Buffer capacity for accommodating machine...

int j prod res 2002 vol 40 no 3 601plusmn624

Bu er capacity for accommodating machine downtime in serialproduction lines

EMRE ENGINARLARy JINGSHAN LIzSEMYON M MEERKOVy and RACHEL Q ZHANGsect

This paper investigates the smallest level of bu ering (LB) necessary to ensure thedesired production rate in serial lines with unreliable machines The reliability ofmachines is assumed to obey either exponential or Erlang or Rayleigh modelsThe LB is measured in units of the average downtime Tdown The dependence ofLB on the reliability model the number of machines M the average uptime Tupand the e ciency e ˆ Tup=hellipTup Dagger Tdowndagger is analysed It is shown that reliabilitymodels with larger coe cient of variation require larger LB and an empirical lawthat connects LB of the exponential model with those for other reliability modelsis established It is shown that LB is an increasing function of M but with anexponentially decreasing rate saturating at around M ˆ 10 Also it is shown thatLB does not depend explicitly on Tup and is a decreasing function of e Based onthese results rules-of-thumb are provided for selecting bu er capacity whichguarantee su ciently high line e ciency

1 MotivationMachine downtime may and often does lead to loss of throughput in manu-

facturing systems To minimize this loss in-process bu ers are used If the capacityof these bu ers is very large the machines are practically decoupled and the systemproduction rate PR (ie the average number of parts produced by the last machineper unit of time) is maximized We denote this production rate as PR1 indicatingthat it is attained when the capacity of the bu ers is inregnite Obviously large bu ersmay lead to excessive inventory long part-in-process time low quality and othermanufacturing ills Therefore it is of interest to determine the bu er capacity whichon one hand decouples the machines so that PR is su ciently close to PR1 and onthe other hand is as small as possible The goal of this paper is to characterize such acapacity and on this basis establish `rules-of-thumbrsquo for selecting the bu ers so thatPR is 95 or 90 or 85 of PR1

In practice both up- and downtime of the machines are random variables evenin the case of the so-called scheduled downtime (for instance tool change time)In this paper we model these random variables by three types of distributionsexponential Erlang and Rayleigh The exponential distribution is intended to

International Journal of Production Research ISSN 0020plusmn7543 printISSN 1366plusmn588X online 2002 Taylor amp Francis Ltd

httpwwwtandfcoukjournals

DOI 10108000207540110091703

Revision received July 2001 Department of Electrical Engineering and Computer Science University of Michigan

Ann Arbor MI 48109-2122 USA Enterprise Systems Laboratory GM Research and Development Center Warren

MI 48090-9055 USA Johnson Graduate School of Management Cornell University Ithaca NY 14853 USA To whom correspondence should be addressed e-mail smmeecsumichedu

model up- and downtime with a relatively large variability (measured by the coe -cient of variation ie the ratio of the standard deviation to the average) Thecoe cient of variation of Erlang distribution depends on the number of exponentialstages involved and is a decreasing function of the number of stages Rayleigh distri-bution is selected to model random variables other than exponential or Erlang

Obviously the capacity of bu ers which accommodate downtime depends onthe average value of this downtime (denoted as Tdown) Therefore the rules-of-thumb sought in this paper give the capacity of the bu ers in units of Tdown Forinstance `k-downtime bu errsquo denotes the capacity of a bu er that is capable ofstoring the number of parts produced or consumed during k downtimes Thenumber k is referred to as the level of bu ering (LB) Note that to accommodatetool change time production line designers usually select the bu ers of one-down-time capacity It will be shown in section 6 that in many practical situations this maylead to 30 loss of system throughput

Intuitively LB may depend not only on the average downtime Tdown but also onthe average uptime Tup and on the machine e ciency e deregned by

e ˆTup

Tup Dagger Tdown

ˆ 1

1 Dagger Tdown=Tup

Finally it may depend also on the number of machines M in the systemTherefore this paper is intended to provide a characterization of the LB necess-

ary to accommodate machine downtime in terms of the following function

LB ˆ F (type of up- and downtime distribution machine e ciency e orTup=Tdown number of machines in the system M average uptime Tupdagger

Roughly speaking the results obtained in this paper are as follows

(1) Production lines with machines having larger coefregcient of variation of thedowntime require a larger LB This implies that exponential machines needlarger LB than other distributions considered

(2) LB for lines with Erlang and Rayleigh distributions is related to LB for lineswith exponential machines according to the following empirical law

bkkA ˆ CVAdown cent kex ˆ frac14A

down

Tdown

cent kex

where A is the distribution of the downtime kex is the LB for exponentiallydistributed downtime bkkA is an estimate of the LB for downtime distributedaccording to A and CVA

down is the coe cient of variation of distribution Aderegned by

CVAdown ˆ frac14A

down

Tdown

(3) Larger machine efregciency e or larger not ˆ Tup=Tdown requires a smaller levelof buffering For example in 10-machine lines with exponential machinesthis relationship is characterized by

kex09hellipnotdagger ˆ iexcl00015not3 Dagger 0068not2 iexcl 1156not Dagger 98504 hellip1dagger

602 E Enginarlar et al

where kex09 is the smallest level of bu ering necessary to achieve

PR ˆ 09PR1 in serial lines with exponential machines(4) LB does not depend explicitly on Tup (ie it depends only on the ratio of Tup

and Tdown akin the machine efregciency e)(5) LB is an increasing function of M However this increase is exponentially

decaying saturating at about M ˆ 10 For instance if e ˆ 09 this relation-ship is described by the following exponential approximation

kex09hellipMdagger ˆ 0045 Dagger 4365 1 iexcl exp iexcl

M iexcl 2

35

micro parasup3 acute M 2 permil2 3 dagger hellip2dagger

Justiregcation of these conclusions and additional results are given in sections 6and 7

The results reported here are obtained using analytical calculations based onaggregation (for exponential distribution) Markov chain analysis (for Erlang withM ˆ 2) and discrete event simulations (for Erlang with M gt 2 and Rayleigh dis-tributions)

The outline of the paper is as follows section 2 describes the relevant literatureIn section 3 the model of the serial production line under consideration is intro-duced The problem formulation is given in section 4 Methods of analysis usedthroughout are outlined in section 5 The results obtained are described in sections6 and 7 Conclusions are given in section 8 Notation is given in appendix 1

2 Related literatureBu er capacity allocation in production lines has been studied quantitatively for

over 50 years and hundreds of publications are available The remarks below areintended to place the current paper in the framework of this literature rather than toprovide a comprehensive review

With respect to the machines production lines can be classireged into two groupsunreliable machines with regxed cycle time and reliable machines with random pro-cessing time This work addresses the regrst group

With respect to the machine e ciency production lines with unreliable machinescan be further divided into two groups balanced (ie the machines have identical up-and downtime distributions) and unbalanced This work regrst addresses the balancedcase and then extends the results to the unbalanced one

Bu er capacity allocation in production lines similar to those addressed herehas been regrst considered in the classic papers by Vladzievskii (1950 1951)Sevastyanov (1962) and Buzacott (1967) A review of the early work in this areais given by Buzacott and Haniregn (1978) In particular Buzacott (1967) showed thatthe coe cient of variation of the downtime strongly a ects the e cacy of bu eringIn addition Buzacott associated the bu er capacity allocation with the averagedowntime and stated that bu ering beyond regve-downtime can hardly be justiregedThese results are conregrmed and further quantireged in the present work

Conway et al (1988) also connected bu er allocation with downtime Theyshowed that one-downtime bu ering was su cient to regain about 50 of produc-tion losses if the downtime was constant (deterministic) They suggested that random(exponential) downtime may require a twice larger capacity to result in comparablegains This suggestion is further explored in the current paper

Another line of research on bu er capacity allocation is related to the so-calledstorage bowl phenomenon (Hillier and So 1991) According to this phenomenon

603Bu er capacity and downtime in serial production lines

more bu ering should be assigned to middle machines in balanced lines It can beshown however that unbalancing the bu ering in lines with downtime coe cient ofvariation lt1 results in only 1plusmn3 of throughput improvement if at all (For furtherdetails see Jacobs and Meerkov 1995b where it is proved that optimal bu ers are ofequal capacity if the work is distributed according to the optimal bowl) Since thisimprovement is quite small the present paper does not consider bowl-type storageallocation and assigns equal capacity to all bu ers in balanced lines and appropri-ately selected unequal bu ering in unbalanced ones

Finally there exists a large body of literature on numerical algorithms thatcalculate the optimal bu er allocation (eg Ho et al 1979 Jacobs and Meerkov1995a Glasserman and Yao 1996 Gershwin and Schor 2000) The current workdoes not address this issue

Thus the present paper follows Buzacott (1967) and Conway et al (1988) andprovides additional results on rules-of-thumb for bu er capacity allocation necessaryto accommodate downtime and achieve the desired e ciency of serial productionlines with unreliable machines

3 ModelThe block-diagram of the production system considered here is shown in reggure 1

where the circles are the machines and the rectangles are bu ers The following arethe assumptions concerning the machines bu ers and interactions among them (ieblockages and starvations)

31 Machines

(1) Each machine m has two states up and down When up the machine iscapable of producing one part per unit of time (machine cycle time) whendown no production takes place

(2) The up- and downtime of each machine are random variables distributedaccording to either of the following distributions

(a) Exponential

f exuphelliptdagger ˆ p exe

iexclp ext t para 0

f exdownhelliptdagger ˆ r exe

iexclr ex t t para 0

)hellip3dagger

(b) Erlang

f ErPup helliptdagger ˆ pEre

iexclp ErthellippErtdaggerPiexcl1

hellipP iexcl 1dagger t para 0

f ErR

downhelliptdagger ˆ r Ereiexclr Ert

helliprErtdaggerRiexcl1

hellipR iexcl 1dagger t para 0

9gtgtgtgt=

gtgtgtgt

hellip4dagger


Figure 1 Serial production line

(c) Rayleigh

f Raup helliptdagger ˆ p 2

Rateiexclp 2Rat2=2 t para 0

f Radownhelliptdagger ˆ r2

Rateiexclr 2

Rat2=2 t para 0

9=

hellip5dagger

The expected Tup and Tdown the variances frac142up and frac142

down and the coe cients ofvariation CVup and CVdown of each of these distributions are given in table 1

(3) The parameters of distributions (3)plusmn(5) are selected so that machine efregcien-cies e and moreover Tup and Tdown are identical for all reliability modelsie

Tup ˆ1

p ex

hellipexponentialdagger

ˆ P

pEr

hellipErlang with P stagesdagger

ˆ 12533

pRa

hellipRayleighdagger

Tdown ˆ 1

r ex


ˆ R

rEr

hellipErlang with R stagesdagger

ˆ 12533

rRa


32 Bu ers

(4) Each buffer has the capacity N deregned by

N ˆ dkTdowne

where dxe is the smallest integer gt x and Tdown is measured in units of thecycle time Coe cient k 2 RDagger is referred to as the level of bu ering

33 Interactions among the machines and bu ers

(5) Machine m i i ˆ 2 M is starved at time t if biiexcl1 is empty and m iiexcl1 failsto put a part into biiexcl1 at time t Machine m 1 is never starved

(6) Machine m i i ˆ 1 M iexcl 1 is blocked at time t if buffer bi is full andm iDagger1 fails to take a part from bi at time t Machine m M is never blocked


Distribution Tup Tdown frac142up frac142

down CVup CVdown

Exponential 1=pex 1=rex 1=p2ex 1=r2

ex 1 1Erlang P=per R=rEr P=p2

Er R=r2Er 1=

P

p1=

R

p

Rayleigh 12533pRa 12533rRa 04292p2Ra 04292=r2

ra 05227 05227

Table 1 Expected value variance and coe cient of variation of up- and downtimedistributions considered

Assumptions (1)plusmn(6) deregne the production system considered in sections 4plusmn6 Insection 7 an unbalanced version of this system is analysed

4 Problem formulationThe production rate PR of the serial line (1)plusmn(6) is the average number of parts

produced by the last machine m M during a cycle time (in the steady state of systemoperation) When the capacity of the bu ers is inregnite the production rate of theline PR1 is equal to

PR1 ˆ minhellipe1 eMdagger

When the bu ers are regnite and selected according to assumption (4) PR isobviously smaller Let E denote the e ciency of the line deregned by

E ˆ PR

PR1 E 2

YM

iˆ1

ei 1

and kE be the smallest level of bu ering necessary to attain line e ciency EThe following are the problems analysed in this work Given the production

system deregned by (1)plusmn(6)

hellipnotdagger Analyse the properties of kE in production lines with identical machines Inparticular investigate the dependence of kE on the machine reliability model e Mand Tup and on this basis provide rules-of-thumb for selecting kE for E ˆ 095 or09 or 085

hellipshy dagger Extend results obtained to production systems with non-identical machineshellipregdagger Investigate production losses measured by hellipPR1 iexcl PRkdagger=PR1 when k ˆ 1

where PRk denotes the production rate of the system (1)plusmn(6) with LB ˆ k This caseis intended to model the current industrial practice used in the design of modernproduction systems where bu er capacity is selected using the òne-downtimersquo rule

Solutions of problems hellipnotdagger and hellipregdagger are given in section 6 Problem hellipshy dagger is discussedin section 7

5 Methods of analysis51 Exponential machines

For M ˆ 2 and exponential reliability model with parameters p i and r i i ˆ 1 2PR of the serial line deregned by (1)plusmn(6) is calculated by Jacobs (1993) to be

PR ˆ

r1r2

hellipp 1 Dagger r1daggerhellipp 2 Dagger r2daggerp 1hellipp 2 Dagger r2dagger iexcl p 2hellipp 1 Dagger r1daggereiexclshy N

p1r2 iexcl p 2r1eiexclshy N

ifp 1

r 1

6ˆ p 2

r2

r22hellipr1 Dagger r2dagger Dagger Nr 1r2hellipp 2 Dagger r2dagger2

hellipp 2 Dagger r2dagger2permilr1 Dagger r2 Dagger Nr 1hellipp 2 Dagger r 2daggerŠ if

p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where

shy ˆ hellipr1 Dagger r2 Dagger p 1 Dagger p 2daggerhellipp 1r2 iexcl p 2r1daggerhellipr1 Dagger r2daggerhellipp 1 Dagger p 2dagger

For M gt 2 no closed formula for PR is available However several approximationtechniques have been developed (Gershwin 1987 Dallery et al 1989 Chiang et al2000) We use here the one developed in Chiang et al since it is directly applicable to


model (1)plusmn(6) It consists of the so-called forward and backward aggregation In theforward aggregation using expression (6) the regrst two machines are aggregated in asingle machine m f

2 deregned by parameters p f2 and r f

2 Then m f2 is aggregated with m 3

to result in m f3which is then aggregated with m 4 to give m f

4 and so on until allmachines are aggregated in m f

M In the backward aggregation m fMiexcl1 is aggregated

with mM to produce m bMiexcl1 which is then aggregated with m f

Miexcl2 to result in m bMiexcl2

and so on until all machines are aggregated in m b1 Then the process is repeated anew

Formally this recursive procedure has the following form

p bi hellips Dagger 1dagger ˆ p

1 iexcl Qhellipp biDagger1hellips Dagger 1dagger r b

iDagger1hellips Dagger 1dagger p fi hellipsdagger r f

i hellipsdagger Nidagger 1 micro i micro M iexcl 1

rbi hellips Dagger 1dagger ˆ 1

Qhellipp biDagger1hellips Dagger 1dagger r b


i hellipsdagger Nidaggerp

Dagger 1

r

1 micro i micro M iexcl 1

p fi hellips Dagger 1dagger ˆ p

1 iexcl Qhellipp fiiexcl1hellips Dagger 1dagger r f

iiexcl1hellips Dagger 1dagger p bi hellips Dagger 1dagger rb

i hellips Dagger 1dagger Niiexcl1dagger 2 micro i micro M

r fi hellips Dagger 1dagger ˆ 1

Qhellipp fiiexcl1hellips Dagger 1dagger r f


i hellips Dagger 1dagger Niiexcl1daggerp

Dagger 1

r

2 micro i micro M

9gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt=

gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt

hellip7dagger

with boundary conditions

p f1hellipsdagger ˆ p r f

1hellipsdagger ˆ r

p bMhellipsdagger ˆ p rb

Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger

and initial conditions

p fi hellip0dagger ˆ p r f

i hellip0dagger ˆ r i ˆ 2 M iexcl 1 hellip9dagger

where function Q is given by

Qhellipp 1 r1 p 2 r2 N1dagger

ˆ

hellip1 iexcl e1daggerhellip1 iexcl iquestdagger1 iexcl iquesteiexclshy N1

ifp 1

r1

6ˆ p 2

r 2

p 1hellipp1 Dagger p 2daggerhellipr1 Dagger r2daggerhellipp 1 Dagger r1daggerpermilhellipp 1 Dagger p 2daggerhellipr1 Dagger r 2dagger Dagger p2r1hellipp 1 Dagger p 2 Dagger r1 Dagger r2daggerN1Š if

p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2

iquest ˆ e1hellip1 iexcl e2daggere2hellip1 iexcl e1dagger



It is shown by Chiang et al that this procedure is convergent and the followinglimits exist

lims1

p fi hellipsdagger ˆ p f

i lims1

p bi hellipsdagger ˆ p b

i

lims1

r fi hellipsdagger ˆ r f

i lims1

rbi hellipsdagger ˆ r b

i i ˆ 1 M

Since the last machine is never blocked and the regrst machine is never starved theestimate of PR denoted as cPRPR is deregned as

cPRPRhellipp 1 r1 p 2 r 2 pM r M N1 N2 NMiexcl1dagger ˆ r fM

p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger

It is shown by Chiang et al that this estimate results in su ciently high precision

52 Erlang machinesFor M ˆ 2 and Erlang reliability PR can be calculated using the method devel-

oped by Altiok (1985) According to this method each stage of the Erlang distri-bution is treated as a state (along with all other states deregned by the occupancy ofthe bu er) Since the residence time in each stage is distributed exponentially astandard Markov process description applies To simplify calculations a discretetime approximation of the continuous time Markov process is utilized Thus accord-ing to this method the performance analysis of system (1)plusmn(6) reduces to the calcula-tion of the stationary probability distribution of a discrete time Markov chain Oncethis probability distribution is found the production rate PR is calculated bysumming up the probabilities of the states where m 2 is up and not starved (Altiok1985)

It should be pointed out that due to the increase of dimensionality this methodis practical only when the bu er capacity is not too large (lt100) For systems withlarger bu ers or with more than two machines discrete event simulations seem to befaster than the method described above even if the Erlang distribution with only twostages is considered

53 SimulationsUnfortunately no analytical calculation methods exist for PR evaluation in

systems with Rayleigh machines For production lines with Erlang machines andM gt 2 the PR calculations are prohibitively time consuming Therefore we analysethese systems using discrete event simulations It should be pointed out that theanalytical calculations are many orders of magnitude faster than the discrete eventsimulation for instance calculation of cPRPR on a PC for a line with 10 exponentialmachines using (7)plusmn(10) takes about 35 sec whereas discrete event simulation takesgt2 h

The simulations have been carried out as follows a discrete event model of line(1)plusmn(6) has been constructed Zero initial conditions for all bu ers were assumed andthe states of all machines at the initial time moment have been selected to be ùprsquoFirst 100 000 time slots of warm-up period were carried out and the next 1 000 000slots of stationary operation were used to evaluate the production rate statistically


The 95 conregdence intervals calculated as explained in Law and Kelton (1991)were lt00005 when each simulation was carried out 10 times

54 Calculation of kE

The level of bu ering kE which ensures the desired line e ciency E (E ˆ 095or 09 or 085) has been determined as follows

For each model of machine reliability PR of line (1)plusmn(6) was evaluated regrst forN ˆ 0 then for N ˆ 1 and so on until PR reached the level of E cent PR1 This bu ercapacity NE was then divided by Tdown (in units of the cycle time) This providedthe desired level of bu ering kE Results of these calculations are described below

6 Results identical machinesHere we assume that all machines obey the same reliability model and the

average uptime (respectively downtime) of all machines is the same Non-identicalmachines are addressed in section 7

61 Two-machine case611 Exponential machines

Expression (6) under the assumption of p 1 ˆ p 2 ˆ p and r 1 ˆ r2 ˆ r leads to aclosed form expression for kex

E Indeed assuming that PR ˆ E cent PR1 ˆ E cent r=hellipr Dagger pdaggerfrom (6) it follows that the bu er capacity NE which results in this production rateis deregned by

NE ˆ2hellip1 iexcl edaggerhellipE iexcl edagger

phellip1 iexcl Edagger

sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger

where as before dxe is the smallest integer gt x Therefore for two-machine lineswith exponential machines LB is given by

kexE ˆ NE

Tdown

ˆ2ehellipE iexcl edagger

1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger

As follows from (12) LB depends explicitly only on machine e ciency e and isindependent of Tup Also (12) shows that kex

E is decreasing as a function of e fore gt 05E and increasing for e lt 05E Since in most practical situations e gt 05 weconsider throughout this paper only the machines with e para 05 gt 05E

The behaviour of kexE for E ˆ 095 09 and 085 as a function of Tup=Tdown (or e)

is illustrated in reggure 2

612 Erlang and Rayleigh machinesUsing Markov chain analysis for the Erlang case (with P ˆ R ˆ 2 and

P ˆ R ˆ 5) and discrete event simulations for the Rayleigh case we calculate kErP

E

and kRaE for E ˆ 095 09 and 085 The results are shown in reggure 3 (for Tup ˆ 30)

and reggure 4 (for Tup ˆ 60) where the exponential case is also included for compar-ison

Results shown in reggures 2plusmn4 lead to the following conclusions

LB for the Erlang and Rayleigh machines akin the exponential case is inde-pendent of Tup (since reggures 3 and 4 are practically identical)



Figure 2 Level of bu ering for exponential machines (M ˆ 2 Tup ˆ 200dagger

Figure 3 Level of bu ering for Erlang Rayleigh and exponential machines (M ˆ 2Tup ˆ 30)

Smaller variability of up- and downtime distributions of the machines leads to

smaller level of bu ering LB (since CVex gt CVEr2gt CVRa gt CVEr5

and thecurves are related as shown in reggures 3 and 4)

Smaller machine e ciency e requires larger bu ering kE to attain the sameline e ciency E

Rules-of-thumb for two-machine lines

If PR ˆ 095PR1 is desired

(not) three-downtime bu er is su cient for all reliability models ife ordm 085 and

(shy ) zero LB is acceptable if e para 094


(not) one-downtime bu er is su cient for all reliability models if e ordm 085and




If PR ˆ 085PR1 is desired zero LB is acceptable for all reliabilitymodels if e para 085

613 Empirical lawAs pointed out above calculation of kE is fast and simple for exponential

machines and requires lengthy discrete event simulations for Erlang and Rayleighmachines It would be desirable to have an èmpirical lawrsquo that could provide kE forErlang and Rayleigh reliability models as a function of kE for exponential machinesFrom the data of reggures 2plusmn4 one can conclude that such a law can be formulated asfollows

bkkAE ˆ CVA

downkexE hellip13dagger

where A is the reliability model (ie the distribution of the downtime ETH either Erlangor Rayleigh) CVA

down is the coe cient of variation of the downtime bkkAE is the esti-

mate of LB for reliability model A and kexE is the LB for exponential machines

deregned by (12)The quality of approximation (13) is illustrated in reggure 5 and table 2 where the

accuracy of (13) is evaluated in terms of the error middotAE deregned by

middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger

As it follows from these data ckAEkAE approximates kA

E with su ciently high precision

Moreover since ckAEkAE gt kA

E selection of LB according to ckAEkAE does not lead to a loss of

performance

Empirical law (13) will be used below for the case M gt 2 as well

62 M-machine case M gt 2621 Level of bu ering as a function of machine e ciency

For various M the level of bu ering kexE as a function of not ˆ Tup=Tdown or e is

shown in reggure 6 for E ˆ 095 09 and 085Polynomial approximations of these functions for M ˆ 10 can be given as

follows

bkk095hellipnotdagger ˆ iexcl00035not3 Dagger 01607not2 iexcl 26492not Dagger 207627



For M ˆ 10 graphs for kErP

E and kRaE and their approximations according to (13)

are illustrated in reggure 7 and table 3 These data indicate that the empirical lawresults in acceptable precision for M gt 2 as well

Based on the data of reggures 6 and 7 we conclude the following

Longer lines require larger level of bu ering between each two machines

As before larger machine e ciency requires less bu ering

Rules-of-thumb for 10-machine lines with exponential machines


If PR ˆ 095PR1 is desired and machine e ciency e ordm 09 seven-down-time bu ers are required for exponential machines and 45-downtime forErlang (P para 2) and Rayleigh machines

If PR ˆ 09PR1 is desired and machine e ciency e ordm 09 four-downtimebu er is required for exponential machines and about 3-downtime forErlang (P para 2) and Rayleigh machines

If PR ˆ 085PR1 is desired and machine e ciency e ordm 09 25-downtimebu er is required for exponential machines and about 2-downtime forErlang (P para 2) and Rayleigh machines

Zero LB is not acceptable even if e is as high as 095 and E as low as 085

622 Level of bu ering as a function of the average uptimeFor M ˆ 2 expression (12) states that kE is independent of Tup No analytical

result of this type is available for M gt 2 Therefore we verify this property using


Figure 5 Level of bu ering kE for Erlang Rayleigh and exponential machines andapproximation kkE using empirical law (13) (M ˆ 2 Tup ˆ 30)

the aggregation procedure of Subsection 51 Calculations have been carried out for

ten-machine lines with Tup ˆ 200 and Tup ˆ 400 Tup=Tdown 2 f1 20g and

E 2 f085 09 095g As it turned out kE for Tup ˆ 200 and Tup ˆ 400 di er at

most by 01 Therefore we conclude that kexE for M gt 2 does not depend on Tup

either

623 Level of bu ering as a function of the number of machines

From reggures 6 and 7 it is clear that kE is an increasing function of M To

investigate further this dependency we calculated kE as a function of M The results

are shown in reggure 8

Clearly although kexE is an increasing function of M the rate of increase is

exponentially decreasing and saturates at about M ˆ 10 This happens perhapsdue to the fact that the machines separated by nine appropriately selected bu ers

become to a large degree decoupled

The curves shown in reggure 8 have a convenient exponential approximation For

instance if e ˆ 09 these approximations are

kex095hellipMdagger ˆ 18 Dagger 6255 1 iexcl exp iexcl hellipM iexcl 2dagger

3

sup3 acutesup3 acute


35



(a) E ˆ 095

Distribution e ˆ 08 e ˆ 085 e ˆ 09 r ˆ 094

Rayleigh 008 007 008 009Erlang 2 006 005 005 006Erlang 10 014 012 011 015

(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08


Table 2 Accuracy middotAE of empirical law (13) as a function of

e hellipM ˆ 2 Tup ˆ 30dagger


375


M 2 permil2 3 dagger

The quality of this approximation is illustrated in reggure 9

Figures 8 and 9 characterize kexE hellipMdagger for the exponential machines Empirical law

(13) can be invoked to evaluate kEhellipMdagger for Erlang and Rayleigh machines as wellThe behaviour of kErP

E hellipMdagger and kRaE hellipMdagger obtained by simulation and kkErP

E hellipMdagger and

kkRaE hellipMdagger obtained from (13) is shown in reggure 10 its accuracy (14) is characterized

in table 4 The conclusion is that empirical law (13) results in acceptable precisionfor M gt 2

Based on the above results we arrive at the following conclusions

Although longer lines require larger level of bu ering the increase is exponen-tially decreasing as a function of M


Figure 6 Level of bu ering for exponential machines with various M

Roughly speaking bu ering necessary for M ˆ 10 is su cient to accommo-

date downtime in all lines with M gt 10

Rules-of-thumb established in Subsection 621 remain valid for Erlang and

Rayleigh machines as well if the level of bu ering is modireged by the coe cient

of variation of the downtime

63 Production losses for k ˆ 1

As it was pointed out above one-downtime rule is often used by production line

designers Performance of 10-machine lines with this bu er allocation is character-

ized in reggure 11 As it follows from this reggure if e ˆ 09 throughput losses are about

30 of PR1 if machine reliability is exponential and about 25 if it is Er2 Thus the

òne-downtimersquo rule may not be advisable if high line e ciency is pursued


Figure 7 Level of bu ering kE for Erlang and Rayleigh machines and approximation bkkE using empirical law (13) (M ˆ 10 Tup ˆ 30)

7 Extension non-identical machines71 Description of machines

Identical machines imply that up- and downtime obey the same reliability modeland the average uptime (respectively downtime) of all machines is the same Non-identical machines mean that either or both of these assumptions is violated Thegoal of this section is to extend the results of section 6 to non-identical machinesassuming however that the e ciency e of all machines is the same This assump-tion is made to account for the fact that in most practical cases all machines of aproduction line are roughly of the same e ciency To simplify the presentation weconsider only two-machine lines here

In this section each machine m i i ˆ 1 2 is denoted by a pair fAhellipp idagger Bhellipr idaggergwhere the regrst symbol Ahellipp idagger (respectively the second symbol Bhellipr idagger) denotes thedistribution of the uptime (respectively downtime) deregned by parameter p i (respect-ively r i) the subscript i indicates whether the regrst or second machine is addressedFor instance fEr5hellipp 2dagger Exhellipr2daggerg denotes the second machine of a two-machine linewith the uptime being distributed according to the Erlang distribution with regvestages deregned by parameter p 2 and the downtime distributed according to theexponential distribution deregned by parameter r2 Obviously in this case the averageup- and downtime of the second machine are 5=p 2 and 1=r2 respectively Note thatin these notations the systems considered in section 6 consist of machinesfAhellipp idagger Ahellipr idaggerg p i ˆ p r i ˆ r 8i ˆ 1 M

72 Cases analysedTo investigate the properties of LB in production lines with non-identical

machines the following regve cases have been analysed


(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





Figure 8 Level of bu ering kexE as a function of M

Figure 9 Approximations of kexE for e ˆ 09

Case 1 Non-identical Tup and Tdown Speciregc systems analysed were

fExhellipp 1dagger Exhellipr1daggerg fExhellipp 2dagger Exhellipr2daggerg

fRahellipp 1dagger Rahellipr1daggerg fRahellipp 2dagger Rahellipr2daggerg

fEr2hellipp 1dagger Er2hellipr1daggerg fEr2hellipp2dagger Er2hellipr2daggerg


p 2 ˆ 2p 1 r 2 ˆ 2r1

Case 2 Non-identical up- and downtime distribution laws Systems considered herewere

fExhellipp 1dagger Rahellipr1daggerg fExhellipp 2dagger Rahellipr 2daggerg



Figure 10 Levels of bu ering kErP

E kRaE and their approximations according to (13) as a

function of M (Tup ˆ 30 e ˆ 095)

fEr2hellipp 1dagger Er5hellipr1daggerg fEr2hellipp 2dagger Er5hellipr2daggerg

p 2 ˆ p 1 r2 ˆ r1

Case 3 Non-identical up- and downtime distribution laws non-identical Tup andTdown Systems studied here were




p 2 ˆ 2p 1 r2 ˆ 2r1

Case 4 Non-identical uptime distribution laws non-identical downtime distributionlaws The systems analysed were

fExhellipp 1dagger Rahellipr1daggerg fEr5hellipp 2dagger Exhellipr2daggerg12533

r1

ˆ 1

r2

fEr2hellipp 1dagger Er5hellipr1daggerg fEr5hellipp 2dagger Er2hellipr2daggerg5

r1

ˆ2

r2

fRahellipp 1dagger Exhellipr1daggerg fEr3hellipp 2dagger Rahellipr2daggerg1

r1

ˆ12533

r2


(a) E ˆ 095

Distribution M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20



M hellipe ˆ 2 Tup ˆ 30dagger

Case 5 Non-identical uptime distribution laws non-identical downtime distributionlaws and non-identical Tup and Tdown The systems investigated here were

fExhellipp1dagger Rahellipr1daggerg fEr5hellipp 2dagger Exhellipr2daggerg12533

r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2

73 Results obtainedWe provide here only the summary of the results obtained More details can be

found in Enginarlar et al (2000)The main result can be formulated as follows The selection of LB for a two-

machine line with non-identical machines can be reduced to the selection of LB for atwo-machine line with identical machines provided that the latter is deregned appro-priately Speciregcally consider the production line fAhellipp 1dagger Bhellipr1daggerg fChellipp 2dagger Dhellipr 2daggergWithout loss of generality assume that the regrst machine has the largest averagedowntime ie Tdown1

gt Tdown2 and the second machine has the largest coe cient

of variation of the downtime ie CVdown1lt CVdown2

Assume that the LB sought isin units of the largest average downtime ie

kE ˆ NE

Tdown1

Then the level of bu ering for the line

fAhellipp 1dagger Bhellipr1daggerg fChellipp 2dagger Dhellipr2daggerg

can be selected as the level of bu ering of the following production line with identicalmachines


Figure 11 Performance of 10-machine lines with kexE ˆ 1 as a function of e

fDhellippdagger Dhelliprdaggerg fDhellippdagger Dhelliprdaggerg

where p and r are selected as follows

EDhelliprdaggerftdowng ˆ EBhellipr 1daggerftdowng

EDhellippdaggerftupg ˆ EAhellipp 1daggerftupg

Here ERhellipvdaggerftg denotes the expected value of random variable t distributed accordingto distribution R deregned by parameter v

Thus selecting LB for two-machine lines with non-identical machines is reducedto the problem of selecting LB for lines with identical machines the solution ofwhich is given in subsection 61

8 ConclusionsBased on this study the following rules-of-thumb for selecting the level of buf-

fering kE in serial production lines as a function of machine e ciency e linee ciency E number of machines M and the downtime coe cient of variationCVdown can be provided

(1) If all machines are identical and obey the exponential reliability model kexE

can be selected as indicated in table 5 If the number of machines in thesystem is substantially less than 10 the level of buffering can be reduced byusing the data of reggure 8

(2) If the machines are identical but not exponential all kexE from table 5 should

be multiplied by the coefregcient of variation of the downtime CVdown Formachines with Erlang and Rayleigh reliability models this leads to about50 reduction of buffer capacity This might justify the effort for evaluatingnot only the average value of the downtime but also its variance

(3) If the machines are not identical the capacity of the buffer between each pairof consecutive machines can be chosen according to

Ni ˆ dkexE cent maxfCVdowniiexcl1

CVdownig cent maxfTdowniiexcl1

Tdownige i ˆ 1 M iexcl 1

where kexE is selected from table 5

It should be pointed out that this paper does not address the issue of which linee ciency should be pursued plusmn 095 090 or 085 However given the data of table 5it is reasonable to conclude that E ˆ 095 might require too much bu ering as far aspractical considerations are concerned (unless the downtime variability is verysmall) E ciency E ˆ 085 might be too low for many industrial situationsTherefore it seems reasonable that the second column of table 5 provides the


e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

Table 5 Level of bu ering kexE as a function of

machine and line e ciency

most important practical information This information deregnes how `leanrsquo a produc-tion line could be to result in a reasonable performance

AcknowledgementsThe authors are grateful to Professor J A Buzacott for valuable advice in con-

nection with his paper (1967) The helpful comments of anonymous reviewers arealso acknowledged The work was supported by NSF Grant No DMI-9820580

Appendix 1 Notationbi ith bu er

CV coe cient of variatione machine e ciency

ex exponential distributionE production line e ciency

ERhellipvdaggerftg expected random variable t distributed according to RhellipvdaggerEr Erlang distributionkE smallest level of bu ering necessary to achieve line e ciency ELB level of bu eringm i ith machineM number of machines in the lineN bu er capacity

NE bu er capacity necessary to achieve line e ciency Ep parameter of the uptime distribution

PR production ratePR1 production rate when the capacity of all bu ers is inregnitePRk production rate when the level of bu ering is k

Q function deregning the aggregation procedurer parameter of the downtime distribution

Ra Rayleigh distributions step of the aggregation procedure

Tup average machine uptimeTdown average machine downtime

frac14 standard deviation and

middot accuracy of the empirical law

References

Altiok T 1985 Production lines with phase plusmn type operation and repair times and regnitebu ers International Journal of Production Research 23 489plusmn498

Buzacott J A 1967 Automatic transfer lines with bu er stocks International Journal ofProduction Research 5 183plusmn200

Buzacott J A and Hanifin L E 1978 Models of automatic transfer lines with inventorybanks a review and comparison AIIE Transactions 10 197plusmn207

Chiang S-Y Kuo C-T and Meerkov S M 2000 DT-bottlenecks in serial productionlines theory and application IEEE Transactions on Robotics and Automation 16 567plusmn580

Conway R Maxwell W McClain J O and Thomas L J 1988 The role of work-in-process inventory in serial production lines Operations Research 36 229plusmn241

Dallery Y David R and Xie X L 1989 Approximate analysis of transfer lines withunreliable machines and regnite bu ers IEEE Transactions on Automatic Control 34943plusmn953


Enginarlar E Li J Meerkov S M and Zhang R 2000 Bu er capacity for accom-modating machine downtime in serial production lines Control Group Report NoCGR-00-07 Department of Electrical Engineering and Computer Science Universityof Michigan Ann Arbor

Gershwin S B 1987 An e cient decomposition method for the approximate evaluation oftandem queues with regnite storage space and blocking Operations Research 35 291plusmn305

Gershwin S B and Schor J E 2000 E cient algorithms for bu er space allocationAnnals of Operations Research 93 117plusmn144

Glasserman P and Yao D D 1996 Structured bu er-allocation problems Discrete EventDynamic Systems 6 9plusmn41

Hillier F S and So K C 1991 The e ect of machine breakdowns and internal storage onthe performance of production line systems International Journal of ProductionResearch 29 2043plusmn2055

Ho Y C Eyler M A and Chien T T 1979 A gradient technique for general bu erstorage design in a production line InternationalJournal of Production Research 7 557plusmn580

Jacobs D A 1993 Improvability of production systems theory and case studies PhDthesis Department of Electrical Engineering and Computer Science University ofMichigan Ann Arbor

Jacobs D and Meerkov S M 1995a A system-theoretic property of serial productionlines improvability International Journal of Systems Science 26 755plusmn785

Jacobs D and Meerkov S M 1995b Mathematical theory of improvability for productionsystems Mathematical Problems in Engineering 1 99plusmn137

Law A M and Kelton W D 1991 Simulation Modeling and Analysis (New YorkMcGraw-Hill)

Sevastyanov B A 1962 Inmacruence of storage bin capacity on the average standstill time of aproduction line Theory of Probability and Its Applications 7 429plusmn438

Vladzievskii A P 1950 The theory of internal stocks and their inmacruence on the output ofautomatic lines Stanki i Instrumenty 21 4plusmn7


624 Bu er capacity and downtime in serial production lines

model up- and downtime with a relatively large variability (measured by the coe -cient of variation ie the ratio of the standard deviation to the average) Thecoe cient of variation of Erlang distribution depends on the number of exponentialstages involved and is a decreasing function of the number of stages Rayleigh distri-bution is selected to model random variables other than exponential or Erlang

Obviously the capacity of bu ers which accommodate downtime depends onthe average value of this downtime (denoted as Tdown) Therefore the rules-of-thumb sought in this paper give the capacity of the bu ers in units of Tdown Forinstance `k-downtime bu errsquo denotes the capacity of a bu er that is capable ofstoring the number of parts produced or consumed during k downtimes Thenumber k is referred to as the level of bu ering (LB) Note that to accommodatetool change time production line designers usually select the bu ers of one-down-time capacity It will be shown in section 6 that in many practical situations this maylead to 30 loss of system throughput

Intuitively LB may depend not only on the average downtime Tdown but also onthe average uptime Tup and on the machine e ciency e deregned by

e ˆTup

Tup Dagger Tdown

ˆ 1

1 Dagger Tdown=Tup

Finally it may depend also on the number of machines M in the systemTherefore this paper is intended to provide a characterization of the LB necess-

ary to accommodate machine downtime in terms of the following function

LB ˆ F (type of up- and downtime distribution machine e ciency e orTup=Tdown number of machines in the system M average uptime Tupdagger

Roughly speaking the results obtained in this paper are as follows

(1) Production lines with machines having larger coefregcient of variation of thedowntime require a larger LB This implies that exponential machines needlarger LB than other distributions considered

(2) LB for lines with Erlang and Rayleigh distributions is related to LB for lineswith exponential machines according to the following empirical law

bkkA ˆ CVAdown cent kex ˆ frac14A

down

Tdown

cent kex

where A is the distribution of the downtime kex is the LB for exponentiallydistributed downtime bkkA is an estimate of the LB for downtime distributedaccording to A and CVA

down is the coe cient of variation of distribution Aderegned by

CVAdown ˆ frac14A

down

Tdown

(3) Larger machine efregciency e or larger not ˆ Tup=Tdown requires a smaller levelof buffering For example in 10-machine lines with exponential machinesthis relationship is characterized by

kex09hellipnotdagger ˆ iexcl00015not3 Dagger 0068not2 iexcl 1156not Dagger 98504 hellip1dagger







M iexcl 2

35


















31 Machines



(a) Exponential


iexclp ext t para 0



)hellip3dagger

(b) Erlang




f ErR




9gtgtgtgt=

gtgtgtgt

hellip4dagger



(c) Rayleigh




Rateiexclr 2

Rat2=2 t para 0

9=

hellip5dagger




Tup ˆ1

p ex


ˆ P

pEr


ˆ 12533

pRa


Tdown ˆ 1

r ex


ˆ R

rEr


ˆ 12533

rRa


32 Bu ers


N ˆ dkTdowne







down CVup CVdown



Er R=r2Er 1=

P

p1=

R

p


ra 05227 05227







E ˆ PR

PR1 E 2

YM

iˆ1

ei 1









PR ˆ

r1r2



ifp 1

r 1

6ˆ p 2

r2



p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where






















Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References



























M iexcl 2

35


















31 Machines



(a) Exponential


iexclp ext t para 0



)hellip3dagger

(b) Erlang




f ErR




9gtgtgtgt=

gtgtgtgt

hellip4dagger



(c) Rayleigh




Rateiexclr 2

Rat2=2 t para 0

9=

hellip5dagger




Tup ˆ1

p ex


ˆ P

pEr


ˆ 12533

pRa


Tdown ˆ 1

r ex


ˆ R

rEr


ˆ 12533

rRa


32 Bu ers


N ˆ dkTdowne







down CVup CVdown



Er R=r2Er 1=

P

p1=

R

p


ra 05227 05227







E ˆ PR

PR1 E 2

YM

iˆ1

ei 1









PR ˆ

r1r2



ifp 1

r 1

6ˆ p 2

r2



p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where






















Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References



























31 Machines



(a) Exponential


iexclp ext t para 0



)hellip3dagger

(b) Erlang




f ErR




9gtgtgtgt=

gtgtgtgt

hellip4dagger



(c) Rayleigh




Rateiexclr 2

Rat2=2 t para 0

9=

hellip5dagger




Tup ˆ1

p ex


ˆ P

pEr


ˆ 12533

pRa


Tdown ˆ 1

r ex


ˆ R

rEr


ˆ 12533

rRa


32 Bu ers


N ˆ dkTdowne







down CVup CVdown



Er R=r2Er 1=

P

p1=

R

p


ra 05227 05227







E ˆ PR

PR1 E 2

YM

iˆ1

ei 1









PR ˆ

r1r2



ifp 1

r 1

6ˆ p 2

r2



p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where






















Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References






















(c) Rayleigh




Rateiexclr 2

Rat2=2 t para 0

9=

hellip5dagger




Tup ˆ1

p ex


ˆ P

pEr


ˆ 12533

pRa


Tdown ˆ 1

r ex


ˆ R

rEr


ˆ 12533

rRa


32 Bu ers


N ˆ dkTdowne







down CVup CVdown



Er R=r2Er 1=

P

p1=

R

p


ra 05227 05227







E ˆ PR

PR1 E 2

YM

iˆ1

ei 1









PR ˆ

r1r2



ifp 1

r 1

6ˆ p 2

r2



p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where






















Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References



























E ˆ PR

PR1 E 2

YM

iˆ1

ei 1









PR ˆ

r1r2



ifp 1

r 1

6ˆ p 2

r2



p 1

r 1

ˆ p 2

r2

8gtgtgtgtgtlt

gtgtgtgtgt

hellip6dagger

where






















Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References








































Dagger 1

r










Dagger 1

r

2 micro i micro M



hellip7dagger



1hellipsdagger ˆ r


Mhellipsdagger ˆ r

s ˆ 0 1 2

9gtgt=

gtgthellip8dagger






ˆ


ifp 1

r1

6ˆ p 2

r 2


p 1

r1

ˆ p 2

r 2

8gtgtgtlt

gtgtgt

ei ˆ r i

p i Dagger r i

i ˆ 1 2





lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References
























lims1


i lims1


i

lims1


i lims1


i i ˆ 1 M



p fM Dagger r f

M

ˆ rb1

p b1 Dagger rb

1

hellip10dagger




















sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References

































sup1 ordm if E gt e

0 otherwise

8lt

hellip11dagger


kexE ˆ NE

Tdown


1 iexcl E if E gt e

0 otherwise

8lt

hellip12dagger







E

























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References









































bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References






































bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References


























bkkAE ˆ CVA







middotAE ˆ

ckAEkAE iexcl kA

E

kAE

hellip14dagger





performance





follows
























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References


































either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References


























either











3



35



(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08





375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References























375







E hellipMdagger and


























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References









































(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References




























(a) E ˆ 095



(b) E ˆ 09

e ˆ 07 e ˆ 075 e ˆ 08 e ˆ 085


(c) E ˆ 085

e ˆ 065 e ˆ 07 e ˆ 075 e ˆ 08












p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References






























p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References



























p 2 ˆ 2p 1 r 2 ˆ 2r1









p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References























p 2 ˆ p 1 r2 ˆ r1





p 2 ˆ 2p 1 r2 ˆ 2r1



r1

ˆ 1

r2


r1

ˆ2

r2


r1

ˆ12533

r2


(a) E ˆ 095



(b) E ˆ 09

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20


(c) E ˆ 085

M ˆ 5 M ˆ 10 M ˆ 15 M ˆ 20






r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References
























r1

ˆ2

r2


r 1

ˆ4

r 2


r1

ˆ 25066

r2







kE ˆ NE

Tdown1

























e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References









































e E ˆ 085 E ˆ 090 E ˆ 095

085 35 5 10090 25 4 7095 15 25 45

















References




































References




































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