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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



([ 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 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



the steady-state



(9.5)

amount of work



times, V is a random variable indicating



in the model



the steady-state



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



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



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



Instead,



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

paradox does not apply).

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.



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(1-p)



(l-p)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 +

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



(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, 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



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 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



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) ’

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

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:

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

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 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



The addition

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.



(adapted



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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2 Cyclic polling: cycle time and conservation law

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