Switch Models for Managing Queue Length Matrices

Switch Model
We consider an N _ N non-blocking, input bu_ered switch.
Figure 4.1: Queueing theoretical account for a waiting line.

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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