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HD28
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no
ALFRED
A
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Unified
Method to Analyze Overtake Free
Queueing Systems
Dimitris Bertsimas
and
Georgia Mourtzinou
WP#
-
3486-92
MSA
October, 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A
Unified Method to Analyze Overtake Free
Queueing Systems
Dimitris Bertsimas
and
Georgia Mourtzinou
WP#
-
3486-92
MSA
October, 1992
A
unified
method
to analyze overtake free queueing systems
Dimitris Bertsimas
*
Georgia Mourtzinou
**
October 1992
Abstract
In this paper
we demonstrate
that the distributional laws that relate the
of customers in the system (queue),
tem (queue), S (W) under the
L
(<?)
first-in-first-out
solution for the distributions of L, Q, S,
tributional laws for both
and the time a customer spends
L and
Q
arrivals
mula
for
under heavy
to the
GI/G/1
discipline lead to
in the sys-
a complete
W for queueing systems which satisfy
dis-
{overtake free systems). Moreover, in such systems
the derivation of the distributions of L, Q, S,
results include a generalization of
(FIFO)
number
PASTA
traffic conditions,
W can be done in
a unified way. Our
to queueing systems with arbitrary renewal
a generalization of the Pollaczek-Khinchin
queue, an extension of the
for-
Fuhrmann and Cooper decomposition
queues with generalized vacations under mixed generalized Erlang renewal ar-
rivals,
new approximate
new exact
results for the distributions of L,
results for the distributions of L,
Q,
S,
S
in
a
GI/G /oo
queue, and
W in priority queues with
mixed
generalized Erlang renewal arrivals.
'Dimitris Bertsimas, Sloan School of
Ma
Management and Operations Research Center, MIT, Cambridge.
02139.
'Georgia Mourtzinou, Operations Research Center, MIT, Cambridge,
'The research of D. Bertsimas was
Ma
partially supported by a Presidential
DDM-9158118 with matching funds from Draper Laboratory. The
02139.
Young
Investigator
research of both authors
supported by the National Science Foundation under grant DDM-9014751.
was
Award
partially
Introduction
1
What
are the laws of electrodynamics? In order to address this question
define the
field
fundamental quantities of electrodynamics, the
E and
first
the magnetic
S. The fundamental laws of electrodynamics are the Maxwell equations. The goal of
electrodynamics
form a complete
is
then to find
set of
E
and
B
is
able to
numerically in a variety of applications.
problem
E
summarized
is
and
B
in
The Maxwell equations
in various applications.
laws in the sense that just starting from them and using the calculus
of partial differential equations one
for
electric field
we should
in the
What
E
compute
is
B
and
important here
either analytically or
is
Maxwell equations, which then lead
that the physics of a
to a complete solution
a unified way.
What
Let us then ask the key question which motivated the present paper.
are the
laws of queueing theory? The fundamental quantities in queueing theory are the stationary
queue and system length (Q, L) and the waiting and system time (W, S) under the FirstIn- First-Out
(FIFO)
discipline.
Of course
there are several other
random
interest (often particular to the application studied), but these are the
The
goal of queueing theory
applications. In
its
is
What
is
most widely used.
then to find the distributions of Q, L, W, S in various
almost a hundred year history queueing theory has addressed a great
variety of problems using a variety of techniques, which solve
others.
variables of
interesting
is
the lack of a unified
Queueing theory research does not
start
from a
way
some problems but
set of well established
uses the particular characteristics of the application to achieve
laws and then
law
law and
its
[13] (see
One
first
It
rather
solution.
would
like
in electrodynamics, lead to
a
candidate for a queueing law
is
Maxwell equations
complete solution of the queueing application.
Little's
its
to our original question regarding the laws of queueing theory, one
to have a set of laws which, similar to
on
to solve a particular application.
proceed to the solution using some well established mathematical techniques.
Coming
fail
the recent review of Whitt [16] which traces the different forms of the
extensions). Let us examine whether Little's law leads to complete solution for
the steady state E[Q], E[L], E[W], E[S] in a
GI/G/s
queue. Let A,
fi,
p
=
^<1
be
the
mean
arrival, service rate
and
Then, from
traffic intensity.
Little's
law in the system
and the queue
E[L]
But, E[S]
= E[W] +
= XE[S], E[Q] = XE[W).
i, while the relation of
Q, L
E[z L = z'E[zQ + Y,
P ( L = "H 2 "
]
)
is
~
z ")'
n=0
from where
»-i
=
E[L)
s
+ E[Q] - Y,( 3 ~ n P i L =
)
">•
n=0
Combining the previous equations we obtain that
£ *-^-P{L = n} =
1
- p,
n=0
which
is
exactly what Little's law would give
the customers in service. For example, in a
P{L =
0}
=
1
—
/>,
but
it
if it
were applied to a service box including
GI /G/l
queue one would be able to find that
would not be possible to
find E[L],
As a
result, despite its
importance, Little's law does not lead to a complete solution for expected performance
measures.
Our goal
in this
Haji and Newell
[7]
paper
is
to demonstrate that the distributional laws
obtained by
are the fundamental queueing laws for queueing systems which satisfy
distributional laws for both the
will call
first
them overtake
number
free systems).
in the
We
system and the number
in the
queue (we
demonstrate that the distributional laws lead
W in overtake
such systems the derivation of the distributions of L, Q,
W can
to a complete solution for the stationary distributions of L, Q, 5,
systems. Moreover, in
free
5,
be done in a unified way. In this way not only we obtain new simple derivations of known
results providing
We
new
insights to old results, but
propose two methods of analysis
An
we obtain
asymptotic (as p
—
>
several
1)
mixed generalized Erlang
arrivals.
results as well.
method which
overtake free systems with arbitrary renewal arrivals and an exact
to overtake free systems with
new
applies to
method which
applies
For the case of Poisson arrivals Keilson and Servi
[10], [11]
found that the distributional
laws have a very convenient form that can lead to complete solutions for some overtake
For the case of mixed generalized Erlang renewal arrivals Bertsimas and
free systems.
Nakazato
[1]
gave another proof of the distributional laws that lead to a very convenient
form of the law. They also proposed a framework to find E[L], E[Q], E[S], E[W]
heavy
traffic for
overtake free queueing systems based on the distributional laws. In this
W
paper we develop a methodology to find the distributions of L, Q, S,
systems with arbitrary renewal arrivals, thus generalizing
is
in
to use asymptotic analysis (which
is
all earlier
for overtake free
work. Our approach
exact in heavy traffic) for the case of arbitrary
renewal processes and exact analysis for the case of mixed generalized Erlang renewal
arrivals.
The paper
Section 3
we
is
structured as follows: In Section 2
method
present an asymptotic
we review
the distributional laws. In
of analysis for overtake free queueing systems
based on the asymptotic properties of the distributional laws and a generalization of the
well
known
result of Poisson arrivals see
time averages (PASTA) to queueing systems with
arbitrary renewal arrivals under heavy traffic conditions. Furthermore,
efficiency of the
method by
deriving the distributions of L, Q, 5,
we
illustrate the
W in GI/G/l, GI/D/s
queues and obtaining new approximate results for the distributions of L, S in a
queue.
Our derivation
unifies the
heavy
the Pollaczek-Khinchin formula to the
act
method
GI/G /l
demonstrates that there
MGEm/G/1
is
it
and leads to a generalization of
queue.
of analysis for overtake free systems with
renewal arrivals and we implement
a
traffic results
in the case of
In Section 4
we present an
MGEm/G/1
queue.
Hilbert factorization. In Section
for overtake free systems,
vacations considered in
5,
in
for the waiting
a direct way without the need for
as another application of the exact
we extend
This section
number of customers
system while our approach reproduces the known results
in
ex-
mixed generalized Erlang (MGE)
a direct closed form expression for the
time involving roots of a certain nonlinear equation
GI/G /oo
method of analysis
the decomposition results for queues with generalized
Fuhrmann and Cooper
[5]
for the
M/G/l
queue to
MGE arrivals.
In Section 6
we propose an algorithm
to find the distributions of L, Q, S,
W in priority
queues with mixed generalized Erlang renewal arrivals, thus we generalize earlier results
Poisson arrivals.
for
compared with the
The
derivations in this section are considerably
results in previous sections.
more complicated
we
Finally, in Section 7
include
some
concluding remarks and indicate directions for future research.
The
2
distributional law
In this section
we first review
the distributional law for arbitrary arrivals
the case in which the arrival process
2.1
A
is
and then consider
a mixed generalized Erlang renewal process.
review of the general distributional law
Consider a general queueing system, whose arrival process
Na (t)
be the number of customers up to time
of the
first
interarrival time has the
Let N*(t) be the
the time of the
number
first
,
1
below.
1
The
(where the time
t
for the equilibrium process
(where
distributed as the forward recurrence time of the
L~ L + (Q~ Q + be
,
Let
distribution as the stationary interarrival time).
)
the
number
in the
an arrival or just after a departure, respectively,
assumptions of Theorem
Theorem
is
a stationary process.
for the ordinary process
of customers up to time
interarrival time
arrival process). Let also
just before
same
t
is
for
system (or
in the queue)
a system that
satisfies the
distributional law can be stated as follows:
(Haji and Newell [7]) Let a given class
C
of customers have the following
properties:
1.
All arriving customers enter the system (or the queue) one at a time, remain in the
system (or the queue) until served (there
is
no blocking, balking or reneging) and
leave also one at a time.
2.
The customers leave
3.
New
arriving class
for previous class
the system (or the queue) in the order of arrival
C
C
customers do not
customers.
affect the time in the
(FIFO).
system (or the queue)
Then, given that they exist in steady
S
(W )
of the class
C
the system (or queue)
state, the stationary
time spent in the system (queue)
customers and the stationary number of the class
L (Q)
C
customers in
are related in distribution by:
l = K(S),
(i)
Q £ N:(W).
(2)
In addition,
L~ = L + =
Na (S),
±Q+
Na {W).
Q-
We
i
define as overtake free systems those systems that satisfy both (1)
and
(2).
Note
that for the general distributional law the arriving process need not be a renewal process.
If
we
function of
[1]
some
consider renewal arrivals, however,
L and
the Laplace transform of
and are reviewed
in
Theorem
S have been proved
number
number
of renewals up to time
of renewals up to time
Theorem
t
t
in
Bertsimas and Nakazato
For the rest of the paper
2 below.
transform of the interarrival distribution, with arrival rate A
the
between the generating
interesting relations
be the Laplace
-l/d(0). Let
Na (t)
renewal process.
2 (Bertsimas and Nakazato [I]) Arrivals of class
C
form
a renewal process
whose interarrival time has a transform a(s). Under the assumptions of Theorem
distribution function Fs(t)
(?£,+ (z) satisfy the
= P{S <
t}
of
S and
be
renewal process and N*(t) be the
for the ordinary
for the equilibrium
=
let ct(s)
the generating functions
1,
the
Gl(z),Gi-(z),
following relations:
G L {z)=
r K(z,t)dF
s (t),
(3)
./o
G L -{z) = G L +(z)=
and
the distribution function
Gq(z),Gq-(z),Gq+(z)
Fw(t) —
f°°
P{W
<
K
i)
(z,t)dFs (t),
of
W
and
(4)
the generating functions
satisfy the relations:
G Q (z)=
f°°
Jo
K(z,t)dFw (t),
(5)
[°°
GQ -(z) = GQ +{z) =
K
(z,t)dFw (t),
(6)
JO
n=0
oo
K
(z,t)='£z n P{Na (t) =
n}.
n=0
T/ie
Laplace transform of the renewal generating functions K(z,t) and
*•(* «) =
K
1
'
/" e- *(*, t) * - i5 - A (1«*(1
7.f- za(s))
"n"
(z,t) are given
-
<
7>
./o
A';(z,,)
=
fe-'
A' o (z,0d<
«(i
For the case of Poisson arrivals K(z,t)
tional laws
become a
relation
A
A
=
A' (z,£)
e -At ( 1-z )
=
.
2 «('»)j
and thus the
between transforms (Keilson and Servi
GL {z) =
2.2
.j"^
-
=
./o
<t>
[10]):
s (\(l-z)).
(8)
vector distributional law
vector generalization of (8) has been proposed in Bertsimas and Nakazato
assumption that the arrival process
is
a mixed generalized Erlang
can approximate any renewal arrival process arbitrarily
of the
MGE distribution
arrival timing channel
Aj, Aj,
...,
Xm
and with
is
presented in Figure
(ATC)
consisting of
afc(<)
in stage k
M
1, i.e.,
.
.
.,
Mth
closely.
notational convenience
Therefore, a(t)
we
will
The
we conceive
(pm =
under the
process, which
stage representation
the arrival process as an
of entering the system after
1)
stage.
be the pdf of the remaining interarrival time
= 1,...,M.
(MGE)
[1]
consecutive exponential stages with rates
probabilities Pi,p?, ...,Pm
the completion of the 1st, 2nd,
Let
distribu-
=
ai(t)
is
Let otk(s) be the Laplace transform of ajt(0-
the customer in the
ATC
the pdf of the interarrival time.
drop the subscript for k
interarrival time.
if
=
1.
Also j denotes the
is
For
mean
-»
X,
p2
= -
a(s)
tiace((Is
+ Ao)
l
Ai),
thus the interarrival pdf becomes
a(t)
-trace(e _Ao *Ai).
=
Note that a mixed generalized Erlang renewal process
ces Aq, Ai. In queueing systems with
we
is
fully characterized
mixed generalized Erlang renewal
by the matri-
arrival processes
introduce:
L + ,Q + = The number
of customers in the system (or queue) immediately after a depar-
ture epoch.
L ~ Qt = The number
of customers in the system (or queue) just before a transition epoch
of the arrival process.
A
t
,
transition includes both arrivals in the system
next exponential stage of the
before an arrival epoch.
ATC. We emphasize
The motivation
that
L^
for considering
the epochs of transition are Poisson distributed and thus
is
Lf
and
not the number of customers
that using uniformization
is
we can apply PASTA.
R + = The ATC
stage immediately after a departure epoch.
R^ = The ATC
stage just before a transition epoch of the arrival process.
P+ = [P{L+ =
^(z) =
We
E~
n n i?+
shifts to the
£~ z»P+, P~ = [P{L r = n n*T = 0]^.
= nnR = i}}^, PL {z) = £~ * n Pn-
= i}%£ P+(z) =
,
Pn =
z»4r,and
[P{L
denote with Pq(z), Pq(z), and Pq{z) the corresponding transforms for the number
of customers in the queue.
The
vector distributional law
is
described in the following
theorem.
Theorem
3 (Bertsimas and Nakazato
[l])
Under
the assumptions of
mixed generalized Erlang interarrival times characterized by
Pl(z)
=
P£(z),
Pq(z)
=
Pq(z),
PL {z) =
A(l
PQ (z) =
A(l - z)P+(z)(A
Theorem
1
and for
the matrices Ao, Ax,
- z)P+(z)(A + zA 1 )~\
+ zA x )~\
(9)
(10)
P+(z)
= ?1 '* s {A + zA 1 ),
P+(z)
= ?1 '$w{A + zA 1 ),
PL {z) =
A(l
- z)ci '* s (i4o + ^i)(Ao + z^i)" 1
PQ
A(l
-
(
=
2)
D
where for any matrix
z)ei
'*w(^o +
we symbolically
2i4,)(i4
+ z^i)
(11)
,
-1
(12)
,
define:
*s(D)= f"
e- Dt
dFs (t).
Jo
The kernel K(z,t)
in (3)
K{z,
is
given by
t)
=
A(l
- z)it
'e-(
A° +zAl »{A +
zA^l,
which leads to
G L {z) =
Once again
A(l
-
z)ei
'*s(A + ^i)(A +
in the case of Poisson arrival the vector
zA^t
forms reduce to scalars and we obtain
(8).
3
An asymptotic method of analysis
for overtake free queue-
ing systems
In this section
satisfy the
L,Q,S,
W
consider overtake free systems with general arrival processes that
assumptions of Theorem
—
oo,
distributions of
In Section 3.1
we
we
and have the property that whenever p
and we propose a unified asymptotic method
L,Q,S, W,
we
1
as well as
L + and Q +
the asymptotic
method
implement
method
to obtain
This section
1,
for the derivation of the
is
structured as follows:
derive the asymptotic form of the distributional law while in Section 3.2
give an asymptotic generalization of the
this
.
—*
PASTA
property. In Section 3.3
we
present
of analysis for overtake free system. Finally, in Section 3.4,
in specific examples,
new asymptotic
i.e,
results.
10
we
GI/G/l, GI/D/s and GI /G/oo queues,
The asymptotic
3.1
The important advantage
orem
distributional law
of the Poisson arrival process
butional law then becomes a relation
mixed generalized Erlang
we
among
K(z,t)
arrivals
renewal arrivals, however, K(z,
tributional laws
=
form K(z,t)
2 has the very tractable
t) is
not
is
e -Mi-*)<.
that the kernel K(z,
transforms,
known
Gl{z) = 0s(A(l -
i.e.,
Theorem
3.
The-
distri-
z)).
For
For arbitrary
in closed form. In order to exploit the dis-
try to understand in this section the asymptotic behavior of K(z,t).
For systems in heavy
traffic (p
—
»
1)
both L, Q,
5,
W tend to
systems with deterministic arrivals and deterministic service,
are interested in the behavior of K(z,t),
Theorem
in
As mentioned above, the
given explicitly in
is
t)
4 Asymptotically, as
t
—
K
(z,t) as
—
oo and z
>
(we need to exclude
result,
we
Theorem 2 behave
as
D/D/l). As a
i.e
-too and
<
1
infinity
—
z
the kernels in
1.
>
follows:
A'(z,
r
)~e-"<*>,
and
K
(z,t)
~
[1
- \(1 -
-
z){c\
1)
+ 0((1 -
z)
2
)]e-"W,
where
/(;)
and
= A(l-z)-iA(l-z) 2 (c*-l),
c\ is the square coefficient of variation of the interarrival process.
Proof
From
(7)
by writing K*(z,
up to second order terms
s
—
>
in the
s)
= j&'L and expanding N(z, s), D(s, z)
in s (note that
—
t
oo in the time
*
as a Taylor series
domain
is
equivalent to
transform domain) we have
(Z,S x>-
2a(0)z - A(l - r)a(O)
{,
where the Taylor
series
Sl
-
+
[za(0)
*>)(*
Z)
+ \\{\ - z)\c\ 11
x V-*) E
W
]s
+
0(s>)
- s 2 )z&(0)
expansion of the smaller root
= _ A (l -
-
si in
1)
terms of
+ 0((1 -
z)
(1
3
),
—
z)
is
o(0)
Using a partial fraction expansion we invert in the time domain. Since we are interested
in the behavior as
a
result, after
t
some
—
As
oo only the smaller root s\ will be asymptotically important.
tedious, but straightforward manipulations
we obtain
that
K(z,t)~{l + 0{l-z) 2 )e'*\
i.e.,
K(z,t)
~
(1
+ 0(1 -
2 )2) c -'(Mi-«)-*Mi-«)V.-D).
and inverting
In a similar way, by expanding K*(z,s) as a Taylor series in terms of s
the time
in
domain keeping only the most important term asymptotically, we obtain that
K
(z,t)
~ [1-
Combining Theorems
i(l - z)(c a2 -
3
1)
+ 0((1 -
*)*)]e-*<*<i-«)-*Mi— Ptt-iM.n
and 4 the asymptotic form of the distributional
Little's
law be-
comes
Theorem
suming
5 In a queueing system that satisfies the assumptions of
that as
p
—
>
1,
L, Q, 5,
W—
*
Theorem
and
as-
—
1:
oo the following asymptotic relations hold as p
GL (z)~Mf(*)),
(13)
<?g(*)~M/(*)).
~
[1
- \{l -
*)(<£
- i)]*s(/(*)).
G Q+ (^) ~
[1
- i(l -
z){c\
-
<?*+(*)
1
l)}4>
w (f(z)),
(
14 >
(
15 )
(16)
with
l
f(z)
= \{l-z)- -\{l-z)\cl-l).
Proof
Substituting in (3), (5) and (4), (6) the asymptotic form of K(z,t) and
previous theorem
we obtain
(13), (14)
and
(15), (16), respectively.
12
K
(z,t)
from the
Although only valid asymptotically,
are relations
among
(13), (14)
we
transforms, which
and
(15), (16) are very useful since they
will further exploit in the section.
=
previous expressions are exact for the Poisson case {c\
further insight
arrivals,
i.e.,
Also, the
In order to develop
1).
some
on the asymptotic expressions of Theorem 5 we consider the case of Ei
=
a(s)
2
-
(yf^;)
K(z,t) =
(
1
Then,
+ >A) 2 -2A(i-^)t _
c
U- V^")
2
c
-2A(i+^)t^
and
K
As
z
(13)
—
>
1
only the
(z t)
first
=
(!
An
Theorem
>/*)
-2A(l-VS)f
series
-
(1
expansions of the
asymptotic generalization of
5 leads to
_
y/z)
2X(l+^)t
of the two exponentials contributes to K(z,
and (15) are the Taylor
3.2
+
satisfies the
Ka (z,
t).
Expressions
exponential in terms of
1
—
z.
PASTA
an interesting generalization of
Consider a queueing system that
first
t),
PASTA
systems in heavy
in
assumptions of Theorem
1.
traffic.
Since in such
systems the number of customers in the system always changes by one (for example a
GI/G/s
that
queue),
L~ = L
L + = L~
in distribution. In the case of Poisson arrivals,
discipline, while the distribution of
however, where Theorem 5
GL -(z)
In particular the
first
is
=
applicable
in
moments
heavy
we have
G L+ (z) ~ GL {z)[l
L does
not. In
heavy
traffic
(p
—
1),
that
- i(l -
z)(cl
-
1)].
(17)
are related by
jb[i-]
which means that
implies
For general arrival processes the distribution of L~ depends
in distribution.
on the queueing
PASTA
traffic,
~
*[£]
+ dzi,
where both E[L~], E[L] are very
large, their difference
asymptotically depends only on the coefficient of variation of the arrival process. Apparently,
a relation similar to (17) holds for the number of customers
13
in the
queue by a similar
.
reasoning.
example,
We
in
a
remark that we need that L~
D/D/l
queue, even
if
—
^
,
L
(or
Q~ Q)
,
go to infinity as
(17) does not hold, since
1,
L~
remain bounded and therefore the assumptions of Theorem 4 are not
An
3.3
Theorem
method
£ (and
»
1.
For
(?~, (?)
valid.
asymptotic method
5 as well as (17) provide us with the necessary analytical tools to
form a unified
be the number of customers in the system and queue respectively, and S and
be the time spent in the system and queue. Let the random variable
time and
let
also
L + (Q + ) be
just after a departure.
follows
—
that solves, asymptotically, overtake free systems.
Q
Let L,
,
/>
We
the
number
W
X denote the service
of customers in the system (or in the queue)
can describe the proposed method in an algorithmic way as
:
Asymptotic method of analysis
1.
Relate the transforms of
L and
S, using the asymptotic
Q
W
form of
the distributional
law (13).
2.
Relate the transforms of
and
,
using the asymptotic form of the distributional
law (14).
S and
3.
Relate the transforms of
4-
Relate the transforms of L and
W
Q
using the fact that
S =
W©X
using the characteristics of the system (see Section
3.4 for further details).
5.
Solve the
4x4
of L, Q, S
6.
system of equations from the previous
from
transforms
andW.
Using the asymptotic generalization of
Q+
4 steps to find the
the transforms of
L and Q
PASTA,
.
14
(17), find the transforms of
L + and
method
Applications of the asymptotic
3.4
The GI/G/1 and GI/D/s queues
As a
first
application
we consider a GI/G/1 queue with a FIFO
service discipline.
Let
1/A, £[A"], c\,c\ be the means and the square coefficients of variation for the interarrival
and
Let <fix{s) be the Laplace transform of the service time
service time distributions.
distribution.
Theorem
6 In a
GI/G/1 queue under FIFO
p —>
as
the Layla.ce transform of the
1
waiting time distribution and the z-transform of the number of customers in the queue are
given by:
^
(5)
MsFTW)
-
(18)
'
and
where f(z)
=
A(l - z)
- \\{1 - x)\t* -
~
2 )(
~P)
*x{f{z))-z>
r (,\ ~GQ{Z)
(
1
1
tia\
(19)
1).
Proof
The
distributional law holds for both
L and Q. Performing
asymptotic method we obtain from (13) and (14), as p —»
GL (z) =
M/(*)),
=
<f>w(f(z)).
Gq(z)
Performing the third step, since S
—
I,
first
steps of the
:
W © X and W, X are independent we obtain
<i>s{f(z))
Finally, performing the fourth step,
1
the two
=
4>
w {f{z))4> x {f{z)).
we obtain the
relation of the generating functions of
Qis
G L (z) = (l-z)(l-p) + zG Q (z).
The previous equations form a system
s
=
f(z) and thus z
= f~
l
{s)
of four equations with four unknowns.
By
setting
and solving the system of equations we obtain (18) and
15
(19), as well as the transforms of the
system time and the number of customers
in the
system.
Remarks:
G^(z)
1.
Using (17) we can also find
2.
In the case of Poisson arrivals,
generalize the well
3.
By expanding
it is
Gq+(z) as p —
or
1.
important to notice that (18), (19) are exact and
known Pollaczek-Khinchin formulae
powers of
<j>w(&) in
=
<t>w(a)
1
-
M/G/l
for the
queue.
we obtain
a
+
•»
2\{\-
)
°( J
)>
with
lr
A =
Then, as p
—
*
(l-^) 2
VX
2
{l-p) 2
P (l+^)
A 2 (l-/>) 2
4
2
2
2
P (1-Cp(l
+
Cl),
A2(l-p)2
1
and
JS[^ 2 =
]
As a
result, the coefficient of variation of
2,4.
W tends to one as p —
>
1,
which
is
with the diffusion approximation for the waiting time in a GI/G/1 queue,
exponentially distributed in heavy
4.
The previous
results for the
consistent
i.e.,
<?/<') is the 5 fold
GI/G/1 system can
also be used in a
GI/D/s queue.
We now
oo
well
known, each customer
sees
same
a GI^''/D/l queue, where
convolution of the interarrival distribution. As a result, the waiting
time in queue in the
The GI/G/
it is
is
traffic.
Since the service times are deterministic, every s customers are served by the
server. Therefore, as
W
GI/D/s queue
is
the
same
as in the
GI^/ D/l
queue.
queue
apply the asymptotic method to find approximate closed form expressions for the
variance of the
number
in a
GI/G/
oo system.
16
=
Theorem
7 In a
GI/G/oc queue
G L (z) »
e
in heavy traffic conditions
(E[X]
—
oo^
-Mi-*)£[*]+*Mi-*) 3 (d-i)/ °°*/i(*)<i*
5
E[L]
= A£[X],
and
Var[L]
n
AJ5[X]
+
(c*
-
/°°
a;/|(z)<fx.
1)
Jo
Proof
In a
GI/G/oc system
rem
1 is
violated
(i.e.,
the distributional law doesn't hold because Assumption 2 in Theothe system allows overtaking). In the special case of the
GI/D/oo
queue, however, the distributional law does hold because, due to the deterministic service
distribution, the customers exit the system in the order they arrived.
Thus we can write
l i n:(S).
Moreover, because of the presence of
5 = X,
thus
fx (t) =
6(t
i.e.,
infinite
the time in the system
- E[X}) and thus from
is
of servers there
is
exactly the service time. But, the pdf of
X
K(z,E[X}).
(20)
—
l,-..,fc-
The customers with
P{X =
is
Xj}
"M = «w f>'-wu - «)'-• ._,:%„
p,
the arrival rate and coefficient of variation for class Cj customers
Xj
2
ca
.
=
service times Xj can be treated as a separate class
Cj of customers with arrival process being a renewal process with Laplace transform
i.e.,
is
decompose the GI/G/oo system into a number of GI/D/oo systems. Suppose
that instead of having a general service distribution the service time
Pji 3
no waiting and
(2)
G L (z) =
We will now
number
= \ Pj
= l + Pj (cl-l).
17
is
.
ctj(s)
If
Lj, j
=
1,.
.
.,
k
is
the
number
of class Cj customers in the system, then
3
The random
pendent
=1
variables Lj are not independent since the arrival processes are not inde-
(in the special case of Poisson arrivals they are indeed independent).
Using the
approximation that they are indeed independent we obtain
G L (z)*f[G L] (z).
Each
class
Cj
sees
GI /D/oo
an
which the distributional law holds. Then applying
for
(20)
GLj {z) = K{z,zj
).
For large Xj the asymptotic form of the distributional law of Theorem 4
is
valid
and thus
Therefore,
G L {z) ~
€- X{1 -'
^"^ p''' + ^ X(1 - z)2{cl -'l) ^^ p
)
Since any general service distribution
distributions
we obtain
is
'
x
'.
the limit of a sequence of mixtures of deterministic
that:
G L (z) ~ e-^
1
-2
^^^?^
1
- 2 '2
^-
00
1
)^
^^)^,
which leads to
= \E[X],
E[L]
and
Var[L]ss XE[X] +
{c\
-
1) f°°
xf\{x)dx.u
Jo
Remark:
For the case of Poisson arrivals (c\
are exact leading to the well
known
=
1) the expressions of the previous
result
g l {z) = e -Mi-*y*m,
i.e.,
L has a Poisson
distribution with rate A2?[.Y].
18
theorem
An
4
we
In this section
Erlang
method
exact
(MGE)
of analysis for overtake free systems
arrival processes that satisfy the
assumptions of Theorem
a unified exact method to obtain the distributions L, Q, 5, W, L + and
,
notation of Section
L + and Q + from
,
2.2. In
an algorithmic form and
of
MGEm/G/1
1
and
Then,
principles.
first
first
in subsection 4.1,
we
finally in subsection 4.2
MGE M /D/s
Under
we
order to accomplish our goal
in
Proposition
mixed generalized
focus our attention on overtake free systems with
1
and we describe
Q + We will use
.
the
derive a relation between
we present
illustrate the
the exact
method
method
in the case
queues under FIFO.
the assumptions of
Theorem
and for mixed generalized Erlang
1
interarrival times characterized by the matrices Ao, A\,
P£(z) = P£(z)*x(Ao + zA l ).
(21)
Proof
Conditioning on the length of the queue and the
the queue
and enters
service
n
we obtain
n >
for
ATC
stage just after a customer leaves
1
M
-oc
P{L + =n,R+=i} = Y,Yl P iQ + =k,R + = ™}
fc=Om = l
a m (t)
*a^- k ~^(t)
*a\(t)
dFx (t)
J°
(22)
And
for
n
=
:
P{L+ =0,R+ = i}=J2
For every pair of matrices
C
HQ +
=
and
D
of full rank
(C + D)- 1 =
0,iZ
+
= m}
of rank
1
+
aUt) dFx (t)
1,
C-^DC'
C -l
/
1
trace(C- 1 D)
Therefore,
ai(«)o*i'(a)
(Is
+ Ao + zAi)- =
1
(Is
+ A
)- 1
+
1
- 201(5)
a M (s)a
19
i
'(s)
which expressed
in real
time gives
a\(t)
...
'
e
-(Ao+zA!)t
_
..
:
/
<(i)
O!(0 \
+£* n
;
n=l
•••
a%{t)
^
a\l(t)
y
(23)
Taking generating functions
Remark
in (22)
and using (23) we prove
:
Equation (21) also follows from Theorem
proof
is
that often in
more general systems
generalize Proposition
An
4.1
Theorem
(21).
exact
3
3.
The reason we have included a separate
(like priority
systems in Section 6) we need to
1.
method
and Proposition
overtake free systems with
1
enable us to present an unified exact
MGE
arrivals
method
under the assumptions of Theorem
for solving
1.
We
will
use the notation of Section 2.2.
Exact method of analysis
/.
Relate the transforms
PL +
2.
Relate the transforms
Pq+ and Pq using
3.
Relate the transforms
PL +
4.
Relate the transforms of
and Pi using
(9).
(10).
and Pq+ using
(21).
Pi and Pq using
the characteristics of the system up to
constant terms and use Little's law to evaluate the constants (see Section 4-2 for
further details).
5.
Solve the 4 X 4 system of equations from the previous 4 steps to find Pi, Pq, P^+
and Pq+
6.
•
Find the transforms of S, and
W, from
20
(11) and (12).
We
are going to illustrate
how
the
method works through an application
in the next
subsection.
4.2
The MGEm/G/I and
We
consider in this subsection a
the arrival process
j4i.
Let a(s)
is
MGEM /D/s
MGEm/G/I
— aN y)
FIFO
service discipline
where
be the Laplace transform of the interarrival distribution where
MGEm/G/I
8 In a
queue,with a
a generalized Erlang process characterized by the matrices Aq and
Q£>(j), ctj^(s) are polynomials of degree
Theorem
queues under FIFO
M and
less
than
M respectively.
FIFO
queue under
PQ (z) = (l-z)M(*x{A + zA 1 )-zI)- 1
PL (z) =
(1
- x)M{* x {Aq + zAt) - z/)- 1 * X {Ao
(24)
,
+ zAi),
(25)
and
where x r
,
r
=
1,
.
.
.
,
M—
1
are the
M—
1
=
1,
vector whose ith component
is
a(-s)4> x (s)
and
H
is
an
M
roots of the equation
Re{s)>0,
Hi = -(1 - \ iPi E[X]) f[(l A
'
Pfe).
(27)
fc=i
Proof
Since this system
is
overtake free
we
will use the exact
the previous subsection. Thus, performing the
(9)
first
method
two steps of the exact method we use
and (10) and we obtain:
PL {z) =
p- (z)
Q
A(l
- z)P£(z)(A + zA x y\
= \(l-z)P+(z)(A + zA 1 )-\
21
of analysis described in
Combining the previous two equations with
$A'(-4o
+
we
(21), third step,
obtain, since the matrices
+ zAi)~ commute,
l
zA\), (A
?l(*)
Applying the fourth
step, the
= Pq(z)^x(A + zA
number
(28)
1 ).
of customers in the queue
and the number of
cus-
tomers in the system are also related as follows
PL (z) =
where
H
is
Combining
M- vector
an
(28)
To complete
with
Hi
Applying the usual
— P{L = 0,R —
M states as
i}.
(25).
= P{L = 0\R = i}P{R =
i}
i}.
law to the server we find that:
Little's
compute
(29)
we next compute H.
= P{L = 0,R =
1
chain with
t
- z)H + zPQ (z),
and (29) we obtain (24) and
the fourth step
In order to
H
(1
- P{L = 0\R =
P{R —
shown
i}
we
=
i}
(\ iPi )E[X}.
represent the
in Figure 2.
ATC
as a continuous time
Markov
Solving for the 9teady-state distribution
we
obtain
l
P{R =
i}
± f[(l- Pk
=
(30)
),
and thus
H
At
this point
{
= ±{l-\uHE[X])l[(l-pk ).
A«
fc=i
we have solved exactly
for
Pl(z) and Pq(z)
the transform of the waiting time distribution (sixth step)
(fifth step).
we combine
In order to find
(12)
and (24) and
obtain
ej '*
We now
it
choose a
can be written
z
w (Ao +
zA x )(* x (Ao + zA x ) -
such that Ao
+ zAi has
zl)
= jH(A + zAx ).
M linear independent eigenvectors and thus
as:
A + zA
x
(31)
= S{z)Q(z)S- 1 {z),
22
)
*.
<>-p,)
M'-p 2
Figure
where 0(z)
for
i
=
1,
.
.
V'-p
)
2:
The Markov chain
of the
is
the diagonal matrix of the eigenvalues of
.,
M.
Bertsimas and Nakazato
[2]
\mO-p*.,)
,
ATC
A + zA\
which we denote by
9i(z)
have shown that the roots of the equation
satisfy:
za(-$i{z))
The columns
=
1,
i'
= l,...,M.
of S(z) are the right eigenvectors of
A +
zA\ which we denote by
£(0,(z)).
Moreover,
$ W (A + zA x = S(z)$ w (Q(z))S-
l
)
$x(A + zA 1 )-zI =
and substituting
to (31)
e\
(z),
S(z)(* x (Q(z)) - zI)S~ l {z),
we obtain
'S(z)i w (Q{z))(* x (*(*)) ~ zl)
= jl}S(z)Q(z)
or
<hv(ei(zm l {9 1 (z)){<l, x (0 1 {z))
with £i(#i(z)) being the
every eigenvalue 0,(z),
i
component of £(#i(z)) (the previous
first
=
1
.
9Wy U "
-z)= jMfaizWiiz),
.
.
M). Since za{-9{{z)) =
1
we have
A(a(-0 1 (z))fo(6> 1 (z))-l)
23
relation also holds for
Sl
U
»'
where the function g{9\ (z)) must have an appropriate form
in
order to maintain the
analytical character of <f>w(6i{z))- Therefore,
=
4>w(s)
9a
K
3
!,~
\
(
X(a{-s)<f>x{s)-
Since <t>w{&)
where x r
r
,
1,
. .
.,
M—
1 are the
M—
1
a(-s)<j>x (s)
and
K
is
(32)
analytic
is
=
9(s).
u
1)
a constant such that
roots of the equation
=
1,
Re{s) >
0.
:
»—»o
which leads to
(26).
Remarks:
1.
Equation (24)
is
to the best of our
knowledge new, while (26)
of the Pollaczek-Khinchin formula for the
M/G/l
queue.
It is
is
a generalization
interesting to notice
that (26) could have been obtained using Hilbert factorization techniques.
It is
remarkable that we were able to derive these formulae just from the distributional
laws.
2.
The previous
queue
5
(see
Remark
4 after
MGEm/G/1
Theorem
The GI/G/1 queue with
In this section
is
results for the
we
consider a class of
we say
vacations
is
that he
denned as
GI/G/1 with
is
generalized vacations
GI/G/1 queueing models with
"on vacation".
MGEm/ D/s
6).
unavailable for occasional intervals of time.
or idle
system can also be used in a
Whenever the
server
a single server
is
who
either unavailable
Formally the GI/G/1 queue with generalized
follows:
generalized vacations
24
Gl. The system
server
may
the
service
mechanism need not be
FIFO
When
order.
the server begins his vacation
We
Zo
is
<j>\r(s).
denote by Zq
when a vacation
of customers present in the system in steady state
interval
determined by the service mechanism.
G3. Each vacation
form
exhaustive.
In particular, as long as the
leave customers behind depending on the service mechanism.
number
starts.
1.
busy, customers are served in a non-preemptive
is
G2. The
he
assumptions of Theorem
satisfies the
We
This system
ered in Doshi
interval
is
distributed as a
assume that the number of
variable
arrivals during
V
V
is
and has Laplace
trans-
independent of Zq.
a generalization of the GI/G/1 queue with exhaustive vacations consid-
is
[4],
random
in
which Z =
vacations considered in
p. 457) in the sense that
Fuhrmann and Cooper
0. It
also generalizes the
Fuhrmann and Cooper
it
allows
more general
system with generalized
(see also the discussion in
[5]
arrival processes. In
however, relax Assumption
[5],
M/G/l
G3
some of
Wolff
[17],
their results
above, allowing the vacation
time to depend on the arrival process. In order, however, to prove sharper decomposition
results they
make
same assumption
exactly the
generalize the results of Keilson
arrivals
and assume exhaustive
Our goal
in this section
is
and Servi
service Zq
(their
two
[11] in
=
Assumption
respects:
6).
Our
results also
They consider Poisson
0.
to illustrate a unified
to solve queues with generalized vacations based
way based on the
distributional laws
on the exact method of analysis from
Section 4.1. Corollaries of our results include the decomposition results established in
[5]
and
results
[11].
In this
way we obtain
[4],
on the extend to which the decomposition
insights
depend on the Poisson assumption.
Examples of the
class of
GI/G/1
queues with generalized vacations that we consider
in this section include:
1.
The standard GI /G/l queue,
2.
The GI /G/l queue with
if all
vacations correspond to idle periods
(i.e.,
exhaustive vacations, in which, whenever the server
he serves the system exhaustively,
i.e.,
Z =
25
0.
V—
»
is
0).
busy,
3.
The GI/G/1 queue with gated
who were
customers,
waiting
tributed according to the
vacations, in which the server accepts only those
when
number
the server returned from vacation,
of customers
who
i.e.,
Zq
is dis-
arrived after the server returned
from vacation.
4.
The GI/G/1 queue with
each
in
5.
up to k customers
limited service, in which the server serves
and then takes a vacation.
visit
Queues served
in cyclic order considered in
Fuhrmann
The vacations associated
[6].
with any particular queue correspond to times when the server
visiting the other
is
queues.
Analysis of
5.1
We
MGEm/G/1 queue
consider the system in steady state and
customers
in the
vacations. Let
is
B
R
and Zq
to be the
present in the system,
C„
We
,
,
is
ATC
ATC
number of
stage of the
the system with generalized
began (the forward recurrence
busy at the time of observation. Obviously
on vacation at the time of observation.
stage of the arrival process and the
when a vacation
= [P{Z =
is
the
nHR
interval starts.
We
number
of customers
define
£
= m\B'}]Z=1 and fo) =
z"<fn
view the vector generating function £(z) as defining the service mechanism. Our main
theorem
is
Theorem
Gl
L v Q v and R„ be
since the last vacation
the event that the server
the event that the server
Let
let
when a random observer observes
V* be the elapsed time
time of V). Let
we
system, the number of customers in the queue and the
arrival process respectively,
B'
with generalized vacations
-
G3
and Ai
,
as follows:
9 In an
that has
MGEm/G/1
system with generalized vacations satisfying Assumptions
mixed generalized Erlang interarrival times characterized by matrices Aq
vacations distributed according to the
26
random
variable
V
and
service
mechanism
characterized by the vector generating function £(z) the vector generating function of the
number
of customers in the queue
PQv (z) =
(l-p)
C(z)
and
*v(Ao + zAx )(l -
*v(A + ^i)(l
PlA*) = (1~P) fc)
system
in the
-
given by
is
z)
{* x (Ao
+ zA )- zl)~\
(33)
1
l
($ X (A + zA 1 )- zI)- * x {A + zA x ). (34)
z)
Proof
Let
Sv
,
W X be the system, waiting and service time of a customer.
Let p be the
v,
intensity.
Because of Gl using the exact method of analysis for overtake
applying (28) for
Qv
is
observer of the system. Recall that
event that the server
P{Q V = n,R v =
p and P{B'}
i}
the event that the server
is
busy and B'
is
on vacation, at the time of observation. By applying
is
P{B} =
B
(35)
1 ).
between Pl v (z) &nd Pq v (z). Consider a random
to establish another relation
to the server
systems and
and L v we obtain
PlAz) = PqAz)* x (A + zA
Our goal
free
traffic
=
1
—
p.
= pP{Q v = n,R v =
By conditioning on
i\B]
+
(1
the event
- p)P{Q v =
Rv =
n,
is
the
Little's
law
B we
obtain
i\B'},
(36)
Conditioning on Zo, Ro, V* we obtain
P{Q V = n,R v =
M
-
n
i\B'}
=
.„„
EE/
P{Qv = n,R v = i\B',V* = t,Zo =
m,R
=
k}
fc=lm=O y °
P{Z = m, R =
M n-l
= y^Y. P{Z
fc=l
oo
= m, R =
m=0
k\B'} /
J°
+ T\ P{Z =
ak (t)
n,
* a^
R =
n-m - l
k\B'} /
V=
t\B'}dt
* a\(t)
dFv .(t)
k,
\t)
ai(t)dFv .(t),
where we used the independence of V* and (Zq,Ro) (Assumption
of queues with generalized vacations).
Taking generating functions
in (36)
Let B(z)
= \^^-oP{Qv =
and using (23) to
(37),
PqM = pB{z) + (1 - p) C(z) * V .{A
27
G3
we obtain
+ zA x ).
n,
(37)
in the definition
Rv =
i\B}z n )™_ l
.
Similarly
P{L V = n,R v =
i}
= pP{Q v =
n-lDR
v
=
i\B}
+
(1
- p)P{Q v = nf)R v =
i\B},
from where, by taking generating functions, we obtain
PLv (z) =
pzB(z)
+ (l-p)
# v .(i4 +
((z)
zAt).
Therefore,
PlAz) = *PqA*) +
-
(1
which combined with (35) gives (34) and
*)(1
-
fc) *v(Ao + zA
P)
x
(38)
),
(33).
Remarks:
1.
Equation (34), as well as (33),
strates,
is
not formally a decomposition result.
It
demon-
however the contributions of the various characteristics of the system to the
system length distribution. The
mechanism
used.
The second term
term
cation, while the third
(1
-
term £(z) represents the
first
$v(A +
p)(l
-
z)
effect of the service
zA-[) represents the effect of the va-
($jr(A
represents the contribution from the underlying
+ zAi) — zl)~ l $x(j4 + *A\)
MGEm/G/1
queue without vaca-
tions.
2.
In the case of Poisson arrivals
PLv (z) =
which
is
we obtain
C{z) 4>v.{\
- Xz)
MX _ Xz) _
.
z
a formal decomposition result obtained in Fuhrmann and Cooper
number
of customers in the system
random
variables:
(1)
The number
is
distributed as the
sum
of customers that are
left
[5].
The
of three independent
in the
system when
a vacation begins, (2) the number of customers that arrive in the system during
a vacation period, and (3) the number of customers in a
vacations.
A
similar relation
is,
M/G/l
queue without
obviously obtained for the queue length distribution.
28
3.
Assumption G3 was only used
(38)
we would obtain
Pl v (z) = zPQv (z) +
where Pl,\B'( z )
1S
PlAz) = Pl t \b-(z)(1 which
is
(1
-
z)(l
- p)PL ,\ B ,(z),
the vector generating function of the
that the server in on vacation.
Combining
p) (1
-
z)
we have been
In the previous subsection
tomers in the system and
in the
queue
for
+*Ai) -
(§jr(A
MGEm/G/1
(39)
in the
a
system given
we obtain
zI)- l * x {Ao
Fuhrmann and Cooper
+
»iti),
[5].
with generalized vacations
able to derive a formula for the
as a function of Q(z). Thus, given that one
number
of cus-
MGEm /G/l queue with generalized vacations
is
able to solve for
length distributions are fully characterized and from
through the distributional laws. In
number
(39) with (35)
the generalization of Proposition 5 in
Applications of the
5.2
Without Assumption G3, instead of
in deriving (37).
this subsection
we
them
C,{z),
the queue and system
the waiting
will consider
and system time
some
specific applica-
tions of the previous analysis that have interesting consequences.
The MGEm/G/1 queue with exhaustive
vacations
For the case of exhaustive vacations Theorem 9 implies the decomposition results of Doshi
W.
Theorem
10 (Doshi
time
sum
is
the
the vacation
[4])
For
the
MGEm/G/1
of the waiting time of a
with vacations
MGEm /G/l
and
the
V
under FIFO, the waiting
forward recurrence time of
V.
Proof
In this case
Z
independent of
PLv (z) =
(1
=
z.
and therefore ((z) —
Then
- p)
A
P{Z = 0,Ro =
*}«i
(34) can be written (since all the matrices
(1
-
z)
(*x(A + zAi)- zI)-^ x (A + zA
29
=
*i
i-
e ->
a vector
commute)
x )
i v .(A + zA x ).
In a regular
MGEm/G/1
queue, however (25) holds,
PL (z) = 8
But Pl. (1)
ATC
is
=
-
z)
(* x (Ao
+ *4i) -
z/)-
1
**^,, + zA
Pl(1)i since the ith component of each vector
in stage
i
which
previous equations
Therefore, in a
(1
i.e.,
is
is
x ).
the probability that the
indepedent of the vacation. Taking limits as z
—
>
1
two
in the
we obtain
MGEm/G/1
PLv (z) = S ^.(^o + ^i)-
1
with exhaustive vacations
(1-2) [$x{A + zAi) - zI)- 1 * x (A + zA r )
^-(Ao + zAr),
(40)
where the vector
$
is
computed
in (27). (40) offers a
complete solution of the
MGEm/G/1
queue with exhaustive vacations.
Following exactly the same approach leading to (32) in the proof of Theorem 8
we
obtain that
\{a{-s)<f>x{s)
-
1)
i.e.,
w
v
i
w e v.
n
The MGEm/G/1 queue with gated vacations
In a gated vacation system our goal
random
variables
is
to find ((z). For this reason
we
define the following
:
Let J be the time the server spends in the system immediately after he returns from
vacation until he starts a
transform of J. Let
Rj be
new
one.
the
ATC
Let Fj(t)
= P{J <
t}
and
stage of the arrival process
<f>j(s)
and
N
be the Laplace
be the number of
the customers that the server finds at the system just after the end of the vacation.
define
Rj = ?{Rj = m}*f= , and N{z) = E{z N
Finally,
we
define also the vectors
30
).
We
rfn
= P{N = n n Rj = m}%=l and
From
the definition of the service
to the
number
who
of customers
£~=0 z n rfn
=
tf(z)
.
Note that Rj
mechanism in a gated system, Zq
arrived during J, thus
is
=
JV*(1).
distributed according
:
oo
53
2
n
P{£o =
Ro = k\Rj = m} =
",
n=0
J
k
f" a m (t)dFj(t) +
J°
which leads to
I" a m (t) * al-
Z»
l
\t) * a\{t)dFj{t),
y°
n=l
:
oo
£z"P{Zo =
n, £(,
=
*}
=
n=0
M
in
<&(*)<tfj(0
/
+
Jo
l
which
°°
/•oo
Y, P{RJ = m}
m=
matrix notation becomes
/-oo
£
~L
a m (0*a<"- 1 )(0*aJ(t)dFJ (f)
z" /
Jo
n=l
:
= ft(l)*j(A + zA
C{z)
(41)
1 ).
Furthermore, the time interval J lasts as long as the server
is
servicing the
N
customers
he finds upon his arrival. So
M»)
Finally
we need
left
tf(*r(«))-
N(z) from the
to evaluate
definition of the gated vacation
server
=
characteristics of the system.
system we see that
behind in the system before starting
arrived during the vacation interval. Therefore
l}
=
^
(41), (42)
for
n >
Recalling the
includes the customers that the
1
customers that
:
Jfc,.Ro
= m}
/
a m (t)
*
a*"-*" 1 )^)
*
a[(t)dFv (t).
:
fi(z)
By combining
,
42 )
t<x
]>3P{Zo =
Taking generating functions
N
his vacation as well as the
M
n
P{N = n,Rj =
(
=
<;{z)*v{Ao
+ zA1 ).
(43)
and (43) we have:
C(z)
=
C*(l)
$ V {A + Ay *j{A +
)
31
zAi),
(44)
where
M»)
Equations (44) and (45)
if
we use Theorem
Remark
:
9
= a4>x(s))^v(A +
we can
fully characterize £(z) as
and the distributional laws we can
(45)
solve for all
moments. Moreover
fully characterize the system.
Notice that in the Poisson case the recursion formula takes the form
C(z)
= C(<M* -
**))
M* - WA -
Az)).
Priority queues
6
communication and manufacturing systems where jobs
Priority queues are important in
of different significance need to be serviced.
priority rules (for
example the so
expected waiting times.
We
of class
It is
In addition, in several applications strict
therefore important to be able to analyze priority queues.
1
which there are two distinct customer
have priority over those of
time for the high priority class
that they are
1
and the low
1
l
{z)S{ {z),and Bo + zBi
matrix of the eigenvalues and Si(z)
1,2).
classes
We
is
mixed generalized Erlang
numbered
1
and
2.
We
priority class 2 respectively.
1
and 2
= S 2 (z)Q 2 (z)S 2
(2)
where 0,(z)
assume
Let (Ao,Ai),
respectively.
arrivals respectively.
~1
Customers
be the pdf of the interarrival
mixed generalized Erlangs of order Mi, M2
= Si(z)Q
=
classes,
class 2. Let a(t ), b(t)
(Bo, B\) be the corresponding matrices for class
zAi
minimize a weighted combination of
called c/i-rule)
consider single server priority queueing systems with
arrivals, in
(t
<j>x(s)A l )l.
is
Then Aq +
the diagonal
the matrix with columns the right eigenvectors
denote with 1/Ai and I/A2 the means of the arrival processes. The two
have different (general) service time distributions with means
and they are served by a
We assume that
E[-X"i]
and EfXa],
single server.
within the same class customers are served in a
priority queues allow overtaking
among
classes, within the
same
take place and therefore the distributional laws are applicable.
FIFO
class
order.
Although
no overtaking can
In this section
we use
the distributional laws to derive the distributions of various performance measures.
results generalize earlier
work of Keilson and Servi
32
[11] for
Poisson arrivals.
Our
We
consider different types of priorities (preemptive repeat, preemptive resume, non-
preemptive).
The type
of priority used does not affect the service time of class
time of
affects the service
queues in a unified way, we define the effective service time, Gi,
regardless of the priority rule used).
spent in a service box.
left
may
or
The
service
start over, but the
customer of class
until the
We
but
develop a generic model to analyze priority
class 2. In order to
from the beginning of service
1,
i
i
=
1,2, as the time
completes service (G\
=
X\,
can visualize the effective service time as the time
may be
customer
interrupted and resumed from where
assumed
is
to stay in the service
box
it
was
until he
is
completely served. In this setting, the time in queue refers to the time from the arrival of
the customer until the customer enters the service box.
The
Takacs
section
[15] for
is
organized as follows. In Section 6.1 we generalize the classical results of
the
M/G/l
generalization, which
class 2
customers
of class
1
in a
customers.
is
queue
for the
busy period distribution to a matrix form. This
also of independent interest,
is
essential since the service time of
preemptive priority system depends on the busy period distribution
In Section 6.2
we
find the effective service time distribution in
various preemptive systems as a function of the busy period matrix. In Section 6.3
we
analyze systems with preemptive priorities, while in Section 6.4 we analyze systems with
non-preemptive
The high
6.1
We
priorities.
denote with
priority customers
ATC\ and ATC2
busy period matrix
the two arrival timing channels. In this section
compute the busy period matrix £1(3) with
[Ei(a)]jj
=
<Tij(s),
the Laplace transform of a sub-busy period interval for class
given that
by the
it
first
started with
ATC\ —
i.
=
1, ..
.
,
(see for
at the beginning of a sub-busy period
idle interval,
a sub-busy period
example Kleinrock
ATC\
[12] p.
can be in any stage.
33
Mi
that ends with
Note that though a busy period interval
customer that arrives after an
whenever a customer enters service
1
i,j
we
will
denoting
ATC\ =
j
is
initialized
is
initialized
210) and therefore
Theorem
11 In a
MGEmJG/1
acterized by the matrices
4>Xi{ s )
we have
queueing system where the interarrival process
Ao and A\ and
is
char-
the Laplace transform of the service time is
that:
M,
Si(-)
where Xj(s) are the
M\
=
E^x (*-*iW)fi(*KiW*))i
1
roots of the equation
=
a{x)4> Xi {s -as)
Re{x) <
1,
for Re{s)
>
0,
and
a l '(x 1 (s))
ZmA s
li(»)
)
a\'{x Ml {s))
Proof
We
will use
a generalization of the
classical
the evaluation of the busy period for the
a busy period
busy
if
is
ends with
M/G/l
queue (Takacs
The duration
[15]).
invariant under the service discipline provided that the server
there are customers present.
discipline.
sub-busy period decomposition argument for
We
m
given that
it
started with
ATC\ =
i.
1
customers that
This definition
the decomposition of the busy period into sub-busy periods. Let i?"' be the
occupied by the customer just after the
Ni(x) be the number of class
on the event
for
n >
U =
{.ft?*
1
customer of the sub-busy period
first
arrivals during x given that
= j,X\ =
x,
always
then use the last-come-first-serve (LCFS) service
Let Bi t7n be the duration of the sub-busy period for class
ATC\ =
is
— n} we
Ni(x)
ATC\ —
i.
is
is
useful for
ATC\
x,
Ni(x)
=
n]
= E[e~'
Then, conditionally
obtain the following decomposition,
{x+
^
»-
= e-"e'i [S 1 (5)re m
34
stage
served. Let
1
E[e- Bi "\R? = j,Xi =
of
.
B
^ +B'^ + - +B'-^}
Unconditioning, we write the previous relation in matrix form
Si(-)
=
«*(*)
...
a\(x)
00 -,-Ja:
dFXl {x)+
J?°e
M
l
*M S*)\
\
ai(x)
+
r^"fl
J0
n=l
In order to
*a[
n - x)
(x)J
compute £1(3) we
compute
will
both parts of the previous equation with
=
Ei(j){(j)
M\ =
'
eigenvalues and eigenvectors. Multiplying
its
((s), the right eigenvector of £1(3) corresponding
and using equation (23) we obtain:
to the eigenvalue u(s),
(Notice that for
a^(x))[M*)] ndFXl {z).
a\(x)
V
1, this
u(s)t(s)
reduces to
= * Xl (*I + A + u(s)A
cri(s)
=
<f>x t
that the transform of the busy period satisfies in a
{s
+A-
M/G/l
Therefore £(s) must be a right eigenvector of $jv, {'I
lently a right eigenvector of
mas and Nakazato
u(s)
=
</>_y,
computed
(s
—
=
is
the equation
queue.)
Aq + u(s)Ai with corresponding eigenvalue —x(s).
Bertsi-
(3))
=
and furthermore from (46)
1
Therefore, the eigenvalues Uj(s) (j
as follows: Uj(s)
which
A<7i($),
(46)
equiva-
have shown that u(s)a 1 (a
x(s)).
)((s).
+ Ao + u(s)Ai) and
,
[2]
1
<f>Xi(s
—
Xj(s)), j
—
1,
= l,...,Mj where
...,M\) of £1(3) are
Xj,(s) are the
Mi
roots
of the equation
a(x)<t> Xl (s
Moreover, £j(s)
is
eigenvalue -Xj(s).
-
x)
=
1,
Re{x) <
the right eigenvector of Ao
The
left
for
+ ^"^(a —
eigenvectors are computed in
Re(s) >
0.
Xj(s))Ai corresponding to the
[2]
and are equal to ai'(xj(s)).
Having characterized the eigenvalues and eigenvectors of £i(s) we can spectrum
compose
it
as follows:
Si(a)
=
£
<f>xA»
~ «i(*))S(*Ki'(«i(«)).
35
de-
where
-i
i
di'(«i(*))
fi(«)
•••
=
toA*)
:
J
oi'(*Af,(«))
Remark: The
transform
<Ti(s) of the
<
busy period distribution
T 1 (s)
=
e'1
given by
is
E 1 (3)f.
Effective service time distribution in preemptive systems
6.2
According to preemptive
priority
customer in
disciplines,
whenever a high priority customer finds a lower
service, he interrupts the service in progress
mediately. Once there
is
no higher priority customer
reentry, the preemptive discipline can be further broken
starts his
own
im-
the system, the interrupted
left in
customer reenters service and depending upon the manner
and
which he
in
down
is
serviced on his
into the following three
categories:
•
Preemptive resume discipline
Under
:
customer continues
this discipline the interrupted
his service
from the point
his service
by resampling.
his service
without resam-
of interruption.
•
Preemptive repeat
Under
•
different discipline
this discipline the interrupted
:
customer continues
Preemptive repeat identical discipline
Under
this discipline the interrupted
:
customer continues
pling.
Each of these three preemptive
class 2 customers.
disciplines
In this section
we
a
define
class 2
random
variables
customer such that
G^,
i,j
ATC\ =
going to affect the effective service time of
calculate the effective service time in all the three
preemptive categories as a function of the
We
is
j
—
class
1, .. .,
1
when
36
busy period matrix.
M\, which
is
the effective service time of
the class 2 customer finishes service given
that
ATC\ =
i
when
transform of G'J and
section
to
is
this class 2
customer started
Let
service.
G^l^) denote the matrix with elements
let
<p
<f>
G ij(s) be the Laplace
G „(s). Our goal
in this
compute the matrix G2(s).
Preemptive resume discipline
Proposition 2 In
the
a single server system with two priority classes each of which satisfies
assumptions of Theorem
terized by matrices Aq,
1
and has mixed generalized Erlang
A\ and Bo, B\
interarrival times charac-
respectively, the effective service time of the class
2 customers for the preemptive resume discipline
G 2 (s) = *x
2
(A +
A
1
is
given as follows:
i: 1
(s)
+
sI).
Proof
According to the preemptive resume
is
discipline,
interrupted, the duration of the interruption
whenever a low priority customer service
is
exactly the duration of a high priority
customer busy period. Furthermore, due to the characteristics of the mixed generalized
ATC\
Erlang arrival process we condition on R\', the
priority customer enters service.
Let
<f>
G ki{s) be the Laplace transform of the
service time of a class 2 customer that ends leaving the
with the
ATCi =
E[e-°?\X2 =
x}
k.
first
ATC\ =
i
given that
effective
it
started
Then
= e—{ot(«) + j:J5 sl [E 1 (*)] 1 j o («)*a}
fc
l
+
where the
stage immediately before a low
k
(«)
Mi
Mi
E
EPi(«)]iji[Si(»)]iA«ii(«)*«i.(«)*«Ji («)
+ —}.
of the right-hand side terms represents the probability that there are no
interruptions during the regular service time of the low priority customer, the second the
probability of having just one interruption, where
we have
stage of the high priority customer at the end of the type
writing the previous formula in matrix notation
37
we
obtain:
to take into account the
1
ATC
busy period, and so on. By
a
<•(*)
a\(x)
ai(aj)
^
* + «— *LES.i
E\e- G >'\X 2 =
x]
(a 1 (*)[E 1 (*)] lll
+ ...+a Ml (*)[Ei(*)]ij*J^
=e-" cl
B
Using (23) we obtain:
G
£[e"' *'|X 2
=
x]
= e-*?k e-( A* +A**W*e
i.
Therefore,
E[e-^'] =
S*fc
$x
(>lo
2
+
i4 1
E 1 («) +
*J)ei,
and hence,
G2
Remark:
(
a)
= * A 2 (^o + ^iS 1 (3) +
is
5 /).
For the Poisson case we obtain
<I>g 2
which
:
in
{s)
agreement with Jaiswal
=
4>x 2
{
x i - \i<r(s)
+
s),
[9].
Preemptive repeat disciplines
di'(«)
Let
3(f)
=
{
ai (t ),...,
fc
(«),...,
a Ml
(«))'
and
4(t)
=
OMi'(*)
Proposition 3
TVie effective service time
the assumptions of Proposition 1
•
is
G2
for the preemptive repeat discipline under
given as follows
In the case of the preemptive repeat different discipline
i
G 2 (s) = j" A(x) e-"fx,{x)dx
jo
•
\l- [°° fx,{*) [' S{y)e-*dydx
L
Jo
Sx (3)
e*,
Jo
In the case of the preemptive repeat identical discipline
G 2 (s) =
J~
A(x)
[/
-
£ a(y)e-»dy %
38
S,(a)
e
"fxA x
)
dx
-
-1
Proof
The underlying experiment
Assume
X^. At the
r.v.
system and
ATC\ =
moment he
There are two
k.
possibilities for the
• either it is greater
• or
the stage of the
it is
less
X
customer arrives the service of the type
1
customer
Xi
r.v.
1
given
customers
remaining time until the
X2 and
in this case
G\ = X2, where
l
and
2
2
at the
customer
as soon as the
moment
is
that the next type
interrupted and
it
X2
for the
starts over
busy period initialized by the type
preemptive repeat identical
So for the repeat different case, conditioning on
E[e- G?\X2 =*] =
a\{x)e->
x
X2 we
fa
+
discipline.
obtain
k (y)e-'ydy
Jo
% ^(s^s)
et
.
Thus,
And
in
[°°
f°°
r
ai(x)e-* fXi (x)dx +
Jo
fXl (x)
Jo
matrix form
G 2 (s) =
fa
k (y)e-*dydx
^
Vi(s)Gi(s) *,
Jo
:
A(x)
e-*fX7 {x)dx
Jo
\l
l
Finally for the repeat identical case
G 2 (s) =
1
over for the preemptive repeat different discipline or with the same
is
value of the r.v.
&?"(«)=
5
i
the low priority finishes service;
than the selected value of
with a new value of the
no type
is
:
than the selected value of
ATC\ when
time
his service
enters service there are
next arrival of the high priority arrival process
is
and
that a class 2 customer enters the service facility at r
by a value of the
in the
the following:
is
J~
A(x)
[/
In the case of Poisson arrivals
4>
-
-
£
%
<l{y)e-*dy
^
(3
x
I
%
^(s)
:
we can obtain the
G ,(*) =
i-i
f
<i(y)e-vdydx
(x)
f°° fXi
Jo
Jo
- -^-(l -
4>
+
e- x fxA*)d*-n
results of Jaiswal
Al)
X2 (s
39
^(s)
+ X^a.is)
[9],
and
namely
1
/•oo
<t>G 7
e
,
-(» + Ai )x
{s
preemptive repeat different and the preemptive repeat identical
for the
discipline, respec-
tively.
6.3
Preemptive
priorities
we analyze a
In this section
generic preemptive discipline in terms of the distribution of
the effective service time. In this
way we
are able to analyze all preemptive disciplines
we
considered in a unified way.
Let Li, Qi, S{, Wi, Ri,
and
ATC
priority
=
i
1,
2 be the system
and queue length, system and waiting time
stage of the arrival process, respectively, of class
customer that
may
t
=
1,2. Notice that the
be in the service box without being served
is
low
not taken into
account in the number of low priority customers in the queue.
Let
Lf Qf and Rf be
and the
Let
the
,
ATC
stage of class
L~ Q^ and R~ be
ATC
and the
stage of class
process of class
2.
A
i
in the system, in the
number
of customers of class
t
respectively, just before a transition
i,
ATC
ATC
We
also define the matrices
stage of class
i,
of customers of class
n+ = [p{[l+ = n nRt =
mnR+ =
P{Li =nnJ2r = mnflJ =
[P{L 2 =
nn
R1 =
J
mn
i
and
=
E
n=0
*
n
queue
shifts to the
next
1
m=i
R 2 = /}C=f
J
~"7
i}]2l™
Z}J
•
n+,
,
/=j
,
H*
n-L2 (z) = f; 2»n-, and n L2 (z) = f; z»n n
n=0
40
queue and
class 1 customer.
and the matrix generating functions
nj,(*)
class 2.
epoch of the arrival
in the system, in the
an arrival of a
respectively, just before
nn =
queue
according to the definitions of Section 2.2.
Q° and R° be the number
the
epoch of
in the system, in the
transition includes both arrivals in the system
exponential stage of the
Let i°,
of customers of class
respectively, immediately after a departure
i,
the
,
number
n=0
.
Exchanging Li with Qi we similarly define the generating functions Tli
(z),
Hq
(z)
and
U Ql (z).
High priority customers
As long as the
discipline
queue. Therefore
Low
We
,
is
preemptive the high priority customers see a usual
Theorem
MGE\f
8 can be used to find the distributions of L\, Qi, Si,
t
/G/l
Wi.
priority customers
will
apply the exact method of analysis of Section
between L\, L^ and Li and L\ and
Q\
4.
We
will first establish relations
that will be used in the analysis of preemptive
systems.
Proposition 4 Let
the pre-transitions
and
Il£ (z), Il£ (z)
and
Tli 7 (z) be the matrix pgf for the post- departures,
the general time probabilities of a class 2
priority system satisfying the assumptions of
Tll 2
Theorem
1.
customer for
a preemptive
Then
n L2 (z),
(z)=
(47)
and
a 2 (i -
z)
n+
(z)
2
= {a + A x y n L ,(z) + n Ll (z) (b + zBx ).
(48)
Sketch of Proof
we apply
First
choose the uniformization constant v
1,
.
.
.
,
Mfc.
The epochs
follows from
we
the uniformization technique to the two phase renewal processes and
=
i/j
+ vi
such that v^
> max
^k,i k for
k
—
1, 2,
=
i
of transitions in both processes are therefore Poisson and thus (47)
PASTA.
and Nakazato
[1]
between post-departures and the pre-transitions probabilities
in
In order to establish (48)
to establish the relation
stochastic processes with
we
follow closely the approach of Bertsimas
random upward and downward jumps. We
first
write
down
the
flow balance equations for all states, where each state has four indexes corresponding to
the two phase type arrival processes, and then
preemptive,
i.e.,
class 2 departures
we use
can only happen
the fact that our priority discipline
if
there are no class
1
customers
is
in the
system. Finally, by taking generating functions in the number of low priority customers in
41
.
we obtain
the system
(48).
The computations
are algebraically involved but conceptually
simple.
Proposition 5 In
a preemptive single server priority system with two classes of customers
each arriving according to a generalized Erlang distribution
:
Mi
n£,(*)
K
=
e,-
£
n+
e*fc
(z)
2
* G »,(2?o + zBt )
(49)
?;.
J
k=i
Proof
Conditioning on the state of the queue that a class 2 customer
left
behind at the moment
he started service and the duration of the effective service time we obtain
p{L+ = n,R+ =
= j} =
i,R+
T,Y,Y, p{Qt = k,Ri=™,Rt =
By
b l (t)*b^-
i}
writing the previous equation in matrix form
Let
Ei,,
customer
=
k
is
that there
number
1,2 be the
a class
Jfc
\t)*b{(t)dFGm 4t).
(49).
of class k customers in queue given that no class k
number
in the service box. Let Afc be the
is
we obtain
k-1
of class k customers in queue given
We
customer in the service box.
introduce the matrix generating
functions
U Ej (z) = f;
= inR 2 = j}f= M j=f>
z"
[P{E 2 =
nnR
z»
[P{A 2 =
it=Mj j=Mj
nnR = inR2 = ;}])=f
jff'
>
l
,
n=0
oo
n A ,( 2
)
=
£
•
1
n=0
Furthermore,
P {Ri =
B\
i
i,
let
R2 =
E be an Mj x
j |Ii
=
0,
L2 =
M
2
matrix and
H 2r
0} and
the Laplace transform of
B\
,-,
o~\
{
(s) is
°^ S) ~
We
also introduce the traffic intensities pi
Pa,
=
/'{one class
t
customer
is
2
1
given by
sE[B hl
r \L\
2
vector such that
=
0,
L\
=
0}
.
in the service box}.
ATC\ =
u
]
=
p\
+ p 2 and we
Our main
result
is
S;,j
=
Finally, let
:
A^fA",], p
42
M
busy period that ended while
l-[Vi(s)]
=
be an
= P {R$ =
be the forward recurrence time of a class
Then
H
define
t.
Theorem
12 In a preemptive queueing system with two priority classes each of which
assumptions of Theorem
satisfies the
1
characterized by matrices Ao, A\ and
of the
number of low
characteristics
1.
and
and has mixed generalized Erlang
Bq,B\
respectively, the matrix generating function
priority customers in queue
calculated as a function of the system
is
the effective service time matrix
Calculate the matrix generating function
n E ,(,)ei = (l-/>i)H«i+Pi
interarrival times
Il£;
3
(
from
the following algorithm:
such that:
2)
J?a'*B v(*o + *£i),
=
t
M
l
2
l
,
(50)
where
=
=-i,3
~
-Pi,iKiE[X~ ]-p"2 j \2, j E[X 2
1
1
i
1
~
,
T
~ Pi
Pi
2?J
t«
,
ATC\ =
and has Laplace transform
i,
A 2 *j*
—
o~\ ^(s)
busy period that ends while
1
''
,e b
\
2.
For
i
=
1,
.
.
.,
M\
£et
11(1 - P2 >r ),
A 2.J r = l
Pl,rJT
Al,(Pl,/=/,r
forward recurrence time of a class
</ie
~
M,i rVl
2w=l Lr = l
and
- *ff
11 "
\i
)
]
solve the system that would give the postdeparture probabilities
n+
2 (
2)
* Gki (B + zB x )
-
z cj
n+
(z)
=
2
fc=i
^~(i-PA
A
3
)[(^o
+ >ii)'n £j (z)+ n E3 (2)(B + 25 1 ),]
(5i)
2
TTie con5<an<
p^ 3
is
calculated
from
the relation
limf
3.
II+
2
(2)1
The general time queue length distribution for
=
1.
class 2 customers
is
calculated by
solving the system
(Ao + Ai)
#
n Q2
(
2)
+
Tl Q2 (z)
43
(Bo
+ zB1 ) =
A 2 (l -
z)
U+
(z).
t
4-
The waiting time distribution for
can be calculated by applying the
class 2 customers
distributional law
VH Qa {z) =
1
'*w2 (Bo + zB!){B + zBi)"
A 2 (l - z)€x
.
Proof
Following the exact method of analysis of Section
queue length distribution
customers
for class 2
and IIq 3 (z) and then solve the underlying
relation
between
whether there
II £ (z)
and Ili
is
4,
our strategy for calculating the
two relations between
to find
we have found
linear system. In (49)
the
first
we condition on
order to find the second relation
(z). In
IIjr„(z)
a class 2 customer in the service box and we obtain that
is
= pa 2 n A2(z) +
n<? 2 (z)
n Li {z) =
z PA2
n A2 (z) +
(i
(i
- pa 2 )
nBa(z ),
- PAi ) n E3 (z).
Hence,
n L ,(z) = zUq
In order to find
Ue2 (z) we
Because of the preemptive
there
is
2
+
(z)
z)(l
- PAi U E2 (z).
(52)
)
use the following argument:
customers are not influenced by the fact that
discipline, class 1
no low priority customer in the service box; so the server serves a
with probability p\ and does not serve class
for a
-
(1
random observer
to see
n >
1
class 2
1
customers with probability
customers given that there
in the service box, he has to arrive during a class 1
R 1=
i,R 2
= j}=
Pl
J2
H^
r=l
—
p\. In order
class 2
if
customer
we denote
initialized the last class 1
period found, upon his arrival, the class 2 customer in stage
n,
no
1
busy period. Therefore,
by #2 r the probability that the high priority customer who
P{E 2 =
is
customer
class 1
br (t)*bt
r,
we have
for
n >
busy
1
n -V(t)*li(t)dF
B;.(t).
J°
Similarly,
P{E 2 =
=
(1
0,
R =
r
i,R 2
=
j}
=
- pi)P{R1 = i,R 2 =
j
|
Xi
= 0,L 2 = 0}+
pi£#
.,
r
44
=l
2r
/
Jo
H(t)dFB .
(t).
where Fb-
,(t) is
with the class
We now
•a
Zij
1
- P{R,
- 4
pip v p
- t,R
2 _ j
Little's
P{L =
X
in stage
i.
busy period that ends
Taking generating functions (50) follows.
0,
{1
also
know
0,
- 0,X
n t _
it x _
8
i
|
L2 =
L2 =
0| J?!
=
P ^ Ll =_0_E_=
0, #i =
#2 =
_________
»',
j}
,.„-.
(53)
.
i,
R 2 = j} = 1 - p^Aj^JVi]
i,
R2 =
- P2j\i,jE[X2],
we have
0,
flx
=
- PuXisEiXi] that
m _-
0}
law to the server we obtain
therefore, using (30)
P{L, =
We
customer being
1
proceed to calculate the constants appearing in (50).
By applying
and
the cdf of the forward recurrence time of a class
P2tJ \ 2iJ
j}
=
E{X 2 }}-^ jj(l A L'
P{L\ = 0,L 2 = 0} =
-
1
pi
r=l
Wir )iL IJ(1 _
A 2.J
ft(P ).
r=l
- p 2 and by
substituting to (53)
we
obtain Eij.
Finally
**
- P{R > -
'
I
______
^ - o,x, - 0>
.
But, because of the uniformization
P{i; =0,_5=
0,J2f
=
l,R°
=
r}
= Xu^iPiL, _ 0,_ 2 =
0,i? x
=
/,_ 2
=
r}
and thus
#
_
Et='l A l,tPl.i"t.r
ZW = 1
Multiplying (52) with (A
using (49)
which
is
we obtain
(51).
+
_-r = l A l./Pl,/-/,r
A\)' from the
left
and with (B + zBi) from the
right
and
Notice that (51) determines lit (2) up to the constant p& 7
calculated from the relation
UmPnj
1
(*)r-i.
45
Having found Ili
(z),
we
find U.q 2 (z)
from
W
(48), while the waiting time
2
can be calcu-
lated by applying the distributional law (11):
VllQa (z) = A 2 (l - z)^'* Wa (B + zB x ){B
+ zBi)- 1
.
D
Remarks:
1.
In the case of Poisson arrival processes (51) gives
nM
l
n - PaJ(1
n W1 n 2 (z) - —(1
which
is
which
+ A 1 (l- r 1 (A 2 -A 2 z))
7—t ;
A 2 (l-z)
(
-r—7T
a>i)
,
exactly the relation obtained in Keilson and Servi [11] using a different
derivation.
1,
,
,
:
The
probability
p^ 2 can be obtained
in this case leads to
p& 7
=
either
by requiring lim,_i IIq
A 2 £^[G 2 or by applying
r
]
(z)
=
Little's
law in the
number
of class 2
service box.
2.
The system time
for class 2
customers
is
52 =
while (52) offers a a
way
found from
W
2
© G2
,
to calculate the distribution of the
customers in the system once the distribution of the number of class 2 customers in
the queue
6.4
is
determined.
Non-preemptive
In this section
cipline,
we analyze
priorities
the single server priority system under a non-preemptive dis-
where an arriving high priority customer that
finds
a low priority customer in
service does not interrupt the service in progress. Therefore, the effective service time for
class 2
customers under a non-preemptive priority discipline
no customer stays
is
in the service
box unless he
is
is
G = X2
2
.
Furthermore, as
actually being served, the waiting time
in this case defined without ambiguity, exactly as in the case of a single
queue.
We
queue and
will first calculate the distribution of the
in the system.
46
number
of class
1,
MGEm/G/1
customers in the
High
Due
priority customers
we do not allow preemption,
to the fact that
the
number
of class
1
customers in
the queue as well as their waiting time are influenced by the possible existence of a class
customer enters
ATC\,
Let R\' be the stage of
2 customer in the service facility.
just before a class 2
service.
Let B{ be the event that the server
busy servicing a class
is
customer at a random time
i
of observation.
Let
Aj be the number of
in service.
We
customers in queue given that there
class 1
is
a class
1
customer
introduce the vector generating function:
PAl (z) =
J z"[P{Ai = n n R, =
t}]|lf >,
z=0
and the
scalar generating function
E
row vectors
and S\, such
that:
E = P{L =
0,
X
t
Finally
we
will use
H
as
Theorem
i}
H
and
lr
=
n}.
We
= P{R\* =
also introduce the
r}.
i_1
r-i-(l
-
assumptions of Theorem
characterized by matrices
number
of class
A
,
Ai and
Xi,iPi, i E[X1 ]) 11(1 -
px.fc).
PQl (z) =
1
and has mixed generalized Erlang
B
,
B\
(1
{l-z)\p2 Hi
-
z)[p 2
H
x
interarrival times
respectively, the vector generating function of
customers in the queue and in the system
1
the system characteristics as follows
PLl (z) =
Ri =
0,
n
Ylt^o z P{Ai
=
in Section 4, i.e
A
=
(z)
1
13 In a non-preemptive queueing system with two priority classes each of which
satisfies the
the
L2 =
denned
Hi
G&
is
given as a function of
:
$x;{A + zA1 ) + E]
i X;(A°+ zA i)+
[* Xl (Ao
^ [ixMo+zM)-
+ zA x ) - zl)~\
zl]~
l
^{Ao+zAt),
(54)
(55)
where,
E,
= {1-
X
,_1
pi,i\ U E[Xi] - \ 2 E[X 2 ]}-r±- 11(1 -
Al >«
47
fc=i
Pi.fc)>
(
5g )
and H\
satisfies
:
*~ X
p 2 P, i x .(A + A^Si
=
A
-^-(1 - Xi^.iElXi]) I](l Al -'
- Ei
Pi,k)
fc=i
Proof
From
the vector distributional law (28)
Pl
We
(z)
1
we have:
= Pq
1
and by conditioning on the events B{ we have,
=
i}
Pl P{Q l
by using the definition of Ai
1
and
for
n =
we
also
P{Q 1 = 0,R 1 =
n >
t|£i}
1
+
Consider a random
busy servicing a
is
class
law to the server P{Bi}
=
t
pi
:
p 2 P{Qi
= n,R =
l
i\B2 },
:
y
i}
+
p 2 P{Qi
=n,R =
r
i\B 2 },
:
i}
+
P2 P{Qi
=0,Rx =
i\B 2 )
+ P{Z =
X
0,
L2 =
0,
R =
1
i}
:
P{Q, =
Furthermore,
Little's
(z).
1
=
PiP{A! =0,Ri =
or equivalently
=n,R1 =
1
have
i}
for
= p l P{A =n,R =
P{Q = n,R 1 =
Pq
B{ be the event that the server
let
customer at the time of observation. By applying
P{Qr = n,Ri =
(57)
1 ).
should establish a second relation between Pia {z) and
observer of the system and
or
+ zA
{z)9 Xi (A
0,
if
customer enters
#j =
*}
= piP{Ai =
we denote by
service,
1
R^ =
i}
+
P2 P{Qx
/f lr the probability that
we have
P{Q 1 = n,R =
0,
i\B 2 )
that for
=
n >
£#ir
/
48
=
0,
ATC\ =
#! = t|P 2 } +
r just before
1:
a r (*)*«
(n ~ 1)
Ef.
(0*«i(0
<***,•(<)•
a type 2
P{Q =0,Ri =
and
By
1
i\B 2 )
= £*?,
H
taking generating vector functions
/
lr
we
number
of customers in the system
PlA z = Pi*P±xi*) +
)
Combining the
last
and £.
zPqi(z)
First note that £,
the server
+
P2(l
and (58) we obtain (54) and
(57)
we
*>2#i
we
also obtain:
*x;{A + zAi) + &.
two equations we have:
PlA 2 ) =
From
:
P2&i *x;{A + zA x ) + &.
)
for the
dFx .(t).
4(t)
get
PqA z = pAi(z) +
Using the same analysis
°°
= P{Li =
- z)S 2 i X;{A
(55). Finally
0,
L2 =
get (56). In order to calculate
S
0,
x
we need
Ri =
we
+ zAx) +
t'},
(1
-
z)M.
(58)
to calculate the vectors
H
x
so by applying Little's law to
recall that in
a regular
MGEm/G/1
queue (25) holds, namely
PL {z) = {l-z)8{$ Xl {A + zAi)-zI)- 1 * Xl (A + zA l
But Pi
the
we
l
=
(1)
ATC
-Pl(I), since the tth
).
component of this vector represents the probability that
of the arrival process of class
1 is in
stage
i.
Thus by taking the
limits as z
—
1,
get
Pi
where Hi
Remarks
•
H
x
i X;{Ao + Ax) =
^(1 - XijPi, B[Xi])lfc} (l - Pi, k
=
l
1
).
M-M,
D
:
Using (54) and (55) as well as the vector distributional law one can easily calculate
the waiting time distributions, as in the case of the single
•
Note that once again
G Ql (z) =
which
is
(1
-
for
MGEm/G/1
queue.
Poisson arrivals (54) take the form:
z)[p2<l>
X;(\i - \ t z) +
(1
-
Pl
-
P2)](*jr,(Ai
the exactly the result obtained in Keilson and Servi
49
-
[11].
Xiz) - z)~\
Low
priority customers
The waiting time
work
in the
of the low priority customer equals in distribution the total unfinished
system at the moment of his arrival subject to generalized Erlang interruptions,
corresponding to class
1
arrivals.
As the work
in the
system as well as the distribution
and duration of the interruptions do not depend on whether we give non-preemptive or
preemptive resume priority to the
class 1
customers we can conclude that the waiting time
distribution for the low priority customer under a
non preemptive policy
waiting time under a preemptive resume policy (see Keilson and Servi
is
is
the same as the
[11]).
However
this
not true for the waiting time in the system because of the notion of the effective service
time that we used
all
in the
preemptive priority analysis.
Nevertheless
we can
calculate
the distributions of interest by using the distributional laws as well as the relation
52 =
W @X
7
Concluding Remarks
2
2
.
We have demonstrated that overtake free systems can be analyzed in a unified way through
the distributional laws, which
we
believe deserve a
More than providing a method
ory.
more prominent place
in queueing the-
of analysis for a class of systems, the paper identified
a subdivision of queueing theory into overtake free systems, which can be analyzed using distributional laws, but are unfortunately a small subset of the systems encountered
in applications,
and systems, which allow overtaking, which are not analyzable
through the techniques of
this paper.
In the case of overtake free systems,
we showed
several insights
be obtained. One which we consider particularly satisfying
results (usually derived using diffusion
unified
way using
directly
is
methods) and exact
the asymptotic and exact
method
and new
results that can
the derivation of heavy traffic
results can
be achieved
in a
of analysis based on the distributional
laws.
The
distributional laws only provide a partial answer (only for overtake free systems)
to the question
we
raised in the
first
section of the paper regarding the laws of queueing
50
theory.
The major open problem
to identify queueing laws for systems that allow
is
overtaking, which lead a complete solution.
problem as
includes well
it
networks,
etc.).
of queues
and
A
is
known open problems
a rather challenging but important
as special cases
solution to this problem will lead, however, to a
likely to
is
This
(GI/G/s, queueing
more complete theory
provide very valuable new insights.
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New
28
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