Last Updated 11 Jul 2018

# Switch Models for Managing Queue Length Matrices

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**523**

Switch Model

We consider an N _ N non-blocking, input bu_ered switch.

Figure 4.1: Queueing theoretical account for a waiting line.

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The input I, has M FIFO waiting lines, qi1

to qiM, where 1 _ I _ N and M _ N. The

length of every FIFO is assumed to be in_nite. N end product ports are divided into

M reference groups each of N=M end products ports. When a package arrives it joins one

of the M group, depending on the its finish. In the system that we consider,

a package from an input I destined for end product port J is put into qij

modM. The

input tra_c is assumed homogenous and with Bernoulli distribution. Packages

91

4.2 Random Selection

are distributed uniformly for all end product ports. Time is assumed to be slotted with

each slot equal to the transmittal clip of a cell. In a cell slot, we have to choose

a upper limit of N cells from MN FIFO waiting lines with non-conicting finish

references. The manner in which these N cells are selected is decided by the cell

choice policy. Di_erent cell choice policies are discussed in the following subdivision.

Here we assume that at most one cell is selected from each input port, destined

to a non-conicting end product.

An e_cient cell choice policy should maximise the throughput and mini-

mize package transmittal hold. It should besides be noted that the programming policy

should be simple for execution. We present here di_erent cell choice poli-

cies.

A Queue length matrix L, of size N _N, is formed from current waiting line length

of FIFO. The current waiting line length of each FIFO is assigned to Lij, where I is

input port and J is the finish port of HOL cell. A 3 ten 3 switch is considered

as an illustration with 3 waiting lines per port

Figure 4.2: Queue length matrix and Indicator Queue length matrix

whose queue length matrix is given in Figure 4.2 ( a ) . An index waiting line length

matrix, K is formed from queue length matrix L by the relation Kij = 1 if Lij & A ; gt ; 0,

else Kij = 0. ( Figure 4.2 ( B ) . )

4.2 Random Selection

In this policy, in a cell slot, one of the random places of the cell is selected.

If the cell is available it will be switched to the end product port. The selected input

port and selected end product port will non contend in farther loops. This procedure is

repeated N times or till no cell is available for switching.There is possibility that

indiscriminately waiting line can be selected for which there is no HOL cell, under such circum-

stances throughput will acquire reduced. Even through switch is con_gured for size of

N X N with M queues/port, still we need scheduling policy to run on N _ N

matrix. No warrant that throughput is 100 % under heavy tra_c i.e. _ = 1.is

92

4.3 Longest Queue Priority choice ( LQPS )

achieved.Implementation of random choice is di_cult in hardware.No unique

solution for same queue length matrix. Following graph shows the throughput

public presentation of MIQ with di_erent switch sizes and fluctuation in figure of waiting lines

per ports. The throughput is dependent merely on value of M when N is greater

than 32.Below N=32 throughput dependant on N and M besides.

Figure 4.3: Impregnation Throughput with Random Policy for assorted values of M

4.3 Longest Queue Priority choice ( LQPS )

In this strategy, precedence is given to the longest waiting line FIFO [ 15 ] . In the waiting line

length matrix L, Lij = 0 indicates that no HOL cell is available from input port

I destined to end product port J. In a cell slot, the algorithm starts with _rst loop

where we select a cell from input port I to end product port Js such that Lij is maximal.

The cells from input port I and cells destined to end product port J are non considered

for choice in all farther loops. From the staying matrix, once more a new

maximal component Lij is found. The algorithm terminates after N loops or

when no cell is available for choice. In Figure4.4, the circled HOL places are

selected cell places. With mention to Fig. 4.4 ( a ) merely three cells are selected

even though there is possibility of choosing more than three cells for exchanging.

93

4.4 Weight Maximum

Figure 4.4: Longest Queue precedence choice

With avaricious attack of maximal queue length choice the packages are

selected for exchanging. As shown in Fig.4.4 ( a ) the VOQ & A ; apos ; s selected for exchanging are

VOQ ( 1,2 ) , VOQ ( 3,1 ) , VOQ ( 4,3 ) , VOQ ( 2,4 ) , where the instantaneous throughput

is non 100 % . There are multiple solutions available as shown in Fig. 4.4 ( B ) . Still

it is non an optimum solution even though the instantaneous throughput is 100 % .

Now see the optimum solution with constrains mentioned earlier which is shown in

Fig.4.4 ( degree Celsius ) .

The programming policy should be such that it should maximise figure of pack-

ets selected i.e. N and at the same clip overall queue length of selected package

should besides be maximal to avoid the cell loss.This is discussed in following subdivision on

longest waiting line precedence choice with pattern fiting ( LQPSP ) . No warrant

that 100 % throughput can be achieved. Multiple solutions are possible. _nding

optimum solution is di_cult. there will be fluctuation in throughput if we consider

amount of queue length of selected waiting lines is maximal. Algorithm becomes more

composite.

4.4 Weight Maximum

In the maximal leaden policy, each HOL cell is associated with a weight,

Wij. Weight Wij is calculated utilizing Indicator Queue length matrix K as follows.

Wij =

_XN

m=1

[ Kim + Kmj ]

_

: Ten

_

Kij

_

( 4.1 )

94

4.4 Weight Maximum

Figure 4.5: Impregnation Throughput with Maximum Queue Length for assorted

values of M

Figure 4.6: Maximum Weighted choice policy ( WMAX )

This weight factor additions with addition in HOL tenancy at input FIFO

and hot-spot tra_c to label end product port. In a cell slot, the algorithm starts

with _rst loop where we select a cell from input port I to end product port Js such

that its weight is maximal in weight matrix W. If the same maximal component

is found at multiple places, one of those is selected indiscriminately or round redbreast

95

4.5 RCSUM Minimum

policy is used among such input ports. Cells from the earlier selected input port

and cells destined for before selected end product port are non selected. This procedure

is repeated till N cells are selected or no cell is left for choice. In Fig.4.6 ( a ) ,

circled HOL place cells are the selected cell places, and the little square

indicates loop figure in which matching cell gets selected. In this instance

merely two cells are selected for exchanging, these are indicated by circles drawn in

Queue length matrix L in Fig.4.6 ( B ) . Merely two cells are selected even though

there is possibility of choosing more than two cells. This decrease in figure of

cells selected occurs because more figure of cells are deleted from competition

at each loop.

4.5 RCSUM Minimum

In this strategy weight matrix generated is the same as in instance of WMAX policy.

The lone di_erence is that here a non-zero minimal value is searched. If it _nds

one such Wij, so cell from matching place is selected for exchanging from

input port I to end product port J. If multiple non-zero lower limit values are available

so one is selected indiscriminately.

Figure 4.7: Minimum Leaden choice policy ( WMIN )

Fig.4.7 ( a ) shows the sequence in which the cells are selected. In Fig. Fig.4.7 ( a ) ,

circled HOL place cells are the selected cell places, and the little square

96

4.6 Cell choice policies with form fiting

indicates loop figure in which matching cell gets selected. Fig.4.7 ( B )

shows the cells selected in Queue length matrix. Fig.4.7 ( degree Celsius ) and Fig.4.7 ( vitamin D ) show

another possible sequence of choice of cells. It clearly shows that more figure of

cells are acquiring selected here than in WMAX policy. In this strategy, choosing non-

zero lower limit from weight matrix will heighten the throughput because in each

choice procedure we delete less figure of cells from the competition in the following

loop. This is precisely opposite of the WMAX choice standards. This work is

published in Canadian Conference on Broadband Research [ 25 ] . But public presentation

graph were non presented.

4.6 Cell choice policies with form fiting

It is seen that there are 2N2 substitution of forms for choosing cells in the

above matrix. However, because of the limitations on cell choice ( in a cell slot

merely one cell can be selected from an input and at most one cell can be switched

to an end product port ) the figure of forms of the matrix suited for choice for

shift is N! if M = N and much less than Nitrogen! for M & A ; lt ; N. We constrain the

form I of the N _ N matrix such that,

XN

j=1

Iij =

XN

i=1

Iij = 1 ( 4.2 )

These forms are substitutions of Identity matrix. Any random form with

above limitation can be generated without hive awaying them into the memory.

4.6.1 Generation of forms

If we have switch size of N _N so we need ( Noˆˆˆ1 ) !

2 distinguishable cell places that

can be used for exchanging. These generate other allowable permuted forms.

Procedure to obtain N! forms is as follows. ( 1 ) Get pattern I and take its

image. This will give two forms. ( 2 ) Shift form I right cyclically. Repeating

measure ( 1 ) and ( 2 ) N times will bring forth N! forms. If we take N = 4, so we

demand three distinguishable forms. To obtain these three form from Indicator matrix,

we have to trade column 2 with column 1 and column 1 with column 4. Repeat

procedure mentioned above to obtain all 24 ( i.e. 4! ) forms. Fig. 6 shows the

procedure of coevals of forms. These forms are favorable forms. These

forms are suited for execution by hardware, as they can be generated

utilizing parallel hardware.

4.6.2 Longest Queue Priority choice with pattern match-

ing

We obtain a soap value matrix X by utilizing the relation X = [

Phosphorus

ij ( Iij: _ Lij ) ] .

Here: _ notation indicates element by element generation. In the illustrated

97

4.6 Cell choice policies with form fiting

Figure 4.8: Form Generation

illustration of 3 _ 3 matrix, a upper limit of six forms will be available. Therefore,

soap value matrix X has six elements. This matrix _nds the lucifer that achieves

maximal aggregative weight under the limitations of alone coupling, i.e. select

form I such that X = [

Phosphorus

ij ( Iij: _ Lij ) ] is maximal and equation ( 1 ) is satis_ed.

The column matrix X indicate the value obtained from di_erent forms as shown

in ( Fig.4.9 ( a ) ) . Select maximal value from X under the restraint of unique

coupling and in bend get the form to be selected for exchanging cells from HOL. In

this instance I6 form is selected, ( Fig.4.8 ( a ) ) . In the selected form, 1 indicates

that cell has to be selected from input I to end product port J. Once the form is

selected so matching cells are deleted from the waiting line. It clearly shows

that 3 cells are selected for exchanging. If multiple entries in X have the same

maximal value, so take any one form indiscriminately. Round robin precedence

may be maintained in choice of forms. This strategy is di_cult to implement

in hardware, as it requires ( N2=2 ) _ R spot adder where R is the figure of spots

required to stand for length of Queue. It gives better public presentation than LQPS.

98

4.6 Cell choice policies with form fiting

Figure 4.9: Longest Queue Priority Selection with form fiting

4.6.3 Random Selection with Pattern Matching

In this strategy, the form I with limitations in equation ( 1 ) , is indiscriminately

chosen among the N! forms. The logical ANDing of I is done with indica-

tor Queue length matrix K. In this strategy, the throughput reduces under non

unvarying tra_c and it will be unpredictable.

4.6.4 Maximal Weight with Pattern Matching

In this method Indicator Queue length matrix K is considered. The sum

weight matrix Z is formed such that Z = [

Phosphorus

ij ( Iij: _ Kij ) ] ( Fig.4.10 ( a ) ) . The ma-

trix Z indicates weight obtained utilizing Indicator Queue length matrix and form

I1 to I6. A maximal value is selected from Z ( hashed elements indicates maxi-

silent value ) . If multiple places have the same maximal value one among them

is selected indiscriminately. In this instance form I6 and I1 get selected. Fig.4.10 ( B ) shows

the place of cells selected from the Queue length matrix. Once the form is

selected so matching cells are deleted from the waiting line. The execution

of this strategy is easy compared to LQPS with pattern matching.

Figure 4.10: Maximum Weighted choice policy with pattern match-

ing ( WMAXP )

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