iIIooIIIIlIIII_"""_" - CS Technion
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
.Computer Science 'QepartmeI!t
OUTPUT,RATE AND OF AN M/G/1
"SERVICE FUNCTIONS" QUEUEING SYSTEM
by Micha Hofri Technical Report April
#148
1979
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Technion - Computer Science Department - Tehnical Report CS0148 - 1979
ABSTRACT The paper considers the output rate of a single server device under
f
a constant (homogeneous Poisson) load ("offline", in network modeling parlance) and nonpreemptible service. dependent on the queue size
(e~cept
This rate is found to be highly
when the service is exponential), the
service distribution and the utilization, in contradiction with some current modeling practices. presented.
A few new analytic results for the M/G/I system are
We conclude that this, "service function"
is actually an
improper characterization of the server, as it can only be assigned values when the input . prbcess is specified. ,)
-
Xeywords and
Phrases:
queueing theofY;
Output rate;
service function;
simulation; operational analysis.
"
e
err'
.s.
9
tr
M/G/I,
- 1
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
A,
lNTRODUCTION
AI. Standard queueing
theo~y
rarely considers the outvut rate of a server,
essentially because it is of little use:
the
unconditio~ed
rate is equal
to the (presumably known) input rate, and conditioning it (e.g. on time since a certain event, or queue size, etc.) Th~
'for which an obvious need existed. been subjected to
intensiv~
simply did
~ot r~sult
with a
quantity
entire~y
output proGess in i ts
has
study, especially since networks of queues
became natural objects of study, but the rate of the process again served no purpose. A2.
This quantity assumed some importance in recent annunciations of
Operational Analysis (see [3] and further references therein)
and its applica
tion in several ·investigations In networks.of~~ueues. Brierly;' ihi~-appreachinvolves there the formulation of (global) bal:ance ;equations for '''s'tate "
occupancies" ities
whfch have the -appearance of those f~r steady state probabil
of a birth and death process.
reciprocal of
'death rate'
former by a quantity named the
fl
The tole
corres~onding
to the
in the latter equations is played in' the
"service fUl1ction", 'S (n), which represents
"mean time between service completions, given that the load on the
server is at level
n".
More precisely, we shall view this quantity as
defined via its sample value definition
lim T(n)/c(n):
which means that
T-+9'>
while the server was observed for
~
duration T, it spent
time with a load level of n, and accomplished
t
within T(n). (This ratio for finite Its reciprocal
definition of Sen).
,.
The analyst who wishes to has. to supply those values,
7
• $
•
u~e
T(n) of
th~t
c(n) ,service terminations
is called the Opefational
r(n)
is the
-write and solve
'completion rate'.)
1;.hese equations
Sen).
S'.
..
'$
•
s'Sbo
trm
- 2
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
A3.
The
abov~
mentioned term
"service function"
suggests that
Sen)
are inherent to the service mechanism, perhaps depending on the load. * ijowever, this last qualification is only reserved for servers which are "online"
i.e., in interaction with other servers through migrating
customers, whereas
w~en
the server is
"offline"
a term normally inter
preted as having the server subjected to an incoming flow
ofserv~ce
requirements which can be represented by a homogeneous Poisson vrocess all
Sen) (i.e., for all values of n) are said to be equal to the
expected service requirement, ;E(s), [1].
This uniformity is a desirable
property, as estimating E(s) is easier (cheaper, for a required level of 'confidence) than that of the stratified quantities
Sen).
On the other
hand, the practice of using it in the balance equations assumes that using E(s) instead of
S(~)
should produce a first" approximation, and
when the impact of the other system components on the server in question -..... ~. ,4"":--.....
is not severe, the approximation sho1,lld prove to, b~~,a"go~'-Prte. Where from does this difference,
M.
b~tween
"onUne" and
"o£fline"
behaviour come? We have not found any satisfactory explication in the literature, and set out then to examine some simple examples.
We were
rewarded with a few surprises: Even for
Sen) •
MIGI 1
depend on n,
1. e. an "Pfflfe"
unless the stationary distribution is geometrical.
More importantly,
sta~e
probabilities;
situation for the server
Sen)
can be written as a function of the steady
an extremely simple one, as a matter of fact.
~ Throughout 'this paper the service mechanisms that are considered are not load dependent (e.g. a standard MIGll service system) .
a
- 3
i.a., that ~he Sen)
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
This means
I (
'J
process as well; mechanism. P
i.e.
For a
are tightly coupled with the input
are not merely a product of the service
trivial example
Sen)
depend criticaly on
= AE(x), the utilization factor, both in value and in functional form.
C and
As we shall see from the development and discussion in Sections these
'surprises'
AS.
can be justified by entirely intuitive argumentation.
B contains a description of the model used
Section
D
a standard
MIGII with a FIFO regime, and some analysis of its steady state distribu Section C derives a simple expression for the service function
tion.
Sen), and discusses some of its properties,
D concludes with
Section
.
some examples and an attempt at drawing conclusions .
We collect here some notation used throughout the paper.
A6.
The rate of the homogeneous input process
(Poisson).
The service requirement of each request (customer;
S
the terms
are interchangeable). The distribution (cdf), density (pdf) ahd Laplace-Stie~tjes
(LS) transform (of
Fs(o)) of the service
duration. B(s) - Mean service
=
p
require~ent;
~
_ l/E(s).
'-E(s) .
= 0,1,2, ...
i
.,..(i)
The steady state distribution of the number
of customers in the system as seen by a departing one (which does not count itself). ....'!"
The joint probability and density of observing the system
p (n,x)
at a. random time
~nd
finding it with n
requests,
units more to go to firlish the Cl,lrrent service.
x time
w~th
For
11:;:;
0 only
x p: 0 is used .
•
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$$
n.
zt
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d
1 •
rtr
t
D
r
,.
- 4
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
N,X - The corresponding state variables.
8.
Sen)
Service function, see A2 and C2.
r(n) -
s-l(n).
MlG/1
STEADY STATE ANALYSIS
The M/G/I
81.
queueing system is one of the best known, and certainly
one of the most widely used. [2];
The most complete reference'
a more accessible one is
[4].
~s,
probably
We shall briefly mention its salient
featur"es: .:.
The input 'Process of single requests is homogeneous, Poisson, .at rate A~l!:; -
The service requirement
.
.
'.
of each arrival has the distribution FsC·),
The sampled' value is independent of the' state of the sY$1,:em" its history, arrival time, etc. - The
select~on
for service mechanism
is not cognizant of ,the required
service durations (or equivalently, this duration selection) .
For nearly all our needs here this
is only knQwrf"post
mech~nism,J~
irrelevant.
For, specificity the reaqev may assume a FIFO (first-in-fifst-out) order. Services are conducted one at a time, and are not interrupted until request completion.
The time between a service
comp1et~Qn
(=departure
time) and the initiation of service for the next request (if one existsl is zero. 82.
Generally, specifying the number of pending requests is not sufficient
to determine the future evolution of the system of the current
servic~*
is also required.
* A fUlly markovian specification
-
the time 'till completion
An exception to this incon
i~
also obtained, if the time since the This approach is taken in [4, ,p, 233] . For our purposes the first version is superior.
be~inning of the service is supplied.
,.
b
P'
$ $
•
tr
...
,. 5
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
venience is
provi~ed
by observing the system at departure epochs;
then
the number of customers provides all the ,required information to (pro babilistically) calculate future developments.
Moreover
it is simple
to show that the distribution of the number of customers at this special set of epochs, denoted by
P(X=n)
= ~(n), is identical to the one
by an incoming request or tallied by a extent, explains why
"random" 9bserver.
'l~mited'
such a
exp~rienced
Thi,s, to some ~
solution enjoys such
wide usage
(- we perforce disregard many ancillary considerations). 83.
The
directly.
wen) are not enough hOWever to determine the output rate The required quantity is
(pdf, really) ;J
..
dur~tion
0t observing
p(n,x)
the system with n
x f.or the current service.
a recurrent markov
pr~cess
we find
fraction of observations, out of
the'
(~teady
.
~ustomers,
state) probability
and a
Using the fact that
relqaining
(n,x}
define
pCn,x) by calculating the expected K,
that would find the system with n
requests and a remaining service duration in the interval (x,x + Ax) .
84.
This is done by conditioning on the number of customers at the last
servi~e
of
~he
completion before the observation took place
subsequent service requirements (-y):
(n;x,x + t.x) observations
Expected number of
= 00
n
pK ~ (TT(i) + ~i,lW(O))
i;:l
(-i), and the length
I
yfs (y)
e·). ( y-x') Ee s ) . l.. y
['C ,n-i dyAx 1\ y;X,.. (n-;L) !
y=x
'
The factor
p
selects those observations that find the system non-empty
and thus the rest of the calculation assumes, that if the last departure terminated a busy period, an arrival has occurred since; inclusion of w(O)
with
~(l).
y~s(Y)/E(x)
hence the
is the pdf of hitting
s
,t
•
-
..
- 6
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
a
I
I.J
service duration of size
y, and
point (- which is uniform) at
x
l/y
is the
from its end;
the probability that arrivals bui.1d up the during
y-x.
Letting
pdf
of the hitting
the last two factors are
"desired"
population
n
The use of the departure state distribution is obvious. K increase we obtain
p(O,O)
= nCO) = 1
- p,
00
n p(n,x) = A ~ [neil i=l
+
o.
In(O)]
1,
I y=x
fs(y)e -ACy-X) [A(Y-X)] n-i I(n-i) !dy, n ;;;;, 1.
This expression cannot be usually much simplified.
(B-1)
.r. s Co) for the
Writing
L8 transform of the service duration distribution we obtain, by immediate summation and integration
.r. (A-AZ) -.r. (5) s 5 f 1 ) = (1- p ) 1 + ( 1- z) AZ ( s _A+ Az) (X ( A~ Az) _Z
N -s X
H( Z , s) = E( Z e
}
(B-2)
S
wh.ere we took advantage of our knowledge of the Khintchin-Pollaczek
B5.
pgf of
n(i)
the
result:
As a check of these note that
H(z ,0) = G(z), which corresponds to
the remark in B2 concerning the coincidence of the distributions observed by a departing customer and a random observer.
Also
H(l,s)
= (1-£ s (sJ)/sE(s),
which is indeed the LS transform of the random modification of the service time, B6.
A "standard" way to obtain the value of state probabilities is to
formulate balance equations in terms of these and solve them. above we could assume the
n(i)
Since in the
known we were spared this route.
ncr
The
- 8
+
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
fs(x)p(n+l,O )
= A[p(n,x)
- (1- o.n, l)p(n-l,x)] n
n =0,
and in a similar manner, for +
p(l,O)
:> 1.
= A(l-p).
(B-3)
It is straightforward to verify that (B-3)
are indeed satisfied by (B-1) .
.'
C.
OUTPUT RATE AND SERVICE FUNCTION
We begin this section by noting that its title is in a sense a mis
Cl. nomer.
It is coined after
processes.
th~
transition rate parameter in
and death
When the service duration distribution is exponential, this is
a natural appelation as the probability of (small) interval
con~luding
a
se~vice
in anr
this being the proper definition of rate
equal to the average number qf service completions per unit will be taken· a's our working definition fQr the case for non-exponential
C2.
bi~th
"output rate".
is indeed
ti~e,
which
This is not
S.
Our main interest lies then in
r(n)
' = S-1 (n),
the aQove average,
over a duration in which the number of customers is n. This is given by lit [p (n,u)du!lit] !P(N=n) , which we- evaluate precisely as the eXp"ression u=O in 86. ~ anp on observing that 11 (N=n) = 1T (n) , ob1: ain
J
+
r(n) = p(n,O l!1T(n), and
(C-l)
(B-1) yielding p(n,O+) = A1T(n-l) finally nets r(n)
= A1T(n-l)/1T(n).
(C-2)
This is the desired result. .,
- 9
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
C3.
MIWI (C-2), gives r(n)
Note immediately that for
= AlP = J.1, constant
as expected. Generally for this invartance to hold the
~(n)
need
pe geometrical, e.g.
the solution of a system of first order difference equations, as in the
M/M/l C4.
system.
We return later to this point.
In view of the apparently unexpected
char~cter
of
explicitly address the 'issue of the dependence of ren) Such a dependence means that giving the value of N
(C-2) we shall (- and Sen)) on as n
conveys
information about the probability of imminent serviGe completion. this condition to be satisfiep
two~
stages of inference
~ust
For
be possible:
(1)
n needs be related to the portion of service already done.
(2)
Th~t
Note:
n.
I
portion has to determine the chance of service completion.
These two stages are distinct when the service mechanism
stipulated are those dealt with in this paper Allowing such a
a~pendence
as we
is not load dependent.
confounds (in the statistical sense)'th,ese
factors, and we want to keep them separate in this discussion, merely for the sake of clarity. CS.
Intuitively, all service distributions that are not memoryless
(i.e. exponential or' geometrical) satisfy the second requirement, to a greater or lesser extent.
The situation of the first one is less clear.
Generally speaking we may associate larger values of n of attained service. and for
I~R
with larger values
(DFR) service'distributions this would
imply larger (smaller) probability for imminent departure. best perhaps to consider (C-2) itself: would be
>1..
n > mode(N).
for
n
less than the mode
For values of
p
if
~(n)
is
It is here
unimod~l,
ren)
of N, and less than -A for
which are small or moderate the mode is
- 10
,
...
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
very often 0, which means that
~
If n(n)
is mu1timoda1 (p ~ 1)
loaded" system
r(n)
r(n)
would be monotonically decreasing. aro~d
will oscillate
A.
For a heavily
N is, approximately normal, with the mode around
(and slightly' less than)
E(N)~
p/ (l-p).
it' is well known that heavily loaded that depend essentially only on
It .is interesting to note that while
M/G/l
systems have waiting times ;
p (the famous exponential approximation),
the completion rate, being dependent on a much finer struot.~e of the queue
MIMI 1 with
ing l?rocess, does not degenerate to ,the one observed in an the same
p
(in such a system N= 0
is always .j
the mode of
N, regardless
of' p).
C6.
One last observation is that for
n
larg~
it is unlikely that a
significant portion of the present customers came during
t~e
current
service,. which would lead us to expect a very mild dependence of on n
for. large
n.
This
~an
be somewhat quantified:
determined by a system of difference equations of a
r(n)
if the TIen)
are
order
as is
f~nite
the case for the extremely rich family of phase-type service' ''dHtributions ' the values of and
TIen)
for large
n
n
are
~w
, where
w is the root of largest modulus (still always
istic equation of these difference equations;
then
is some -constant
a
~l)
of the character
lim r(n) = A/W. n-+o
D. 01.
EXAMPLES, DISCUSSION Work is now performed to try and characterize these rate functions
r(n), in
ter~s
of both the service regime and distribution, and the
arrival process.
Preliminary work
mainly simulation so fat
indicates considerable v.ariation of the with the general discussion of Section C.
$
r(n)
with n, in conformance examp~e:
For
b
in M/Er/i,
3
t
9
t
,I'
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
- 11
when
Er
and
A = 1) r(n)
where
stands for
p =
r = 4,
p
0.9,
Erlang distribution of order
!f(n)::: 0.83, 1,014,
~.03,
1~09,
extreme values for
1.13.
of r(n) on a wider range
~f
p
functions,
for
r
produces more
tends to spread the variation
values of n.
ev.aluatin~,
Sen)
the changes wrought in the service
from their value under a homogeneous load
ditions such as a finite source, instaneous feedback
And then it was
fo~d
"offline"
under
we elected to
that those
conditions.
d~vote
Sen)
"worse"
"online"
~nd
con
several others.
Due to the surprise value of this discovery
a separate paper ot its investigation.
The crucial item in that surprise is not that
What is
the
have a non trivial structure even
this is unfortunate, from the point of view of
Sen)
estim~tion
depend on
n',
of parameters.
is that it was found necessary to demote i t from the
role of exogenous endo~enous
variable of the system (such as the service distribution) one, such as waiting time, that can. only be known after
the participating processes are solved for. that even the functional shape of Sen) D4.
T~2
0.8,
p:::
"offline" situation, to the value obtained under several
to an
= 0.5,
The work reported in this paper was started with an aim towards
understanding, perhaps
D3.
p
1,04,.",1.09 (n= 10);
Further increase of
r(n), and higher
where
0.87, 1.08, 1.17. 1.18;
= 0.5, r(n) = 0.83, 1.22, 1.38. 1.59;
r(n) = 0.75, 1.01,
D2,
valu~s
had the first four
r,
This is evident when we note
depends on
p
One could summarize by saying that a service facility cannot be
parametrized by its service function except under very ances (e.g. Sen) = 11M,
$
all
n
~
1).
s~ecia1 circ~mst-
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
- 12
REFERENCES [1]
B~lbo,
G., P.J. Denning:
Queueing Network~.
Homogeneous Approximations of General
Tech. Rep. CSD-TR 290, Department of Computer
Sciences, Purdue University, November 1978. [2]
Cohen, J.W.: The Single Server Queue. Company, 1969.
[3]
Denning., P.J., J.P. Buzen: Network Models.
Compo Surv.,
North-Holland Publishing
The Operational Analy~is of Queueing
!£,
pp. 225-262, 1978. ,
[4]
Kleinrock, L.:
Queueing Systems, Vol. I.
Jbhn Wiley
&'Sons,
1975.
J...
.Computer Science 'QepartmeI!t
OUTPUT,RATE AND OF AN M/G/1
"SERVICE FUNCTIONS" QUEUEING SYSTEM
by Micha Hofri Technical Report April
#148
1979
•
~'---
L
TECHNION - Israel Institute of Technology
iIIooIIIIlIIII_"""_""""_"""-"" st •• '
t. stetz
$"-
""';.a.,..-
li...oO:oI
t
...
'"H
piS
• , •
..
\a
-.. _ • •.. .. ...
I
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
ABSTRACT The paper considers the output rate of a single server device under
f
a constant (homogeneous Poisson) load ("offline", in network modeling parlance) and nonpreemptible service. dependent on the queue size
(e~cept
This rate is found to be highly
when the service is exponential), the
service distribution and the utilization, in contradiction with some current modeling practices. presented.
A few new analytic results for the M/G/I system are
We conclude that this, "service function"
is actually an
improper characterization of the server, as it can only be assigned values when the input . prbcess is specified. ,)
-
Xeywords and
Phrases:
queueing theofY;
Output rate;
service function;
simulation; operational analysis.
"
e
err'
.s.
9
tr
M/G/I,
- 1
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
A,
lNTRODUCTION
AI. Standard queueing
theo~y
rarely considers the outvut rate of a server,
essentially because it is of little use:
the
unconditio~ed
rate is equal
to the (presumably known) input rate, and conditioning it (e.g. on time since a certain event, or queue size, etc.) Th~
'for which an obvious need existed. been subjected to
intensiv~
simply did
~ot r~sult
with a
quantity
entire~y
output proGess in i ts
has
study, especially since networks of queues
became natural objects of study, but the rate of the process again served no purpose. A2.
This quantity assumed some importance in recent annunciations of
Operational Analysis (see [3] and further references therein)
and its applica
tion in several ·investigations In networks.of~~ueues. Brierly;' ihi~-appreachinvolves there the formulation of (global) bal:ance ;equations for '''s'tate "
occupancies" ities
whfch have the -appearance of those f~r steady state probabil
of a birth and death process.
reciprocal of
'death rate'
former by a quantity named the
fl
The tole
corres~onding
to the
in the latter equations is played in' the
"service fUl1ction", 'S (n), which represents
"mean time between service completions, given that the load on the
server is at level
n".
More precisely, we shall view this quantity as
defined via its sample value definition
lim T(n)/c(n):
which means that
T-+9'>
while the server was observed for
~
duration T, it spent
time with a load level of n, and accomplished
t
within T(n). (This ratio for finite Its reciprocal
definition of Sen).
,.
The analyst who wishes to has. to supply those values,
7
• $
•
u~e
T(n) of
th~t
c(n) ,service terminations
is called the Opefational
r(n)
is the
-write and solve
'completion rate'.)
1;.hese equations
Sen).
S'.
..
'$
•
s'Sbo
trm
- 2
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
A3.
The
abov~
mentioned term
"service function"
suggests that
Sen)
are inherent to the service mechanism, perhaps depending on the load. * ijowever, this last qualification is only reserved for servers which are "online"
i.e., in interaction with other servers through migrating
customers, whereas
w~en
the server is
"offline"
a term normally inter
preted as having the server subjected to an incoming flow
ofserv~ce
requirements which can be represented by a homogeneous Poisson vrocess all
Sen) (i.e., for all values of n) are said to be equal to the
expected service requirement, ;E(s), [1].
This uniformity is a desirable
property, as estimating E(s) is easier (cheaper, for a required level of 'confidence) than that of the stratified quantities
Sen).
On the other
hand, the practice of using it in the balance equations assumes that using E(s) instead of
S(~)
should produce a first" approximation, and
when the impact of the other system components on the server in question -..... ~. ,4"":--.....
is not severe, the approximation sho1,lld prove to, b~~,a"go~'-Prte. Where from does this difference,
M.
b~tween
"onUne" and
"o£fline"
behaviour come? We have not found any satisfactory explication in the literature, and set out then to examine some simple examples.
We were
rewarded with a few surprises: Even for
Sen) •
MIGI 1
depend on n,
1. e. an "Pfflfe"
unless the stationary distribution is geometrical.
More importantly,
sta~e
probabilities;
situation for the server
Sen)
can be written as a function of the steady
an extremely simple one, as a matter of fact.
~ Throughout 'this paper the service mechanisms that are considered are not load dependent (e.g. a standard MIGll service system) .
a
- 3
i.a., that ~he Sen)
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
This means
I (
'J
process as well; mechanism. P
i.e.
For a
are tightly coupled with the input
are not merely a product of the service
trivial example
Sen)
depend criticaly on
= AE(x), the utilization factor, both in value and in functional form.
C and
As we shall see from the development and discussion in Sections these
'surprises'
AS.
can be justified by entirely intuitive argumentation.
B contains a description of the model used
Section
D
a standard
MIGII with a FIFO regime, and some analysis of its steady state distribu Section C derives a simple expression for the service function
tion.
Sen), and discusses some of its properties,
D concludes with
Section
.
some examples and an attempt at drawing conclusions .
We collect here some notation used throughout the paper.
A6.
The rate of the homogeneous input process
(Poisson).
The service requirement of each request (customer;
S
the terms
are interchangeable). The distribution (cdf), density (pdf) ahd Laplace-Stie~tjes
(LS) transform (of
Fs(o)) of the service
duration. B(s) - Mean service
=
p
require~ent;
~
_ l/E(s).
'-E(s) .
= 0,1,2, ...
i
.,..(i)
The steady state distribution of the number
of customers in the system as seen by a departing one (which does not count itself). ....'!"
The joint probability and density of observing the system
p (n,x)
at a. random time
~nd
finding it with n
requests,
units more to go to firlish the Cl,lrrent service.
x time
w~th
For
11:;:;
0 only
x p: 0 is used .
•
$
b
C,'
$$
n.
zt
•
d
1 •
rtr
t
D
r
,.
- 4
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
N,X - The corresponding state variables.
8.
Sen)
Service function, see A2 and C2.
r(n) -
s-l(n).
MlG/1
STEADY STATE ANALYSIS
The M/G/I
81.
queueing system is one of the best known, and certainly
one of the most widely used. [2];
The most complete reference'
a more accessible one is
[4].
~s,
probably
We shall briefly mention its salient
featur"es: .:.
The input 'Process of single requests is homogeneous, Poisson, .at rate A~l!:; -
The service requirement
.
.
'.
of each arrival has the distribution FsC·),
The sampled' value is independent of the' state of the sY$1,:em" its history, arrival time, etc. - The
select~on
for service mechanism
is not cognizant of ,the required
service durations (or equivalently, this duration selection) .
For nearly all our needs here this
is only knQwrf"post
mech~nism,J~
irrelevant.
For, specificity the reaqev may assume a FIFO (first-in-fifst-out) order. Services are conducted one at a time, and are not interrupted until request completion.
The time between a service
comp1et~Qn
(=departure
time) and the initiation of service for the next request (if one existsl is zero. 82.
Generally, specifying the number of pending requests is not sufficient
to determine the future evolution of the system of the current
servic~*
is also required.
* A fUlly markovian specification
-
the time 'till completion
An exception to this incon
i~
also obtained, if the time since the This approach is taken in [4, ,p, 233] . For our purposes the first version is superior.
be~inning of the service is supplied.
,.
b
P'
$ $
•
tr
...
,. 5
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
venience is
provi~ed
by observing the system at departure epochs;
then
the number of customers provides all the ,required information to (pro babilistically) calculate future developments.
Moreover
it is simple
to show that the distribution of the number of customers at this special set of epochs, denoted by
P(X=n)
= ~(n), is identical to the one
by an incoming request or tallied by a extent, explains why
"random" 9bserver.
'l~mited'
such a
exp~rienced
Thi,s, to some ~
solution enjoys such
wide usage
(- we perforce disregard many ancillary considerations). 83.
The
directly.
wen) are not enough hOWever to determine the output rate The required quantity is
(pdf, really) ;J
..
dur~tion
0t observing
p(n,x)
the system with n
x f.or the current service.
a recurrent markov
pr~cess
we find
fraction of observations, out of
the'
(~teady
.
~ustomers,
state) probability
and a
Using the fact that
relqaining
(n,x}
define
pCn,x) by calculating the expected K,
that would find the system with n
requests and a remaining service duration in the interval (x,x + Ax) .
84.
This is done by conditioning on the number of customers at the last
servi~e
of
~he
completion before the observation took place
subsequent service requirements (-y):
(n;x,x + t.x) observations
Expected number of
= 00
n
pK ~ (TT(i) + ~i,lW(O))
i;:l
(-i), and the length
I
yfs (y)
e·). ( y-x') Ee s ) . l.. y
['C ,n-i dyAx 1\ y;X,.. (n-;L) !
y=x
'
The factor
p
selects those observations that find the system non-empty
and thus the rest of the calculation assumes, that if the last departure terminated a busy period, an arrival has occurred since; inclusion of w(O)
with
~(l).
y~s(Y)/E(x)
hence the
is the pdf of hitting
s
,t
•
-
..
- 6
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
a
I
I.J
service duration of size
y, and
point (- which is uniform) at
x
l/y
is the
from its end;
the probability that arrivals bui.1d up the during
y-x.
Letting
of the hitting
the last two factors are
"desired"
population
n
The use of the departure state distribution is obvious. K increase we obtain
p(O,O)
= nCO) = 1
- p,
00
n p(n,x) = A ~ [neil i=l
+
o.
In(O)]
1,
I y=x
fs(y)e -ACy-X) [A(Y-X)] n-i I(n-i) !dy, n ;;;;, 1.
This expression cannot be usually much simplified.
(B-1)
.r. s Co) for the
Writing
L8 transform of the service duration distribution we obtain, by immediate summation and integration
.r. (A-AZ) -.r. (5) s 5 f 1 ) = (1- p ) 1 + ( 1- z) AZ ( s _A+ Az) (X ( A~ Az) _Z
N -s X
H( Z , s) = E( Z e
}
(B-2)
S
wh.ere we took advantage of our knowledge of the Khintchin-Pollaczek
B5.
pgf of
n(i)
the
result:
As a check of these note that
H(z ,0) = G(z), which corresponds to
the remark in B2 concerning the coincidence of the distributions observed by a departing customer and a random observer.
Also
H(l,s)
= (1-£ s (sJ)/sE(s),
which is indeed the LS transform of the random modification of the service time, B6.
A "standard" way to obtain the value of state probabilities is to
formulate balance equations in terms of these and solve them. above we could assume the
n(i)
Since in the
known we were spared this route.
ncr
The
- 8
+
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
fs(x)p(n+l,O )
= A[p(n,x)
- (1- o.n, l)p(n-l,x)] n
n =0,
and in a similar manner, for +
p(l,O)
:> 1.
= A(l-p).
(B-3)
It is straightforward to verify that (B-3)
are indeed satisfied by (B-1) .
.'
C.
OUTPUT RATE AND SERVICE FUNCTION
We begin this section by noting that its title is in a sense a mis
Cl. nomer.
It is coined after
processes.
th~
transition rate parameter in
and death
When the service duration distribution is exponential, this is
a natural appelation as the probability of (small) interval
con~luding
a
se~vice
in anr
this being the proper definition of rate
equal to the average number qf service completions per unit will be taken· a's our working definition fQr the case for non-exponential
C2.
bi~th
"output rate".
is indeed
ti~e,
which
This is not
S.
Our main interest lies then in
r(n)
' = S-1 (n),
the aQove average,
over a duration in which the number of customers is n. This is given by lit [p (n,u)du!lit] !P(N=n) , which we- evaluate precisely as the eXp"ression u=O in 86. ~ anp on observing that 11 (N=n) = 1T (n) , ob1: ain
J
+
r(n) = p(n,O l!1T(n), and
(C-l)
(B-1) yielding p(n,O+) = A1T(n-l) finally nets r(n)
= A1T(n-l)/1T(n).
(C-2)
This is the desired result. .,
- 9
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
C3.
MIWI (C-2), gives r(n)
Note immediately that for
= AlP = J.1, constant
as expected. Generally for this invartance to hold the
~(n)
need
pe geometrical, e.g.
the solution of a system of first order difference equations, as in the
M/M/l C4.
system.
We return later to this point.
In view of the apparently unexpected
char~cter
of
explicitly address the 'issue of the dependence of ren) Such a dependence means that giving the value of N
(C-2) we shall (- and Sen)) on as n
conveys
information about the probability of imminent serviGe completion. this condition to be satisfiep
two~
stages of inference
~ust
For
be possible:
(1)
n needs be related to the portion of service already done.
(2)
Th~t
Note:
n.
I
portion has to determine the chance of service completion.
These two stages are distinct when the service mechanism
stipulated are those dealt with in this paper Allowing such a
a~pendence
as we
is not load dependent.
confounds (in the statistical sense)'th,ese
factors, and we want to keep them separate in this discussion, merely for the sake of clarity. CS.
Intuitively, all service distributions that are not memoryless
(i.e. exponential or' geometrical) satisfy the second requirement, to a greater or lesser extent.
The situation of the first one is less clear.
Generally speaking we may associate larger values of n of attained service. and for
I~R
with larger values
(DFR) service'distributions this would
imply larger (smaller) probability for imminent departure. best perhaps to consider (C-2) itself: would be
>1..
n > mode(N).
for
n
less than the mode
For values of
p
if
~(n)
is
It is here
unimod~l,
ren)
of N, and less than -A for
which are small or moderate the mode is
- 10
,
...
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
very often 0, which means that
~
If n(n)
is mu1timoda1 (p ~ 1)
loaded" system
r(n)
r(n)
would be monotonically decreasing. aro~d
will oscillate
A.
For a heavily
N is, approximately normal, with the mode around
(and slightly' less than)
E(N)~
p/ (l-p).
it' is well known that heavily loaded that depend essentially only on
It .is interesting to note that while
M/G/l
systems have waiting times ;
p (the famous exponential approximation),
the completion rate, being dependent on a much finer struot.~e of the queue
MIMI 1 with
ing l?rocess, does not degenerate to ,the one observed in an the same
p
(in such a system N= 0
is always .j
the mode of
N, regardless
of' p).
C6.
One last observation is that for
n
larg~
it is unlikely that a
significant portion of the present customers came during
t~e
current
service,. which would lead us to expect a very mild dependence of on n
for. large
n.
This
~an
be somewhat quantified:
determined by a system of difference equations of a
r(n)
if the TIen)
are
order
as is
f~nite
the case for the extremely rich family of phase-type service' ''dHtributions ' the values of and
TIen)
for large
n
n
are
~w
, where
w is the root of largest modulus (still always
istic equation of these difference equations;
then
is some -constant
a
~l)
of the character
lim r(n) = A/W. n-+o
D. 01.
EXAMPLES, DISCUSSION Work is now performed to try and characterize these rate functions
r(n), in
ter~s
of both the service regime and distribution, and the
arrival process.
Preliminary work
mainly simulation so fat
indicates considerable v.ariation of the with the general discussion of Section C.
$
r(n)
with n, in conformance examp~e:
For
b
in M/Er/i,
3
t
9
t
,I'
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
- 11
when
Er
and
A = 1) r(n)
where
stands for
p =
r = 4,
p
0.9,
Erlang distribution of order
!f(n)::: 0.83, 1,014,
~.03,
1~09,
extreme values for
1.13.
of r(n) on a wider range
~f
p
functions,
for
r
produces more
tends to spread the variation
values of n.
ev.aluatin~,
Sen)
the changes wrought in the service
from their value under a homogeneous load
ditions such as a finite source, instaneous feedback
And then it was
fo~d
"offline"
under
we elected to
that those
conditions.
d~vote
Sen)
"worse"
"online"
~nd
con
several others.
Due to the surprise value of this discovery
a separate paper ot its investigation.
The crucial item in that surprise is not that
What is
the
have a non trivial structure even
this is unfortunate, from the point of view of
Sen)
estim~tion
depend on
n',
of parameters.
is that it was found necessary to demote i t from the
role of exogenous endo~enous
variable of the system (such as the service distribution) one, such as waiting time, that can. only be known after
the participating processes are solved for. that even the functional shape of Sen) D4.
T~2
0.8,
p:::
"offline" situation, to the value obtained under several
to an
= 0.5,
The work reported in this paper was started with an aim towards
understanding, perhaps
D3.
p
1,04,.",1.09 (n= 10);
Further increase of
r(n), and higher
where
0.87, 1.08, 1.17. 1.18;
= 0.5, r(n) = 0.83, 1.22, 1.38. 1.59;
r(n) = 0.75, 1.01,
D2,
valu~s
had the first four
r,
This is evident when we note
depends on
p
One could summarize by saying that a service facility cannot be
parametrized by its service function except under very ances (e.g. Sen) = 11M,
$
all
n
~
1).
s~ecia1 circ~mst-
Technion - Computer Science Department - Tehnical Report CS0148 - 1979
- 12
REFERENCES [1]
B~lbo,
G., P.J. Denning:
Queueing Network~.
Homogeneous Approximations of General
Tech. Rep. CSD-TR 290, Department of Computer
Sciences, Purdue University, November 1978. [2]
Cohen, J.W.: The Single Server Queue. Company, 1969.
[3]
Denning., P.J., J.P. Buzen: Network Models.
Compo Surv.,
North-Holland Publishing
The Operational Analy~is of Queueing
!£,
pp. 225-262, 1978. ,
[4]
Kleinrock, L.:
Queueing Systems, Vol. I.
Jbhn Wiley
&'Sons,
1975.
J...
