:iB
;*:
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A DISTRIBUTIONAL
J.
GTE
FORM OF LITTLES LAW
Keilson*
Laboratories Incorporated and Mass.
GTE
Inst,
of Technology
and
L. D. Servi*
Laboratories Incorporated
#1965-87
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
S1988
A .^.T/
f
FORM OF
A DISTRIBUTIONAL
Laboratories Incorporated and Mass.
GTE
LAW
Keilson*
J.
GTE
LITTLE'S
Inst,
of Technology
and
L. D. Servi*
Laboratories Incorporated
itl965-87
ABSTRACT
For many system contexts for which
form of the law
is
also valid.
Little's
Law
is
valid a distributional
This paper establishes the prevalence of such
system contexts and makes clear the value of the distributional form.
Keywords: Queues,
Little's
Law,
priority,
December,
*
40 Sylvan Road, Waltham,
MA
1987
02254
limit
theorems
FEB
8 1988
RECEIVED
Introduction
1.
Over
the twenty-five years since Little's
Law
simplicity and importance have estabhshed
its
it
first
appeared
(Little [11]),
as a basic tool of
queueing
theory (cf e.g. the survey paper by Ramalhoto, Amaral and Cochito [12]
Little's
Law
equates the expectations of two variates in a system. For
the systems to
which
Little's
Law
applicable, a stronger relation
is
two variates
the distributions of the
is
available.
The
The
many
).
of
between
setting required
form of
is
Little's
law
has been observed previously in special contexts, (Fuhrmann and Cooper
[2],
described in the theorem in Section
2.
distributional
Servi [13], Svoronos and Zipkin [14]) but the generahty of the setting and
importance for queueing theory has not been
Suppose
customers
system with Poisson arrival
in the
N and
is
rate X.
there
Let
is
a
N be
class
C
of
the ergodic
system in that class and let T be the ergodic time
system spent by a customer in that class. Let k^^(u) = E[u^] be the p.g.f. of
number of customers
in
set forth.
some ergodic queuing system,
that, for
its
let
known
i.e.
(Xp^Cs)
in the
= E(e"^^) be the Laplace-Stieltjes transform of T. Suppose
for class
C
it
that
7lj^5(u)= a^s(?i-?LU)
(1)
N=d K^j
(2)
that
Here
Kq
is
a Poisson variate with parameter
.
In words, the ergodic
number of customers in that class in the system is equal in distribution to the
number of Poisson customers arriving during an ergodic time in system for
members of that
class.
distribution holds
A
may
customer class for which the equality
be said to satisfy the distributional form of
Law or to be an LLD class.
sees that LLD implies Little's Law,
Little's
converse
page
1
is
in
not true as wiU be seen.
If
i.e.
one differentiates
E[N] = A,E[T] for
(1)
at
u =
that class.
1,
one
The
For many of the system contexts for which
distributional
form of the law
The
also valid.
is
Little's
Law
is
vahd the
,
object of the paper
is to
demonstrate the prevalence of such system contexts, and to make clear the
value of the distributional form.
2.
The prevalence
known
It is
(
LLD
of
Classes
the customers of an
c.f.Kleinrock [10]) that
FIFO
with infinite queue capacity and
M/G/1 system
discipline satisfy (1).
prevalence of the distributional form of
Little's
Law
is
A
broader
suggested by the
following two theorems.
Prop. 1
(
Keilson and Servi
[7])
Consider an M/G/1 type system with two classes of customers,
infinite
queue capacity and FIFO preempt-resume discipline for the low priority
class.
The
distributional
form of
Little's
Law
is
valid for the
low
priority
customers.
Prop.
2.
Every class of customers
in
an M/G/1 type priority system for which
classes have Poisson arrivals and
lower priority classes
is
an
LLD
The
totality
as a single class
of
all
is
sees only the customers with higher
customers with higher priority
having a Poisson arrival
arrival rates with higher priority
which
discipline over
class.
Proof. Each class of customers
priority.
FIFO preempt-resume
may be
rate equal to the
sum
regarded
of
all
the
and an effective service time distribution
a weighted mixture of the service times with higher priority.
general result then follows from that for two classes.
These examples suggest an even broader
theorem is given next.
page 2
all
validity.
A more
general
The
Theorem.
Let an ergodic queueing system be such that for a given class
customers,
system
a)
arrivals are Poisson of rate X;
b)
all
C
of
arriving customers enter the system, and remain in the
no blocking, balking or reneging;
until served i.e there is
the customers remain in the system until served and leave
c)
the system one at a time in order of arrival (FIFO).
Then
form of
the distributional
Little's
Law
is
valid for that class
C
of
customers.
The proof is based on two lemmas.
Lemma
A.(Cf. Cooper [1] for the result due to Burke and Takacs) Let N(t)
be a time homogeneous process
continuous time on the
in
negative integers with changes of ±1
departure epochs
n
]
= lim
j
Lemma
_^
c>o
(T'^j)
P[N(T'^,-)
at
when
]
B. (Cf. Wolff [17],
j _^
oo P[N(T^j-i-)
(T'^'j).
Let N(t) be an ergodic population
[1])
at
successive
Then
lim
t
_^
<n
ooP[N(t)
Proof. Suppose that
at t=0, the
of the k'th customer
in class
]
= lim
system
C, and
is
let
j
^ ooP[N(TA +) < n
empty. Let
T^^,
]
T^ be the arrival epoch
be the departure epoch for that
customer. Let N(t) be the number of customers of class C in the system
time t. Let T^ be the time spent in the system by the k'th customer, i.e. let
=
Tpj^
-
T^
at arrival
.
Then N(Tdj.+) = ^
by the
entered service,
page 3
<
these limits exist.
counting process in continuous time with Poisson arrivals
epochs
of non-
sequences of successive arrival and
and (T^j) respectively. Then lim
<n
lattice
k'th
all
K^-j-j^
.
customer have
This
is
left the
true, since all
at
Tj^
customers found
system before that customer
customers arriving during his
time in system are
still
of such systems
may
server
(
Keilson and Servi
cf.
[8],
Harris and Marchal [3])
the
,
take vacations after servicing a customer with a probability
dependent on queue length. The server might not end a vacation duration
until exactly
integer)
(
N
Yadin and Naor
cf.
N
customers are waiting (where
Heyman
[16],
is
[4]
a prespecified positive
Little's
)
law holds
in
distribution for the customers in such systems provided they are served in
order of arrival.
C.
M/G/1 systems with preemptive interruptions
Keilson and Servi
service times
Little's
is
[6]
If a
).
Poisson stream of ordinary customers with
preempted by other
law holds
at clock ticks (Cf.
iid tasks arriving at
in distribution for the time to service
clock ticks,
iid
then
completion of the
ordinary customers.
D. Cyclic service systems (Cf. Takagi [15])
queue
at service sites
and a single server moves
Suppose Poisson customers
cyclicallly
either serving the customers at the sites to exhaustion,
schedule (Keilson and Servi
before moving on.
Then
[5]
),
between such
employing
or serving at most
K
the customers at any site are an
Bemoulli
a
customers
LLD
sites,
at a site
class provided
they are served in order of arrival. The changeover time of the server
between
E.
sites
Tandem
does not disturb the
LLD property.
server systems
M/G/G/
tandem each with
infinite
(
.
.
/G).
Consider
queue capacity and each having
K
iid service
for successive arrivals. If arrivals to the system are Poisson,
discipline
maintained throughout the system, then
is
distribution relates the time spent in the system to the
servers in
times
and FIFO
Little's
law
number of customers
in
in
the system at ergodicity.
4.
The value
Apart
in
are
(3),
from the relations
the distribution
listed
page 5
of the distributional
below.
form of
between
Little's
Law.
means and variances given
information has other benefits, some of which
Stochastic bounds
well
It is
with
e. It
the
It
is
When
number
Kq
Poisson variate
that the
N
then follows from (2) that
increases.
system.
known
sometimes easy
increases stochastically
increases stochastically
to obtain a
distribution
bound
T
when
for a time in
one can do so one automatically has a distribution bound for
in the system.
Heavily loaded systems.
Suppose
approximation.
Little's
Law
it
known
is
T
that
exponentially distributed to good
is
then a direct consequence of the distributional form of
It is
that for
LLD
an
system
N
is
geometrically distributed.
The
then available from the
first
single parameter needed for the distribution
is
moment. For heavily loaded systems, knowledge
exponential
distribution
with
FIFO
implies that
in distribution
.
An easy example
is
N is
that
asymptotically
geometric
asymptotically
number
that of the
T is
in the
in
system for M/G/1
discipline.
Decomposition Results.
For a broad
the ergodic
number
two random
variables,
of customers in the system
one of which
ordinary M/G/1 queue.
LLD
class for
Fuhrmann and Cooper
class of customers,
which
This class
N
is
is
the ergodic
decomposes has
The
LLD
system and
to the
Little's
time in queue.
number
the proof of the theorem
to a
application of the theorem,
from the requirement
M/G/1,
page 6
in the
Law
It is
the time in queue and the
in
number
in the
is
in
that
sum of
system of an
LLD class.
Every
decomposition for T. This
[2] via separate
property for the number
The ordinary form of
a
proved
distributed as the
a proper subset of the
decomposition has also been observed in
5.
is
[2]
reasoning.
queue.
applicable both to the
time in
LLD
result for
natural to try to find an
queue by transferring the ingredients of
subsystem consisting of a queue only. Direct
however does not work. The
theorem
say, arrivals bypass the
difficulty arises
that the arrival process
queue when the server
is
be Poisson
idle
.
The
.
For
arrival
process to the queue (as against to the system)
with censored epochs
the time in
i.e. is
is
then a Poisson point process
not Poisson. Nevertheless
queue and the number
in
for an
,
queue do obey
M/G/1 system,
Law
Little's
in
distribution.
This
may be
first
verified
the pgf of the ergodic
from the familiar M/G/1
number
in the
queue and
the ergodic time in queue. Let ^s^^^) ^^ ^^^
system and
let ajs(s)
results
P§^ °^
We
Let %q(u) be
o^qCs) be the transform of
let
^^
ergodic
be the transform of the ergodic time
be the transform of the service time.
.
first
number
in system.
Let a^(s)
note that ji^qCu) has
contributions from the idle state and from the states where the server
One
in the
is
busy.
then has
W^^
From our theorem, %s(u)
71^S(U)-(1-P)
=
^
u
=aj^{'k-Xu)
^^'P^
•
= a^Q{X-Xu)aj(X-7M). One has
—
ajQ{X-Xu)aj(X-X\i) -(l-p)(l-u)
^nq(u) =
Since the time
in
we may employ
queue
at
•
ergodicity coincides with the ergodic waiting time
the Pollaczek-Khintchine
Law
,
for o^qCs) to see that
(l-p)(l-u) aT-a-Xu)
^-(l-p)(l-u)
ay(?L-?wU)-u
aj(X-Xu)
-,
('-p>"-">tM^:;^ -'
'
(i-P)(i-u)
as required.
The vahdity of Little's Law
in distribution
could also have been obtained
One could
from our theorem with the help of the following
artifice.
M/G/1 system by an M/G/1 vacation system
takes a vacation of duration D whenever it finds
(e.g., [1])
the
page 7
the
queue
replace
where the server
idle
.
In this
way
the
queue
gives the
is
treated as a subsystem with Poisson arrivals
LLD
durations Dj
tlie
limit for
sequence of such subsystems with vacation
considered, and Dj -^ 0, LLD holds for each j and hence in
result.
is
When
and the theorem
a
M/G/1
The same
artifice or a similar artifice
customer classes described
available for the ergodic
in
could be applied to
Section 3 to conclude that the
number
in
queue and time
in
all
of the
LLD property
is
queue.
Acknowledgment.
The authors wish
to
thank S. Graves for his careful reading of
for the reference to the paper
page 8
by A. Svoronos and
P. Zipkin.
this
paper and
References
Cooper,R, Introduction
N.Y., 1981.
[1
]
to
Queueing Theory, 2nd edition.North Holland.
and R. Cooper, "Stochastic Decomposition in the M/G/1
Queue with Generalized Vacations", Operations Research, 33, 1117-1129
[2]
Fuhrmann,
S.
(1985).
[3] Harris,
C.
M. and W. G. Marchal,"State Dependence
Vacation Models", to appear
[4]
Heyman,
in
in
M/G/1 Server-
Operations Research, 36, (1988).
D., "Optimal Operating PoHcies for
M/G/1 Queuing Systems",
Operations Research, 16, 362-381 (1968).
and L. D. Servi, "Oscillating Random Walks Models for
GI/G/1 Vacation Systems with Bernoulli Schedules" Journal of Applied
Probability, 23, 790-802 (1986).
[5]
Keilson,
J.
and L. D. Servi, "A Distributed Poisson Approximation for
Prempt-Resume Clocked Schedules", submitted to IEEE Trans, on Comm.,
[6]
Keilson,
J.
(1987).
[7] Keilson, J. and L. D. Servi,
Feedback", pending.
"A Two
Priority
M/G/1 Queue
with
J. and L. D. Servi, "Blocking Probability for M/G/1 Vacation
with
Systems
Occupancy Level Dependent Schedules", to appear in
Operations Research., (1988).
[8]
Keilson,
Keilson, J. and F. W. Steutel, "Families of Infinitely Divisible
Distributions Closed under Mixing and Convolution", r/ie Annals of
Mathematcial Statistics, 43, 242-250 (1972).
[9]
[10] Kleinrock,L.
,
Queueing Systems. Vol
1
.
John Wiley
& Sons, New York,
1975.
[11] Little,
J.,
"A proof of
383-387 (1961).
page 9
the
Theorem L = XW", Operations Research,
9,
Ramalhoto, M.F., J. A. Amaral, and M. T. Cochito, "A Survey of
51, 255-278 (1983).
Little's Fomiula", Int. Stat. Rev.
[12]
J.
,
"Average Delay Approximation of M/G/1 Cyclic Service
Queues with Bernoulli Schedules", IEEE Journal on Selected Areas in
Comn-iunication,4, 813-820 (1986).
[13] Servi, L. D.,
Svoronos and P. Zipkin, "Evaluation and Optimization of 1-4-1
Replenishment Policies for Multi-echelon Inventory Systems". Research
Working Peper 86-9 Columbia Business School, Columbia University
[14]
,
(1986).
[15] Takagi, H.
[16] Yadin,
,
Analysis of Polling Systems,
M. and
P.
1986.
Naor, "Queueing Systems with a Removable Service
Station", Opnl. Res. Quarterly, 14,
[17]
MIT Press,
393-405 (1963).
Wolff, R., "Poisson Arrivals See Time Averages", Operations
Research., 30, 223-231
page 10
(1982).
i.
53
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