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2 Cyclic polling: cycle time and conservation law

# 2 Cyclic polling: cycle time and conservation law

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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

inter-visit

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 so-called

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 inter-visit

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

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 left-hand

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 steady-state amount of work in a model in which the service

order is FCFS (the right-hand

side of (9.4)).

If we have only one station

(N = 1) and zero

switch-over times, we obtain a normal (work conserving) M]G]l queue, and the right-hand

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 switch-over times, we obtain a

queue with multiple

server vacations.

When the model is not work conserving,

that is, when the switch-over

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 so-called pseudo-conservation

law still does hold. This

pseudo-conservation

law is based on the principle

of work decomposition:

V=V+Y,

where v is a random

with positive switch-over

variable

indicating

(9.5)

amount of work

times, V is a random variable indicating

in the model

amount

9.2 Cyclic

of work

polling:

cycle

time

and conservation

in the model when the switch-over

law

179

times are set to 0, and Y is a random

vari-

able indicating the steady-state amount of work in the model at an arbitrary

switch-over

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 switch-over times decrease, if the visit

order becomes more efficient or if the scheduling becomes “more exhaustive”.

In particular,

for polling models with non-zero switch-over times (with cyclic, tabular

ordering) a pseudo-conservation

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(l-PI

-

2P(l

where E, G, L, and D are the index sets of the queues with

a l-limited

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 pseudo-conservation

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 l-limited

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 2-station

Consider

an asymmetric

and station

asymmetric

polling

2 l-limited

model

scheduling.

0.4, E[\$]

= 0.32,

Xi =

conditions

are satisfied

with

so finite

pseudo-conservation

law, yielding

relation

9.3

When

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

count-based

customer

system

customer

in service

tomer

E[S”]/2E[S].

with

from

one queue

that

Sc2) denotes

will,

the variance

of. the switch-over

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 switch-over

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

2. Similarly,

results

since in a fully

scheduling

an arriving

cyclic

all the scheduling

closed-form

so that

operation.

where

1 has exhaustive

directly;

relation

symmetric

to one another

expected

times

and E[W2]

the following

can be drawn

pseudo-conservation

use this

station

have to lie on this line.

Count-based

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

switch-over

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 Count-based

symmetric

3. An arriving

switch-overs,

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

4. Finally,

181

models

of queues N is a constant,

the waiting

at the

time

of a customer, the steady-state 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.

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(1-p)

(l-p)E[W]

+ E[W]

We can rewrite

+ NE[N,]E[S].

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 +

W-d

*

‘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 l-limited

scheduling

= 26

for the decrementing

E[~D]

+

discipline

S(2)

E[JC,]

Finally,

(9.13)

= 26

+ 6(N + p)

W-P)

(9.14)

n

we have:

NXE[S2]

+

scheduling

+ 6(N + p) + NJd2)

2(1-pNM)

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, l-limited,

and decre-

scheduling

From these explicit

they correspond

formulae,

the earlier derived stability

to those traffic conditions

From these expressions,

conditions

where the right-hand

can also easily be seen;

denominator

becomes zero.

it can also be observed that

and

E[WG] > E[WD]

Example

9.2. Symmetric

polling

In Table 9.1 we have tabulated

and E[WG] < E[WD]

models:

influence

the average waiting

(9.18)

of scheduling.

times for symmetric

polling

models

with exhaustive, gated, l-limited,

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 l-limited

and decrementing

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 l-limited

case where every service is

effectively lengthened with the succeeding switch-over 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 Count-based

disciplines

asymmetric

cyclic

might be considered

polling

models

183

better since they prevent one station from totally

hogging

the system.

Example

9.3. A 2-station

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 l-limited

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+ o-32

2(1 - 0.4)

2Jwq 20 - P>

We thus have: 0.317 < E[IVi]

3.9 5 E[W2] 5 6.766.

9.4

Count-based

(9.19)

= Oe317*

5 1.75 and, using the pseudo-conservation

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 pseudo-conservation

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

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) ’

inter-visit

(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 k-limited

=

(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 three-step 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 inter-visit

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 switch-over

9.4 Count-based

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

1-P

’ (1 - p)p + CfLl

pAc2)

+ \$q

2A

-

and more accurate

c

kE[\$‘]

pz

+ &

\$PiCl

i-l

solution

EIWE,iI

approximation

= (1 - Pi)E[RI,

E[WG,i]

+ pi))

*

(9.24)

2-l

procedure

for asymmetric

been proposed by Groenendijk

[112, Section 7.2.31.

For models in which a mixture of exhaustive, gated, and l-limited

exists, the following

has been

[28]:

’ + pi

1 - p - AiA

X

An iterative

l-limited

has been proposed by Groenendijk

= (1 + Pi)E[R],

E[WL,i]

l-limited

scheduling disciplines

[ill].

= ll~~pp~Ap~E[fil,

i

where E[R] approximates the mean, scheduling dependent, residual

ing this result in the pseudo-conservation

law yields:

models has

First

set

(9.25)

cycle time. Substitut-

(9.26)

Example

Reconsider

we calculate

9.4. A a-station

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 pseudo-conservation

but also within

Finally,

the bounds we established

for k-limited

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 (l-limited)

or Ic = 00 (exhaustive).

For asymmetric ki-limited

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 count-based

cyclic

IBM

token

ring

server models for the analysis of

timed-token

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

timed-token

access mechanism by means of (approximate)

k-limited 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

Timed-token

in his

81.

access mechanisms

In this section a brief operational explanation is given of the timed-token

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

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

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:

switch-over

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

At the moment Qi starts transmitting

data packets on the ring, it also starts its (local)

token holding timer (THT), which operates as a count-down 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 non-preemptive.

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 (so-called

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 timed-token

access mechanism

We cannot directly

model the above timed-token ring access mechanism in terms of countpolling

models. However, wc can approximate it by a cyclic (asymmetric)

based polling

model with a k-limited

scheduling

strategy

as follows.

Consider a timed-token

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 round-trip

delay, depending on which token scheme is used [35].

Whenever

be transmitted

l-limited

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

l-limited

2-limited

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 k-limited

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 k-limited scheduling

strategy

seems reasonable to assume, with

Ic = round

Example

(

1+

(9.28)

g).

of 1 comes from the fact that the timer expiration

9.5. A 3-station,

timed-token

ring

system

is non-preemptive.

from

[112]).

Consider a timed-token

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 timed-token

l-limited

mechanism

and a 2-limited

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 switch-over

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 2-limited 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 l-limited

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

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