HD 28
.M414
\»ST.
OCT 20198.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
M/G/oo with Batch Arrivals
GTE
J.Keilson
Laboratories and Mass.
Inst, of
Technology
and
A.
Seidmann
University of Rochester
WP1932-87
September 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
M/G/oo with Batch Arrivals
GTE
J.Keilson
Laboratories and Mass.
Inst, of
Technology
and
A.
Seidmann
University of Rochester
WP1932-87
September 1987
Abstract
Let Pc„(n) be the distribution of the number N(oo)
in
the
ergodicity for
systems with an infinite number of
batch arrivals with general batch size distribution
and
general holding times. This distribution is of impotance to a variety
of studies in congestion theory, inventory theory and storage
systems. To obtain this distribution, a more general problem is
addressed
In this problem,
each epoch of a Poisson process gives
rise to an independent stochastic function on the lattice of integers,
which may be viewed as a stochastic impulse response. A continuum
analogue to the lattice process is also provided.
system
at
servers,
.
OCT 2
->
,937
RECEIVED
O. Introduction.
The
number
distribution of the
number
infinite
general problem
is
of
importance to a variety
and storage systems. To obtain
theory, inventory theory
process gives
batch arrivals with general batch size distribution
of servers,
and general holding times
addressed.
is
rise to
the system at ergodicity for systems with an
in
In this
of studies in
this distribution,
each epoch
problem,
congestion
a more
a Poisson
of
an independent stochastic response function on the
lattice
of integers.
The
simplicity
and importance
known, but the authors have not been able
method
employed
is
probabilistic
suggest that they may be
of the results
to find
them
in
the
and succinct and may be
The
literature.
of
independent
interest.
A more
1.1
general problem
The work described has been motivated by M/G/oo batch
arrival
system
needs. The results however relate simply to a more general system context
and may have value elsewhere.
For the M/G/^o batch system, customers arrive at Poisson epochs of rate X
random batch
with
infinite
number
system,
is
-4
oo
.
and random
of servers
To
wanted.
problem where,
stochastic
size
for
.
The
i.i.d
p.g.f. at
each
arrival
epoch
The system occupancy
level N(t) is
we
of of the
the
number
in
obtain the
p.g.f. of N(t«) for
Poisson character
as a superposition
of
M
the
consider the more general
Poisson stream
,
there
is
as
a
t
then given by
oo
X
k =
page1
t,^
,
system having an
response function Nk*(t) having integer values with Nk*(t) -^
N(t) =
of the
ergodicity of N(t),
find this distribution
k = +
To
service times to a
N
*(
t -
t
J
-oo
the general problem, one notes that because
of the arrival process,
one may regard the
independent sub-streams.
6/1 5/0
input stream
Each sub-stream has
12:35
PM
Poisson arrivals
N*(t)
X/M and has the same stochastic response
of rate
and each sub-stream may be thought
subsystem. The response function
after a
the level
stream
M
be assumed
large, the probability that
is
a subsystem
present simultaneously
in
interval of length D,
the
let
as associated with
of
p.g.f. of
initially to
number
the
as
o(}t/M)
is
own
its
terminate at
Because each sub-
of finite duration D.
service interval
when
thin
is
N*(t) will
function
two or more batches are
M
-^
During the service
oo.
the subsystem present at time
in
y after the service inception be
g(u,y)=
(1)
where
is
r^{y)
= P[N*(y) = n]
Xn
Because the
.
rnCy)
u"
process
arrival
for
each subsystem
Poisson, the length of the intervals between the termination of service on the
previous batch and the the start of service on the next batch
distributed with
mean
M/?t.
In (y)
If
at time y, the fraction of time
spent
in
exponentially
the indicator function for being at state n
is
level n at ergodicity is given for n >1
by
D
D
E[j
is
dy
I„(y)
]
r^(y) dy
f
"o
X
X
It
follows that the p.g.f of the
number
—
71
(u
,
-)
=
next permit
latter limit is
From the
page2
M
to
+
g(u,y)
_
j
J
lim
,^
for fixed D,
+ 0(1)
,
M
-^
oo
+
infinite
and then D
to
become
infinite in turn.
appropriate since the duration of the holding times
familiar
is
M
dy
5__
|_
become
the system at ergodicity
D
D
We
in
^^ (1+^)^
=
e'^,
may be
The
infinite.
one has
6/1 5/0
12:35
PM
D
71 (
=
u,oo)
=
lim
^ ^ exp
^
lim^^ ^exp[ -X
X D {
-
[
D{1J
—
-
1
1
(
dy
g{u,y)
j
dy
-
g(u,y)
-
g(u,y)}dy
)
}
]
}
]
.
I.e.
oo
(2)
H
7i:(u
A more
exp [-X
=
J{1
].
formal demonstration of (2) can be obtained from a continuous
product representation of .k{u
Note that
for the ordinary
time c.d.f A(x)
infinite
,<>=).
K
M/G/°o system with batch size
=1 and service
one has
,
A
g(u, y) = A(y) + u
(3)
(y)
Hence
and one obtains from
(See
for
(2)
A
=
1- g(u, y)
the classical result
example Tijms(1986).) Equation
Theorem.
{1-u)
(y)
For a process N(t)
with
7i:(u ,oo)
(2) implies at
= exp[
once
system
number
is
of servers, the
ergodic distribution of the
a compound Poisson
The basic
result
continuum case
for
(2)
XE[T] (1-u)
]
.
that:
batch poisson arrivals of rate X,
independent identical random impulse response g(u,t
infinite
-
)
identical
each batch and
number
distribution with p.g.f. given
can be extended by an
for
by
of items in the
(2)
argument
.
to the
which
(4)
k = +
X(t)
=
X
k =
pages
CO
^
X
- T
^( t
J
.
-oo
6/1 5/0
12:35
PM
Here the summands are
infinity.
v|/(s,y)
deterministic
(1959).
in
= E[exp{-s
[ - ?i
X,^*(y)}]
and exponential
one
]
.
In
this result
An extensive discussion
of the
goes
to
has
{1
f
t
\|/(s,y)}
-
dy
]
the special case where X^*(y)
coincides with
J.
Keilson
general deterministic case
&
is
D. Mirman,
may be found
S.O.Rice(1945). The result extends to multivariate processes of the form
1.2
(4).
The ergodic Mean and Variance
From
(2)
E
If
exp
=
,00)
(t)(s
where
stochastic functions which go to zero as
= E[ exp{-sX(°o)}
(^{s ,c«)
If
i.i.d.
one has
[
N(oo)
N*(t) is the
at
]
once
=
^
J
g^(1,y) dy
decreasing number
in
=
^
^
n r^{y)
dy
.
the system associated with each Poisson
epoch, then
00
E[NH)] =X
(5)
In
the
same way, one
jE[N*(y)]dy
finds that
00
Var[NH]=
X
J
[
g^ud
.V) +
SuC^
.y)
1
dy
so that
Var[NH] =X
(6)
1.3
j
E[N*2(y)] dy
Mbatch/M/00
Until
now no assumptions have been made about
batches are served independently. For
page4
M'^^^'^'^/M/00
6/1 5/0
,
N*(t) other
than that
each customer
is
served
12:35
PM
independently and lifetimes are exponentially distributed.
that batch size
is
suppose
Let us also
exactly K. Then, as for (3)
^
g(u,y) = [ {1-e-ey) + ue'^V]
so that for
(2)
one must evaluate
oo
^k(") =
j
[
(1-e-^y) + ue-ey]
"^
dy
From the Binomial expansion, one must then evaluate
oo
h(K,r) =
J
(l-e-Qy)'^''" e" '^v
from the integral representation
of the
dy =
Beta function..
er
[
It
^C,
]-'^
follows that
(7)
e
This
may be seen
to coincide with the p.g.f.
obtained for M^/M/oo by analysis
via a birth-death process for general batch size distribution.
and
Pn(t)
= P[N(t) =
n]
.
One
.
Let a^ =
P[K=n]
then has from the forward Kolmogorov
equations
^Pn(t) =
The use
of
-
(^ + n^)Pn(t) + A (Pn(t))*(an) +
(n+1)nPn+i(t)
generating functions then gives
3Y7t(u,t) = -?i[1-a(u)] 7i(u,t) + u(1-u) 3^:t(u,t)
,
I.e.
(1-u)
^log^Cu) =-[1-a(u)]
U
One has
finally
u
(8)
This
7:(u)
= exp
may be seen
pages
[
-
1-w
to coincide with (7)
°^
J
= ..
when a(u) u
6/1 5/0
K
12:35
PM
MK/G/oo
1.4
For constant batch size and general holding times, one has
g(u,y) =
[
^
A
+ u
A(y)
(y)]
so that
oo
7i(u
= exp
[
^
J
exp
=
,00)
[
{[A(y)
X
+ u
J
A
[g(u,y)
(y)]"^
-
g(1,y)
-
[A{y)+ A
]
dy
(y)]*^ }dy]
.
Hence
7r(u ,00)
(9)
exp{ -0[1
=
-
p(u)
]
}
where
(10)
e=
XE
[
max (T,,T2, ..,1^)] = X
j
[1-Ak(y)] dy
and
(11)
X
Mu)
u'C,
Note that only when K =
pages
1
Note also that
is
[
1
-
Aj^(y)
]
dy
the distribution of N(=«) independent of the lifetime
for constant batch size
K one has from
differentiation
E[N(H1 = ^K E[T]
(12)
(13)
A^^-^y) dy
=
j
distribution.
A';(y)
J
Var[N(oo)] =
XK
E[T] + ?iK2
6/1 5/0
J[1-A(y)]2dy
12:35
PM
in
agreement with Tijms
N(oo)
from
(9)
[
1986]
The numerical evaluation
I
e-9
-p(u)l }=
n-fold convolution of the lattice distribution
from the compound Poisson character
distribution
with finite
distribution of N(oo)
mean, when K
becomes normal
batch size K alone since
normality
will
substantial part by the
fixed
is
.
e=
This study
Wm.
The authors are indebted
E.
to S.
ip
pn(u)
for P(u) with itself.
of (9) that for
and ^becomes
the K'th power of a
XE
was
max
[
initiated
Simon School
Graves,
(
L
T^, Tg,
at
is
clear
.
.
large, the
with increasing
in
p.g..f.
,
T^
.
) ]
and supported
of the University of
Servi, U.
It
any holding time
Normality does not set
7r(u ,«>) is not
be present when
Acknowledgment:
helpful
of the distribution of
can be obtained algorithmically from
exp{-e[1
and
.
The
is
large.
in
Rochester.
Sumita and D. Nakazato
for
comments.
References
[1]
J.
Keilson and N.D.Mirman (1959)
"
The second order
integrated shot noise" IRE Transactions
[2]
[3]
on Information theory" pp 75-77
S.O.Rice (1945) Mathematical Analysis
Technical Journal
Henk
Sons,
page7
distribution of
of
Random
Noise. Bell
.
System
Vol. 23
C. Tijms (1986)
Stochastic Modelling and Analysis, Wiley &
New York
6/1 5/0
12:35
PM
Date Due
MAY 1 2 20(1
3
TDflD
DOS 131 MIT