Efficiency (statistics): Difference between revisions

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{{notability|date=November 2010}}
When it comes to selecting a �[http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=SOG+Specialty+Knives%2C+Inc.+is+more+generally+referred+to+as+SOG+and+has+established+itself+as+a+agency+favourite+among+tactical+knife+consultants.+They%27re+best+known+for+his+or+her+famous+copy+of+the+army+SOG+knife.+Nonetheless%2C+the+company+produces+a+full+line+of+knives+and+different+instruments+and+is+effectively-respected+both+for+its+prime+quality+builds+and+progressive+designs.+At+the+moment%2C+SOG+Knives+gives+every+part+from+military-model+knives+to+everyday+folding+fashions+and+past.%0D%0A%0D%0AOverall+the+SOG+Gov-Tac+is+a+superb+knife+for+combat+and+only+obtained+better+as+we+used+it+throughout+this+review.+In+case+you+are+in+search+of+a+knife+that+provides+you+with+dependability%2C+consolation%2C+and+is+straightforward+to+use+we+are+able+to+definitely+suggest+the+Gov-Tac.+Now+we%27ve+had+an+opportunity+to+play+with+two+different+models+designed+by+SOG+%E2%80%93+the+SOGzilla+and+the+SOG+AutoClip.+Does+the+SOGzilla+stay+as+much+as+its+tough+title%3F+What+does+%E2%80%9CAutoClip%E2%80%9D+actually+mean%3F+SOG+Knives+Put+to+the+Check+Years+later%2C+in+1986%2C+that+special+knife+impressed+a+younger+designer+named+Spencer+Frazer+to+discovered+SOG+Specialty+Knives+His+mission%2C+to+breed+the+original+SOG+Bowie+Knife+and+pay+tribute+to+the+special+ops+unit+that+created+it.+I+believe+its+fingers+down+one+of+the+best+survival+knives+we+have+ever+seen.+Blade+Kind%0D%0A%0D%0AI%27ve+had+this+knife+for+two+years.+Never+once+has+it+failed+me+in+sharpness+or+motion.++If+you+have+any+thoughts+with+regards+to+exactly+where+and+how+to+use+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fsog-knives-review%2F+Http%3A%2F%2FWww.Thebestpocketknifereviews.Com%2FSog-Knives-Review%5D%2C+you+can+get+in+touch+with+us+at+the+web-page.+I+carry+it+daily+because+it+just+down+proper+functional.+My+clip+recently+broke+off+when+I+caught+it+on+a+metal+table+edge.+I+used+to+be+so+dissapointed+in+myself.+All+I+wanted+to+do+was+go+buy+another+immediately.+So+I+logged+on+to+Sog+on-line+and+to+my+amazement+they+provide+a+FREE+clip+substitute+by+mail.+May+I+stress+FREE%3F%21+It+arrived+at+this+time.+Very+fast+shipping+and+even+with+a+brand+new+screw+for+meeting.+I%27m+a+lifetime+buyer.+THANK+YOU+SOG%21%0D%0A%0D%0AThe+SOG+Flash+II+folding+knife+seems+to+be+like+most+abnormal+folding+knives+until+you+open+it.+The+Flash+II+features+SOG+Assisted+Know-how+which+springs+the+blade+absolutely+open+after+just+a+bit+help+out+of+your+thumb.+The+result+is+a+swap-blade-esque+impact%2C+with+out+the+whole+%E2%80%9Cillegal%E2%80%9D+half+that+goes+together+with+switchblades.+The+assisted+expertise+is+addictive.+You+may+find+yourself+opening+and+closing+the+blade+for+no+apparent+cause%2C+other+than+juvenile+amusement.+Due+to+this+addictive+quality%2C+we+advocate+you+restrain+your+SOG+Flash+II+experimentation+to+non-public+settings.+No%2C+we+can%27t+bail+you+out.%0D%0A%0D%0AThis+is+principally+the+same+review+I+wrote+for+the+non-black+customary+Flash+model.+Nonetheless%2C+the+difference+is+in+the+look+of+the+knife+total+and+not+in+the+%5Bhttp%3A%2F%2FAnswers.Yahoo.com%2Fsearch%2Fsearch_result%3Fp%3Doperate%26submit-go%3DSearch%2BY%21%2BAnswers+operate%5D.+I+just+like+the+seems+of+this+all+black+model+very+a+lot.+Aspect-by-aspect+I+might+select+the+Black+version+over+the+standard+Flash%2C+nonetheless%2C+they+are+both+good+knives+and+it+is+a+matter+of+choice+when+you+can+solely+select+one.+The+black+handle+and+black+blade+of+this+version+makes+it+fairly+interesting.+Even+the+clip+is+black+with+this+model%2C+which+is+a+nice+touch.%0D%0A%0D%0AI+carry+this+knife+daily+and+it+is+great+for+just+that+goal.+Doesn%27t+have+a+steel+liner+within+the+handle%2C+but+that+helps+preserve+weight+down+and+the+knife+nonetheless+appears+a+lot+stiff+and+tough+for+an+each+day+carry+multi-objective+knife.+You%27ll+be+able+to+open+the+blade+with+the+assisted+opener%2C+or+manually+should+you+like+or+if+the+opener+had+been+to+interrupt.+This+was+my+first+assisted+opener+kind+knife+and+it+took+a+couple+of+tries+to+get+the+dangle+of+it+%28watch+out+the+first+few+times%21%29%2C+however+now+I+like+that+characteristic.+It+also+has+a+safety+lock.+Extremely+beneficial%21&create=Create � Finest] Folding Knife ’, we will’t just ignore Gayle Bradley and go forward to select something else. Gayle Bradley is one of the main gamers in making customized made folding knife since 1988. Later they partnered themselves with Spyderco. Partnership of those two brought the revolution as Spyderco are well-known to supply fine knives and gives greatest of completion to every knives. That’s how Spyderco Gayle Bradley Folder knife was born. These two knives maker come together and manufactured this inflexible, medium framed folding knife in USA.<br><br><br><br>Pocket knives play an important position in man’s on a regular basis lives. They function man’s hand tools for varied functions and can be utilized as weapon when hazard arises. These knives may be small in size however they possess numerous traits which distinguish them from other sorts of knives. The most effective pocket knives are recognized for being useful which anybody can carry from one place to a different as they'll simply fit in one’s pocket. 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Despite the fact that you had initially bought the knife for camping functions, you'll finally use this [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%0D%0A%0D%0ATo+each+his+%28or+her%29+personal%2C%E2%80%9D+is+among+the+most+relevant+phrases+yow+will+discover+with+regards+to+deciding+on+ANY+defense+or+utility+tool.+Nearly+everyone+has+totally+different+preferences.+That+mentioned%2C+SOG+Knives+makes+a+range+from+which+nearly+anybody+ought+to+be+capable+of+find+a+knife+that+matches+their+preferences.+For+me%2C+the+Trident+Elite+comes+pretty+shut+%28for+a+folding+knife%29.%0D%0A%0D%0AThe+SOG+SlimJim+is+simply+that%2C+slim.+Actually+slim.+I+did+not+realize+simply+how+skinny+the+knife+was+until+I+obtained+it+out+of+the+package.+I+wasn%27t+particularly+looking+for+a+slim+profile+however+once+I+carried+it+around+I+fell+in+love+with+it.+I+had+two+main+features+I+used+to+be+looking+for+in+a+knife+to+hold+with+me+day+to+day.+The+first+was+an+assisted+opening+system.++In+case+you+adored+this+informative+article+in+addition+to+you+desire+to+acquire+more+details+concerning+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fsog-knives-review%2F+SOG+Knives+Review%5D+generously+go+to+the+internet+site.+The+second+was+a+low+mount+clip+so+it+will+sit+deep+in+my+pocket.+This+knife+has+both+these+options%2C+in+all+actuality+loads+of+knives+have+each+these+features.%0D%0A%0D%0ATesting+included+some+primary+fire+constructing+and+bush-craft+tasks+as+well+as+getting+used+for+a+wide+range+of+duties+around+the+home.+Onemy+favorite+test+is+to+use+a+knife+to+cook+dinner+dinner+and+bone+out+a+whole+hen.+This+is+a+highly+relevant+check%2C+as+a+result+of+if+a+knife+design+makes+it+tough+to+prepare+dinner+food+within+the+luxuries+of+dwelling%2C+cooking+at+camp+or+in+a+survival+state+of+affairs+will+suck.+The+blade+shape+made+the+food+prep+for+our+bone+in+Herbes+de+Provence+Hen+and+broiled+broccoli+with+Parmesan+and+bacon+occasion-much+less.+Definitely+a+great+standard+handy+blade+shape.%0D%0A%0D%0AI+agree+with+your+overview+of+those+throwing+knives.+For+the+price+these+knives+are+great.+They+come+sharp+from+the+factory%2C+great+stability%2C+and+are+pretty+straightforward+to+throw.+I+used+to+be+disapointed+within+the+throwing+knives.+The+knives+are+made+in+China+from+stamped+steel.+The+steel+may+be+very+hard+which+helps+these+knives+hold+thier+edge+for+a+very+long+time.+With+that+being+mentioned%2C+the+metal+is+so+onerous+to+the+point+that+it+is+brittle.+This+could+be+fine+for+a+normal+use+repair+blade%2C+but+not+a+throwing+knife.+I+had+these+knives+for+per+week+when+I+dropped+one+knife+and+three%2F4%22+of+the+tip+broke+off.+A+throwing+knife+ought+to+retain+the+power+to+withstand+some+abuse%2C+afterall+nobody+is+ideal.%0D%0A%0D%0AFounded+in+1769%2C+Puma+is+Germany%27s+eighth-oldest+trademark.+First+out+there+in+the+U.S.+starting+in+1957%2C+Puma+knives%E2%80%94especially+the+pre-Rambo-era+White+Hunter%2C+maybe+the+primary+bushcraft+knife+to+attract+the+fancy+of+Individuals%E2%80%94have+been+de+rigueur+in+the+backwoods.+Puma+knives+aren%27t+on+as+many+hips+these+days%2C+but+the+firm%27s+thought+of+merging+Outdated+World+craftsmanship+with+put+up-trendy+globalism+may+change+issues.+To+convey+down+the+price+of+its+German+steel+blades%2C+Puma+introduces+the+SGB+line.+These+knives+nonetheless+carry+the+German+blades+solid+in+famed+Solingen%2C+however+they+are+assembled+in+Asia.+The+result%E2%80%94a+traditional+German+Puma+knife+at+a+considerably+lower+price.%0D%0A%0D%0AOn+the+first+edge+the+SOG+scores+275%2B%2F-+34+g+on+thread%2C+on+1%2Ffour%22+%5Bhttp%3A%2F%2Fsearch.usa.gov%2Fsearch%3Fquery%3Dpoly%2Bunderneath+poly+underneath%5D+a+a+thousand+g+load+it+can%27t+make+acut%2C+below+a+1500+g+load%2C+a+reduce+requires+1.62+%2B%2F-+zero.09+centimeters+ofedge+The+serrations+on+the+SOG+just+bust+the+thread+%28650+-+750+g%29.On+the+primary+edge+the+MPK-Ti+cuts+the+thread+with+200+%2B%2F-+14+g+of+force.+Ittakes+2.15+%2B%2F-+0.27+%5Bhttp%3A%2F%2FWww.Adobe.com%2Fcfusion%2Fsearch%2Findex.cfm%3Fterm%3D%26centimeters%26loc%3Den_us%26siteSection%3Dhome+centimeters%5D+of+edge+to+make+a+cut+on+1%2F4%22+poly+under+a+1000g+load.+The+serrations+on+the+MPK-Ti+cut+the+thread+with+305+%2B%2F-+6+g&create=Create software] for many different causes inside the house. The knife is helpful whereas doing the common activity at residence. You will hardly find a residence which doesn't want this knife for normal duties. Be it a easy job like slicing a wire or opening a bottle, the device comes in use for every little factor in the course of the day. Let us look at the areas where we will use the knife.<br><br>Selecting the very best pocket knife is tough sufficient if you find yourself a knife fanatic, but for a novice there are so many things to study. One query that I find most newcomers (and a few professionals) ask me is which metal is the most effective for the blade of a knife? It's a actually good query as a result of a blade is only nearly as good because the steel it's made from. The first thing I tell individuals who ask this [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=Mini+Vulcan%2C+VG-10%2C+Zytel+Handle%2C+Plain+%E2%80%93+SOG+Knives+evaluation+with+large+buy+off+constrained+time+only+from+Sports+%26+Outside+Evaluate+%26+sale+off+overview+therapy+proper+care+switched+to+help+you+to+the+actual+atmosphere.+Most+of+us+varieties+discover+that+from+the+face%2C+a+worldwide+well-known+medical+medical+health+insurance+along+with+techniques.%0D%0A%0D%0A%0D%0A%0D%0AThe+SOG+SlimJim+is+just+that%2C+slim.+Actually+slim.+I+did+not+%5Bhttp%3A%2F%2Fwww.Reddit.com%2Fr%2Fhowto%2Fsearch%3Fq%3Drealize+realize%5D+just+how+skinny+the+knife+was+until+I+received+it+out+of+the+bundle.+I+wasn%27t+particularly+searching+for+a+slim+profile+however+as+soon+as+I+carried+it+around+I+fell+in+love+with+it.+I+had+two+major+options+I+was+searching+for+in+a+knife+to+hold+with+me+each+day.+The+primary+was+an+assisted+opening+system.+The+second+was+a+low+mount+clip+so+it+could+sit+deep+in+my+pocket.+This+knife+has+each+these+features%2C+in+all+actuality+a+whole+lot+of+knives+have+each+those+options.%0D%0A%0D%0AIntroducing+the+gorgeous+new+Fielder+XL.+The+contrast+of+the+vertical+grain+wooden+in+opposition+to+the+excessive+luster+bolsters+creates+a+really+appealing+visible+tension.+It+is+a+knife+that+is+snug+as+a+daily+carry+or+out+within+the+wilds.+Utilizing+a+confirmed+liner+lock+system+provides+reliability+in+addition+to+one-hand+opening+of+the+Fielder+XL%27s+%5Bhttp%3A%2F%2Fcapable.net%2F+capable%5D+blade.+The+Fielder+XL+will+catch+your+eye%2C+but+not+your+pocket+ebook%21++In+case+you+loved+this+informative+article+and+you+would+like+to+receive+much+more+information+with+regards+to+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fsog-knives-review%2F+Www.Thebestpocketknifereviews.Com%5D+kindly+visit+the+internet+site.+The+SOG+Escape+has+a+3.4%22+blade+produced+from+9Cr18MoV+metal.+It+is+a+sort+of+martensite+stainless-steel+made+in+China+that+has+been+in+comparison+with+ATS-34+due+to+its+excessive+carbon+content.+The+Escape+is+razor+sharp+right+out+of+the+box+and+the+blade+is+half+serrated.+This+particular+mannequin+had+a+black+non-reflective+coating+on+the+blade.+The+Parry+Blade%0D%0A%0D%0AOkay+so+I+simply+purchased+a+griptilian%2C+and+I+love+it.+Nevertheless%2C+it+is+just+a+little+massive+for+some+tasks+and+may+freak+some+folks+out+in+public.+I+wish+to+buy+a+small+lightweight+knife+to+carry+for+smaller+tasks.+After+a+little+bit+of+research+I%27ve+actually+come+to+love+the+SOG+Flash+1+and+the+SOG+Aegis+mini.+Does+anybody+have+expertise+with+these+knives%3F+Are+there+other+decent+%28preferably+assisted+open%29+knives+that+compare+to+the+Flash+1+within+the+weight+and+measurement+categories%3F+Price+isn%27t+actually+a+problem%2C+however+I%27d+love+to+pick+one+up+for+%7E%2450.%0D%0A%0D%0AA+terrific+characteristic+of+this+knife+is+the+built+in+carbide+sharpener+in+the+back+of+the+handle.+Genius%21+I%27ve+participated+within+the+gutting+%2F+skinning+of+an+elk+and+know+full+properly+that+it+will+be+one+superb+knife+to+make+it+by+means+of+the+entire+ordeal+without+having+to+stop%2C+change+out+a+knife+and+move+along+the+current+knife+for+a+quick+sharpening.+Having+the+sharpener+constructed+in+the+handle+is+nice+-+particularly+for+back+nation+searching+the+place+light+weight%2C+practical+tools+are+invaluable.%0D%0A%0D%0AAfter+all+as+we%27ve+identified+%2C+it%27s+not+essential+to+sharpen+your+knife+on+a+regular+basis%2C+2+or+3+occasions+a+yr+might+be+quite+enough%2C+supplied+you%27ve+gotten+a+honing+metal+to+keep+that+edge+straight.+So+in+the+event+you%27re+planning+on+buying+the+Sog+knife+sharpener+we+advise+you+additionally+put+money+into+a+steel+to+avoid+the+need+of+using+the+sharpener+too+usually.+-+Discontinued+knives+that+pop+up+once+every+few+months+on+the+%22For+Sale%22+part+of+boards+or+on+ebay.+Still+can+be+discovered+at+a+number+of+online+retailers+though.+-+Knives+that+want+a+number+of+researching+to+seek+out.+Not+available+anymore+in+on-line+knife+retailers+and+pop+up+%28should+you%27re+fortunate%29+%3E4+months%C2%A0for+one%C2%A0at+a+time.+The+reversible+bayonet+clip+permits+you+carry+the+SOG+Flash+II+in+your+pocket%2C+your+bag+and+even+in+your+purse.&create=Create question] is that not all steel is created equal. In truth, there is a enormous difference between a premium metal and a budget metal.<br><br>We have now achieved all of the arduous work and reviewed every knife with the honest assessment that may cover all the information of the particular knife. We have now offered the assessment very clearly so as to understand and compare it with other critiques in our website. We extremely advocate you that [http://www.Malgefragt.net/mwiki/index.php?title=Sog_Knives_Flash_Ii_Review undergo] our web site step-by-step to get essentially the most use out of it. Guide me to determine to chose Greatest Pocket Knife ! It’s very important to know all of the factors before deciding to buy a finest pocket knife. 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The '''Three-detector problem'''<ref>Daganzo, Carlos. 1997. Fundamentals of transportation and traffic operations. Oxford: Pergamon.</ref> is a problem in traffic flow theory. Given is a homogeneous freeway and the vehicle counts at two detector stations. We seek the vehicle counts at some intermediate location. The method can be applied to incident detection and diagnosis by comparing the observed and predicted data, so a realistic solution to this problem is important. Newell G.F.<ref>Newell, G. F. 1993. "A simplified theory of kinematic waves in highway traffic. Part I, General theory". Transportation Research. Part B, Methodological. 27B (4).</ref><ref>Newell, G. F. 1993. "A simplified theory of kinematic waves in highway traffic. Part II. Queuing at freeway bottlenecks". Transportation Research. Part B, Methodological. 27B (4).</ref><ref>Newell, G. F. 1993. "A simplified theory of kinematic waves in highway traffic. Part III. Multi-destination flows". Transportation Research. Part B, Methodological. 27B (4).</ref> proposed a simple method to solve this problem. In '''Newell's method''', one gets the cumulative count curve (N-curve) of any intermediate location just by shifting the N-curves of the upstream and downstream detectors. '''Newell's method''' was developed before the variational theory of traffic flow was proposed to deal systematically with vehicle counts.<ref>Daganzo, Carlos F. 2005. "A variational formulation of kinematic waves: solution methods". Transportation Research. Part B, Methodological. 39B (10).</ref><ref>Daganzo, Carlos F. 2005. "A variational formulation of kinematic waves: basic theory and complex boundary conditions". Transportation Research. Part B, Methodological. 39B (2).</ref><ref>Daganzo, Carlos F. 2006. "On the variational theory of traffic flow: well-posedness, duality and applications". Networks and Heterogeneous Media. 1 (4).</ref> This article shows how '''Newell's method''' fits in the context of variational theory.
 
== A special case to demonstrate Newell's method ==
'''Assumption.''' In this special case, we use the Triangular Fundamental Diagram (TFD) with three parameters: free flow speed <math>v_f</math>, wave velocity -w and maximum density <math>k_j</math> (see Figure 1). Additionally, we will consider a long study period where traffic past upstream detector (U) is unrestricted and traffic past downstream detector (D) is restricted so that waves from both boundaries point into the (t,x) solution space (see Figure 2).
 
The goal of three-detector problem is calculating the vehicle at a generic point (P) on the "world line" of detector M (See Figure 2). '''Upstream.''' ''Since'' the upstream state is uncongested, there must be a characteristic with slope <math>v_f</math> that reaches P from the upstream detector. Such a wave must be emitted <math>\tau_1=L_U/v_f</math> times unit earlier, at point P' on the figure. ''Since'' the vehicle number does not change along this characteristic, we see that the vehicle number at the M-detector calculated from conditions upstream is the same as that observed at the upstream detector <math>\tau_1</math> time units earlier. ''Since'' <math>\tau_1</math> is independent of the traffic state (it is a constant), ''this result is equivalent to'' shifting the smoothed N-curve of the upstream detector (curve U of Figure 3) to the right by an amount <math>\tau_1</math>.
 
'''Downstream.''' Likewise, ''since'' the state over the downstream detector is queued, there will be a wave reaching P from a location <math>P_2</math> with wave velocity <math>-w<0</math>. The ''change'' in vehicular label along this characteristic can be obtained from the moving observer construction of Figure 4, for an observer moving with the wave. In our particular case, the slanted line corresponding to the observer is parallel to the congested part of TFD. This means that the observer flow is independent of the traffic state and takes on the value: <math>k_j(-w)</math>. Therefore, in the ''time'' that it takes for the wave to reach the middle location, <math>\tau_2=L_D/(-w)</math>, the change in ''count'' is <math>\delta=k_j(-w)\tau_2=k_jL_D</math>; i.e., the change in count equals the number of vehicles that fit between M and D at jam density. ''This result is equivalent to'' shifting the D-curve to the right <math>\tau_2</math> units and up <math>\delta</math> units.
 
'''Actual count at M.''' In view of the Newell-Luke Minimum Principle, we see that the actual count at M should be the lower envelope of the U'- and D'-curves. This is the dark curves, M(t). The ''intersections'' of the U'- and D'- curves denote the shock's passages over the detector; i.e., the times when transitions between queued and unqueued states take place as the queue advances and recedes over the middle detector. The ''area'' between the U'- and M-curves is the delay experienced upstream of location M, ''trip times'' are the horizontal separation between curves U(t), M(t) and D(t), ''accumulation'' is given by vertical separations, etc.
 
'''Mathematical expression.''' In terms of the function N(t,x) and the detector location (<math>x_u</math>, <math>x_m</math>, <math>x_d</math>) as follows:
: <math>
    N(t,x_m)= \min\{\ N(t-L_U/v_f,x_u)\ ,\ N(t+L_D/w,x_d)+k_jL_D\ \}  \qquad (1)
  </math>
where <math>L_U=x_m-x_u</math> and <math>L_D=x_d-x_m</math>.
 
== Basic principles of variational theory (VT) ==
'''Goal.''' Suppose we ''know'' the number of vehicles (N) along a boundary in a time-space region and we are ''looking for'' the number of vehicles at a generic point P (denoted as <math>N_p</math>) beyond that boundary in the direction of increasing time(see Figure 5).<ref>Daganzo, Carlos F. Lecture notes: Operation of transportation facilities. Compiled by Offer Grembek</ref>
 
Suppose, again, that an observer starts moving from the boundary to point P along path L. We know the vehicle number the observer sees, <math>N_L</math>. We then break the path of the observer into small sections (such as the one show between A and B) and note that we also know the maximum number of vehicles that can pass the observer along that small section is, <math>C_{AB}</math>. The relative capacity formula tells us that it is: <math>C_{AB}=r(v^0)\Delta{t}</math>. For TFD and using <math>v_{AB}</math> for the slope of segment AB, <math>C_{AB}</math> can be written as:
: <math>
    C_{AB}=r(v_{AB})\Delta{t}=q_0\Delta{t}-k_0\Delta{x}=q_0(t_B-t_A)-k_0(x_B-x_A); for\ v_{AB}\in[-w,v_f] \qquad (2)
  </math>
So, if we now add the vehicle number on the boundary to the sum of all <math>C_{AB}</math> along path L we get an upper bound for <math>N_p</math>. This upper bound applies to any observer that moves with speeds in the range <math>[-w,v_f]</math>. Thus we can write:
: <math>
    N_P \le N_L + \sum_L(C_{AB}),\    v_{AB} \in [-w,v_f] \qquad (3)
</math>
Equations (1) and (2) are based on the relative capacity constraint which itself follows from the conservation law.
 
'''Maximum principle.''' It states that <math>N_P</math> is the largest possible value, subject to the capacity constraints. Thus the VT recipe is:
: <math>
    N_P =\min_L\{N_L+\sum_L(C_{AB})\}\qquad (4)
</math>
Equation (4) is a shortest path(i.e., calculus of variations) problem with <math>C_{AB}=r(v_{AB})</math> as the cost function. It turns out that it produces the same solution as Kinematic wave theory.
 
==Generalized solution==
  '''Three steps:'''
  1. Find the minimum upstream count, <math>N_U</math>
  2. Find the minimum downstream count, <math>N_D</math>
  3. Choose the lower of the two, <math>N_P=\min\{N_U\ ,\ N_D\}</math>
 
===Step 1===
All possible observer straight lines between the upstream boundary and point P have to constructed with observer speeds smaller than free flow speed:
: <math>
  C_{QP}=q_0\Delta{t}-k_0\Delta{x}=q_0(t_P-t_Q)-k_0(x_P-x_Q) \qquad (5)
</math>
where <math>\Delta{t}=t_P-t_Q=\frac{(x_M-x_U)}{v_{QP}}</math> for <math>v_{QP}\in[0,v_f]</math> and <math>\Delta{x}=x_M-x_U</math>
 
Thus we need to minimize <math>{N_Q+q_0(t_P-t_Q)-k_0(x_M-x_U)}</math>; i.e.,
:<math>
  N_U=\min_{t_Q}\{N_Q+q_0(t_P-t_Q)-k_0(x_M-x_U)\}  \qquad (6)
</math>
Since <math>dN_Q/dt \le q_0</math>, we see that the objective function is non-increasing and therefore <math>t_Q^*=t_{P_1}</math>. So Q should be placed at <math>P_1</math> and we have:
:<math>
C_{QP}=C_{P_1P}=q_0\left(\frac{x_M-x_U}{v_f}\right)-k_0(x_M-x_U)=0  \qquad (7)
</math>
Thus, <math>N_U=N_{P_1}</math>
 
===Step 2===
We have:<math>N_D=\min_{Q^'}\{N_{Q^'}+C_{Q^'P}\}=\min_{Q^'}\{N_{Q^'}+q_0\Delta{t}-k_0\Delta{x}\}</math>
So repeat the same steps we find that <math>N_D</math> is minimized when <math>Q^'=P_2</math>. And at point <math>P_2</math> we get:
:<math>
  N_D=N_{P_2}+q_0(\frac{x_D-x_M}{w})-k_0(x_D-x_M)  \qquad (8)
</math>
Since the FD is triangular, <math>\frac{q_0}{w}+k_0=k_j</math>. Therefore (8) reduces to:
:<math>
  N_D=N_{P_2}+(x_D-x_M)k_j  \qquad (9)
</math>
 
===Step 3===
To get the solution we now choose the lower of <math>N_U</math> and <math>N_D</math>.
:<math>
N_P=\min\{N_U\ ,\ N_D\}=\min\{N_{P_1}\ ,\ N_{P_2}+(x_D-x_M)k_j\}  \qquad (10)
</math>
 
''This is Newell's the recipe for the 3-detector problem.''
 
==See also==
*[[Fundamental diagram of traffic flow]]
*[[Kerner’s breakdown minimization principle]]
*[[Microscopic traffic flow model]]
*[[Microsimulation]]
*[[Newell's Car Following Model]]
*[[Road traffic control]]
*[[Rule 184]]
*[[Three-phase traffic theory]]
*[[Traffic bottleneck]]
*[[Traffic congestion: Reconstruction with Kerner’s three-phase theory]]
*[[Traffic counter]]
*[[Traffic flow]]
*[[Traffic wave]]
 
== References ==
<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist}}
 
[[Category:Road transport]]
[[Category:Road traffic management]]
[[Category:Transport engineering]]

Revision as of 20:18, 4 March 2014

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