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ALFRED
A Two
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Priority
M/G/1 Queue with Feedback
by
GTE Laboratories
J.
Keilson
Incorporated and Mass.
Inst,
of Tech.
and L.D. Servi
GTE Laboratories Incorporated
WP#1994-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
**'f*»!»i(Wf
A Two
M/G/1 Queue with Feedback
Priority
GTE Laboratories
by J. Keilson
Incorporated and Mass.
Inst,
of Tech.
and L.D. Servi
GTE Laboratories Incorporated
WP#1994-88
0^ \m-
ABSTRACT
An M/G/1
the
low
queueing system with tow priority classes of
priority class is studied.
ergodic time to completion, the
the transform
domain and
traffic
and Bernoulli feedback for
For the low priority customers, the distributions of the
first
pass time, and the
the first
moments
number
are displayed.
approximation introduced previously by the authors
is
in the
system are found in
The
distributed Poisson
them employed
to
analyze systems
with a preempt-resume high priority clocked schedule and low priority traffic with
feedback.
AUG
2 B 1981
1.
Introduction
The following system which we
consists of
two
that task feeds
the
i)
M/G/1 system,
ii)
a
two
to again await service.
class
example, by Keilson and Sumita
priority as considered, for
Many
analysis.
of the
M/G/1 system with
and
[9],
system with feedback considered, for example, by Takacs [11] and Disney
of priority and feedback interact
service times;
i.i.d.
upon completion of service of any
back with probability 9 to the end of the C2 queue
This system contains as special cases:
prempt-resume
studied: a) the system
is
classes of traffic, Cj and C2. ^^ch having Poisson arrivals and
b) the C] customers have preempt-resume priority over C2; c)
C2 task,
system
will refer to as the feedback
[1],[2].
iii)
an M/G/1
The elements
feedback system and compound the effort required for
in the
its
formulae obtained, however, can be given a structurally simple and
informative form.
The study was motivated by concern
priority
and feedback
Servi [7] and the results
which
is
the p.g.f. of the
mean time
Kooharian
4. In
distributed Poisson approximation introduced in Keilson and
obtained for our feedback system permit exact analysis of a system
a fair approximation to one such telecommunications switch
In Section 2, notation
the
The
featiires.
is first
in the system for the
[6] the
system
in the feedback
system
the busy period
is
is
at the
epochs
number
first
in the
system
pass time and
which low
at
examined. Finally,
approximation introduced analysis
at the
its
priority
and
end of the
The
in
expected value.
is
priority
pass time. This
In Section 6 the
customers and the
of Section 2
is
applied to preempt-
priority customers using the distributed Poisson
results are then validated numerically.
system for low priority customers.
notation will be helpful for reference.
input stream
first
customers leave the system. In Section 7
in Section 8 the analysis
in [7].
The mean time
The following
whose Poisson
found as well as
used to find the transform of the joint distribution of the duration of
resume clocked schedules with feedback of low
2.Notation.
is
related to that of Keilson
found of the joint distribution of the sojourn time of the low
in the
system
equations for the state- space motion are found in Section 3 and solved in Section
leads to the Laplace transform of the
is
in the
C2 customers. Using an approach
the first pass time in the system and the
transform
.
introduced. Next, using analysis of a system without feedback,
number of C2 customers
Section 5 this analysis
number
for the performance of a telecommunications switch with
the class listed in the
page 2
first
Each row describes an M/G/1 system
column.
This should be contrasted with the feedback system of focal
interest
where the C2 customers
return to the back of the line with probability 6 and C^ has preempt-resume priority over
The
C2
follov/ing representative notation will also be employed.
number of C2 customers
N(t)
the
T
the ergodic time in system for
system
in the
time
at
C2 customers
t.
in the
feedback
system.
9
the probability of feedback for
Tgpi
the busy period for
aBPi(s)
E[exp(-sT3pi)]
<t>PKi2(s)
E[exp(-sWj2)],
,
C2 customers.
Cj
in the absence of
Cj customers
the busy period transform for
C^ customers.
the waiting time Pollaczek-Khinchin transform
for the priorityless C12 system.
E[exp(-sWs)],
4>PKs(s)
the waiting time Pollaczek-Khinchin transform
for the priorityless
An
upper case asterisk will always designate forward recurrence time,
*
With
-
1
«T12(s) =
The
1:
^Ar
-
priority
a^-cCs)
(2.4)
E[Ts]s
arrival
=
'-A
pi2a^j2(s)
low
priority
1
no feedback
of a tagged
customer delay
which
(6
is
-
(2.5)
psa.p5(s)
= 0)
exhibited there.
sets the stage for the
C2 customer
to that of a priorityless system of
c.d.f.
(})pKs(s)
system with no feedback has been treated by Keilson and Sumita
this distribution is given next
times
"tS^s) =
and
priority system with
distribution of the
The
1 -
*
^'^
e.g.
this notation
1
The
a-p,9(s)
E[Ti2]s
<t>PKi2(s)=
Case
Cg system.
in steady state finds
Cj and C2
of Ajj(x) and Ay2(x).
traffic
All subsequent
customer.
However subsequent Cj
each interruption time equal
arrivals
to that of
difficult
simpler derivation of
feedback case.
an ergodic backlog distribution equal
2.1 the
effect
L-S transform of
this
backlog
is
on the delay time of the tagged C2
modify the delay time
as interruptions of rate Xi with
Tbpi, the Cj busy period.
page 4
and the
having arrival rates Xj and X2 and service
From TABLE
C2 customers have no
<()p^j2(s)-
more
A much
[9]
It
follows
at
once
that the
C2
waiting time distribution has an Laplace transform given by
Using
similiar reasoning the time in the
=
<h-i2(s)
Case
2:
<t>PKi2(s
+ ^1
<t>pKi2(s
+ ^i
-^i<7bpi(s)) [4], [5].
system has a Laplace Transform
-^i<7bpi(s)) ct^{s
+ ^j
-A^jObpiCs))
(2.6)
•
The feedback system.
The process
N(t)
insensitive to a certain variant
is
discipline. This insensitivity enables
one
to
employ
Little's
on the underlying order of service
Law
[10] to evaluate the expectation
EfT^y^] directiy.
Theorem
2.1:
a)
The system
at ergodicity, £(»=), is given
b)
The
stable
is
by
1 -
when p^ = p j + pj (l-O)""" <
=
and the
idle state probability
p^.
probability generating function of N(oo)
h(u)
1
(l)pKs(C(u))
is
given by
(2.7)
aT2EFF(C(u))
where
C(u)
=
?i2(l-u)
^^
and aj2EFF(^) ^"^ ^KS^^^
c)
+ Xi[
1 -
aBpi(X2(l-u))
(2.8)
],
defined in (2.2) and (2.5).
E[N(oo)] =X2E[T5y3],
(2.9)
and
d)
For a low
priority customer, the
(l-9)^lE[T,2]
E[T '^' ] =
mean time
in the
system
is
given by
+?i2E[T22] + 2E[T2](1-Pi-P2)
2(i-Pi){(i-e)(i-Pi)-P2}
Proof: The ergodic number of low priority customers in the system, N(«»), would be the same in
distribution if the
low
rather than to the
back of
priority
System behavior under
in
which the low
customers that feed back were to feed back to the front of the tine
the line.
Hence, h(u) must also be the same under either schedule.
the front of the line schedule
priority customers
have an
is
equivalent to that for a two priority system
arrival rate
page 5
X,2
and a service time T2EFF having the
Laplace transform given
priority customers of a
Using the reasoning of
in (2.2).
two
priority
M/G/1 system with
(2.6) the time in
arrival rates
system for the low
X^ and X2 and service times
Tj and T2£pp is given by
E[exp(-sT5y5)] =
where
From
<l)pKs(s)
and
cjgpj(s) are
the distributional
•"
<})pKs(s
defined in
form of
distribution of N(o°) and T^
^1 -^i^^BPi^^)) «t2EFf(s
Little's
•"
(2. 1 1)
^1 -^i<^Pi(s))
TABLE 2.L
Law
[8],
provided for conveniencein Appendix A.l, the
are related to each other according to
h(u) =E[uN(-)]
=
E[exp(-(X2(l-u))T3y,)]
(2. 12)
so (2.7) follows.
Equation (2.9) follows from either
Little's
Law
,
in
TABLE
-^
1^
T2Epp and pi are defined
and the Pollaczek
(2. 1 1)
(2.13)
2.1.
Equation (2.10) then follows from (2.13)
Khintchine formula,
-
E[WS1 =
where Ts and ps
From
-pl
1
W5
differentiation of (2. 12).
=B^^^dl^^l2^
H[T3,]
where
from
[10] or
are defined in
TABLE
(2.14)
2.1. (Alternatively (2.10) follows
from
(2.7)
and the
observation that E[N(oo)] = h'(l).)*
Remark: Note
queue.
When
that
X^ =
when Xj =
and
0, (2. 10) agrees
[11, equation (35)].
9=0,
(2.10) coincides with E[Tsys] for the ordinary
with E[Tsys
When 6=0, agreement
M/G/1 queue with two classes of traffic given
In the case of no high priority customers,
h(u)
for the
]
is
found with E[Tsys
i.e.,
Xj =
0,
equation (2.7)
is
equivalent to
0x2^2(1 -u))
-
so
-
E(c«)
=
-(l-P2(l-e)-0u(l-aT2(M^-")))
page 6
by
for the preempt-resume
]
euaj2(^20-")) + (l-6)aT2(^2(l-"))
h(u)
as given
in [9, equation (3.8f)].
(l-u)(l-e-p2)
=
M/G/1 queue widi feedback
M/G/1
"
which
is
consistent with [11, equation (20)]
The motion
3-
To describe
of the system on
state space
its
the feedback system, the following notation will be employed.
Let N(t) be the number of low priority customers in the system.
Let X(t) be the virtual time to availability of service for C2 customers
at time
t,
where B j(t)
the
is
C^ (high
C2 customer who has been
The reader
at
a
Bi(t)
+
interrupted and
is
is
R2(t),
t
and R2(t)
,
a)
a
at
V
=
if
any
not bivarate markov for the following reason.
.
On
When
the other hand,
Cj service completion epoch (when no C2 customers having been
respectively, distinguishes
J(t)
the residual service time of
awaiting resumption of service.
served in the current busy period), N(t) does not change.
V
is
C2 service completion epoch, N(t) decreases by one
becomes zero
X(t)
=
priority) backlog at time
will note that [N(t), X(t)]
X(t) becomes zero
and
are not in service
i.e. let
X(t)
when
who
the server
is
between such system
busy and,
A third process J(t) taking two values S
histories. Specifically:
in the current
busy period, no C2 service has been
provided;
b) J(t)
=S
if the
server
is
busy and some C2 service has been provided during the current busy
period.
Finally, let
With
space
S
E
be the idle
this notation
N = V + S +E
={(n,x,S)
;
1
<n
it is
state.
clear that [N(t), X(t), J(t)]
<oo,
< X < «};
N induced by arrivals,
enumerated in TABLE 3.1.
space
a multivariate
Markov process on
the state
with
V
Let Tj be the service time of a type
state
is
= {(n,x,V);0 < n
i
<oo,
<x <
E
00);
customer having density ajj(x),
i
=
={E}.
1,2
Jumps on
the
busy period terminations and low priority service completions are
page 7
Cj
E
Changes from V or
Epoch
(n,x,V) -->
arrival
(n,
Changes from S
x + Tj, V);
x + Tj,
(n,x,S) -->
(n,
(n,x,S) -->
(n+1, x, S);
S);
E -> (l.Ti.V)
Cj
(n,x,V) --> (n+1, x,
arrival
E
=
;
->(1,T2,S)
(n,0,V) --> (n,T2,S)
termination of the
J(t)
V)
V mode
C2 service completion
(n,0,S) --> (n-l,T2,S)
with no feedback
(1 ,0,S) -->
C2
1
E
feedback
TABLE
When
3.1
underlying service time distributions are absolutely continuous the distribution on the state
space A^ can be described
fvn(x,t)
E(t)
The
n >
(n,0,S) --> (n,T2,S)
service completion
v.'iih
;
in
terms of densities on
= ^Prob[ N(t) =
= Prob
[
System
is
n, X(t)
<x
in the idle state
equations of motion on the state-space
,
S and
J(t)
E
V.
Let
= V]
at
may be
time
t]
written
down
in the
customary way
(see, e.g.,
[6]).
One
^
=
finds that
-
(?Li+?L2) E(t)
+
fvo(0,t)
+
(l-e)fsi(0,t)
page 8
(3.1)
^vo(x,t)
3fvo(x,t)
3i
5x
=
^fvn(x.t)
3fvn(x.t)
5t
5x
=
afg
+ ^1^(0 aTi(x)
-^.jfvo^'^'O
+
-X2fvn(x,t)
(x,t)
-
Xifv„(x.t)
-
>^fvn.i(x,t)
^ifvo(x.t)
+
+
Xifvo(x,t)*aTi(x),
(3.2)
Xifvn(x,t)*aT-i(x),
n^l,
(3.3)
n >
(3.4)
afs (x.t)
5x
5t
=
-
Vsn^'^'t) + Vsn-l(X'0
+
fvn(0,t) aT^(x)
^fsi(x.t)
3fsi(x,t)
5t
oix
=
-X2fsi(x,t)
+
-
^lfsn(X'0 + >.ifsn(x,t)*aT.i(x)
+ (l-0)fs„^i(O,t)aT^(x) +
X2E(t)aTV2(x)
+
-Xifsi(x,t)
fvi(0,t)aT2(x)
+
efs„(0,t)aT2(x),
1
Xifsi(x,t)*aji(x)
+ (l-e)fs2(0,t)a^(x) +
efsi(0,t)aT^(x)
(3.5)
and
E(0) =
Equation (3.1) for example
the Cj and
C2
arrivals
(3.6)
1.
due
to losses associated with
and gains from busy period terminations. Equation
(3.4) says that the time
rate of change of fY„(x,t) seen
{(n,x,S)} due to arrivals of
states that the rate
of change in E(t)
is
by an observer moving with velocity
C2 customers, gains from
losses and gains on{(n,x,S) due to Cj
arrivals, gains
- 1
consists of losses from
{(n-l,x,S} due to
from (n,0,V)
C2
arrivals,
internal
for termination of busy
periods initiated by a Cj customer arrival, and service completion of C2 customers. The other
equations have a similar probabilistic interpretation.
4.
Ergodic analysis
In the next theorem the transform of the joint distribution of N(o«), the
customers
in the
system
at ergodicity,
number of low
and of X(oo) are obtained from Equations
page 9
(3.1)-(3.6).
priority
Let
oo
(I>Voo(u,x)
oo
fvn(x,-) u"
X
n=0
=
;
<I>Voo(u.w)
=L
n=0
oe
00
(t)s^(u,x)
£
=
fs„(x,oo) u"
Os^(u.w) = L
;
[
n=0
(where L[]
is
5; fvn('^.~) ""]
[
^ fj^Cx.-) u"]
n=0
the Laplace Transform operator) and
let
^(u) be the solution of the functional
equation
C(u)
=
?i2[l-u]
+
>.i
[1
-
OtiCCCu))
This functional equation plays a key role in the solution.
equation (4.1) has the unique solution given in
Theorem
4.1
..
^~
As shown
in
Appendix A.2, the functional
(2.8).
(The Ergodic Distribution on the State Space)
The ergodic
^
(4.1)
]
on
distribution of [N(t), X(t), J(t)]
u E(oo) C(u)
^
(
its state
aT2(w)
-
space
is
described by
aT2(C(u))
)
[w->.2(l-u)-Xi(l-aTi(w))][u-a-r2(C(u)) + eaT2(C(u))(l-u)]
'
^
1 c.
<I>V~(u.w) = ?.iE(oo)
I
^
aji(C(u))-aTi(w)
^/'
]\ ^
-—-
(4.3)
.
and
E(oo)=l-ps=l-pi-p2(l-e)-l.
(4.4)
Proof: Equations (4.2)-(4.4) are obtained from the equations of motion, (3.1)
methods employing
provided
in
p.g.f.'s,
-
(3.6),
by standard
Laplace transforms, regularity conditions and algebra. Details are
Appendix A.3.
Remark: By
definition h(u)
and some algebra, equation
=
<I>5„(u,0)
+
<I>y^(u,0)
+
E(oo).
Hence, from equations (4.2)-(4.4)
(2.7) follows. Details are provided in
page 10
Appendix A.4.
The
5.
first
pass time
We define the first pass time of a Cj customer
system
until the
completion of that customer's
at ergodicity is a
to be the time
arrival of that
customer
The expected value of the
service.
first
from the
measure of the performance of the system.
Its distribution is
first
to the
pass time
also an important
ingredient for the calculation of the total time in the system-
Theorem
5.1
(Ergodic First Pass Time and the
transform of the joint distribution of V. the duration of the
,
of C2 customers
T^i(u,s)
in the
= E[u^l*
= ot^^Cw)
[
system
at
the end of the
pass
first
is
Number
first
in
the System):
The
pass time, and of N,, the number
given by
e-sVl]
<I>voo(z.w)
+
+ <Ds^(z,w)
E(oo)]
(5.1)
Here
z=
z(u,s)
= (l-e+Gu) aiv2(w(u,s))
(5.2)
and
w = w(u,s)
=
s
+ ^2(l-u) + X\-Xi
CTbpj(s
+
X.2 (1-u)
(5.3)
).
Hence,
i&l(u,s)
=
(l-(pj+p2))aj2(w)
irc/
[E(~)
N
*^
[
]
6p2 , X (u-z) ,,,,
L_h(z)-4p^]/(l-(p
,
.
H
(1-e)
l-(Pi+P2)a.j,j2(^)+e(l-u)aT^(w)/w
^^l^2J
,,
+P2))
,,
-,
(5.4)
*
where
a
^(w) and h(z) are defined in (2.4) and
(2.6).
Proof: Equation (5.1) follows from a generalization of the argument of Theorem 2.1. Equation
(5.4) follows
from algebra involving
The Laplace transform of the
(4.2), (4.3),
(5.1). Details are
provided in Appendix A.5.
distribution of the first pass time at ergodicity can
evaluating (5.4) at u=l. This transform
Corollary
and
5.2. (Ergodic First
is
given in the next corollary:
Pass Time
The Laplace transform of the ergodic
first
)
pass time distribution
page
11
is
given by
be found by simply
=
+
<l>pKi2(^(s)) aT^(w(s)) [p
(1-p)
h(a^(w(s))) a^(w(s))
(5.5)
]
where
w(s) =w(l,s)= s+X,i-XiaBpi(s),
p = E(oo)
and
<}>pKi2(^) ^^
The mean
E[Wi] =
is
given in
(I-P1-P2)
/
=
(l-pi-p2(l-e)-i)
/
(I-P1-P2),
(2.5).
given by
-4^'i(0)
((l-e)(l-pi)-ep2){ X2E[T22]+(l-e)?ciE[Ti2]-2p2E[T2]
^
)
2(i-pi)2(i-e) ((i-e)(i-pi)-p2)
E[T2](l-9)
^
^^
^^
(i-e)(i-pi)-p2
Proof; Equations (5.5) and (5.6) follow from (5.4).*
Corollary 5.3 (Joint Distribution under no feedback)
ife=o
=
i^i(u,s)
where w(u,s) and
<t)pKi2(^)
^^
aj2(w(u,s))
<})pKi2(w(u,s))
(5.7)
defined in (5.3) and (5.6).
Proof: Equation (5.7) follows from (5.4).*
Remark:
In
thfe
case of no feedback
»Fi(s)=
which
is
*Fj(s)
=
Littie's
,
i.e.,
i3i(l,s)
6=0, from
=
Law,
[10],
and
(2.6)
(5.8)
<lvri2(s)
equivalent to equation (3.5) and (3.7) of
<)>pKi(^) 01^2(5)
(5.7)
[9].
K6
=
and X\=
this
equation simplifies to
as expected.
can be used to relate the average number of customers in a system to the average
time in the system. The following corollary verifies the distributed form of this law for the case of
9=0.
page 12
Corollary 5.3 (A Distributional form of
Ke
=
Law):
Little's
0. then
'¥y(k2
-
>.2U)
=
Oi(l,>.2
-
^2u) = t5i(u,0) = h(u).
(5.9)
Hence
E[N(oo)] = >.2E[Tsys]
(5.10)
Var[T^3] = Var[X2N(oo)] + E[N(oo)].
(5.11)
and
Proof:
If
e =
from
then,
^l(X2-X2u)=
But
if
=
(2.8), (5.3)
=
there
(5.7),
l5i(l,X2->^2u)=l»i(u,0)=
then cXy2EFF(s)
follows. If
and
is
=
aj2(s), <t>pKi2(^)~
no feedback so the time
<t)pKi2(C(u))
^KS^^^ ^°
Hence
6.
equals h(u) so (5.9)
in the system, Tsys, is equal to the first pass time,
V*. Therefore "¥{(0) = E[Tsys], ^i"(0) = E[Tsys2],
E[N(oo)].
O^iW)^°™ (^-^^ ^^
hXD =
E[N(oo)], and
h"(l)
= E[N(~)2]
-
(5.10) and (5.11) follow from (5.9).*
Total time in system
Theorem
remaining
6.1:
Let x(u,s) be the transform of the joint distribution
in the
system
at
of: a) the
number of customers
departure and b) the total elapsed time from arrival to the system to
departure from the system. Then x(u,s) satisfies
X(u,s)
where
Proof:
=
z(u,s), w(u,s)
(l-0)i^i(u,s)
and
+ 0aT-2(w(u,s)) x(z(u,s),w(u,s)))
t3i(u,s) are defined in equations (5. 2), (5. 3)
Using an argument similar
otp2(w(u,s))i5j(z(u,s),w(u,s))
where
to
Theorem
(5.1)
i^j(u,s) is the joint
passes and the number of C2 customers in the system
(6.1)
and (5.4)
one can show
that 'dj^.i(u,s)
transform of the duration of the
at the
end of the
ith pass.
But
00
^
(l-0)0'"^dj(u,s)
so equation (6.1) follows. Details are provided in Appendix A.6.*
i=l
7.
The busy period
page 13
first
x(U'S)
=
i
=
The following theorem
Theorem
7.1:
gives a generalization of the Takacs busy period equation.
(The Busy period)The busy period has a Laplace transform
<^BP(S)
=
^Cy*BPl(s) +
:
''"
1
"''
1
(7.1)
<7*BP2(S)
.
,
2
2
where
a*BPi(s)
=
aT.i[ s
<^BP2(S)
=
0^p2EFf[
+ X,
S
>.jCr*Bpi(s)+ X^
-
+ ^1
-
-
X^^s^^^^is)
XjO*gpi(s)+ X^
'
(7.2)
],
^2'^BP2(s)
C^-^)
1'
TABLE 2.1,
a^j(s) and aj2EFF(s) are defined in
and
p.^P2('-e)-'
E[T3,i=
l-Pi-P2(l-9)-i
_1_^EI±
X,U2
,,.4,
1-Ps
Proof: See Appendix A.7.
Note
that if Xj or
Xj is
set equal to zero then equations (7.2)
and
(7.3)
reduce to the familiar Takacs
busy period equation.
8.
Clocked schedules
Consider a system
Suppose further
the
which the
priority tasks arrive with deterministic interarrival times
that: a) the service
every integer N, Ti
variables; b)
in
is
low
time of the priority tasks, Ti
distributed as the
sum of
N
is infinitely divisible, i.e.,
for
independent and identically distributed random
priority tasks arrive with an exponentially distributed interarrival time
have a service time with a general
kA.
distribution; c) the
low
priority tasks feed
and
back with probability
e.
The
distributed Poisson approximation of [7]
may be used
sequence of Poisson streams parametrized by the variable
having Laplace Transform
a in
(s)
= aTi^'^(s).
page 14
to
model
N with rate
the high priority traffic as a
^in = N/A and
service time
As
in the classical proof of
de
Finetti's
Theorem
aTi(s) = limj,^_^„[exp
i.e.
{
-
[3],
A
>.in
one has
[1
-
am (s)
the distribution of the high priority tasks arriving in
A
Poisson model with arrival rates ^in and service times aiisKs)
be shown
}
(8.1)
],
time units under the approximate
is
exactly ajiCs).
From
(8.1)
it
can
that
limN_^^{>.lNAE[TiN]} =
lim
N_,^
{
XiN A E[Tin2]
}
E[Ti],
(8.2)
= Var[Ti],
(8.3)
and
Pl = lim
Equation
limit, in
N^oo
(8.1) suggests that the
{
^'IN
E[Tin]
}
= E[Ti]
/
A.
(8.4)
clocked schedule with feedback could be accurately modeled as the
N, of a system with two classes of Poisson
traffic
having arrival rates ^in and X2, service
times with Laplace transforms aiN(s) and a2(s) and having a feedback parameter 9. In
this distributed
fact, in [7]
Poisson approximation has been shown analytically as well as numerically to be
highly accurate for the case of 6=0. The
mean
time to completion for the limiting system can
now
be readily obtained:
Theorem
8.1
(Mean Time
to
Completion of the Distributed Poisson Approximation
to the Clocked Schedule with Feedback)
The mean time
to completion
is
given by
(l-e)Var[Ti]/A ^X^EIT^^] + 2E[T2](1-Pi-P2)
'^'
,.
2(i-Pi){(i-e)(i-Pi)-P2)
Proof: Equation (8.5) follows immediately from equations (2.10),
Results
page 15
(8.2), (8.3)
and
(8.4).
^,
To
test the
CASE
1:
accuracy of (8.5) the following three examples were compared with simulations:
X-i
=
Ti has an Erlang-3 distribution with a mean of
1;
distribution with a
CASE 2:
X,i
=
1;
mean of
Xi
=
1;
In
.05;
X2
=
Five values of 9 were examined: 0=0,
mean of .2; Eight
1;
T2 has an exponential
.2;
5;
.75.
T2 has an Erlang-2
.1, .2, .3, .4,
X2
=
.1;
and
.5.
T2 has an Eriang-5
values of 9 were examined: 9=0, .25, .50, .75, .90, .95, .96,
.97.
TABLE 8.1, for each case the value of E[T
of EnTgys], E[Tg
of E|T
from
]
g]
.
,
(8.5),
- E[T,^J,.
IE[T,^J
sys^
sys-'sim
is
given with a
E[T
.
*-
.2;
Ti has an exponential distribution with a mean of
distribution with a
and
mean of
Xj =
Four values of 6 were examined: 9=0,. 25, .5, and
Ti has an Eriang-2 distribution with a mean of
distribution with a
CASE 3:
.15;
.2;
'^
]^^
^"^
,
is
95%
]
is
computed via
simulations.
The simulated value
confidence interval. In addition, the theoretical value
given along with the relative error,
I
theory
EtTsysJsin.
page 16
CASE
e
E[T,y,l,,^^^y
^^^sysJslm
ETT—
T~^
sys^slm
•^
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
3
3
00
b)
all
arriving customers enter the system,
and remain
in the
system
until
served
c) the customers leave the system one at a time in order of arrival
d) arriving customers do not affect the time in the system of previous customers
then N(oo) and Tgys, the ergodic number in the system and the time in the system, are related
according to equation (2.12).
A.2 Proof that equation
Let Q(\) = V + Xi(l
-
(2.8) is
C7gp2(v))
a solution to equation
and V(w) =
w
-
^i(l
(4.1):
ajj(w)) where cy3pj(v)
-
is
the Laplace
transform of a busy period of an M/G/1 queue with arrival rate Xi and service time Laplace
transform ajj(w).
domain,
It
can be easily shown that Q(v) and V(w) are inverses on the appropriate
i.e,
V(Q(v)) = ri(v)-?ii(l-aT^i(Q(v))}
=v+
?ii
{
l-OBpi(v)}->.i{l-aTi(f2(v))} = V
because the Takacs equation
is
equivalent to cJbp2(v)
= ajj(Q(v)).
Hence, on the appropriate domain
w
= Q(V(w))
= V(w) + ?ii(l-aBpi(V(w)))
= w - Xi ( 1 - a^i(w)) + Xj ( 1
-
aBpi(w
-
Xi
( 1 -
aT-i(w))
).
This implies that
aji(w)
Evaluating (A.l)
at
= agpi(w
from
Xi
( 1 -
aTi(w))
(A.l)
).
w = ^(u) gives
=OBpi(C(u)-)ii[l-aTi(C(u))]
aTi(C(u))
so,
-
)
(4.1),
aTi(C(u)) = aBpi(?t2[l-u]).
(A.2)
Therefore equation (2.8) follows from (4.1) and (A.2).
A.3 Proof of Theorem
From
4.1:
equations (3.3) and (3.4) the Laplace transform with respect to the variable x and the
probability generating function with respect to the variable n
t
approaches
<)v^(u, 0)
<«.
-
is
found and
its
limit is evaluated
Then
w Oy<^(U,W
)
= -X2(l-u)Ovoo(u.w) -XiOy^(u,w) + XiOv^(u,w)aTi(w) + XiE(co)aji(w),
page 18
when
so
From
)
-—
-
^i(l
at
-
w
(4.3) then follows
Also, from (3.1),
+
(Xl
dE(oo)
}
>.2)E(~)
=
^(u). Regularity in
.
One
w
in the half-plane {w:
then has from equation (A.3)
(A.4)
>.iE(oo) aTi(C(u)).
from (A.3) and
=
=
=
(A.3)
aji(w))
requires that the numerator also vanish there.
(>V^(u, 0)
Equation
/.2(1"U)
-
denominator of (A.3) vanishes
(4.1) the
Re(w) >0
w
>.iE(oo)aTi(w)
-
({V^(u,0)
^eo(u,w) =
(A.4).
Hence
fvo(0.~) + (l-e)fsi(0,oo)
(A.5)
Equations (3.4) and (3.5) imply that
-
3(t)c^(u,x)
= A2(l-u)Os Ju,w)
+
<tVoo(u.O)
[
+
From
,
^
,
=
Xi(l
-
fvo(0'~)
(l-0)[ <l)soo(u,0)u-i
(A.4), (A.5)
^
<Pr^(u,W)
-
-
aTi(w))
i
-
Og Ju.w)
aT2(w) + ^2E(~)aT2(w)u
fs,(0,co)
]aj2M
+
e(l)s^(u,0)aT2(w).
(A.6)
and (A.6) one obtains,
—<t>s«(u.O)
[
l-eaT2( w)-(l-9)u-iaT2(w)
—
]
w-X2(l-u)-Xi(l-aTj(w))
E(oo)aT2(w)
[
^,(1- Obpi(>>2(1-"))) +>-2(^-")
w-X2(l-u) -Xi(l-aTi(w))
page 19
]
(A.7)
From
denominator of (A. 7) vanishes
(4.1) the
{w:Re(w) >0
=
)
[
1
eaT^(C(u))
-
-
m
=
1 -
[
(4.3)
in the half-plane
then has from equation (A.7)
]
X^( l-aBpi(X2(l-u))) +X2(l-u)
—
eaT2(C(u))
and the condition
-
]
:
and
(A. 8)
:
(l-e)u-iaT2(C(u))
(2.8), (A.7)
and (A. 8). Equation (4.4) follows from equations
that <I)s<^(l,0)
A.4 Derivation of h(u) from equation
(4.2)
One
w
E(oo)aT.2(C(u)) C(u)
Equation (4.2) then follows from
From
^(u). Regularity in
(2.7), implies
^
r
(bo^(u,0)
"^^^
and
=
(l-e)u-laT2(C(u))
-
+ E(oo)a^(^(u))
(4.2)
w
requires that the numerator also vanish there.
<t)soo(u,0)
which, from
at
+ OycoCLO) +
£(«>)
=
1.
(4.2)-(4.4):
(4.3)
O3^(u,0) =
-E(oo) C(u) (1
-
aT2(C(u))
)
^
X2(l-u)[
1
-
eaT2(C(u))
-
(l-e)u-iar2(C(u))
1
and
^
m
<I>V<«(u>0)
.
^r
E(oo),
=
T..
N
E(oo)
+
Xi(l-aTi(C(u)))
+
i
ti—
^
Ml-")
^
/^(l-u)
Then from
(4.1)
<Dvoo(u,0)
+
E(co)
=
E(~)C(u)
But
h(u)
so, after
some
= Os^(u,0) + Ov^(u,0) +
E(oo)
simplification,
h(u)
-E(oo)C(u)(l-9)aT2(C(u))
-
(^ 9^
X2( u
-
euax2(C(u))
-
(l-e)aT2(C(u))
)
-E(oo)C(u)(l-9)aT2(C(u))
X2(l-aT2(C(u))) ->^(l-u)
page 20
(
l-eaT.2(C(u))
)
which, from
(4.1),
-E(oo)C(u)(l-e)aT2(C(u))
>.2(l-aT2(C(u))) +[?ti(l-aTi(C(u)))
which, from
C(u)
-
]
(
l-eaT2(C(u))
(l-e)aT^(C(u))
(Xi+>^)(1- (Xts(C(u)))
1 ^
"
^^
l-eaT^(C(u))
(l-e)a-i^(C(u))
l^Ps
^
aTs(C(u))
l-8aT^(C(u))
C(u)E[Ts]
(2.7).
A.5 Proof of Theorem
The
C(u)
(2.3),
E(oo)C(u)
equals
-
5.1:
joint transform of V*, the duration of the first pass,
the system at the end of the
Cj or C2 customers
first
arrive
E[u^Vs^*
The
first
If J
=S
,
wait conditioned on the value of (N,X,J) upon arrival,
in
if no future
is
N=n,
I
and N*, the number of C2 customers
X = X, J = V]
= (1-e+eu)" aT2"+Hs) e-"
(A.IO)
.
term reflects the fact the each of the n C2 customers will feed back with probability 9.
the right
hand side of (A. 10)
is
modified to reflect the fact that a C2 customer
is
in service
so
ELu'^VsV* N=n,
I
X = X, J = S]
=(l-e+eu)"aT2"(s) e-",
(A.ll)
and
E[u^Vs^*
I
system
is
idle
i.e.,
N(oo)=0,
X = 0, J = E]
= a-n(s).
(A.12)
Hence,
00
00
E[u^VsV*]= X
f
(l-0+0u)"aT2""'ks)e-" fvn(x,~)dx
n=0 s=0
00
00
+
I
f
(1-0+eu)" aT2"(s) e-" fsn(x,«>)dx + aT2(s)E(-o).
n=Os=0
page 21
(A.13)
Therefore,
E[u^VsV*] = a^(s)
(Dv^[(l-e+eu)aT2(s),s] -KDs^[(l-e+eu)aT^(s),s] + aT^(s)E(oo) (A. 14)
In the presence offuture Cj arrivals, the
the
Np,,
first wait,
is still
each interruption time
is
number of C2 customers
V
N*. However, the Ci arrivals modify
in the
system
at the
end of
by interruptions of rate Ki with
equal in distribution to that of Tbpi, the Ci busy period. Hence, in the
presence of future Ci arrivals the joint wait transforms
of,
V,
,
the time to completion of the
C2
*
customers and Npj
is [4], [5]
E[u^Pi*e-s^i*]
where
= E[ u^* e<^ +
CJgpj(s) is defined in
^1 -^i«^bpi(s))V*
(A.15)
]
TABLE 2.1.
In the presence of both future Cj arrivals and future C2 arrivals the time to completion C2
customers, Vi,
first wait,
Npi,
is still
is
V.. However, the
number of C2 customers
in the
system
at the
end of the
a modification of Np. which reflects the Poisson stream with rate ^2 that arrives
*
during V,
.
Hence
the joint distribution of
= E[u^Pi
i^l(u,s)
Combining
e-sVi]
[
=
[
+
^ ->^u)Vi*
aT2(w)E(°o)(w
'"
M=
E(~)]
T..
w
<^Soo(z.w)
(^
j
(^
16)
(4.3),
.
^r^
.
X
o^(w)
Oy^(z,w)
+
(4.2)
= E[u^Pi*e-
is
(A. 14), (A.15) and (A. 16) gives (5.1)-(5.3).
Fix)m (2.8) and
From
Npi and Vj
-
^2(1-2)
and (A. 17),
+ aT2(w)
[
Oy^(z,w) +
IM
E(oo)]
,
w-X2(l-z)-Xj(l-a-p2(w))
page 22
-
Xi(l
-
-
C(z))
—-
aTi(w))
(A. 17)
z C(z)
^
z
From
(aT2(w)
aj2(C(z)) +
-
aT2(C(z)))
-
,
,,
^,
,,
aT2(C(z))( 1 -z)
(5.2)
E(oo)a-jv2(w)
=
ii^
[
]
w-X2(l-(l-e+eu)aT2(w))-Xi(l-aj.i(w))
C(z)(z-(i-e+eu)aT2(C(z)))
—
[
z
-
+
w
., ,
C(z)
-
],
aT2(^(z)) + eaT2(C(z))(l-z)
which, from (2.1) and (2.4)
E(oo)a^(w)
=
eC(z) (z-u) aT2(C(z))
*
I
—
[
J
z
w-(Pi+p2)wa.j.j2(w)+e(l-u)aj2(w)
-
+
w
J
ay2(C(z)) + 6aT2(C(z))(l-z)
which, from (A.9),
E(co)a^(w)
=
[
-h(z)X2e (z-u)
-^
J
]
[
w-(pi+p2)wa.j-j2(w)+e(l-u)a^(w)
which
+
w
]
E(oo)(l-0)
simplifies to (5.4).*
A-6 Proof of Theorem
6.1:
Using
the
argument of Equation (A. 10) one finds that
joint distribution of the duration of the first
i
sj'stem at the end of the ith pass if no other Cj or
E[u^VsV* Npi=ni, Vi =
I
vj]
Cj customers
argument of (A. 15)
to
= (l-0+eu)"i
cy3pi(s)
is
system
at the
the
in the
(A. 18)
aT2"i"^'(s) e-^^l
end of the
is
ith pass,
Npj
,
first
is
i
passes, Vj*
defined in
= E[ u^*
e-(s
+
h -Mobpi(s))V*
TABLE 2.1.
page 23
]
,
found using the
be
E[u^P2*e-sV2*]
where
in the
V*
arrive then
In the presence offuture Cj arrivals, joint transform of the duration of the
and number of C2 customers
if
passes and N* be the number of C2 customers
(A. 19)
In the presence of both future Cj arrivals and future
Vj
,
number of C2 customers
argument of (A. 16),
at the
end of the
ith pass,
Npi
,
is
i
passes,
found using the
e-sV2]
= E[u^P2*e-
(^
+ h. -^u) V2*
]
(A.20)
(A. 1 8), (A. 19) and (A.20) gives
t52(u,s)
where
system
arrivals, the duration of the first
be
= E[u^P2
T^2(u,s)
Combining
to
in the
C2
z(u,s)
= aT2(w(u,s))di(z(u,s),w(u,s))
and w(u,s) are defined
Using a similar argument,
i^i+i(u,s)
for
=
i
=
in equations (5.2)
and
(5.3).
1,2,...
we uncondition on the number of passes then we find that
C2 customers in the system and the total elapsed time is
If
X(u,s)
=
y
(A.21)
aT2(w(u,s))T3j(z(u,s),w(u,s)).
the joint distribution of the
number of
(A.22)
(l-e)ei-li3i(u,s)
1=1
which equals
=
(i-e)di(u,s)
+ e
£ (i-0)ei-idi^i(u,s)
i=l
which, from (A.21),
CO
=
(i-e)di(u,s) +
eaj2(w(u,s))X
(i-e)ei-ii»i(z(u,s),w(u,s))
i=l
which, from (A.16) equals (6.1)*
A.7 Proof of Theorem
If CJgpj(s) is the
arrivals,
7.1:
Laplace transform of an busy period of Cj customers assuming no C2 customer
i.e.,
aBpi(r)
=
otri(r
+ X^
-
X^(5^^^{r)).
page 24
(A.23)
Define
(y*Bpi(s) to
be the Laplace transform of a busy period that begins with a Cj customers
presence of both Cj and C2 arrivals.
in the
Then, o*Bpj(s) corresponds to the duration of a Tbpi
interrupted at rate X2 by a interruptions having a duration with
Laplace Transform cr*Bp2(s).
Hence,
«^BPi(s)
=
The busy period of the C2 customers
assumed, as was done
in
+ ^2 - ^2^BP2(s))-
C^BPi^s
Theorem
is
(A.24)
independent of the order of service. Therefore
2.1, that the
it
will
be
customers feed back to the front of the line rather
than the back of the line and therefore have an effective service time given in equation (2.2).
If this service time is
modified due to the interruptions of the Poisson stream of Cj customers
having busy periods with a Laplace transform Ogpj(r) then the modified service time has a Laplace
transform,
«T2EFfW = "t2EFf(^ +
and a busy period
Cy*BP2(s)
From
(A. 23)
cy*Bpi(s)
=
starting wiUi a
= aT2EFF(s
and (A.24),
aq<i(s
+ X2
-
C2 customer is given by
^2
"^l^BPl^^^
=
?i2a*BP2(s)
= OjjEFF^S * ^2
"
which from (A.26) implies
s
+ Xj
+ ^i
-
-
(A.26)
^
X2CT*Bp2(s),
^i<7bpi(5
"•
^2
'
^2^BP2(^)))
(7.2).
(A.25) and (A.26), setting
Cy*BP2(^)
"
setting r
which from (A.24) implies
From
"*"
(A.25)
^i^piW)
^1
r
=
s
^2'^*BP2(^)
+
"^
^.j-
^1
^2'^BP2(^)
'
^l^BPl^^
"^
^2
"
^2*^BP2(^))
(7.3).
Equation (7.1) follows from the definition of a3p(s), cy*BPi(^)' ^^^ ^bp2(^) ^"^
differentiations of (7.1), (7.2)
and (7.3)*
ACKNOWLEDGMENT:
The
which was used
to generate
authors are grateful to A.
TABLE
8.1.
page 25
Khoo
for
tiie
^'^•^^
^^°"^
programming support
REFERENCES
[I]
Disney, R. L. 1984. "Stationary Queue-Length and Waiting-Time Distributions in Single-Server
in Applied Probability. 16. 437-446.
Feedback Queues". Advances
[2] Disney, R. L. McNickle, D. C. and B. Simon. 1980. "The M/G/1
Bernoulli Feedback" JVava/ Research Logistic Quarterly. 27. 635-644.
[3] Feller,
W. "An
and Sons,
New
Introduction to Probability Theory and
instanteous
Applications", Vol. 2, John Wiley
York, 1971.
[4]
Gaver, D. P. 1962. "A Waiting
the
Royal
[5]
Keilson,
Time
with Interrupted Service, including Priorities", Journal of
Statistical Socieity, Ser. B. 24. 698-720.
1962. "Queues Subject to Service Interruptions". The Annuals of Mathematical
J.
1314-1322.
Statistics. 33.
[6]
Its
Queue with
Keilson,
and A. Kooharian. 1960.
J.
Mathematical
Statistics.
"Time Dependent Queueing Processes". Annals of
31,104-1 12.
[7] Keilson, J. and L. D. Servi, "A Distributed Poisson Approximation for Preempt-Resume
Clocked Schedules", submitted to IEEE Trans, on Communication.
[8] Keilson, J. and L. D. Servi, "A Distributional Form of Little's Law", submitted to Operations
Research Letters.
[9]
Keilson,
Priority
J.
Queue
and U. Sumita, "Evaluation of the Total Time in System in a Prempt-Resume
Advances in Applied Probability, Vol. 15, 1984,
via a Modified Lindley Process,
pp. 840-859.
[10] Little,
J.,
"A Proof of the Theorem L = >.W", Operations Research,
9,
383-387
(1961).
[II] Takacs, L.
"A Single-Server Queue with Feedback", The Bell System Technical Journal,
March, 1963, pp. 505-519.
page 26
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