Switch Models for Managing Queue Length Matrices

Switch ModelWe consider an N _ N non-blocking, insert bu_ered switch. condition 4.1 standing theoretical account for a delay rakehell.The input I, has M first in first out time lag lines, qi1to qiM, whither 1 _ I _ N and M _ N. Thelength of every FIFO is mis acceptn to be in_nite. N repeal product bearings be divided intoM reference groups each of N=M end products manners. When a packet boat arrives it joins oneof the M group, depending on the its finish. In the system that we consider,a software system from an input I bound for end product porthole J is put into qijmodM. Theinput tra_c is assumed homogenous and with Bernoulli distribution. Packages914.2 Random Selectionare distributed uni mannikinly for all(prenominal) end product ports. age is assumed to be slotted witheach slot equal to the transmittal clip of a carrell. In a carrel slot, we feed to choosea upper limit of N cells from MN FIFO waiting lines with non-conicting finishreferences. The manner in which these N cells are selected is decided by the cell plectrum policy. Di_erent cell prize policies are 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 excerption 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 plectron poli-cies.A adjust length ground substance L, of size N _N, is patterned from received waiting line lengthof FIFO. The current waiting line length of each FIFO is appoint 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 encipher 4.2 Queue length matrix and Indicator Queue length matrixwhose queue up length matrix is given in Figure 4.2 ( a ) . An index waiting line lengthmatrix, K is formed from queue lengt h matrix L by the relation 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 places of the cell is selected.If the cell is lendable it will be switched to the end product port. The selected inputport and selected end product port will non contend in farther loops. This result isrepeated N times or till no cell is addressable for switching.There is possibility thatindiscriminately 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 ofN X N with M queues/port, still we need scheduling policy to run on N _ Nmatrix. No sanction that throughput is 100 % under heavy tra_c i.e. _ = 1.is924.3 Longest Queue Priority choice ( LQPS )achieved.Implementation of random choice is di_cult in hardware.No unique rootage for same queue length matrix. Following graph shows the throughputpublic intro of MIQ w ith di_erent switch sizes and fluctuation in dactyl of waiting linesper ports. The throughput is dependent only if on value of M when N is greaterthan 32.Below N=32 throughput dependant on N and M besides.Figure 4.3 fecundation Throughput with Random Policy for mixed values of M4.3 Longest Queue Priority choice ( LQPS )In this strategy, precedence is given to the longest waiting line FIFO 15 . In the waiting linelength matrix L, Lij = 0 directs that no HOL cell is available from input portI destined to end product port J. In a cell slot, the algorithm starts with _rst loopwhere 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 consideredfor choice in all farther loops. From the staying matrix, once more than a newmaximal component Lij is found. The algorithm terminates after N loops orwhen no cell is available for choice. In Figure4.4, the circled HOL places areselec ted 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 charge 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 twofold solutions available as shown in Fig. 4.4 ( B ) . Stillit is non an best solution even though the instantaneous throughput is 100 % .Now see the optimum solution with constrains mentioned preliminary which is shown inFig.4.4 ( degree Celsius ) .The programming policy should be such that it should maximise attribute of pack-ets selected i.e. N and at the same clip overall queue length of selected packageshould besides be maximal to bar the cell loss.This is discussed in following s ubdivision onlongest waiting line precedence choice with pattern fiting ( LQPSP ) . No warrantthat 100 % throughput can 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 leaden policy, each HOL cell is associated with a system of weights,Wij. Weight Wij is calculated utilizing Indicator Queue length matrix K as follows.Wij =_XNm=1 Kim + Kmj _ Ten_Kij_( 4.1 )944.4 Weight MaximumFigure 4.5 Impregnation Throughput with Maximum Queue Length for assortedvalues of MFigure 4.6 Maximum Weighted choice policy ( WMAX )This weight factor improvers with addition in HOL tenancy at input FIFOand 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 weig ht 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. cubicles 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 wholeindicates loop figure in which unified cell gets selected. In this showcasemerely two 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 subside in figure ofcells selected occurs because more figure of cells are deleted from competitionat each loop.4.5 RCSUM MinimumIn this strategy weight matrix set aboutd 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 _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 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 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 sequence 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 le ss figure of cells from the competition in the followingloop. This is precisely opposite 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 the to a higher place matrix. However, because of the limitations on cell choice ( in a cell slotmerely one cell can 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 forshift is N if M = N and much less than northward 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 substitutions 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 distingui shable cell places thatcan be used for exchanging. These generate other allowable permuted forms.Procedure to obtain N forms is as follows. ( 1 ) Get pattern I and take itsimage. This will give two forms. ( 2 ) Shift form I right cyclically. tellmeasure ( 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 affirmatory 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 obtain 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 fiti ngFigure 4.8 Form Generationillustration of 3 _ 3 matrix, a upper limit of sextette 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 obstacle 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 cells 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 ro bin 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 spots necessitate 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 indiscriminatelychosen 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 maximum 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 valu e 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 )