank and
full
of rank
I,
(C + D)
^
=
time we obtain
^
b'l'it)
D
^
bi{t)
,-(Bo + uB,)(
*b[''-'\t)*(^b\{t)
...
b-l'{t))
n= l
••
As a
result, (14)
u{s)^{s)
=
as in claim
1
If
\ b.Mit) J
becomes
^''(s)£{e-'S° + "(''5>)'a,(0}
Therefore, since ai(s)
{Bo + u{s)Bi).
b',',(t)
—:{s)
a rational function, ^(s)
is
is
= ^{s)a,{Is + Bo +
must be
a
u(s)5i).
row eigenvector of
the corresponding eigenvalue, following the
same technique
we have that
ti(s)/?,(r(s))=l.
Since Qri(s)
Comparing
is
(15)
a rational function of
s,
we get from
u(s)
=
a,(s--'(s)).
(16) and (17)
we observe that
as x(s) (equations (6)) and therefore :{s)
(16)
(15) that
(17)
:(s) satisfies exactly the
=
x(s), u(s)
=
same equations
w(s) and ^(s)
—
3\{x{s)).
Having characterized the eigenvalues and eigenvectors of T[s) we can spectrum
decompose
it
under the distinctness assumption:
M
(X{s)r = X;(a,(s
-
x.(s)))"0r(s)/?i (x.(s)),
r=l
18
(18)
where
3i (r,(5))
3{s)
=
0\t{s)
•••
<^i(s)
01 i^.sfis))
After the characterization of the r(s) we take the Laplace transform of (13).
Using (18) and after similar manipulations with the analysis of the queue build up
time,
we obtain
r
=l
/
=
\
,3i(ri(5))-l
-1
",(^(5))
/?l(xM(a))-l
Since for any non singular matrix
.4
we know that
f .4
^y
=
1
—
find that
,-f
^1
<t(s)=
1-
3i(xi(s))-l
»-r,l5)
1
det(/3(s)-
)•
det(^(5))
^ 3i(rAf(0)-l
1
But
for all
r=
1,...,A/
S-X.Vf(s)
^j ,//f
.
we
=
1
-det(diagonal(
T":)) ^«'t(/5
r-T'--s-x\f[s)
s-xi(s)
Denominator Ji(s) — Numerator Ji{s)
+
Bi
+
Bq)
^2i';'(l-Q^)/^0^•-•^l^^))
Although the analysis used some rather heavy machinery from
it
linear algebra,
used direct probabilistic arguments by considering the dynamics of the system.
The reward
of this analysis
a very simple expression for the transform of the busy
is
period distribution, which as we show
tional advantages. In addition,
it is
in
Section 6 offers very important computa-
not hard to
compute
moments
closed form
in
of
the busy period distribution by repeated differentiation of the Laplace transform.
5
The waiting time
distribution under
In this section
we derive an expression
arriving time.
Our
for the conditional waiting
strategy for the analysis
bution of the number of customers
in the
on the number of customers found upon
and
finally
we
FCFS
is
the following;
system
arrival,
we
time pdf given the
the distri-
first find
at an arrival epoch; conditioned
we then
find the waiting time
pdf
find the (unconditioned) waiting time pdf.
form expression
In order to obtain a closed
assumption that the system
is
initially
in
the transform domain,
empty and futhermore, the
we make the
initial
probability
distribution has a very special form.
Assumption
We assume
that the initial probability vector Po{Q)
In principle, this
assum[)tion
is
for the solution.
Without
1
this
closed form expression both
not necessary
if
we take
assumption, however,
in real
time and
20
in
it
=
Aai(O).
a pure numerical
is
approach
not possible to obtain a
the transform region.
In the next
theorem we prove a
critical
consequence of assumption
already reached steady state from the beginning,
Po,it)
Proposition
first
customer
=
Pr[Rait)
=
(]
=
If the initial condition satisfies
IS
the
forward recurrance time of
life
time of the renewal interval.
the
Kolmogorov equation:
=
=
1
Po,.(0).
assumption
J,
the arrival time of the
the tnterarrival time,
-P'{t){Ao
P'[t)
=
i]
Furthermore, Po(0)
±P\t) =
the arrival process has
i.e.,
Pr[i?a(0)
1
1;
+
is
i.e.
the residual
the stationary solution of
A,)
1,
that describes the arrival process.
Proof
The transform pdf
of the interarrival time T^ of the
first
customer
is
given by.
I ai(5) ^
a'{s)
=
E[e''^]
=
P^{0)
V
^U«)
/
^ ai(5) ^
Since ai(0)
=
=
^"^
e*i/lo
{Is
+
.4o)~'(-.4ie-i),
we
find after
simple algebraic manipulations the transform of the forward recurrance time under
assumption
1
as follows:
(x'{s)
=
Aa;(0)-(/s
=
\^^{A^)-\U^A^)-\-Axe^)
=
Ae",! ((^o)-'
=
-(l-ai(s)).
+
.4o)-'(-.4ie-'i)
-
s
21
{Is
+
.40)-') (-.4ie-i)
To obtain
we observe that
the stationary distribution,
because of the structure of
.4i
,
we know that
.4i
=
1
i4ie*i.
(.4o
+
Thus we
=
--^i)!
and
find
l^-Ao^Aiex,
(19)
or equivalently
We
are
now ready
ai(0); for this
to prove that the stationary probability vector
show
suffices to
it
q'j(0)(.4o
+
=
.4i)
0'.
is
proportional to
Since q'i(O)
=
^^Aq^, we
have
q'i(0)(.4o+
We
=
e-*i.4o'(.4o
=
e*i
=
0'
complete the proof by showing j
a-[{s)
— — e*i(/s +
.4o)~'.4ie'i
a;(o)-
.4i)
=
+
e*i
/Iq
1-
<^i(0)
+
'
^
^i)
1
Utilizing (19)
and the
definition
we obtain
r
=
-a',{0)A-' Aiei
=
-\[ms-,oe'iAQ\ls
=
lim,_oie-\((/s
_
i
+
+
Ao)~'^Aie'i
.4o)-i-.4o-').4,e-,
Therefore, we have proved that the ergodic solution to the above Kolmogorov equation
is
A
Aq,(0)
=
Po(0).
corollary of the theorem
is
that the expression for
22
Dr
in
theorem
1
further
simplifies to
°- =
7
ii"(x,(.))
"Wn
(20)
, .(„_,.(„
since
^
I] Poa(OW(s) =
We
will
r
+ dr)
and the pre-arrival
In the following proposition
We
its
(1
-
define the event .4.40
Laplace transform
=
probabilities: P^ti''')
we
a,(s))
—
the system
in
Arrival about to occur in
Pr['^'(^)
= "
^a('')
=
i|/1^0]-
find the pre-arrival probabilities.
Proposition 2 Under assumption
and
= -
next find the distribution of the number of customers
seen by an arriving customer.
(r,
A
a'(s)
1
the vector of the pre-arrival -probabilities
is
is
'^
^n{s)
=
1
{^D.Ji(x.(s))(q,(5-x.(5)))" {n>
1}
^.=1
^0{S)
=
TY,Dr3]{Xr{s)).
^.1
Proof
_
ELi
= nDRAr) = inR^jr) =
Pr[ujL,i?a(r) = /n.4.40]
Pr[uf^i(:V(r)
Pr[.4.4C'|.V(r)
^
n H /?,(r)
ELi
= H
i
fl,(r)
Pr[.4.40|/?a(r)
=
=
/]
/]
/) fl
.4.40]
= nH
Pr[fla(r) =
Pr[.V(r)
_
/il.fr)
= n
i
/i:,(r)
/]
But
Pr[.4.40|.V(r)
= nnRs{r) = in
Ra[r)
=
/]
= Pr[AAO\Ra{T) =
and
Pr[/t,(r)
=
/]
=
Po,/(0)
23
=
Acti(O),
I]
=
Xipidr
=
/]
from proposition
this
it
1.
It
=
assumption Pr[/?a(^)
independent of
is
formula
since
for
^i-i
^ would be a function of
and therefore
r.
assumption
at this point that
is
it
becomes
1
critical.
Without
wiuie under assumption
t.
would not be possible
to find a closed
1
form
the transform of the pre-arrival distribution. Therefore, we obtain
=
^ipio:[{0)
fti(O)
=
1.
Therefore, using vector-tensor notation we have;
pr(^) = {^;(o{(-.4,e-,)®/}.
In the
transform region, using
r
=l
we obtain (the derivation
for ttq^s)
is
similar)
M
=
^n(-0
-^Dr/?i(r.(5))(«i(5-r.(.s))r
{n>
1},
r=l
=
^o"(«)
^l]a/?J(r.(s)).n
r=l
We
are
Theorem
now ready
3
bution under
Under assumption
FCFS
theorem of
to prove the central
(he Laplace transform, of the waiting time distri-
1
is
M
f
this section.
e- Prmr) < t]dr =
Jo
M
^
~; ^"X'^u'
1
i
+
S
S(3-^'{lr{s))
rj5)
\
.^r{s)t
11
\t=\ ^r{s)- Xk{s)
I
Proof
Given there are exactly n customers
arrived and the
STC
is
in
stage
i,
the system including the customer just
then waiting time
c.d.f.
is:
when n >
3
/o T.'jL\ bi{t)qjfijdt
when n =
2
U{t)
when n =
1
/J 6.(0 * 6l"-''(0
,
in
Ejl,
24
b\{t)q,fi,dt
where U{t)
is
By conditioning on
unit step function.
found the system, and using the expressions
the state the arriving customer
for the pre-arrival probabilities
from
Proposition 2 we obtain:
e-'^PT[lV{T)<t]dT
f^
h\(t)
I
+
Er=i(ai(s-x.(s))
^
ti(0
in+l
*'''r''(0*(
QlMi
di\
6}(0
V ^.\K<) /
21)
We
have observed
h\{t)
-{So + uS])(
the analysis of busy period from the previous section that
in
...
^
h\\i)
_
6i(0
^
+E""
n=
*fc';~''(0*(
,.\/
h\'[t)
61(0
l
\ bxtit) I
fe:l{(0
Substituting this to (21) we obtain
/'
e-''PT[W{r) <t]dr
=
Jo
-
2
Dr3[{xr[s)) i^ed'it)
Since 3\{^Xr{s))
and ai(s
—
is
j^
a,{s
-
x.(s))e-(^°+"'(^-^^(^''^'"(-5ie-i)(i<
a row eigenvector of [Bq-\- ci\{s
Xr(s)),'^i(-Cr(s))(
/o^
+
— Sie*!) =
e-^^ Pr[ir(r)
<
=
\
=
} E.^I. aC'-(t)
=
iE.'ii
YM.
1,
)
with eigenvalue
we obtain
<](ir
DrUit) (^l(r.(5))e-
ar(o
— Zr[s))B\
+
^i^)
^)
+ ^)
{3[{xrisW - ^^^^f^).-: +
(-^,J;(x.(s))floe'i
25
^
— Xr(s)
Substituting expression (20) for D^ and using (11) we finally obtain
-
1
-Vf
(-l)V'J,vr(0)
,
r,{s)
e'^'<'''.n
A
corollary from the previous theorenn
Jo
Note that
this expression
quantity linij^o
s<t>(s,u;) is
G//G/1
i.e.
queue,
that
/" /" e-'^-^'-^
=
$(s,-.')
is
is
Pr[W{T) < t]dtdT
en
Jo
a synnmetric function of the j;r(s)'s.
In addition the
the solution of the steady state Lindley equation for the
the transform of the steady state waiting time distribution.
Numerical Results
6
previous sections we have derived explicit expressions for the Laplace trans-
In the
forms of the queue length, the waiting time and the busy period distributions. In this
we
section
will
transforms.
remove the "Laplacian curtain", by numerically inverting the Laplace
The numerical
inversion of the Laplace transform
not completely solved problem
Bartholdi
[11]
overall algorithm
developed by Wolfram
polynomial equations
[13]
is
is
#P
study.
1.
We
[7]
Platzman, Ammons,
to
complete, that
is
a hard computational problem.
written using the software package of Mathematica,
and works
2is
follows.
(6) for selected s values.
functions of Mathematica to find
of [11] and
fact,
a well studied but
show that the problem of numerically inverting the Laplace transform
of a probability distribution
Our
numerical analysis. In
in
is
compute the
all
We
first
compute the
roots of the
For this purpose we use the build in
the roots of (6).
We
then use the algorithms
inverse Laplace transform of the distributions under
used two algorithms to invert numerically the Laplace transforms:
The algorithm
of
Platzman
This algorithm works
et.
al.
[11]
for distributions that are defined over finite regions.
26
We
used this algorithm combined with fast Fourier transform for the inversion of
the busy period distribution.
region (0,oc),
3Var[Bp])
in
Although the busy period takes values
order to apply the algorithm we used the region
as the region on which the busy period
details of this algorithm the reader
2.
The algorithm by Hosono
Hosono
[7]
is
is
different
(0,
E[Bp]
from
proposed an algorithm
not well-known
We
lished in Japanese.
referred to [11],
We
Laplace transforms which
for inverting
namely:
small
memory
used this algorithm for numerically inverting
<i>{s)
(t>{s).
it
It
is
primarily pub-
a very robust algorithm.
easy to program and control the error.
It
Moreover,
it
has
requirements, short computational times and can be used for a
We
We
briefly introduce the
algorithm below.
choose a precision p {significant digits} so that error of numerical
is
less
than
The algorithm works
Find
it
10~^'''^|/(<)|.
Let
as follows:
so that
E :^-
<
k-\
p-i
n=l
r=0
r=0
(b)
is
it is
The
be the input function. Let f{t) be the inverse Laplace transform of
inversion
(a)
the western literature since
found however that
wide variety of problems.
Let
in
is
necessary conditions which an ideal fast algorithm should sat-
satisfies all the
isfy,
For
0.
the busy period, the waiting time and the queue length distributions.
is
+
[7]
quite robust and accurate.
algorithm
the
in
Evaluate f{t):
27
.2
Hosono claims that
Comments on
All the
algorithm works when f{t)
computation was done
a Macintosh
in
smooth'.
and
II,
program
the
all
is
written
For computing the transient queue length and the waiting time
we assumed that the
distribution,
sufficiently
is
the numerical results
Mathematica.
in
this
arrival
time of the
customer
first
is
the forward
recurrence time of the interarrival distribution.
From our preliminary experience we can
say that the algorithms, in particular the
algorithm by Hosono, are robust and run very
MGE2o/MGE2o/\.
an
fast.
The
largest
example we ran was
which was solved by the algorithms without any
difficulty.
Unfortunately, we did not have any other numerical results to compare with except
the ones for the A//A//I queue queue, whose solution
of modified Bessel functions (see Gross and Harris
compare the
stability
and robustness of the algorithms we present
The
6.1
We
A//A//1
start our
is
A
CDF
=
1
and service completion rate
of the busy period
Fn needs
for sufficiently large n, 1/2
•
when n
It
A"^
illustration of the
in section 6.2
an example of
(The mean
is
known
E[Bp]
=
traffic intensity
/i
=
=
0.75.
4/3. In order to
solution
3
p
we computed
and the
The
compare
in
table
I
coefficient of variation
to satisfy the following conditions:
•
where
61 we
queue
the accuracy of the two algorithms with the
the
As an
In section
terms
queue.
examples with an A//A//1 queue with
interarrival rate
explicititly in
p. 143).
[6],
results of this algorithm to the exact results.
MCE3/MCE2/I
the
known
is
—
oo, fn,
denotes the
r
< \F„+JFr,\ <
Af„, A^fn,
•
•
I.
converge monotonically to
0,
th difference.
can be shown that the violation of these conditions results
appears at points of discontinuity of
f{t).
28
in the
Gibbs phenomenon, which only
is
Cg =
The algorithm by Hosono
7.)
gives identical results with the the exact
solution.
In figure 2
of
5
^
we
In figure 3
we
plot the first
we
plot the waiting time distribution as a funtion oft. In figures
4,
t.
The MGE3/MGE2/]. queue
6.2
Merely as an illustration of the algorithms we chose a
the following distributions:
Qi(s)
=
2
0.5-
with mean E[T]
The
2+s
=
+
The
mean
^^[A']
traffic intensity is
By
2
4
+
2+s4+s
2
4
6
0.5 x 0.3;
2+s4+s6+s
0.683333 and coefficient of variation
Cj-
is
—
0.70.
=
5
0.8-
5+s
+
5
3
0.2-
5+s3+s
=
0.266667 and coefficient of variation
p
= 039.
is
C^ =
differentiating the transform of the busy period distribution
mean
Cg =
first
queue with
service distribution has Laplace transform:
with
the
MCE3/MGE2/I
interarrival distribution has Laplace transform:
0.5 X 0.3-
/3ifs)
is
as a funtion
and second moments and the distribution of the queue length as
plot the first
a funtion of
and second moments of the waiting time
of the busy period
2.327.
In figure 6
is
we
E[Bp] =
0.
1.125; thus the
we found
that
398444 and the coefficient of variation
plot the busy period
CDF,
in
and second moments of the waiting time as a funtion of
the waiting time distribution as a funtion of ^ In figures
9,
10
figure 7
t.
we
In figure 8
we
plot the
plot the
we
plot
first
and
second moments and the distribution of the queue length as a funtion oft.
7
Concluding Remarks
In this
for
paper we attempted
to
demonstrate the power of root finding techniques
problems which were considered intractable
29
like
the queue length and waiting
time distribution
in
MCEil MGE\i/\
the transient
domain and the busy period distribution
queue. Using direct probabilistic arguments combined with tech-
niques from linear and tensor algebra, we succeeded
for the
for the
deriving explicit expressions
in
Laplace transforms of the distributions under study.
Algorithmically our approach offers a method for finding these distributions
time domain through the numerical inversion of the Laplace transforms.
in
the
Our
ex-
periments with the method are very encouraging since our experience with the
gorithms we used was very positive, since they are very
fast,
al-
robust and very easily
programmable.
References
[1]
Abate,
J.
and Whitt,
\V. (1988). 'Transient behavior of the
M/M/l
queue via
Laplace transforms'", Adv. Appl. Prob., 20, 145-178.
[2]
Bertsimas, D. (1989). "An analytic approach to a general class of
G/G/s
queue-
ing systems", to appear in Operations Research.
[3]
Bertsimas, D., Keilson,
J.,
Nakazato, D., and Zhang, H. (1989)." Transient
and busy period analysis of the GI/G/l queue: Part
problem", submitted
[4]
[5]
Camb.
Phil.
Soc,
in
the theory of stochastic
51, 313-319.
Grassmann, VV.K. (1977). "Transient solutions
Markovian queueing
sys-
Gross, D. and Harris, C. (1985). Fundamentals of queueing theory, Wiley,
New
tems", Comput. and Oper. Res.,
[6]
solution as a Hilbert
for publication.
Cox, D.R. (1955). "A use of complex probabilities
processes", Proc.
II,
4,
in
47-53.
York.
[7]
HosonoT., (1981) "Numerical inversion of Laplace transform and some
cations to wave optics", Radio Science, 16, 1015-1019.
30
appli-
[8]
Keilson,
ate
[10]
and Zachmann. M. (1988). "Homogeneous Row-Continuous Bivari-
Markov Chains with Boundaries". Journal
ment
[9]
J.
Vol.
25A (Celebration of Applied
of Applied probability Supple-
Probability), 237-256.
Kieinrock, L. (197-5). Queueing systems: Vol.
1:
Theory. Wiley.
New
York.
Xeuts. M. (1981). Sfatrii-geomeinc solutions in stochastic models; an algorith-
mic approach. The John Hopkins University Press, Baltimore.
[11]
Platzman.
L..
.\mmons.
algorithm to compute
J.
tail
and Bartholdi,
J.
(1988).
"A simple and
efficient
probabilities from transforms'", Oper. Res., 36, 137-
144.
[12]
Ramaswami. V.
(1982).
"The busy period of queues which have
geometric steady state probability vector". Opsearch.
[13]
Wolfram.
S.
19.
a matrix-
238-261.
(1988). Mathematica: a system of doing mathematics by computer.
Addison Welslev,
New
York.
31
t
2<
Figure
2:
The
first
StOivdnfty)
*
StDlv
•
Mean(lnfty)
O
Mean
and second moments of the waiting time of an A//M/1 queue as
a function of time
33
1.00
75
o
u
50
25
00
'
*
5tDiv(inrty)
StDlv
riean(lnfty)
B
Figure
4:
The
first
and second moments of the queue length of an
as a function of time
35
Mean
M/M/l
queue
1
00-
a
u
Figure
6:
The busy period
CDF
37
of an
MCE3/MCE2/I
queue
30
20
«
»
•
a
StDlvClnfty)
itOlv
neandnfty)
-^ean
10
00
Figure
7:
The
first
and second monnents of the waiting time of an
queue as a function of time
38
MGE3/MGE2/I
050
t
=
000
t
=
08
t
=020
= 045
= 250
t
= infty
t
t
25
00
I
0.0
I
I
02
1
I
04
I
I
I
;o
I
Z i
I
I
t
:
I
2
Wait
Figure
8:
The waiting time
distribution of an
of time
39
MCE3/MCE2/I
queue as a function
»
5tDlv(lnfty)
*
5tDiv
"
nean(lnrty)
Mean
Figure
9:
queue as
The
first
and second moments of the queue length of an A/G£'3/A/G£'2/l
a function of time
40
=
=
=
20
O-iS
2 50
= inrty
Q -engtn
Figure
10:
The queue
length distribution of an
of time
41
MCE2/MCE2/I
queue as a function
26b"/;
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Date Due
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MIT
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