MIT LIBRARIES
3 9080 01917 7614
WMWM
/^l
H i
'-I
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
The Distributional Form of Little's Law
and the Fuhrmann-Cooper Decomposition
J.
Keilson
L.D. Servi
September 1988
WP #
2083-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
M.I.T.USl?APfF<J.nr^ flfC\/
The Distributional Form of Little's Law
and the Fuhrmann-Cooper Decomposition
J.
Keilson
L.D. Servi
September 1988
WP # 2083-88
ABSTRACT
For certain classes of customers, the ergodic number of customers in
system (queue) and the ergodic time spent in system (queue) are related
by a distributional form of Little's Law [6]. Classes for which Little's
Law holds in distribution will be called LLD classes and will be said to
have the LLD property. Because LLD classes have a simple
LLD
property is
characterization permitting their quick recognition, the
Pollaczeksystem,
for
example,
the
powerful
tool.
For
the
M/G/1
a
Khinchin distribution drops out
conditions
(cf.
in a
few
lines.
Moreover, under simple
2) the pgf of the ergodic number in queue is
a structurally simple decomposition of the form of
Theorem
shown to have
Fuhrmann and Cooper
[2].
This decomposirion is implemented for
is shown thereby to provide a degree of
many queueing systems and
unification to queueing theory.
1.
Introduction
In a previous paper, Keilson
customers
(cf.
Section
2), the
[6]
showed
that for certain
"LLD"
classes of
number Ns of customers in system, i.e.
ergodic time Tg spent in system by customers of
ergodic
or in the service box, and the
were related by a
and Servi
distributional
form of
Little's
Law. Specifically
it
in
queue
the class
was shown
that the
two descriptive functions
TC^Cu)
^s^"^~ Ops^^"^")-
(1.1)
Equation (1.1)
is
equivalent to the statement that
X during an
interval of duration Ts,
(1.2)
is
number
in
LLD classes
queue and the time
in
The prevalence of such
were discussed
arrivals
M/G/1 system,
LLD
property
in
queue
for an
LLD
class
and
corresponding result
their
quick recognition.
for example, the Pollaczek-Khinchin distribution is found
factors, the first being of the
many
A
classes
It is
then
provides an analytical tool of some power. For
algebra. Moreover, under the simple conditions of
number
in [6].
LLD
queue was also given.
have a simple characterization permitting
useful to observe that the
unifies
number of Poisson
equals the
to the equality in distribution
a Poisson variate of parameter 0.
the simple consequences of the law (1.1)
for the
i.e.
Ns
^s-^XTs
where Kg
the
E(e"^'^S),
by
are related
at rate
= E[u^S] and 0^5(5) =
Theorem
2, the
by simple
pgf of the ergodic
has a decomposition into two structurally simple
Pollaczek-Khinchin form. The decomposition of Theorem 2
of the results of queueing theory, and provides a quick derivation for other
systems as well.
Many
of the results for priority queues
simple structural form. The results for
Fuhrmann and Cooper
conditions close to that for
[2]
Head of the Line
fall
out quickly, and acquire a
discipline are especially attractive.
have demonstrated the decomposition of Theorem 2 under
LLD
classes.
They employ
page 2
a
somewhat longer and more
indirect
argument and they provide
Their theorem
2.
LLD
given
is
in
Section
implementation for concrete cases of
little
interest.
4.
Classes
For ease of reference, the definition of an
LLD
class
and the basic theorem of the
earlier paper are given next in slightly different form.
Definition
2.1
Let an ergodic queueing system S have a subsystem S*.
customers entering S* such that:
b)
c)
The
class
C
will then be called
an
then has the following theorem
Theorem
Law
Little's
C
be a class of
customers from C enter S* in a Poisson stream of rate X',
the customers of C in S* leave S* one at a time in order of arrival;
for any time t, the entry process into S* from C after time t and the time
spent in S* by any customer in C arriving before time t are independent
a)
One
Let
Let
1.
is
C
be an
valid for the ergodic
time Tj, spent by customers of
C
in
LLD Class
for S*.
[6].
LLD
class for S*. Then the distributional form of
number N„, of customers of C in S* and the ergodic
S*
,
i.e.
one has for k^,(u) = E[Ns»] and Ojg.Cs)
=
E[Ts.].
7i3.(u)= oc^s.(X-Xu).
(2.1)
If the
is
subsystem S*
is
the
queue and the service box then
(2.1) reduces to (1.1). If
S*
simply the queue then one has
^q(")= Ot-qCX-Xu)
(2.2)
where
tIqIu)
=
is
the p.g.f. of the
number
in the
time in the queue.
page 3
queue and
(Xj-q(s) is the
transform of the
Proposition
distributional
between
For a counting variate
1.
form of Little's
Law
given
N
and an asscKiated time
in (2.1) or (2.2)
T governed by the
one has the following relations
moments:
the successive
E[N] = E[?iT]
(2.2a)
E[n2] = E[(XT)2] + E[XT]
(2.2b)
E[N^] = E[(>.T)^] + 3 E[(>.T)2] + E[XT]
(2.2c)
E[N^] = E[(kTf] + 6 E[(>.T)^] + 7 E[(XT)2] + E[XT]
(2.2d)
E[N^] = E[(XT)^] + 10 E[(XT)''] + 25 E[(XT)^] + 15 E[(XT)^] + E[XT]
(2.2e)
Proof: Equations (2.2a)
with the help of
-
(2.2e)
were obtained by successive differentiation of (2.2)
MACS YMA.
Remark: Measurement
distribution of
N
approximate the
of the mean, variance, skewness and kurtosis from an empirical
permits one to solve for the corresponding moments of T and to
distribution of T by selecting it from some four parameter family of
distributions.
For most of the examples
in [6], the distributional
both the number of customers of
in
C
in
form of
queue and the "number
Litde's
law was valid for
in system",
i.e.
the
number
queue and service box.
The
applications of interest are vacation models, priority service systems and cyclic
service systems. In
queue
all
of these systems one
may
visualize a server in a class as joining the
box and
for that class, entering the server
initiating service.
The
service
interrupted and resumes where left off or starts over. In this setting, the time in
refers to the time
from
the arrival of the
customer
to the
service.
The
until the
customer completes service. Until service
effective service time, Teff. refers to the time
regarded as being
Theorem
If the
and both
queue
in the service box,
whether or not
it
is
is
until the
may
be
queue
customer begins
from the beginning of service
completed, the customer will be
being served.
2.
time in queue of a customer in class
(1.1)
and (2.2) are
C
satisfied, then
page 4
is
independent of the effective service time
(2.3)
=
7tQ(u)
.
(1-Peff)(^-")
^
.
n^iu)
,.
^P-—
^-Peff
=
7tR(u)
where
=
7Cq(u)
pgf of the ergodic number of class
the
given that no class
a^pp(s)=
in the service
in
queue
box,
E[e-sTEFF],
~
Peff
C customer is
C customers
^E[T£pp],
and
^^^^
sE[Tepp]
Moreover,
Nq i
(2.4)
with Nm/g/i
^d Nb
+ Nb
Nm/g/1
independent. Here
Nq
=
the ergodic
number of class
Nm/g/1
=
the ergodic
number
Nb
=
in the
X and
the ergodic
number of class
no
C customer is
class
the queue,
queue of an M/G/1 queue with an
arrival rate
that
C customers in
a service time Teff.
C customers
in the
queue given
in the service box.
Proof:
To
prove the theorem some additional notation
Pb
=
P[ a customer
KqcCu)
=
the
is in
TtgCu),
i.e.,
Pb u tTqbCu)
the
queue has
the service box]
a
C customer is in
pgf of '^OB^'^^
KqCu) = pBJtggCu) + (1-Pb)
+ (l-pe)
(2.5)
If the time in
T^^i^i)-
7t5(u)
queue
is
needed. Let
pgf of the ergodic number of class
given that a class
The number in
is
in
queue
the service box.
^ ^ customer is
TtgCu).
C customers
Using
in the service
a similiar
box and,
argument,
Tt^Cu)
=
Hence,
= ;:q(u)u + (l-pB)(l-u)jrg(u).
independent of the effective service time then
otjQ(s)a-j^pp(s). Therefore
if not,
from
(1.1)
and
(2.2),
pages
a.j.o(s)
=
TigCu)
(2.6)
=
tTqCu) Oj-^ppCX-Xu).
Equating the right hand side of equations (2.5) and (2.6) and solving for 71q(u) gives
= 71q(u)[(1-pb)(1-u]
71q(u)
[aq.£pp(X-Xu)
pB = Peff =
(2.7)
and equation
first
/
(2.3) then follows.
factor of the right
Remark: Equations
service
is
-
and
u)].
From
71^(1)
=
1
and LHopital's Rule,
^Ot-effXO)
From classical queueing
hand side of
(1.1)
-
theory (or from case
1
below) the
(2.3) is E[u'^M/G/l] so (2.4) follows.
and hence
(2.2),
(2.3)
and
FIFO. However, for any service order which
service times, the distribution of the
number
in the
(2.4), are true
is
when
the order of
not dependent on the individual
queue
is
the same.
Hence, (2.3) and
(2.4) are valid for this larger class of service orders.
Corollary
If the
1:
time in queue of a customer in class
C is
independent of the effective service time and
both (1.1) and (2.2) are satisfied then the pgf of the number in the system
(2.8)
*
^s^^
^
the transform of the time in system
(2.9)
0^3(5) =
-Peff«teef(^-^">
is
(^-Peff)«teff(^)
1
CL^is)
is
'^
1
^^([X-s)A]).
-Peff«teef(s)
and the transform of the waiting time
(2.10)
7^3(11).
71b([X-S)A]).
-PEFFttTEEF^s)
Proof: Equations (2.8)-(2.10) follow from
(1.1), (2.2)
page 6
and (2.3).^
is
.
3.
Special cases.
CASE
M/G/1 Queue
1:
Here
ot^-^ppCs)
=
(tj-Cs)
and
Nb =
so that from (2.3)
(l-p)(l-u)
CASE
For
is
=
Kq(u)
(3.1)
2:
.
M/G/1 with vacations and exhaustive
this discipline, e.g., [1],
inactive,
/
'J"':
(i.e,
is
It is
if
assumed
the system is empty.
that
V
is
Otherwise the queue
system must be on vacation.
Nb
then
is
arrived since the last vacation began.
a
Jto(u)
^
3:
But the time
is
in progress then the
at ergodicity since the last
vacation
a vacation time are equal in distribution.
^-^
-
[E[V]s]
/
a*(?L-Xu)
a j(X-Xu)
p
M/G/1 queue with two
classes
and preemptive
Consider a schedule with two classes of Poisson
customer service time transforms be
traffic respectively.
no service
(2.3),
=
1
If
pgf n^iu) = ay(X.-Xu) where a^Cs) = [l-a^Cs)]
and ay(s) = E[e-s^]. Hence, from
(3.2)
again served
equal to the number of customers that have
began and the forward recurrence time V* of
Hence Nb = Kxv, and Nb has
is
independent of the arrival process.
Again, Teff = T, the service time of the queue.
CASE
served exhaustively. Then the server
"on vacation") for a duration V. At the end of a vacation period another
vacation period begins
exhaustively.
an M/G/1 queue
service
cx_.(s),
traffic
priority.
with arrival rates X\
,
X2 and
<x~Js) for the high priority and low priority
Let the transform of the high priority busy period be agpj(s). Suppose
the high priority traffic preempts the
low
priority traffic.
The
cases of preempt resume,
preempt-repeat and possible hybrid modes compatible with the
considered simultaneously.
page?
LLD
requirement are
The pgf of
the
number Nj
in
queue of the high
Pollaczek-Khinchin form since low priority
number
in
queue of the low priority
traffic,
priority traffic has the classical
traffic is ignored.
one needs
To
find the pgf of Nj, the
pgf KgCu)
the
in
Theorem
2.
As we
see next, for any preemptive case, one has
where OgpiCs) =
=
-
pi
+ pi Ogpj(X2-X2U)
1,2.
Ej^
the
be the event that no
transform of the backward recurrence
Cj^
customers are
Let A'' be the event not A. Then Ej =
box
at
EjEj + E2E^ and
= P[ EjEjIEj ] E[ u^2(~)|E2Ei
The
in the service
=E[uN2(°*)|E2]
7ig(u)
(3.4)
is
busy period.
traffic
see this let
ergodicity, k
1
[l-c^BpjCs)] / [E[Tgpj]s]
time of the priority
To
=
TTgCu)
(3.3)
]
+ P[ EjEjIEj
events Ej and Ejare independent since P[ EjlEj] = P[Ej]. This
is
]
E[ u^2(~)|E2E^
because the Cj
customers have preempt priority over C2 customers and hence the C2 customers are
"invisible" to the Ci customers.
that the idle state
I
One
then has P[ EjEj
= P[Ej]
]
PfEj].
One
also verifies
= Ej Ej Hence
.
A):
P[EiE2lE2] = P[E,] =
B):
P[E2E^IE2] = l-P[E2EilE2] =
C):
E[uN2(~)|E2Ei] = E[u'^2(«')|l] =
1
-
Pj
,
Pi,
and
To
see (3.3) from (3.4) and A),B),C)
D):
the pgf of the
we need only
E[ uN2(~)|E2EJ
number of Poisson
arrivals
]
l.
establish that
= a*pj(?i2-X2u).
of C2 customers during the backwards recurrence
time of a Ci customer busy period.
page 8
]
The event EjE. corresponds
service box.
Such an event
is
Cj customer
to a
and the duration of the sojourn on the
from renewal theory applied
by the
initiated only
set
E2E.
to the recurrences
Using simple algebra one can show from
is
and no Cj customer
in service
arrival of a
Cj customer
a Ci busy period.
The
in the
to an idle server
result then follows
of such initiations.
(3.3) that
.B(u)=li^eii5iHL
,3.5,
where
(3.6)
C(u)
Case 3a Preempt
The
-
= X2-X.2U+Xi->.iaBPi(X2-X2U).
resume:
effective service time of a
leaves the queue until
interruptions).
One
it
then has
and from
customer
is
the time
(after possibly
from when
one or more
it
first
priority
[3], [5],
"^
^1
"
^i'^bpKs))
(2.7)
P2EFF=7^-
(3-8)
Hence, from
(2.3), (3.3), (3.7)
71q(u)
(3.9)
and
(3.8),
^^ta
=
1
-
[
P2EFFaT2EFF(^2->^2U)
(3.6),
(3.10)
which
priority
completes service
"t2EFf(^^ ^ "72^^
(3.7)
From
low
7c„(u)
^
is
=
(^-P^-P2n^^
X2[aT2(^("))
consistent with [7, equation (A. 9)].
page 9
-
"]
1
-
pi
+ pi a*pj(X2-X2u)
]
.
Using algebra and (3.10) one can also show
7:q(u)
(3.11)
that
I-P1-P2
=
1
(pi-p2)aTj2(C(u))
-
where
1
which
is
-
o-Tn^s)
consistent with [8, equation (1.6)].
Case 3b Preempt-repeat
Here, whenever the interruption of low priority customer ends a
begins after the interruption.Two cases could be considered
interruption a
second case
new
service times
is
after the interruption
[4].
In
service time
one case
randomly selected (Preempt-Repeat
the service time remains the
new
after the
different). In the
same (Preempt-Repeat
Identical).
First
Preempt-Repeat different will be considered: The Laplace Transform of T2EFF
can be found either
found
in [4,
in the appendix.
(3.12)
Chapter IV, equation (2.34)] or by using the simpler derivation
It is
a-nEFF(s)
=
-
0x2(5 +^i)
i— (l-aT2(s+>.l))OBPl(s)
s
Hence,
+X
1
(3.14)
=
tTqCu)
'-^^
[
]
where p^pp Pa and
PA+
(1-
PA)a*pi(X2-X2u)]
^(u) are given in (3.13), (2.7), and (3.6).
,
The
[
P2EFF aT2(^("))
-
1
distribution of the effective service time, T2EFF. for preempt-repeat identical
obtained as follows.
One
evaluates the distribution for preempt-repeat different with the
service time T2 having a deterministic value x and then weights the result by
distribution of T2.
From
is
the
one then has
(3.14),
00
(3.15)
aT2EFF(s)
r_
=
exp(-(s -hXQx)
dAT2(x)
^(l-exp(-(s Ui)x))obpi(s)
+K\
1
S
where At2(x)
Case
the cdf of the
C2 customer
M/G/1 queue with two
4:
For
and low
is
service time.
classes
this service discipline, there are
priority.
traffic, but a
low
The lower
priority
priority
customer
C2
in
and Head of Line
priority
again two classes of Poisson
traffic is interrupted
traffic
with high
by the higher priority Ci
progress completes service before
C2
service
is
interrupted.
Let B(t), the combined backlog process for
and C2 work
in the system.
Let p(s)
is
this schedule,
C2
i.e.,
Little's law, the p.g.f.
defined
aQ2(s) = P(s + X\
of the number
-
XiaBPi(s)).
in the
queue
is
equal
is
ergodic backlog with Poisson interruptions of rate X\ having
period,
total
amount of C\
the transform of the ergodic bacldog B('»).
for preempt-resume discipline, the waiting time of
Ci busy
be the
iid
From
Then
in distribution to the
durations equal to that of a
the distributional
form of
aQji^l-'^l^) = P(C(")) (where ^(u)
is
in (3.6).
Backlog processes, however,
are independent of the order of service. In particular
P(s) for preempt-resume and head-of-line
(HOL)
page
1
service disciplines are the same.
For
OqjCs) for both disciplines, moreover,
of Ci busy periods.
the
It
backlog
this
is
subject to the Poisson interruptions
follows that the pgf. of the number of Ci customers in the queue for
HOL and the preempt- resume service disciplines are the same (and given in
(3.10) and
(3.11)).
Of
course, the effective service time of a C2 customer for a
Transform 072(5) (and not
service. Thus, the
(3.10)
from
or (3.1 1)
(
)
number of C2 customers
the distributional
is
of case
As
3:
ergodicity.
One needs
=
the
so P[ El E2
.
As
I
Ej
is
]
[
in the
E1E2IEJ
= I-P1-P2
I
.
|
]
queue
found using an argument similar to
is
would have been
u'*^!^""^
= Prob
EiEj] =
].
box
at
Again we have
[
EJ
= Prob
,
At ergodicity
P[ EjE^I Ej
E1E2
]
=
]
E[ u^i^^'^lEiE^
]
Prob [Ej]
/
1-pi so that P[
.
But the
E1E2
lEj
[
E^ EJ =
I
1
-
P[ E2
c
E^, that a Cj service
EJ =
under
is
this service time,
I
]
idle state
I
= (l-pi-p2)
]
Nj was
the elapsed time since the
Cj
p2
way
=
/
/
(1-pi).
The
with a service
zero, or Cj service
service
distributed as the forward recurrence time for Tj with transform a^^is)
s]
that
1.
At the beginning of
started.
Ej
I
+
]
Clearly, P[ Ei
equivalent to the event
time distributed as Tj.
E[T2]
7t(-.(u) is
(2.1) [6].
E[ u'^i^^'^lEjEj
before, E[u^2(~)
c
given by (2.6) where
]
]
For EjE^, Prob [EjE^
event EjE^
is
aT2(s) and the time in the queue or the system follow
pgf 7Cg(u)= E[
P[ EjEjIEj
For El E2 one has Prob
(1-pi)
system
before, let Ej^ be the event that no C|^ customers are in the service
(3.16) 7:b(u) = E[ uNi(~)
E1E2
in the
form of Little's Law,
The number of Ci customers
has Laplace
because the C2 customers are not interrupted during
(3.7))
and 0Cp2£pp(s)
HOL
was
initiated is
= [1-0^2(5)]
/
[
.
Hence E[u
'^'"^
I
EjE
]
=
a.j,2(^i"^iu)-
on semi-Markov processes reaches
the
^
more formal and long winded argument based
same conclusion. From
page 12
(3.16)
we have
finally
nM=^^^^^^^
B
(3.17)
-^a;,(X,-X,u)
^'
1-pi
1-pi
In
from
HOL the
(2.3)
Ci are not interrupted so aji£pp(s) =
the p.g.f. of the
and (3.17)
4.
The Fuhrmann
-
[2]
which need not have exhaustive
Theorem
The decomposition
service.
It
given by
^a;,iX,-X,u)]
Theorem
applies to
2 but with
somewhat
M/G/1 vacation models
could be applied for example to N-policies,
M/G/1 queues with gated
vacations, limited service queueing model,
systems.
(Fuhrmann and Cooper
3.
is
Hence,
have given an interesting and important decomposition for
different conditions listed below.
static priority
queue
['-^ .
'-^
vacation models in a general setting similar to that of
and
in the
Pj.
Cooper Decomposition.
Fuhrmann and Cooper
cyclic service queues,
and Pj^^p =
number of Ci customers
V")=
(3.18)
0Lj.j(s)
[2])
Let the following conditions hold.
1:
Customers arrive
to the
system according
to a
Poisson Process of rate
X>0 with
identically distributed service times independent of each other, independent of the arrival
process and independent of the vacation periods that precede
2: All
3:
customers are evenuially served.
Customers are served
4: Service
service
"it is
is
a simple
The
maner
Then
an order that
is
independent of the services times.
to appropriately 'inflate' the service times
They give no
rules that govern
anticipate future
in
non-preemptive. (Fuhrmann and Cooper observe that for preemptive
the [preempt] interruptions".
5:
it.
jumps of
when
...
to
account for
details.)
the server begins
and ends vacations do not
the Poisson Arrival process.
the decomposition of Equation (2.3) holds with the
page 13
summands independent
If further
6:
The number of customers
number of customers
present in the system
KqCu) =
(4.1)
that arrive
[
during a vacation
when
the vacation
This can be seen as follows, using the arguments of
is in
at the
instant in a vacation.
began
this paper:
progress then the system must be on vacation. Hence,
system during an arbitrary
independent of the
^ov(") av.(X-?iu).
]
^Z"*^^?""^
is
But
this
If
Nb
no effective service time
equals the
number
in the
equals in the number in the queue
beginning of a vacation, Nqv, plus the number of arrivals during the vacation
backward recurrent time,
(4.1) follows
from
i.e.,
Nb = Nqv + Kxv*
so 7ig(u)
= KQy(u) av*(X-Xu). Hence,
(2.3).
Remark: Fuhrmann and Cooper do not require FIFO order of service because of their
exclusive concern with queue population. As shown in the remark under Theorem 2,
Theorem 2 has comparable requirements.
Acknowledgment: The
calculation of Proposition
Proposition
authors wish to thank Hongtao Zhang for his succinct
1 via
and Professor R. Larson for encouraging
MACS YMA
1.
page 14
REFERENCES
Doshi, B. T, "Queueing Systems with Vacations
pp. 29-66, 1987.
[1]
1,
-
A
Survey", Queueing Systems, Vol.
S. W. and R.B. Cooper, "Stochastic Decompositions in the M/G/1 Queue
Generalized Vacations", Operations Research, Vol. 33, No. 5, September-October,
1985, pp. 1117-1129.
[2]
Fuhrmann,
v^ith
[3]
Gaver, D.
of Royal
P.,
"A Waiting Time with
Statistical Society, Series B,
[4] Jaiswal, N. K., Priority
[5]
Keilson,
J.,
Interrupted Service, including lYiorities, Journal
Vol 24, pp. 73-90.
Queues, Academic Press,
New
York, 1968.
"Queues Subject to Service Interruptions", Annals of Mathematical
No. 4, December, 1962, pp. 1314-1322.
Statistics, Vol. 33,
[6] Keilson, J. and L. D. Servi, "A Distributional
Operations Research Letters, Vol. 7, No. 5, 1988.
Form
of
Little's
Law",
to
appear
in
and L. D. Servi, "The Preempt-Resume M/G/1 queue with Feedback
Schedules
and Clocked
", submitted to Performance Evaluation, 1988.
[7] Keilson,
J.
Keilson
J.
[8]
Resume
and U. Sumita, "Evaluation of the Total Time in System in a PreemptQueue Via a Modified Lindley Process", Advances in Applied Probability,
Priority
Vol. 15, 1984, pp. 840-856.
page 15
APPENDIX:Derivation of Teff
For
simplicity,
assume
that all
for the Preempt-Repeat-Different discipline
of the random variables are absolutely continuous. In the
case where they are not absolutely continuous an argument could be constructed using a
limiting argument.
Let
be the time of the first interruption minus the time of the fu-st C2 service
initiation.The interarrival times of interruptions are exponential distributed with a
ti
mean lAi.
be the duration of the busy period of a high priority customer. This equals the
tBPi
duration of an interruption. Let sbpi(x) and ObpiCs) be the density function and
Laplace transform, respectively, of
tepi.
be the service time of the low priority customer. This equals the time that the first
low priority service would have ended if there were no interruptions. Let arzCx) and
tx
aT2(s) be
its
density function and Laplace transform, respectively, of tepi- Let
AT2(x) = Prob {tT>x}.
of the interval from the fu^t service initiation until either the end of a
service time or until the end of the n''^ interruption (whichever comes first). Let
aT2EFF(n)(x) be its density function.
t2EFF(n) t>e the duration
The key observation
fA^^
(^^)
Note
is
that
_
-
f
^2EFF(n+l)
/tT
iftT<ti
lti+tBPl+t2EFF(n)
if tj
>
that
-^
dx
(A.2)
< X and
<
{Prob
[
tT
Prob
[
ti+tBPi+t2EFF(n)
tT
ti ]
)
=
e-^i'^
aT2(x)
and
^
(A.3)
{
=
From (A.l)-
< X and
tj
>
ti
]
}
Xie-'^l''AT2(x) * SBPl(x) • aT2EFF(n)(x)
(A.3),
page 16
ti
aT2EFF(n+l)(x)
As
=
e-^l'<aT2(x)
n approaches infinity, aT2EFF(x)
+ >.ie-^l^AT2(x)
*
Sbpi(x)
aT2EFF(n)(x)
= c-h^zjjix) + Xich^ At2(x)
so (3.12) and(3. 13) follow.
page 17
•
sbpi(x)
.
*
aT2EFF(x)
S37k
26
Date DueEC
iu
Lib-26-67
MIT LIBRARIES
3 9080 01917 76