2 Cyclic polling: cycle time and conservation law
Tải bản đầy đủ  0trang
178
9 Polling
models
This equation follows from the fact that for stability reasons, on average, everything that
arrives in one cycle at station i, must be servable in one cycle. For the average time between
the departure of the server from station
intervisit
time Ii, we have
i and the next arrival
E[IJ = E[C]  E[P,] = (’ lv~~”
Important
to note is that upon the arrival
the server reaches that station
i, the socalled
e
of a job at station
is not E[Ii]/2.
at station
(9.3)
i, the average time until
Th is is due to the fact that Ii is a random
variable, and we thus have an example of the waiting time paradox. The average time
until the next server visit therefore equals the residual intervisit
time E[1:]/2E[Ii].
Notice
that, in general, an explicit expression for E[1!] is not available.
The (cyclic) polling models we address are generally not work conserving,
that is, there
are situations
but in which
in which there is work to be done (the queues are non empty)
the server does no real work since it is switching from one queue to another.
When
the switching times are zero, the polling model would have been work conserving and
Kleinrock’s
conservation
law would
apply
([ 1601; see also Chapters
5 and 6 on M]G] 1
queues) :
N
x pi~[~i] = pCEl xiE[s3.
w
i=l
Because piE[Wi]
= XiE[Si]
x E[N,,i]/Xi

= E[AJq,i]E[Si],
(9.4)
P>
the lefthand
side of (9.4) is often
called the amount of work in the system. Independent of how the queues are visited, this
amount always equals the steadystate amount of work in a model in which the service
order is FCFS (the righthand
side of (9.4)).
If we have only one station
(N = 1) and zero
switchover times, we obtain a normal (work conserving) M]G]l queue, and the righthand
side of (9.4) is just the expected waiting time in the M]G]l model. When we have only
one station with exhaustive service but now with positive switchover times, we obtain a
queue with multiple
server vacations.
When the model is not work conserving,
that is, when the switchover
times are positive,
Kleinrock’s
conservation
law does not hold anymore.
It has, however, been shown by
Boxma et al. [28, 27, 1121 that a socalled pseudoconservation
law still does hold. This
pseudoconservation
law is based on the principle
of work decomposition:
V=V+Y,
where v is a random
with positive switchover
variable
indicating
the steadystate
(9.5)
amount of work
times, V is a random variable indicating
in the model
the steadystate
amount
9.2 Cyclic
of work
polling:
cycle
time
and conservation
in the model when the switchover
law
179
times are set to 0, and Y is a random
vari
able indicating the steadystate amount of work in the model at an arbitrary
switchover
instance. The principle of work decomposition
is valid for cyclic polling models as well
as for polling
models with Markovian
of the scheduling
discipline,
routing
or a polling
whereas Y and therefore
table. V is totally
independent
? are dependent on the scheduling.
Intuitively,
one expects Y and ? to decrease if the switchover times decrease, if the visit
order becomes more efficient or if the scheduling becomes “more exhaustive”.
In particular,
for polling models with nonzero switchover times (with cyclic, tabular
ordering) a pseudoconservation
law of the following form applies:
When we are dealing with a cyclic polling
C
;E[wi]
+ c
iEE,G
;
(I
=)
XiE[Sf]
2(1

E[Wi]
+ c
;
(1  A$
I;‘“)
Ac2)
n(&J  Cz,
=
pf) + n
C~EG,L
d
n
and a decrementing
c&D
~ix~E[s~l
,
2P(l

P>
P(lPI

2P(l
where E, G, L, and D are the index sets of the queues with
a llimited
E[Wi]
iED
+2a+
P>
visit
order, one has:
iEL
= Cf!,
or Markovian
scheduling
discipline,

an exhaustive,
respectively.
(9.7)
P>
Clearly,
a gated,
the pseudo
conservation law expresses that the sum of the waiting times at the queues, weighted by
their relative utilisations
(for E and G directly and with more ccmplex factors for L and
D) equals a constant.
The pseudoconservation
law does not give explicit expressions for the individual mean
waiting times since it is only one equation with as many unknowns as there are stations.
Nevertheless, it does provide insight into system operation and in the efficiency of scheduling strategies. Also, it can be used as a basis for approximations
or to verify simulation
results (see below).
It is interesting to study the stability conditions for cyclic
server models with an exhaustive or a gated service discipline
condition
is p < 1. For models with
a llimited
server models. For cyclic
a necessary and sufficient
service strategy,
a necessary condition
can be derived as follows. The mean number of customers arriving at station i per cycle
equals XiE[C]. Th is number must be smaller than 1, as there is only 1 customer served
per cycle. Using the fact that E[C] = n/(1  p), th e necessary stability condition equals
p + xin
< 1, for all i. For models with a decrementing
stability
condition
of the form p + Xi(l  pi)A
scheduling
strategy
< 1, for all i, can be derived.
a necessary
180
9 Polling
Example
9.1. A 2station
Consider
an asymmetric
and station
asymmetric
polling
2 llimited
model
scheduling.
0.4, E[$]
= 0.32,
Xi =
conditions
are satisfied
with
so finite
pseudoconservation
law, yielding
relation
9.3
When
we address
independent,
models
+ kE[W2]
parameters
1,2.
Clearly,
do exist for both
E[Si]
We are
we can apply
E[W,]
=
the stability
stations.
however,
between
scheduling
apply:
the
and E[W2]:
(9.8)
= 3.7.
in the E[IVi]E[W,]
pl ane; the exact
solutions
for E[Wi]
0
in which
we can obtain
law,
approach
we will
polling
model
waiting
times
the expected
countbased
customer
system
customer
in service
tomer
E[S”]/2E[S].
with
from
one queue
that
Sc2) denotes
will,
the variance
of. the switchover
queues
all the average
by using the
waiting
in (9.7).
insight
into the actual
as presented
in Chapters
the expected
not
system
exhaustive
the lines of the proofs
model,
times
We will
for the
5 and 6.
waiting
time for
of 4 components:
remaining
in switchover
time here).
property,
find
p. The remaining
1  p an arriving
The
times
as follows:
probability
to another.
are station
time in a fully symmetric
following
due to the PASTA
with
probability
us much
polling
to consist
can be understood
some queue)
equals
and related
waiting
only one unknown
waiting
way,
symmetric
can be thought
the 4 components
1. An arriving
derive
in the MIG]l
For an exhaustive
moment
symmetric
models
and parameters
for the average
we are left with
in an operational
polling
disciplines
here, since it does not provide
Instead,
2. Similarly,
results
since in a fully
scheduling
an arriving
cyclic
all the scheduling
closedform
so that
operation.
where
1 has exhaustive
directly;
relation
symmetric
to one another
expected
times
and E[W2]
the following
can be drawn
pseudoconservation
use this
station
have to lie on this line.
Countbased
are equal
(I).
the following
waiting
E[IVi]
E[W,]
linear
2 stations:
Furthermore,
average
to compute
This
model
1, Si = 0.05 and Ji2) = 0 for i =
not in the position
and E[W2]
polling
models
customer
will
switchover
times,
another
service
time
customer
(at
of this
cus
find the server switching
time
whereas
equals
62 denotes
S2/2S
(notice
the second
9.3 Countbased
symmetric
3. An arriving
switchovers,
cyclic
polling
job arrives at any queue with equal probability,
each of expected
so on average (N  1)/2
length 6, are needed for the server to arrive
particular
queue (since the number
paradox does not apply).
4. Finally,
181
models
of queues N is a constant,
the waiting
at the
time
of a customer, the steadystate amount of work in the system
equals NE[N,]E[S].
I n eq ui 1i b rium, this amount of work should be handled before
the randomly arriving customer is served.
Adding
upon arrival
these 4 components,
we have:
JqWl = (I PI2 +
Using Little’s
law to rewrite
(9.10)
w21
 Puwl ’
(9.11)
E[N,]
= XE[W],
=
(I~)$+~c~+P~
=
s2
+ N  ’ 6 + NXE[S]
26
2(1p)
(lp)E[W]
+ E[W]
We can rewrite
+ NE[N,]E[S].
the first two additive
we obtain
w
terms as follows:
(N  p)6
(9.12)
2(1  P> ’
so that we have
h(2)
NXE[S2] + 6(N  p)
= 26 +
Wd
*
‘E’ is added to indicate that the formula is valid for exhaustive
E[WT]
The subscript
Along the above lines, one can also derive the mean waiting
strategies
scheduling.
time when all scheduling
are of gated type:
c5c2) NXE[S2]
E[~G]
For the llimited
scheduling
= 26
for the decrementing
E[~D]
+
discipline
S(2)
E[JC,]
Finally,
(9.13)
= 26
+ 6(N + p)
WP)
(9.14)
n
we have:
NXE[S2]
+
scheduling
+ 6(N + p) + NJd2)
2(1pNM)
discipline
’
(9.15)
we have:
dc2) NXE[S2](1  XS) + (N  p)(6 + xhC2))
= 26 +
2(1  p  M(N  p))
’
(9.16)
182
9 Polling
P
E[WE]
E[WG]
E[WL]
J?@%]
0.05
0.6000
0.6053
0.6085
0.6031
p
@[WE]
0.55
E[WG]
1.711
B[WL]
1.833
models
E[b]
2.089
1.931
0.15
0.7176
0.7353
0.7485
0.7302
0.65
2.314
2.500
3.070
2.793
0.25
0.35
0.8667
1.062
0.9000
1.115
0.9310
1.179
0.8953
1.119
0.75
0.85
3.400
5.933
3.700
6.500
5.286
15.00
4.690
12.27
0.45
1.327
1.409
1.535
1.428
0.95
18.60
20.50


Table 9.1: E[W]
menting
in symmetric
polling models with exhaustive,
gated, llimited,
and decre
scheduling
From these explicit
they correspond
formulae,
the earlier derived stability
to those traffic conditions
From these expressions,
conditions
where the righthand
can also easily be seen;
denominator
becomes zero.
it can also be observed that
and
E[WG] > E[WD]
Example
(at low load),
9.2. Symmetric
polling
In Table 9.1 we have tabulated
and E[WG] < E[WD]
models:
influence
the average waiting
(at high load).
(9.18)
of scheduling.
times for symmetric
polling
models
with exhaustive, gated, llimited,
and decrementing scheduling strategies for increasing
utilisation
(established by increasing the arrival rate). The other parameters are: N = 10,
s = 0.1, JC2) = 0.01, E[S] = 1.0, and E[S2] = 1.0. The above inequalities can easily
be observed.
Also notice that for p = 0.95, the llimited
and decrementing
already overloaded.
systems are
0
The fact that the exhaustive scheduling discipline is the most efficient can easily be
understood.
It simply does not spoil its time for switching purposes when there is still
work to do, i.e., when the queue it is serving still is not empty. The gated and decrementing discipline, however, sometimes take time to switch when there are still customers in
the current queue. This counts even more for the llimited
case where every service is
effectively lengthened with the succeeding switchover time. The fact that the amount of
switching overhead per customer is smallest with an exhaustive scheduling strategy does
not necessarily
imply
that it is also the best.
From a fairness
point of view, the other
9.4 Countbased
disciplines
asymmetric
cyclic
might be considered
polling
models
183
better since they prevent one station from totally
hogging
the system.
Example
9.3. A 2station
asymmetric
polling
model
(II).
Reconsider
the asymmetric
scheduling
customers.
strategy, station 1 profits from this as it receives more opportunities
to serve
In fact, E[IVi] should be smaller in the mixed scheduling case than when both
stations
exactly
station
polling model addressed before. Since station 2 uses a llimited
would have exhaustive
scheduling.
In this latter symmetric
case, however, we can
compute the average waiting times: E[VVr,J = 1.75. So, in the asymmetric case
1 is expected to perform better, i.e., E[IVr] 5 1.75. This implies, by the pseudo
conservation law derived for this example, that in the asymmetric
more, i.e., EIIVZ] 2 3.9.
We can improve on the above bounds by considering
case station 2 will suffer
the case where the arrival
rate of
station 2 is set to zero. In that case, the model reduces to an M]G]l queue with exhaustive
service and multiple vacations (as seen from station 1) because at station 2 no jobs arrive.
Thus, after the queue in station 1 empties, the server switches to queue 2 and directly
back
to queue 1. This switching can be interpreted as a vacation with average length 0.1 (two
switches of length 0.05). The variance of the switching (vacation) time is 0, so that we can
compute E[IVi]
using (5.38) as follows:
qJ/j/q_ Jfw21I WS21 =Oil+ o32
2(1  0.4)
2Jwq 20  P>
We thus have: 0.317 < E[IVi]
3.9 5 E[W2] 5 6.766.
9.4
Countbased
(9.19)
= Oe317*
5 1.75 and, using the pseudoconservation
asymmetric
cyclic
polling
law again,
cl
models
The analysis of asymmetric cyclic server models is much more complicated than the analysis
of symmetric models. We will present the exact analysis of an asymmetric cyclic polling
model with exhaustive
service in Section 9.4.1 followed by a number of approximate
derived by using the pseudoconservation
9.4.1
Exhaustive
In the exhaustive
i has the following
service:
asymmetric
results
law in Section 9.4.2.
exact analysis
case the average waiting
form (see also (5.38) for the M]GJl
time E[IVi]
queue with
perceived
at station
server vacations
in
9 Polling
184
Chapter
5):
E[WE,i]
JW3
= 2E[Ii]
where we recognise as the first term the residual
costs for the server to arrive at the station.
models
XiE[Sf]
+ 2( 1  pi) ’
intervisit
(9.20)
time, i.e., the average time it
The second term is simply
the MlGll
waiting
time applied for station i only. Since the scheduling discipline is exhaustive, once the
server is at station i, we can analyse station i in isolation as if it were an MlGll queue.
The second term can readily be computed.
E[li] d irectly follows from (9.3). The only
problem
we have in evaluating
by Ferguson and Aminetzah
E [W E,+.] is the determination
of E[I,f]; it has been derived
[88] as:
E[I?]2 = E[Ij22 + &!)I + ’  pi c
P
where the coefficients
ri,j
(9.21)
7
j#i
ri,j (i, j = 1,. . . , N) follow from a system of N2 linear equations:
j
ri,j
solution
9.4.2
solution
i,
j > i,
=
j
A similar
<
exists for gated systems.
For klimited
=
(9.22)
i.
and decrementing
systems such
schemes do not exist. For these cases, one has to resort to approximate
Some approximate
For most asymmetric
is a wide variety
solutions.
results
system models, exact results do not (yet) exist. For these cases, there
of approximate
results, some of which we will present here. The pseudo
conservation law plays an important role in the construction
the use of the following threestep approach:
1. The expected waiting
time for queue i, E[Wi],
of these approximations,
by
is expressed in terms of the expected
residual cycle time E[jZi] = E[If]/2E[&]
(for th e congestion due to traffic at other
queues) and some local parameters (for the congestion at queue i, e.g., in the form of
an MlGll result). As a result, E[W.] is expressed as a function of known parameters
and the unknown
parameter
E[Ri],
similar
to (9.20).
2. It is then assumed that the residual intervisit
i.e., E[&] = E[R], f or all i. This assumption
when the system becomes more asymmetric
times are equal for all the stations,
has been shown to be less accurate
and when the variance of the switchover
9.4 Countbased
asymmetric
cyclic
polling
185
models
times increases. The resulting expressions for E[Wi] are substituted
in the pseudoconservation
law in which, due to the assumption just made, only one unknown
remains, being E[Q . This yields an explicit
3. The results of Steps 1 and 2 are combined
expression
for E[R] .
to obtain explicit
expressions
for all the
E[W].
Using this approach, for a mixture of exhaustive
following result has been derived by Everitt [86]:
E[~E,G,~]z (1 f pi)&
and gated scheduling
disciplines,
the
(I  p)d2) + Cj"=, X$[S;]
1+
A
A2
>) *
(9.23)
For the gated scheduling discipline the + sign should be taken, whereas for the exhaustive
scheduling discipline the  sign holds.
Similarly,
for an asymmetric
derived by Boxma and Meister
E[WL,+]
FZ
model, the following
approximation
1P
’ (1  p)p + CfLl
pAc2)
+ $q
2A

and more accurate
c
kE[$‘]
pz
+ &
$PiCl
il
solution
EIWE,iI
approximation
= (1  Pi)E[RI,
E[WG,i]
+ pi))
*
(9.24)
2l
procedure
for asymmetric
been proposed by Groenendijk
[112, Section 7.2.31.
For models in which a mixture of exhaustive, gated, and llimited
exists, the following
has been
[28]:
’ + pi
1  p  AiA
X
An iterative
llimited
has been proposed by Groenendijk
= (1 + Pi)E[R],
E[WL,i]
llimited
scheduling disciplines
[ill].
= ll~~pp~Ap~E[fil,
i
where E[R] approximates the mean, scheduling dependent, residual
ing this result in the pseudoconservation
law yields:
models has
First
set
(9.25)
cycle time. Substitut
(9.26)
Example
Reconsider
we calculate
9.4. A astation
the asymmetric
asymmetric
polling
model
(III).
polling model addressed before. Using the above approximation
E[R] = I.028 and consequently
EIWl]
= 0.617 and E[W2] E 6.167. As can
186
9 Polling
models
be observed, these values not only lie on the line described by the pseudoconservation
but also within
Finally,
the bounds we established
for klimited
systems, Fuhrmann
before.
has derived the following
law
cl
bound for the symmetric
case [loll:
(9.27)
This bound is so tight
that it can be used as an approximation.
Note that the equal
sign holds if either Ic = 1 (llimited)
or Ic = 00 (exhaustive).
For asymmetric kilimited
systems, Fuhrmann and Wang have also provided an approximation
based on the pseudoconservation
9.5
law [ 1001.
Performance
evaluation
of the
In this section we discuss the use of countbased
cyclic
IBM
token
ring
server models for the analysis of
timedtoken
ring systems such as the IBM token ring. We briefly touch upon the IBM
token ring access mechanism in Section 9.5.1. Then we discuss an approximation
of this
timedtoken
access mechanism by means of (approximate)
klimited cyclic server models in
Section 9.5.2, and discuss the influence of the token holding time on the system performance
in Section 9.5.3. The methods presented here have been developed by Groenendijk
Ph.D thesis [112, Chapter
9.5.1
Timedtoken
in his
81.
access mechanisms
In this section a brief operational explanation is given of the timedtoken
network access
mechanism as used in the IBM token ring. We do not address the priority mechanism. For
more details, we refer to the survey paper by Bux [35].
In a token ring system a number of stations,
denoted Qi through QN, are connected to a
ring shaped medium. On this medium, a special pattern of bits circulates, the token. There
are two types of tokens: busy tokens and free tokens. A busy token is always followed by
an actual data packet. At the beginning of the data packet is the header which contains,
among others, a field of bits reserved for the indication of the destination address (the
address field).
Whenever a token passes a particular
station Qi, there are two possibilities.
When
the passing token is of the busy type, Qi checks whether the address field matches its own
address. If so, it starts copying the trailing data packet(s) in its input buffer. If the address
9.5 Performance
evaluation
of the IBM
field does not match the station’s
address, Q; simply
packet pass.
When a free token passes the station,
data packets
downstream
token
ring
187
lets the busy token and the trailing
there are again two possibilities.
to send, it just does nothing
station which will take a certain
to the token:
switchover
it simply
If Qi has no
passes it to the next
time. If Qi has data packets to
send, it grabs the empty token, changes it to a busy token and puts it on the ring again,
directly followed by the data packets it wants to send, of course preceded by the correct
header.
At the moment Qi starts transmitting
data packets on the ring, it also starts its (local)
token holding timer (THT), which operates as a countdown timer with initial value thti.
Now, Qi continues
to send until either all its data packets have been transmitted
THT expires, whichever comes first. After finishing the transmission
one of the two above reasons, the station issues a new free token.
Important
to note is that the expiration
of the THT
that is in progress. Because data packets have a maximum
maximum time a station will hold the token.
Upon receipt
station
of a data packet by a station,
of data packets, by
is nonpreemptive.
that once the THT expires: Qi is still allowed to finish the transmission
or the
This means
of the data packet
length, one can calculate
the
there are two ways to go. The receiving
might change the busy token in a free token (socalled
destination
release) or it
forwards the token to the sending station, which then releases a free token (source release).
Note that destination release is more efficient, although it might incur unfairness between
stations
9.5.2
as a truly
cyclic polling
Approximating
order is not guaranteed.
the timedtoken
access mechanism
We cannot directly
model the above timedtoken ring access mechanism in terms of countpolling
models. However, wc can approximate it by a cyclic (asymmetric)
based polling
model with a klimited
scheduling
strategy
as follows.
Consider a timedtoken
ring system, in which for station i the token holding time is
thti and the average packet transmission time is E[Si]. E[Si] and E[Sf] should reflect the
transmission
time of the packet, plus possibly the propagation delay on the medium
even the total roundtrip
delay, depending on which token scheme is used [35].
Whenever
be transmitted
llimited
thti is much smaller
per visit
scheduling
to station
or
than E[Si], in most cases no more than 1 packet will
be modelled as a
i. This situation can conveniently
discipline.
If, on the other hand thti is much larger than E[Si], in most cases all queued packets
188
9 Polling
simulation
llimited
2limited
p
E[W,]
E[W2,3]
E[wl]
E[W2,3]
E[W,]
0.3
0.421
0.347
0.525
0.370
0.420
0.322
0.5
1.12
0.747
1.51
0.775
1.12
0.670
0.8
10.6
2.33
55.7
2.31
9.67
2.23
0.3
0.442
0.444
0.317
0.558
1.20
0.399
0.989
0.506
0.5
1.38
1.16
0.861
1.38
0.8
7.52
5.60
10.72
8.30
6.91
6.22
Table 9.2: Comparison
of the expected waiting
results and approximate
E[W2,3]
time for a timed token protocol;
results using klimited
cyclic server models (results
at station i will be served. This can be modelled
models
as an exhaustive
scheduling
simulation
from [112])
discipline.
Clearly, interesting modelling problems arise whenever thti z E[S,]. Noting that, on
average, t&/E[Si]
packets “fit” in a single service period at station i, a klimited scheduling
strategy
seems reasonable to assume, with
Ic = round
The addition
Example
(
1+
(9.28)
g).
of 1 comes from the fact that the timer expiration
9.5. A 3station,
timedtoken
ring
system
is nonpreemptive.
(adapted
from
[112]).
Consider a timedtoken
ring system with only three stations. For comparison purposes,
the average waiting times E[Wi] h ave been derived in two ways: (1) by simulation, using
the timedtoken
llimited
mechanism
and a 2limited
in all its details, and (2) by using cyclic server models with a
scheduling
strategy.
The value of the token holding time in station i is equal to the average packet service
duration, i.e.,th& = E[Si]. Th e switchover
times & are assumed to be 0.1, deterministically, i.e., bi2) = 0. All packet lengths are assumed to be exponentially
distributed.
According to (9.28), we should use a 2limited scheduling discipline when these parameters
apply (Ic = 2).
In the upper half of Table 9.2 we present the results for Xi = 0.6, X2 = X3 = 0.2,
and E[Sr] = E[S2] = E[S3]. In the lower half of Table 9.2 we present the results for
x1 =
x2
=
x3
= l/3,
and E[Si]
= 3E[S2] = 3E[S3].
Th e results for the llimited
cyclic
server models have been obtained by the Boxma and Meister approximation
(9.24) for
small utilisation
(p = 0.3) and by an improved approximation
for the higher utilisations