C9
IJAN
121990
'•=^«»»i
CENTER FOR COMPUTATIONAL RESEARCH
IN ECONOMICS AND MANAGEMENT SCIENCE
Transient and Busy Period Analysis of the GI/G/1
Queue: Part II, Solution as a Hilbert Problem
ty
Dimitris
J.
Bertsimas, Julian Keilson,
Daisuke Nakazato, and Hongtao Zhang
Sloan W.P. 3099-89-MS
December, 1989
SLOAN SCHOOL OF MANAGEMENT
MASSACHUSETTS
INSTITUTE OF TECHNOLQGY
CAMBRIDGE, MASSACHUSETTS 02139
ALFRED
P.
Transient and Busy Period Analysis of the GI/G/1
Queue: Part II, Solution as a Hilbert Problem
by
Dimitris
}.
Bertsimas, Julian Keilson,
Daisuke Nakazato, and Hongtao Zhang
Sloan W.P. 3099-89-MS
December, 1989
Transient and busy period analysis of the
queue: Part
Dimitris
J.
Solution as a Hilbert problem
II,
Bertsimas
GIjGjX
*
Julian Keilson
Hongtao Zhang
November
Daisuke Nakazato
^
^
^
20, 1989
Abstract
In this paper
we
find the waiting time distribution in the transient
and the busy period distribution of the G//G/1 queue.
problem
as a
two dimensional Lindley process and then transform
factorization problem.
the
GI / R/\,RIG/\
We
We
R
is
form formulcie
to a Hilbert
obtain simple closed form expressions for the Laplace
FCFS when
the busy period distribution. Furthermore,
for the first
Key words. Transient
it
the class of distributions with rational
transforms of the waiting time distribution under
empty and
formulate the
achieve the solution of the factorization problem for
queues, where
Laplace transforms.
initially
We
domain
the system
we
is
find closed
two moments of the distributions involved.
analysis,
busy period, Lindley equation, Hilbert factoriza-
tion.
'Dimitris Bertsimas, Sloan School of
bridge,
for
Ma
02139.
The
research of the author was partially supported by grants from the Leaders
Manufacturing program
at
MIT
'Julian Keilson, Sloan School of
Ma
Management and Operations Research Center, MIT, Cam-
and from Draper Laboratory.
Management and Operations Research Center, MIT, Cambridge,
02139.
MA 02139.
MA 02139.
'Daisuke Nakazato, Operations Research Center, MIT, Cambridge,
^Hongtao Zhang, Operations Research Center, MIT, Cambridge,
1
Introduction
1
work (Bertsimas and Nakazato
In the first part of this
perform transient and busy period analysis
MCE
the
is
for the
[1])
we presented a method
MGEi/MCE\i /\
the class of mixed generalized Erlang distributions.
method of
to
queue, where
Our
analysis used
stages combined with the separation of variables and root finding
techniques together with linear and tensor algebra.
We
found simple closed form
expressions for the Laplace transforms of the queue length and the waiting time
distribution under
distribution.
FCFS when
the system
is
We
first
mensional Lindley process and then transform
of
empty and the busy period
we extend and generalize these
In this paper
queue with arbitrary distributions.
We
initially
results to the
formulate the problem as a two
queues, where
As a
nal Laplace transforms.
result,
R
we
is
di-
to a Hilbert factorization problem.
it
problem
are able to solve explicitly the underlying factorization
GI/R/l and R/G/l
G//G/1
for the cases
the class of distributions with ratio-
find closed
form formulae
for the
Laplace
transforms of the waiting time and busy period distribution.
Formulations of queueing problems as Hilbert factorization problems can be
traced back in Lindley
the
G//G/I queue
is
in
[6],
which the steady state waiting time distribution of
derived via a spectral factorization of the underlying Hilbert
problem. For other examples of the method see Keilson
The paper
is
[2,3].
organized as follows. In the next section, which
paper we formulate the transient behavior of the
CI/C/l queue
as a
is
central in the
two dimensional
Lindley process, derive the key formula of the transient and busy period dynamics
and then transform
it
to a Hilbert factorization problem.
the factorization problem for the
solution for the
GI/R/l
previous two sections are
GI/M/l
queue.
in
R/G/l
queue, while
In Section 5
in
In Section 3,
Section 4
we observe how the
agreement with the known results
we achieve
contains
some
closing remarks.
its
results of the
for the
M/G/l
queues and consistent with the results of Bertsimas and Nakazato
final section
we solve
[1].
and
The
System Formulation
2
In this section
we formulate the
CI /C/X queue
transient behavior of the
as a
two
dimensional Lindley process, derive the key formula of the transient dynamics and
then transform
it
to a Hilbert factorization problem.
notion of a busy interval which
is
Our
analysis will focus on the
defined as the busy period plus an immediately
following idle period. In Subsection 2.1 we define the notation we will use, in Subsection 2.2
we derive the key formula
we transform the problem
2.3
dynamics and
in
Subsection
to a Hilbert factorization problem.
Notation and Assumptions
2.1
In this subsection
are using.
We
arriving time
is
for the transient
is
we define the random variables and establish the notation we
assume that the system
initially
is
and the
idle
first
customer's
the forward recurrence interarrival time. Although this assumption
restrictive for the waiting time distribution,
it is
not restrictive for the busy period
distribution, since the busy period regenerates.
We
first
Xn
T
B[
we
will use as follows:
—
1
th
and n th customer.
the arriving time of n th customer. Note that Tn
:
:
variables
the interarrival time between n
:
r„
random
the service time of n th customer.
'
Tn
define the
the arriving time of a
ti
+ ^^=2
'^k-
random customer.
the duration of a busy interval,
:
=
i.e.
the interval between the initiating epoch
of a busy period and the initiating epoch of the next busy period.
Bp
W^
W^
the duration of a busy period.
:
:
:
the waiting time in the queue of n th customer.
the waiting time of a
random customer.
We
a{t)
the interarrival time probability density function (pdf).
:
q(s)
J
use the following notation:
will
the Laplace transform of a(i).
:
=
E[Tn]
Cj =
= — a(0)
the
:
Va,r[Tn]/ E[Tn]^
a'{t)
the
:
first
'
mean
interarrival time.
the squared coefficient of variation of the interarrival time.
customer's arriving time pdf (because of our assumption
it
is
the
forward recurrence time of the interarrival time).
a*(s)
b{t)
i
=
=
E[Arn]
=
j(l
—
a(s)).
:
:
'
the
;
:
mean
service time.
the squared coefficient of variation of the service time.
the traffic intensity.
the busy interval pdf.
:
sp{t)
= —P{0)
Var[A'„]/ E[A'„]^
-
Slit)
cr(s)
a'(s)
the Laplace transform of 6(().
:
C\ =
p
i.e.
the service time pdf.
:
(3[s)
the Laplace transform of a*(<),
:
the busy period pdf.
the Laplace transform of sp{t).
In addition,
we define
/(x,y)
=
=
^Pt[W+
,.
<y\T =
x]
]^ELiafePr[rn<r,[C<y]
lim
^E^=iPr[r„<x]
(1)
1
Transient Dynamics
2.2
we derive
In this subsection
ics
CI/G/l
of the
0, 1, 2,
.
,
.
.
n
in
tlie
key formula that describes the transient dynam-
For notational convinience we enumerate customers by
queue.
We
the order of arrival.
analyze the case,
We
arrives at the busy period initiated by k th customer.
r
and observe
(see Figure 1) that
if
in
which the n th customer
let
=k+ l
Wn+k <
and Wr
>Oforr =
+ ^r) =
Bl
jfc-|-l...n
+
Ar—
then
k+n
T=k + l
k+n-\
k+n
E
Bp =
X. =
E
{Tr
+
Wr^ + k-
(2)
r=*+l
r=fc
Similarly,
if
Wr>0
Summarizing, the
for
r
critical
= k+l...n +
observation
is
k,
that
that inunediately follows the busy period Bp,
Wn+k >
0,
then
Wn+k
is
waiting time of
track of the busy interval
+
IV;^^,^
= W^+k-
if
Wn+k <
is
—Wn+k\ on
0,
(2)
the other hand,
n th customer. Therefore,
and
(3)
then the idle period,
B[ and the quantity Wn+k, then we can
busy period and the waiting time from
now
it
then
(3) respectively.
if
we keep
find both the
For this goal we
consider the joint densities:
A(x,y)
=
^^Pr{Tn<x,^n<y},
U{x,y)
=
^-^Pr{rn+k-rk<x,W;^_^.i^<y,Wr>0,r = k+l...n +
/o(x,y)
=
6(x)6(y),
oxoy
if
k},
—WkHX
\*\
T"k +
-WkM
Wk.2
^k + 2^
1.+
k+1
2
k+3
kM
\*3
Tk.
Br
Figure
where 6{x)
is
Transient dynamics
1:
the Dirac delta function. Note that that
and /n(x,y) has positive support
in y,
A(x,y)
nonnegative support
in
is
independent of u
i and
is
independent
oik.
Since r^+k+i-Tk
1
...
n
+
t
where (7(y)
+
is
1
= r^+k-Tk+T^+k+i
and
we obtain the recurrence
is
= ^V^+kHn+k+i
=
6{x)6iy)
/i(x,y)
=
A(x,y)C/(y)
/„+i(x,y)
=
[/„(i,y)*A(x,y)]t/(y),
fn{i,y)
*
A(x,y)
=
'^^^Vr
>
0, r
=
relations:
/o(x,y)
a unit step function and
lution sign that
IV'^+fc+i
we denote
jT'^o So
fn{x
(4)
"*" as the 2-dimensional convo-
-u,y-
t;)A(u, n) ducfv.
We
also
k
+
define
rnir,y)=
^^Pr{B[
axay
< x,lV„ + k < y,Wr >
Note that r„(z,y) has nonpositive support
it is
independent of
The motivation
=
0,r
k
+
I
.
.
.
n
+
k
-
l,iy„+fc
<
0}.
y and nonnegative support in x and
in
k.
above definitions
for the
quantities of interest
in
is
of the
terms of the functions r„(z,y). Clearly
-Pr{5/<x}=
=
00
*Q
1
5;(x)
we can express the pdf
that
^r„(x,y)dy,
/
(5)
n=l
and using
(2)
Using
From
(2)
(4)
and
and
(3)
(7)
=
00
-0
J
spix)
—Pr{Bp<x}=
dx
we obtain
^r„(x-y,y)cfy.
J-00 n =
\
(6)
,
in
a similar
way
as before
ri(x,y)
=
A(x,y)(l-f/(y))
rn+i(x,y)
=
[/n(x,y)*A(x,y)](l-C/(y)).
we obtain the key formula
for the
G//G/1
(7)
transient dynamics in
real time:
+
/n+i(-c, y)
2.3
r„+i(x, y)
=
/„(r, y)
A(x,
y).
(8)
Formulation as a Hilbert Problem
In this subsection
we
will
work
in
the transform domain, where the solution of (8)
equivalent to a Hilbert factorization problem.
^+{s,u)=
/
/O
yoo
/
-00 -'0
We
is
introduce the Laplace transforms:
e-"-v^/„(x,y)(fx<iy,
•»
e-'^-^v^r^x.yjcfxdy.
„,
n=l
Note that
/CO
fOO
-co
The
superscript
half of the
is
+
is
^0
employed
complex w plane.
analytic in the
left
to designate that $+(5,0;)
Similarly, the superscript
half of the
By taking transforms
in (8)
$•"(5,0;)
+
complex
uj
—
is
analytic in the right
designates that p~{s,io)
plane.
we obtain
p"(s,w)
=
1
+
a(s -a;)/3(w)<J>''"(s,w),
or equivalently
=
<I>+(s,w)(l-a(s-w)/?(a;))
(9)
is
a Hilbert factorization problem in
$•(5,^)
p~[s,u))
The
is
is
uj
l-p-{s,u;).
with fixed
analytic in Re(u;)
>
and Re(s)
>
analytic in Re(w)
<
and Re(s)
>
s,
(9)
where
0.
following additional boundary conditions complete the description of the fac-
torization problem:
a{0)<m
Once p
{s,lli) is
(^P<1)
found, we can use (6) to obtain the Laplace transform of the busy
period:
A
f°°
a{s)=
e-'^sp{x)dx
=
p-{s,s),
(10)
Jo
and similarly from
(5)
r
e-'^si{x)dx
=
p-{s,0).
Jo
The transform
of the conditional waiting time (transform variable
queue of a customer whose arriving time (transform variable
ui)
s) is given,
in
the
can be
found from
From
$"''(s,a;) as follows.
(1)
we
find that (the convolution "*"
is
with
respect to x)
=
/(r,y)
|-Pr[PF+ <y|r =
x]
oy
=
-a-(x)*^s'/>(x)*^/„(x,y),
since
(11)
n=0
r=0
we assumed that the arriving time of the
first
customer
the forward recur-
is
my
rence interarrival time and thus from the renewal theorem (or
simply taking
Laplace transforms) we have
oo
oo
-iX:Pr[r„<z] = a-(x)*^a(")(^) =
dx
A,
n=0
n=l
and moreover
oo
i2
-^ J2
Pr[rn
<
X,
w;t <y]
=
a'{x)
^^*^y n=l
By
*
J2
*'/'(^) *
E
/"(^' y)-
n=0
r=0
defining
roo
^s,u;)=
in (11)
^(^-)
e-'^-^yf{x,y)dxdy
/
/
Jo
Jo
and taking transforms
f-co
we obtain that
=
i(r^^^^(--)
=
lllf^.
(12)'
s$+(s,0)
^
Therefore, we can express both the transforms of the busy period and the waiting time distribution in terms of
<I>"'"(s,a;)
and p~{s,u>).
As a
result,
we reduced
the problem of obtaining the transforms of the busy period and the waiting time
distribution to the solution of the Hilbert problem (9).
In its full generality,
distributions,
solution.
it
is
not
i.e.,
with completely arbitrary interarrival and service time
known whether
In special cases, however,
the Hilbert problem (9) has a closed form
when one
of the distributions has a rational
Laplace transform, then we can solve the factorization problem
the next sections we solve (9) for the
R/C/l and CI/R/l
the class of distributions with rational Laplace transforms.
in closed
respectively,
form. In
where
R
is
The Solution
3
Problem
of the Hilbert
for the
R/G/l
Queue
In this case a{s)
afyj{s)
is
—
"'^
(l
where a£)(s)
,
is
a monic polynomial in
s
of degree
L and
a polynomial of degree less than L.
For fixed
with Re(s)
s
>
let z
0,
—
ir(s), (r
=
1
.
.
.
L) be the L roots of the
equation:
a(s-z)/?(r)=
The proof
is
Re(2)>0.
1,
of this follows along the lines of claim 3 of
[1].
(13)
Once the number
established through Rouche's theorem, we simply follow the
by Keilson
[3,2].
Now,
of roots
methods pioneered
(9) can be written as
l-^-(s,w)
<I>+(s,a;)
(14)
a£i(3-u/)
aC)(3-u<)-0!jv(j-u;)/3(ui)
By observing that the expression
Re(ai)
>
Re(a;)
<
and the expression
and using
Liouville's
be equal to a function of
the function
is
in
s
in
the rhs of the equation (14)
the Ihs of the the equation (14)
is
is
analytic for
analytic for
theorem we conclude that both expressions should
From
the boundary conditions of (9)
a constant function
1.
To complete
we
easily find that
Liouville's theorem,
we need the
following proposition.
Proposition
1
The ezpnssions
equation (1 ^) are bounded.
tn both sides of the
Proof
Let Re(s)
>
0.
For the
Ihs,
out) that the denominator
with Re(u;)
is
>
0, it is easily
bounded away from
0,
and thus
nLi(x.(5)-u>)
aois -
We
u^)
then check that the numerator
|a)+(s,u;)|
<
is
ocn{s
also
-
ui)(3[Lj)
bounded;
$+(0,0)
10
seen (since the zeros cancel
>
e.
for
some
f
>
0;
TTn Uo
-'0
DO
<
1
fc
E
+
^'•i
n=l
Since p
<
1
£'[4r]
<
(Vr).
that there exists a constant 6
As a
<
1
result,
<
analogous way the denominator of the
0, i.e., for
some
f
>
—
<
rhs,
\\-p-{s,^)\
Liouville
's
<
\
<
l+/>-(0,0)
<
oo.
<
0}
i5",
and thus
with Re(u;)
<
0, is
bounded away
>
e.
<i>^{s,u)
(s,^)
=
=
+
is
seen as follows;
\p-{s,u)\
=
l
+
a(0)
=
2
a
theorem we conclude that the unique solution
the Hilbert factorization problem (9)
p
>
^DO.
boundness of the numerator of the rhs
Thus by applying
0}
0;
nL,(^r(5)-a;)
In addition the
>
^^
applying the Chernoff bound, we obtain
\
from
E
r=k+l
such that -P'"{IIr=?+i ^r
|<I>"''(S,W)|
In an
+n
is:
aD{s — w) — a;v(s I
to
w)/:^(w)
- nL,(^r(s)-u;)
ao^s -oj)
Hence we get from (12)
<I>(s,u;)
=
s{aD{s - w) - ayv(s -w)/?(w)) n
;^Jj
Xr{s) -U!
(15)
iris)
and from (10)
(16)
aD(0) ^Vi
11
.
The reward
of our analysis
a simple closed form expression for the transform
is
of the busy period and waiting time distribution.
form expressions
Moreover, we can find closed
two moments of the waiting time and busy period
for the first
The
distribution by differentiating the corresponding transforms.
following formu-
were derived using the symbolic differentiation routine of the software package
lae
Mathematica on a Macintosh
computer.
II
L
w
s
-«
C7W
l-a(s)
V^ Xr{s)
aois)^
f)2
e-'^E[{W+)^\r = x]dx^ lim
.
—
—
-<t.(s,u;)
Ou)^
is)
-I-
E[flp]
Var[Sp]
^Z^r=l JTpy
=
—
^—_nx,(0)
as
y
p)aD(O)
a(-z.(0))/?(x.(Q))
P
(1
-
-
^t1 Q(-x,(0))^(i-.(0))
(i-pfA.„(o)
where w, used ^(, -,,(,))
a{3
^-^
-
lim-^log((T(5))
^
^(s -
^^
l-a(3)
d
= -lim-a(s) = -
s
and
-t-
Xr(s))
=
a(s
=
-
n
''(")
-
-
q(-x,(0))/:?(x,(o))
n^fO)
'^
.y
n '^(0)
i(i-rt„„(o,j
„,._„,.,g— l^lg'^^',,!.!,,,.,,
Xris))P{xr{s)) x
- Xr{s))a(, - Xr{3))3(rr{3))^-26{3 -
Xr(3))^ i3{lA 3)f + 6(,
-
Xrjs))^ l3(jCrl 3))0{lr(s))
(a{3 - Tr{3))0{Tr(3))-a{3 - X. (5))/3(r. (5)
))
The formula
for the first
two moments of the busy period was simplified using the
observation that there exists a unique root such that xi{0)
12
=
(see Keilson
[4]).
As an additional check of the algebra we can
L
=
1
verify that for the
the formula for E[Bp] becomes E[Bp]
= —^.
M/G/l
The Solution
of the Hilbert
i.e.
Finally, note that the roots
Jr(0) are precisely the roots that appear in the steady solution of the
4
queue,
Problem
R/G/l
queue.
GI/R/l
for the
Queue
In this case f3{s)
/3jv(s) is
As
be the
=
x4f)
where I3d{s)
,
a monic polynomial in s of degree
is
M
previous section, for fixed
s
and
M.
a polynomial of degree less than
in the
M
with Re(s)
>
0, let z
=
Xr{s) (r
=: 1
.
.
.
A/)
roots of the equation:
a(s-z)p{z) =
The unique
l,
Re{z)<0.
solution to the Hilbert problem can be found in a similar
way
as in the
previous section to be:
<I>+(s,u')
=
rAf
n;^4,(c.-x.(s))
Note that the connection with the results of the previous section
R/R/l
is
in
the case of
established by noticing that
L+.\f
(-1)^
n
'^'^
~
-^rls))
=
o:d{s -lj)I3d{u)) -aA'(s-a;)/?Af(a)).
r=l
Hence we get from (10) and (12) that
*(.,.)=
As an accuracy check we can
results for the
Mil
n_^
easily check that (17)
MGEi/MCE\f/\
queue obtained
13
(17)
and (18) are identical with the
in part
I
of this study (Bertsimas
and Nakazato
the
moments
As
[1]).
the previous section we can find closed form formulae for
in
of the distributions involved as follows:
/•oo
Jo
1
/
s
\fr{ Xt{
1
E[5,]
=
(zi)!:M2)fi
Var[5p]
=
(-1)'''/?d(0)
\yZ12_±l^Y.
^
^
+
fct^
(-l)'^^(l
a(-rfc(0))/?(x,(0))
x,(0){a(-x,(0))/?(x,(0))
+ Ci)%(0)
/?d(0)\
^
)
-
T^
1
L\
^r{0)
a(-x,(0))/?(xfc(0))}
//?D(0)V-i^
V
M V*
1
M
The M/G/1 and GI/M/l Queues
5
In this section
M/G/1
—
xi(s)
verify
CI/M/l
xi(s)).
=
we
and generalize
well
known (Takacs
[7])
known
GI/M/l and
results for the
queues.
For the
a{s
1
By
^{w{s)
For the
—
=
is
letting A/
1),
M/G/1
satisfies cr(s)
it
[3(s
we
we observe that from
A
in (18)
1
same
find the
queue
+
=
—
it
is
well
=
+
=
w{s)
=
1,
i.e.
expression.
known
1
= ^^^~^, where
and observing that w{s)f3{xi[s))
(see Kleinrock [5]) that the
In order to see
A<7(s)).
(16) cr[s)
that (t{s)
Iz^ilii^
how we can
busy period
derive this from (16)
from where xi(s)
=
+
s
X
—
Xcr{s).
Since xi(s) satisfies from (13)
a(s
-
xi{s))f3{x,{s))
=
-—
A
we can now
—f3{s + A - Xa{s)) =
—
/?(s
+
A
—
A<t(s)).
of the waiting time can be expressed
follows:
S-W +
1
$(s,w) =
s
1,
xi(s)
easily derive the desired relation a{s)
The time-dependent behavior
cr(^s) eis
+s-
A(1-«t(s))
+ A(I-<7(5))s-w + A(1-a?(w))
14
in
terms of
a solution to the well
known Takacs
This
is
rock
[5]
6
Concluding Remarks
In this
or Takacs
integrodifferential equation (see Klein-
[8]).
paper we attempted to demonstrate the power of direct probabilistic
guments
for the
waiting time distribution
in
the transient
We
ar-
domain and the busy
period distribution for the
G//G/1
the transforms and the
two moments of these distributions. Algorithmically our
approach
offers a
first
method
queue.
found closed form expressions
for finding these distributions in the
time domain through
the numerical inversion of the Laplace transforms. In Bertsimas and Nakazato
we reported numerical
sient
results for finding numerically the
queue length and the waiting time distributions
in
a
for
[1]
busy period, the tran-
MGE/MGE/l
queue,
by inverting numerically the corresponding Laplace transforms.
References
[1]
Bertsimas, D. and Nakazato, D. (1989). "Transient and Busy period analysis of
the
[2]
G//G/1
queue; Part
Keilson, J. (1961).
I,
the
method
of stages", submitted for publication.
"The Homogeneous Random Walk on the Half-Line and
the Hilbert Problem", Bulletin de L'tnstttut international de statistique, 33ed
session,
[3]
113, 1-13.
Keilson, J. (1962). "General Bulk
Royal
[4]
Paper#
Queue
as a Hilbert
Problem", Journal of
Statistical Society (series B), Vol. 24. No. 2, 344-358.
Keilson, J. (1969).
"On
the Matrix Renewal Function for
Markov Renewal
Process", Annals of Mathematical Statistics, Vol. 40, No. 6, 1901-1907.
[5]
Kleinrock, L. (1975). Queueing systems; Vol.
15
1:
Theory, Wiley,
New
York.
[6]
Lindley, D. (1952).
Phil.
[7]
single server", Proc.
Cambridge
Soc, 48, 277-289.
Takacs, L. (1962). Introduction
New
[8]
"The theory of queues with a
to the
theory of queues, Oxford University Press,
York.
Takacs,
L.
(1962).
"A Single-Server Queue with Poisson Input", Operations
Research, Vol. 10, 388-397.
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