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A-R126
545 THE OUTPUT
OF M(T)/G(T)/INFINITY OUEUES(U)
ARMY
1/1
INVENTORY
RESEARCH OFFICE PHILADELPHIA
PR M KOTKIN
JUL
82 USAIRO-TR-82/2
UNCLASSIFIED
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MICROCOPY
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MICROCOPY RESOLUTION
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.4
TECHNICAL
REPORT
USAIRO-IR-82/2
AMSAA
THE
OUTPUT
OF
M(t)/
G(t)
0
CUEUES
INVEN
TOI
1
RESEAICH
OFFI
CE
July
1982
-
US Army
Inventory Research Office
US Army Materiel
Systems Analysis
Activity
800 Custom
House,
2d
&
Chestnut
Sts.
Philadelphia, PA 19106
82
20
0 8
-
8/18/2019 Ada 120545
4/24
Information
and
data
contained
in
this
document
are based
on
the input
available
at
the
time
of
preparation.
The
results
may be subject
to
change
and
should
not be
construed as
representing
the DARCOM
position
unless
so
specified.
A
-
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8/18/2019 Ada 120545
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SECURITY CLASSIFICATION
OF THIS
PAGE (When
DO*
SnIM_
REPORT
DOCUMENTATION
PAGE
______ __
C-___M__M
1.
REPORT
NUMKII1
GOVT ACCESSION
NO. 3. RECIPIENT'S
CATALOG
NUMBER
DO-4Lor
5
_________
4.
TITLE
,d t u*)
S.
TYPE OF REPORT
&
PERIOD COVERED
Technical Report
TE
OUTPUT O 1(t)/G(t)/- Queues
S.
PERFORMING
ORG. REPORT NUMBER
7.
AUTHOR(,)
S.
CONTRACT
OR
GRANT MUMER(. )
9. PERFORMING ORGANIZATION
NAME
AND
ADDRESS
IO.
PROGRAM ELEMENT.
PROJECT, TASK
, AREA WORK UNIT NUNSEIRS
US
Army
Inventory
Research
Office
Room
800
-
US
Custom
Rouse - 2nd & Chestnut
Ste.
Philadelphia,
PA
19106
II. CONTROLLING OFFICE NAME
AND ADDRESS I2.
REPORT
DATS
US Army
Materiel Development
&Readlness
Commsnd nv12
5001
Eisenhower
Avenue
1-.
-#I-JER OF PAGES
Vlezandris,
A
22333
18
14.
MONITORING
AGENCY NAME
&
AOORESSKU
ElHmo t
m
C
aWIltoI
O0160)
IS.
SECURITY
CLASS.
(ofiiV
.pmd)
Army Materiel
Systems Analysis Activity
Aberdeen Proving
Ground,
M
21005
UNCLASSIFIED
I"a DE FSIICATIONfOWNGRADING
N.,ISTRIBUTION
TATEMENT (of Old Rhpt
Approved
for
Public
Release;
Distribution
UnlimitedV0
17.
DISTRIBUTION STATEMENT
(iiiu
468 susentmie
n =00 . It
tgbUuE bO
tm)
IL
SUPPLeMENTARY
NOTES
Information
and
data
contained
In
this document are b88
input avail-
able at the
time of preparation. Because
the
results
may
be subject
to
change,
this
document
should not be
construed
to
represent
the
official
position of
the US Army Materiel
Development & Readiness
Command
unless so stated.
it. KEY WORDS
(Gwktw.
M revee asi
If
neeeam
En i~i or Weeka-ber)
tNon-Eomoseneous Poisson Process
Infinite
Server
Queues
Departure/Output
Process
Laplace
Transforms
-
We study an
Infinite server queue
In which the
arrival process is a Non-
Homogeneous
Poisson
Process
(UHPP) and in which
the service
tines of customers
may depend on
the time service was
initiated. We establish
a splitting
theorem
for
XHPP similar to
the well known splitting theorem
for
Stationary Poisson
Process. Wlth
this
theorem,
we
show
that
the departure
process of th e
M t)/0 t)/l queue is
a
IP,
and that the number
of departures during
any -
.
OD
W3 E3IMNTWS
9 a
s
ES
USUTa
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1UfTWTW1M
TV
LMPICATWU
Or
To-i
PAGMN .
noe. hM.
NTU
(COT)
interval
Is Independent of
th ni.t'of
remaining
customers at the
end of
the
interval.
The splitting
theorem
can
also
be
used show that
the nuner
of busy
servers
at any
tIme
Is Poisson
distributed.
0
A?
SE~IIRuNITY CLAW1IiCATIONI OF
r
THIS
P A Q K ~ r M
D418
RAW
A... . . .
.
_
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. .
. .
.
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La p -
Th3U
7
1WWh
2.
11o- eo.uu
Polsou 1oce
in...............................
2
3.
A Spllttfto
Thsmoao.....
..............
o...
o....
.........
...
3
4. The
Output of
an X t)/G t)/o
QU ..................
.........
8
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M16
D --
Innv17
4
Dist
Sp..1aC3
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oe
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se
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* *
1.* Introduction
We
consider an Infinite
server queue to which
customers arrive
accordiu,
to a
no-
o..gsasus
Poise=n
Process (UP) with known Intensity
X
t)
9 t >
o.
A
customer
arriving
at
tim
a
immediately
eaters service
and
with
probability
1(s,w) will
complete
service by
time
w
where, of course. 7(a~v)
- o for
w < a. Service times for
customers are assumed to be statistically
Indepen-
*
dent
of each other. Such a
queue
will be referred
to an an N(t)/G(t)/o
queue
to
Indicate
that the arrival process is
a
UIPP,
that a customer' a service
tim may depend
on the tim
he
entered
service
and
that
there are an
Infinite
nuber of servers. If 7(sqw)
G
-)
Is a function
only
of
the
difference
was and X t) -
A
for
all
t.
we
haves
as a special
cases the
familiar
K/G/.
queue.
An Important
stochastic
process
of
Interest
In
queueing
theory
Is
the
counting
process (D(t) *
t t~
o)
where
D(t) is
the
number
of departures
from
the
queue
by tim
t.*
In this note
we establish
and use a splitting
theorem
for NUP? to show that
for
the X)/tIjeu.
Dt)t
o) Is
s.
NM.
Furthermore,
we
show
that the
nuber of departures;
froa
the queue during any
finite
tim Interval Is Independent of the
wAer of
customers still in service
* at
the
end of that Interval.*
As
special
cases of
this result
we show that
the
transient
output
of an
X11010 queue to
a
IMP
and
that in steady state
the
output process of an N/G/w
queue Is a
stationry
Poisson Process with rate
equal
to the rate of the
Poisson
input process.*
These
results for the K Gb
queue
were
first proved
more laboriously by Virasol.
14].
Hillestad
ad
Carillo
[11] have shown that
the
number
of
busy
servers at
tim
w In the
)f(t)/G(t)/.o queue
Is
Poisson distributed
with sm depending
on w
and
the
formi of
the
service
time distributions This result
can be simply
obtained
by
judiciously using
the splitting theorem
established In Section
3.
To
the author's
knowledge, no
study
has bee made of
the
output
process
of
the
)f t)IG t)/40
or X(t)IG(t)/s queues.
In a
smequel
to
this
report,
we
shall study the output
process of 11(t)
IN/s
queues.
2.e
o-ooeeu Poisson Processes
Definition
2.1. Let
(N~t)
,t
~ )be
a
stochastic counting
process
and
q let
a. N ()
-o
b.
(N(t)v t o)
have
Independent Increments.
2
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c. Pr(2 or move
events In (tot+h))
O h).
d. Pr(exactly one
event in (tt~h)) A
t)h + O(h).
Then{N t)
0 t
>
o)ie-.
said to be
a no1-lMogene2ous
Poisson Process
with
*
intensity
function
X t).
Let 1(t)
-fl(s)ds. Then
It Is
straightfoward
to
show that
Definition
2.*1
in equiLvelgnt to
Definition
2.2.
Definition 2.2.
(1(t), t o)is said
to be
a UEVP with
mean
value func-
tion
OW) X(t)
if
a'.
N(o)
o
bW
N1 t),
t
>
o) bes
idenetIncrements.
c' for also
Pr(N(t)
-
N s)
a
)
6-(K(t)-Ks)
()-
ni
in this report,
we shall
assume
the WV
of
all
NP to be
completely
differentiable.
mhe proofs presented
here can be extended
to
th
ca
where
the MV
Is not
completely differentiable
but this
would
add
unnecessary
technical
complexity
to the
discussion herefn.
3. A Splitting
Theorem
it
is well
known
(see
15] or
16]) that If
(N(t), t > o) s
a stationary
Poisson
Process
with rate
I end en
event that occurs at time
t
Is
classified
of
type
I
with
probability
P.,
1
...
.
Independently
of
the
classification
of
other
events
ZP~
1),
then the stochastic
Processes
(Ni~t),, t >0), iuml,...,
n
are
independent
stationary Poisson
Processes
withirates
XPI , I -l,..
one
In
this
section
we prove
en
analogous
result for INO?.
Theorem,
1:
,Let'
(N t),s
t > o) be
a
NOPP
with
Intensity X (t) and
MY 1(t) .
An event
that occurs
at time w 31
Is classified as
a
type
1 event with
probability
Pl(v)
and
as
a type
2
event
with
probability
P
2
V)
-
1
- Pl(w).
Classification of each
event
Is
Independent of
the
classification
of
other
events.
Let
NiL t)
be
the
number
of
type
I
events
that
have
occurred
by
time t. The
(Nict), t
>
o)
Is
a
NEP?
with Intensity X(t)PiL(t) and Kyl
N
1
(t)
A s)P
1
(s)ds,
12
ute
oe
{N(t)
t )'
and
(N
2
(t)9t
1
are
satistically Independent.
Before proceeding with the
proof of
Theorem
1,
we need
to
establish the
* following Lines
which Is given an
en exercise In both lose
[6,p2g]
and
Parson
15,p1433.3
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Lms 1-:_Let t),
t
o)
be
a NIP? with
Intensity I (t)
and WI X(t).
Given
that(t) -
a, the
joint
distribution
of the a epochs
at wlich the events
occurred, T. ,
.2...,n
v
Is
the ms
as If
they were
the
order
statistics
corresponding to
a
Idep t random varables yl Y2
"'yn with comn
distribution function
i s)
-
0
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Since the
Intervals Iw ±41i] do not overlap,
we have
from (b)
of Definition
2.2 and (4)
that
(3)
become af
ter soms
reduction
*-K t)
,(w±4hiL) -~:)
-1X(t)
1 t)n
mn1
n
Pr(wjyicviL4hi,
imi.
n 1 t)-n)
-t
n
1(wihi)
-
(w
1
)
Thi1
Now
take the limit of both
side s
the
hi
oo,opI-... n uniformly.
Since
1I.a
X(w,+h,)
-
Wdi
d
*
h,1
hi
j
and since the
MY~s
are
differentiable,
the limit
of
the
right
had
side
of (5)
exists
which
therefore
implies the
limit on
the
left
side
of equation
(5)
lso
azeists ad
it Is, In
fact, an
ordinary probability
denity function. Thus,
f (w
1
...
owa-jN(t)
-
n) - ha
Pr wci~ih,1.mleoemnjN(t)-n)
hi0
which
is
precisely
the
probability
density function
(2).
Lem
1
states
that
given
N(t)
-
no the
n event
times,
Tl9T
2
9**sTn
when
considered as unordered random variables
have
the sme
joint
distribution as
n
idependent random variables
with comen distribution 7(s) given
by
(1).
Therefore, as
a consequence of Lin 19
gIven
that
we know an
event
occurred
In (not), the actual
time that the event occurred
has
distribution
function
F(s)
* and Is Independent of
the time of
the
other
events occurring
In (ot).
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*
Proof of Moeores
1. lb
prove Theorem
1 by first
shoving
that
(N (t) ,t
>
0)
1 1,2 satisfy
Definition 2.2
for
131?.
We
shall show that
Ni(t) Is Poisson
distributed with
mean Jx.(s)p
1
L(*)ds by
calculating the genrating
function
of
11(t)s
I 192.
The Independence
of the
two
processes
is
established
by
cal-
culating
the
joint
generating function
of
Ni(t)
and
N
2
(t)
and
shoving
that
this
joint
generating
function
is
the product of the
Individual
generatift
functions.
Clearly,
N1(o)
N
(o)
- o so that
(a) of
Definition
2.2 holds.
For
1
1l,2,
t
,o
and laIi<
let
giL(z
t)- [ I
be
the
generating
function
for Nict).
By
conditioning
on N(t)
and applying
the
Law of Total
Probability
we
have
that
Ni(t)
g
1
z,t)
-'t'
1()](6)
From
Lea
1, given that
an
event occurred In o,t
J,
the probability
it
occurred
at
time z, o
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But (8)
Ia just
the generating
function of a Poison
random
variable
with mean
t
0
Thus, N1 t) i
- 1,2 Is Poisson
distributed
with mean given
by
(9)
for
all
t
> o. By
a
similar argument
one
can show
that
fgr
any
interval (tl
1
t
2
],
Ni(t
2
) -
N
1
(ti)
is
Poisson
distributed
with m ?X(z)p(x)dx
since
given
that
an
event
occurred
In (t
1 5
t
2
J, it is
type i with
probability
.,
w± (t.t€
2
)
-
J
i
1
.(z)
(td
)
--
i-l,.)
n t
Z)
dz
1-
1,2
ti
Independently
of the other events
occurring
in (tlt
2
j.
That
each process
has Independent
Increments
follows
straightforwardly
from the
fact that
(N(t). t
31 has
It
nrments
and that
each
event
is typed Independently
of
other
events.
If we consider
any
finite number
of
non-overlapping
time
i
ntervals, the
nmber of type
±
events
that
occur
in
each
interval depends
only on
the number
of events
of the process
(N(t),
t
> o)
that occurred
In that Interval
and the typing
mechanism
since
only
events of
the
process
t), t 3_-}
occurring
In a particular
Interval
can
be
classified
and
counted
as
a
type
I
event
In
that Interval.
Since
(N(t),
t >
o)
has
Independent
Increments,
the fact
that the processes N t),
t >
o) I
-
1,2
have Independent
Incrments follows
Imediately
by
conditioning
on the
number
of events
of
{O(t).
t
>
o) that
occurred
In
each
Interval.
All the
conditions
of Definition
2.2
have been
satisfied
and
therefore
Ni t),
t >_.
o)
1m,2 are NHP.
All
that rmins
now In
proving
Theorm
1
In to show
that Nl t)
and
N
2
(t)
are Independent for
all t. Consider
the
joint
generating
function
Z~z
E~zN
(t)
a
N
2
(t)
1
2
,t)
1
(2
-
-z Is n jA ri(t)
- ,
2
(t)
- 1N(t)
n + a)
nwo swo
Pr(N(t)
- a
+ a))
*
from our
previous discussions
we realize
that
Pr(Nl(t)
ano
N
2
(t).
l1t
+
2)
74)
F
7
r
'" * ,
;,., .,,,:,- - " -: • "
" -
. .
-
8/18/2019 Ada 120545
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-- -- . ,~.- .m - .
since
given
N(t) a + n. there are
ways
of choosing
u
events
to
be
type
1 (and thereby choosing
the
z to be
type 2). Thus,
g(z
1
z
2
t
z z
3
ft_) ~
,z2,)
12
n
1
2
-MtYl~~
E
-~tY
22 1(t),
where we
have
used
the fact that
e7*M(t) -
*(t)T
1
-14(t)Y
2
since
Yl+ Y
2
-1
Thus,
7
g(5
1 9
z
2
,t)
-
(zlt) g(z
2
,t)
Since
this holds
for
all z
1
,z
and
t, N
1
t ad
2
(t) are independent
an d
this completes
the
proof.
Note that it
is trivial to extend Theorem 1
to the case
when
an event
at
tine t can be
classified, Independently of other
events,
into
one of
u
categories lith the probability
that
the event is
classified as
type i
being
Pi t)
with E piL(t)
-
1 for all t.
This
splitting theorem
for
141FF
is useful
in proving
many Interesting properties of
NEPP. In particular,
we
shall
use
It
to
show
that
the
output process of the
X(t)/G(t)1
queue
is
indeed
a
NEPP.
4.
The
*Output of
an
N(t)IG(t)1
Quee
In this section
we
will determine
the distribution of
the number
of depar-
tures In an arbitrary
Interval
(t,
+T)
of
length
T
In the
)1(t)/G(t)I- queue.
Recall
that In this queue, service
tine are
independent
of each other
and
that If a customer arrive@
at time s,
the
probability he
will
complete
service
by tine w is
1 sw),
with
d
s,w))
a 1
Let
V~s,w)
- 1 - F(s,w). We shall classify any
arrival Into
one of three
types:
Type, 1: if the arrival
will
depart the
system in
(t,t4T]
Type 2: If the arrival
will
depart
the
system
in (t4T,w)
Type
3: If the
arrival will
depart the system
in
o,tJ.
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Let
pi(x) be the
probability
that t customer
arriving at tine
x
Is
classified
as a
Type
±
customer, ± - 1,2,3.
Then, for all x
> o
plCx) - (zlt+T)
-7 xlt)
P(
-
N(,t--T)
I
r
(a
zt4T)
P
3
(x)
-
l~z,t)
Vhere
3
we have, of course, adopted the convention
that ?(=,y) o
if
z
y Note
-:that I
pi(x)
-
1
for all x and
since, by assumption,
each
arrival
Is
classified
indepindently of other
arrivals,
the conditions
of Theorem 1 have
been
satisfied.
Therefore,
if
NI(v)
- the
number of
customers
who have arrived by tim
v end
are categorized
as Type
i, we have from
Theorem 1 that {Ni(wv v
2,
) ,
i
1,2,3
are
NHPP with NW fX(s)p
1
(s)ds,
1 192,3, and
are
utually
independent.
We recognize
tdat {N
1
(w), w >
o) Is simply
the counting process
describing
departures
from the queue
In (t,t+-T].
In
particular, 1
1
(t+T)
Is
the number of arrivals
by
tOT who have departed In
(t~t+TJ
and
N
1
(t4'r) Is
independent
of N
2
(t4-T)
which is
the number of arrivals by
t4 T that are still
*
in the system at tim
t4T
* Thus, We have the
Important
and
Interesting result
*
that
for
an
)1(t) IG(t) /- he
number of
departures In any Interval
Is imdedent
* of
the
number
remaining
in
the system,
at
the end
of
the Interval. Tkis
generalizes Mirasol'
s
result
for N/Qle queues. urtheruore, N
1
(t+T) Is
I"de-
pendent
of
N
3
(t+T)
-
N
3
(t)
which
Is the
numer
of
departures
in
(o~tJ.
In
fact,
*if
we take any countable
-mimber
of nn-verlappin
Intervals,
using the
same analysis as above,
It is
straightfoward
to shmw that
the
departures during
any of these
Intervals
is Poisson
distributed
and Is Independent of departures
during any other
Interval.
(If
there are
N non over-lapping Intervals le t
pn
(z)
e the probability that an arrival at tim
z will complete
service In
the nth < N interval.*
Then argue
as above).
Clearly
then,
the conditions
of Definition
2.2
are
satisfied,
and
by letting tno
we
sea
N
1
(T)
- D(T)
so -
that
we
have established
the
fact that
(D(w)9
w > o) Is
a
NUP and for
all
T
.
o,
D(T) Is
independent
of
the
number still
In
service
at tim
T.
Also,
we
have
as
a by-product that N
4t+T)
Is
Poisson distributed so that
the nuber
of busy
soriers
at
time
t
in the X(t)/G(t)/ft queue Is Poisson
distributed
with
Mean
f
(X)
i z,t)dx.
For all t
l.o
then,
D(t) is
Poisson
distributed with
14VF
t
14 (t)
-/A(z) 1 zot)dz (10)
0
-
8/18/2019 Ada 120545
16/24
ad
Intensity
function
d
t
(t)
%(~t)
jr A~z) f (zot)dz
+
()
tt
il~~
(t
,€:
.
(,,,)
(U )
w e e f(X. t)
m and I(t,t)
o
only
if
the service
distribution
at
Ot
time t has
an
atom
at o.
R tturning
to
the
process {N
1
(w), v
*
0)
Ve note
that for all t,T
> o
u[1g tT)
I
I
rnumber
of
arrivals
by
t+T
who
1 departed
the
system
in (t
t4T)
- [wmber of departures in (t, t4T] ]
t4'r
t4,T
. I )(x)pl(z)dz
.1 (xZ)1(Xgt4-T) - ? xt)ldx
0
0
t
,- ,
(Z) [1rt+n)
-
1Cxt)]
dz
0
O+T
+
X(z) 7(Xtt+T)d
(12)
t
The f irst term n (12)
can
easily be seen to be
the
number
of
arrivals
in
(ot]
who
depart
In (t,tT] and
the second
tern Is the
number of
arrivals
in
tt+T] who
dpwrt
in
(t.t+T].
(12)
can
also be obtained by
noting
that
I[w
1
(t+T)J
-
D(z)dz.
t
5.
Sp9ecial,
Cases
a.
WOWI@I am u
for w a lot
IP(,v)
*
G(v-e) be
a
function
only
of the difference
v-s.
This Is
just the lUnacrkjl
definition
of
service times
in queues.
Prom
(12)
(or equivalently, from
(11)) ve have
that
X
1
(t+T)
Is
Poisson
distributed with
mean
t
t+T
A )10) [
(t+T-z)
-
o(t-z)ldz
+
A() G(t+T-z)dz
0
t
t4T
t
A z) O(t4,-'z)dz
- /1(z)G(t-'x)dz
(13)
o
0
b. WE/. Qum
TAt
the service
tine distribution
be s
n Case
a. and let
X(t) -
10
-
8/18/2019 Ada 120545
17/24
for all tq
o.
Fron (13) we see that
t+T
ZEN (t')]
- 1 X(y)dy
14)
t
From (14) we note that
t4'?
Ila
E[N(t+T)]
lin
I
%G(y)dy
-AT
t-W
t-i t
so that
in
steady
state,
the expected
number of
departures
over
an
interval
of
length T is
simply XT. Thus, we would
expect that
for the 141GI-o
queue,
the
steady
state
output
rate
would equal
the Input rate A. This
is verified
directly from (11) since the Intensity function of the departure
process is
t
- (t) -
0 Xg(t-x)dz
+
AG(o)
.'. - Aolt)
end
l"a
t)
t44,
This interesting
result
for 14/G/
queues
was
first
obtained by
Niareol [51.
c. M(t)/D/- QuEe
: f if t
<
x + T
Let
F(z,t) -
for
all
x represent the - distributiona
Vr
'a-determ1:ni-stic service
tUse of length
T.
From
(10),
ND(t)
$ X(z)dz -
the expected number of arrivals In (t-Tt)
t-T
end from
(11)
XD (t) - x(t-T)
That
i, the
departure Intensity at t
Is simply
the
arrival
Intemity
T time
units
previous.
This
Is
as
It
should be since
for
h
small,
Pr(departure in t,t+h))
- Pr(arrival In (t-T, t-14)
- A(t-T)h +
O(h)
by
our assumptions on the arrival process.
N11
Po
.
-
8/18/2019 Ada 120545
18/24
d. Quenes
With
DiiIZat
A MIP
with
intensity
function A(t)
is said
to
be dyin
if
(1)
o
_X(t)
'
for an t
Z
o
(2)
It1(t)
-
o
Thus, a MIP
is
dying
If
its
IntOnSity
function
to dying out
with
time
and
If
during
any
finite Interval
the epcted
umber
of
events of
that
process that
occur
is finite.
Intuitively, ve
vould
expect that if
the Input
process to
a
queue Is
dying
that the
output
process
should
be dying
as wal.
Theorem
2,
a Tauberia
type
theorsal,
verifies
our
Intuition.
Theorem
2: ,or
the
X(t)/o(t)/4
quue,
the departure
process
is dying
If
ad
only
If
the
arrival
process
Is
dying.
oof:
sall
prove
Theorm 2 for
queues whose
arrival
nt ns ty
function
and
Its derivatives
are Laplace
Trmfocmnble
for
all
Re(&) o.
(s" 17)). Lot XA(t).
XD(t)
be the
Intensity
function
of the
arrival and
departure
processes, respectively.
from
(1)
IS
I '
+
az A(t)(t:)
since
o <
ls.
A t)l tt)
I U
2
. A(t)
- . T'herefore,
to
study the
lsiting
behaviour
IF\(t)
it
sufficer.
su
the
lIting
behaviour
of
the function
t
b
(t) -
0 IA
6
X)f(Zot)dz
(16)
0
A (),)
'-ot% (t)dt
D
0
D s)
.
..
.
.
0
*1
-
8/18/2019 Ada 120545
19/24
Taking
the
Laplace
Trafor.
(LT)
of
both
sides
of
(16)
s aee
at
t
b(a) =
dtA(x)
=)
dt,
0
0
0AC~LfcItzt7d
"
I
A )f ZS)dz
(17)
0
Let
(s)
-
cap
f(z,s)
I
y(a)
-
m n f (ZOa)
Recall
that
for
fixed
z,
I
f xy)dy
-
1
since
It represents
the (proper)
service
time
distribution
of
a
cuitamer
entering
service
at
t1me
x.
Therefore,
for
all o
-
8/18/2019 Ada 120545
20/24
and sine 13t
(a)
iu-
(a)
- I we
have, by applying
the Final
Value: ~it
,o,.
+
+
i ~
•
Its
b (it)< 0
an
hus
Il-
b(t)
-Il
19
(t)
-0o
t4.
t4
ft
Prom the
left
Inequality :in (19) we
also
see
that If la
XD(t) - o it
must
be
that
Ila- o
and the
theorem
Is
established.
The
tecnqus
developed in the
proof of
Theoes
2 lead to
an
interesting
result
for general Laplace transfomble
functions
that can
be represented
as the
convolution of two
functions.
Theorel 3:
Lot b t) be
Laplace
Tramsformable and le t
t
b t)
* I
g z)h t-z)dz (20)
0
be
the
convolution
8
f two
non-negatjve,
non-identically zero, proper functions
£
and h such
that $
g(x)dz -r- h(z)dz.
Then 1la
b t)
-
o If and
only
If
0
0
Ila g(s)
- Ur
h(s) -
0
Proof: Since
b t) is a convolution#
AA
where
b(s), g(s) h(s)
are the LT of
bp S9 and h,
respectively.
Then
by the
Final Value Theorem
we
have
la sb(s)
-(13Z
g(s))(Ulu
h(s))
(1,.
,(,))(
(s))
Sincehla
&(s) - s(z)dz
a a a iu
h(s) - h(z)dz
3* 00
0
Theorem
3
follows
Iindiately.
Inspection
of
the
convolution integral
(20)
Intuitively shows
why
Theorem
3
Is true. If
both functions
S(x) and h(z)
did not go
to
zero
se
significantly
positive
vass voud be
acclated
when Integrating from
o to t and heace
b t)
would
hove
a positive
limiting value.
-w
:
.
.. * ..
. - ..
-
8/18/2019 Ada 120545
21/24
We
note that
in
Theorem 2, since 7 zlt)
is asumed to
be a proper
- probability
distribution function, It is trivially
true that lim f xt)
3tt
In
this
report
we
have
established a splitting theorem for NOP
similar
to the splitting theorem
for
stationary Poisson Processes. Using this
powerful result
we
shoved that
the
output process
of an
X(t)
IG(t) I queue
*Is a NIP? and
that
the number
of departures from the queue,
during
a
finite
interval
are Independent of the number of
cuetomers, ramining in
service at
the end of the Interval. Furthemore,,
using the splitting theorem we
showed
that
the
nueber
of
busy
servers
In
the X(t)
IG(t)/I
queue
Is
Poisson,
distributed
with
mean
f X(z)i(x~t)dz.
These
results provide a
tool
for analyzing
queues
*
for
which
he arrival proces and service time
are
non-stationary.
15
-
8/18/2019 Ada 120545
22/24
. .Z
.*
° -°
-*
o
°
-
.
. ,
.
, . -
.
.
.
. • . o
.
' . . _ -
-
•.. .
1 ilnlestad,
1.
J.
and X. J.
Cariflo, "*Idels
and Techniques
for
Recoverable
Ite Stockage
When
Demand and the
Repair Procass
are
Non-Stationary
-
Part
1:
Performance
measureas
UND
Note
N-l48Z-AIF
Nay
1980.
2 Ilenrock.
L.,
Queuelug Systsms*
Vol 1:
Theory.
John Viley &
Sons,
1975.
3 Kitchen,
J.
W.,
Calculus
of
One
Variable,
Addi
n-Vssely
Publishing
Co., 1968.
4
Nirasol, N.
M.,
The Output
of an U/G/-
Queueius System
is Poisson,
Letter to
the
Editor,
Operations
Research
(11:2) pp282-284.
5 Parsen,
I.,
Stochastic Processes,"
Eoldem-Day,
In. , 1962.
6
loss, S.
X., Applied
Probability
Nodels with
Optimization
Applications,
NO1de-Days,
Inc.,
1970.
7 Vidder,
D.
V., The Laplace Transfera.
Princeton
University
Press, 1946.
.?16
-
8/18/2019 Ada 120545
23/24
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