Switch Models for Managing Queue Length Matrices

Switch ModelWe consider an N _ N non-blocking, arousal signal bu_ered switch. attribute 4.1 queue uping theoretical account for a postp mavinment line.The input I, has M FIFO delay lines, qi1to qiM, w present 1 _ I _ N and M _ N. The continuance of every FIFO is simulated to be in_nite. N barricade carrefour sorts be divided intoM reference groups each of N=M closing products expressions. When a package arrives it joins peerlessof the M group, dep residualing on the its finish. In the system that we consider,a package from an input I doom for end product appearance J is put into qijmodM. Theinput tra_c is assumed homogenous and with Bernoulli distribution. Packages914.2 Random Selection atomic number 18 distributed uni word constructly for whatsoever end product ports. Time is assumed to be expansion slotted witheach slot equal to the transmittal clip of a carrelular phoneular phonephone. In a electric cellphone slot, we have to choosea stop number limit of N cells from MN FIFO postponement lines with non-conicting finishreferences. The manner in which these N cells atomic number 18 selected is decided by the cell pickax policy. Di_erent cell woof policies atomic number 18 discussed in the following subdivision.Here we assume that at most one cell is selected from each input port, destinedto 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 policyshould be simple for execution. We present here di_erent cell choice poli-cies.A queue up distance ground substance L, of size N _N, is functioned from menstruum waiting line durationof FIFO. The current waiting line length of each FIFO is assigned to Lij, where I isinput port and J is the finish port of HOL cell. A 3 ten 3 switch is consideredas an illustration with 3 waiting lines per port puzzle out 4.2 waiting line length ground substance and indicat or waiting line length matrixwhose queue length matrix is stipulation in Figure 4.2 ( a ) . An index waiting line lengthmatrix, K is organise from queue length matrix L by the singing Kij = 1 if Lij & A gt 0,else Kij = 0. ( Figure 4.2 ( B ) . )4.2 Random SelectionIn this policy, in a cell slot, one of the random spaces of the cell is selected.If the cell is for sale it will be switched to the end product port. The selected inputport and selected end product port will non contend in further twines. This procedure isrepeated N times or till no cell is available for switching.There is possibility thatindiscriminately waiting line gage be selected for which there is no HOL cell, under such circum-stances throughput will pull ahead reduced. Even through switch is con_gured for size ofN X N with M queues/port, still we need scheduling policy to run on N _ Nmatrix. No warrant that throughput is 100 % under level-headed tra_c i.e. _ = 1.is924.3 Longest Queue priority choice ( LQPS )achieved.Implementation of random choice is di_cult in hardw ar.No uniquesolution for homogeneous queue length matrix. avocation represent shows the throughputpublic presentation of MIQ with di_erent switch sizes and fluctuation in trope of waiting linesper ports. The throughput is dependent merely on cheer of M when N is greaterthan 32.Below N=32 throughput dependant on N and M besides.Figure 4.3 Impregnation Throughput with Random Policy for assorted values of M4.3 Longest Queue Priority choice ( LQPS )In this strategy, priority is given to the longest waiting line FIFO 15 . In the waiting linelength matrix L, Lij = 0 indicates that no HOL cell is available from input portI destined to end product port J. In a cell slot, the algorithmic rule starts with _rst loopwhere we select a cell from input port I to end product port Js such that Lij is supreme.The cells from input port I and cells destined to end product port J are non consideredfor choice in all farther loops. From the staying matrix, once more a new supreme component Lij is found. The algorithm terminates after N loops orwhen no cell is available for choice. In Figure4.4, the circled HOL places areselected cell places. With mention to Fig. 4.4 ( a ) merely three cells are selectedeven though there is possibility of choosing more than three cells for exchanging.934.4 weight MaximumFigure 4.4 Longest Queue precedence choiceWith avaricious attack of maximal queue length choice the packages areselected for exchanging. As shown in Fig.4.4 ( a ) the VOQ & A apos s selected for exchanging areVOQ ( 1,2 ) , VOQ ( 3,1 ) , VOQ ( 4,3 ) , VOQ ( 2,4 ) , where the fast throughputis non 100 % . There are nonuple solutions available as shown in Fig. 4.4 ( B ) . Stillit is non an optimum solution even though the instantaneous throughput is 100 % .Now see the optimum solution with constrains mentioned earlier which is shown inFig.4.4 ( degree Celsius ) .The programming policy should be such tha t it should maximise designing of pack-ets selected i.e. N and at the same clip overall queue length of selected packageshould besides be maximal to avoid the cell loss.This is discussed in following subdivision onlongest waiting line precedence choice with ensample fiting ( LQPSP ) . No warrantthat 100 % throughput trick be achieved. Multiple solutions are possible. _ndingoptimum solution is di_cult. there will be fluctuation in throughput if we consideramount of queue length of selected waiting lines is maximal. Algorithm becomes morecomposite.4.4 Weight MaximumIn the maximal arduous policy, each HOL cell is associated with a tip,Wij. Weight Wij is calculated utilizing Indicator Queue length matrix K as follows.Wij =_XNm=1 Kim + Kmj _ cristal_Kij_( 4.1 )944.4 Weight MaximumFigure 4.5 Impregnation Throughput with Maximum Queue Length for assortedvalues of MFigure 4.6 Maximum plodding choice policy ( WMAX )This weight factor additions with addition in HOL tenancy at input FIF Oand hot-spot tra_c to label end product port. In a cell slot, the algorithm startswith _rst loop where we select a cell from input port I to end product port Js suchthat its weight is maximal in weight matrix W. If the same maximal componentis found at multiple places, one of those is selected indiscriminately or round redbreast954.5 RCSUM Minimumpolicy is used among such input ports. Cells from the earlier selected input portand cells destined for before selected end product port are non selected. This procedureis 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 squareindicates loop exercise in which twin(a) cell gets selected. In this instancemerely 2 cells are selected for exchanging, these are indicated by circles drawn inQueue length matrix L in Fig.4.6 ( B ) . Merely two cells are selected even thoughthere is possibility of choosing more than two cells. This drop-off i n figure ofcells selected occurs because more figure of cells are score outd from competitionat each loop.4.5 RCSUM MinimumIn this strategy weight matrix generated is the same as in instance of WMAX policy.The lone di_erence is that here a non-zero marginal value is searched. If it _ndsone such Wij, so cell from matching place is selected for exchanging frominput port I to end product port J. If multiple non-zero lower limit values are availableso one is selected indiscriminately.Figure 4.7 Minimum Leaden choice policy ( WMIN )Fig.4.7 ( a ) shows the while in which the cells are selected. In Fig. Fig.4.7 ( a ) ,circled HOL place cells are the selected cell places, and the little square964.6 Cell choice policies with form fitingindicates 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 ) showanother possible chronological succession of choice of cells. It clearly shows that more figure ofcells 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 eachchoice procedure we delete less figure of cells from the competition in the followingloop. This is precisely frigid of the WMAX choice standards. This work ispublished in Canadian Conference on Broadband Research 25 . But public presentationgraph were non presented.4.6 Cell choice policies with form fitingIt is seen that there are 2N2 substitution of forms for choosing cells in theabove matrix. However, because of the limitations on cell choice ( in a cell slotmerely one cell batch be selected from an input and at most one cell can be switchedto an end product port ) the figure of forms of the matrix suited for choice for pouch is N if M = N and much less than Nitrogen for M & A lt N. We constrain theform I of the N _ N matrix such that,XNj=1Iij =XNi=1Iij = 1 ( 4.2 )These forms are substitution s of Identity matrix. Any random form withabove limitation can be generated without hive awaying them into the memory.4.6.1 Generation of formsIf we have switch size of N _N so we need ( No1 ) 2 distinguishable cell places thatcan be used for exchanging. These generate other allowable permuted forms. action to guard N forms is as follows. ( 1 ) Get pattern I and take itsimage. This will give two forms. ( 2 ) Shift form I right cyclically. Repeatingmeasure ( 1 ) and ( 2 ) N times will bring forth N forms. If we take N = 4, so wedemand 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. Repeatprocedure mentioned above to obtain all 24 ( i.e. 4 ) forms. Fig. 6 shows theprocedure of coevals of forms. These forms are favorable forms. Theseforms are suited for execution by hardware, as they can be generatedutilizing parallel hardware.4.6.2 Longest Queue Priority choice with pattern match-ingWe o btain a soap value matrix X by utilizing the relation X = Phosphorusij ( Iij _ Lij ) .Here _ notation indicates element by element generation. In the illustrated974.6 Cell choice policies with form fitingFigure 4.8 Form Generationillustration 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 achievesmaximal aggregative weight under the limitations of alone coupling, i.e. selectform I such that X = Phosphorusij ( Iij _ Lij ) is maximal and equation ( 1 ) is satis_ed.The column matrix X indicate the value obtained from di_erent forms as shownin ( Fig.4.9 ( a ) ) . Select maximal value from X under the restraint of uniquecoupling and in bend get the form to be selected for exchanging cells from HOL. Inthis instance I6 form is selected, ( Fig.4.8 ( a ) ) . In the selected form, 1 indicatesthat cell has to be selected from input I to end product port J. Once the form isselected so matching ce lls are deleted from the waiting line. It clearly showsthat 3 cells are selected for exchanging. If multiple entries in X have the samemaximal value, so take any one form indiscriminately. Round robin precedencemay be maintained in choice of forms. This strategy is di_cult to implementin hardware, as it requires ( N2=2 ) _ R spot adder where R is the figure of spotsrequired to stand for length of Queue. It gives better public presentation than LQPS.984.6 Cell choice policies with form fitingFigure 4.9 Longest Queue Priority Selection with form fiting4.6.3 Random Selection with Pattern MatchingIn 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 nonunvarying tra_c and it will be unpredictable.4.6.4 Maximal Weight with Pattern MatchingIn this method Indicator Queue length matrix K is considered. The sumweight matrix Z is formed such that Z = Phosphorusij ( Iij _ Kij ) ( Fig.4.10 ( a ) ) . The ma-trix Z indicates weight obtained utilizing Indicator Queue length matrix and formI1 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 themis selected indiscriminately. In this instance form I6 and I1 get selected. Fig.4.10 ( B ) showsthe place of cells selected from the Queue length matrix. Once the form isselected so matching cells are deleted from the waiting line. The executionof this strategy is easy compared to LQPS with pattern matching.Figure 4.10 Maximum Weighted choice policy with pattern match-ing ( WMAXP )