Need to understand interaction of macroscopic measures

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Darcy Hawkins
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1 CE 322 Transportation Enginring Dr. Ahmd Abdl-Rahim, h. D.,.E. Nd to undrstand intraction o macroscopic masurs Spd vs Dnsity Flow vs Dnsity Spd vs Flow Equation 5.14 hlps gnraliz Thr ar svral dirnt orms I you hav modl or u vs can stimat q Givn u vs u u 1 j q vs and q vs u ar thn 2 q u j u q j u u 2 Givn u u 1 j Calibration paramtrs u = r low spd (75 mph) j = jam dnsity (60 vpmpl Input paramtr = dnsity (45 vpmpl) q = u x Solv or q What is q cap? 1

2 Macroscopic rlationships and analyss ar vry valuabl, but Larg portion o traic analysis occurs at th microscopic lvl Elapsd tim btwn th arrival o succssiv vhicls (i.., tim hadway) Vhicl quus Wait tim Simplst approach to modling vhicl arrivals uniorm spacing Givs dtrministic, uniorm arrival pattrn constant tim hadway btwn all vhicls Assuming uniorm pattrn is usually unralistic vhicl arrivals typically ollow a random procss Nd a modl to rprsnt a random arrival procss What is mant by random? For a squnc o vnts to b considrd truly random, two conditions must b mt: 1. Any point in tim is as lily as any othr or an vnt to occur (.g., vhicl arrival) 2. Arrival tims ar indpndnt o ach othr oisson distribution its this dscription Th oisson distribution: Discrt distribution (not continuous) Rrrd to as a counting distribution Rprsnts th count distribution o random vnts tim 2

3 Equation or oisson dist. is: ( t) n) n! n t (Eq. 5.23) (n) = probability o xactly n vhicls arriving in a tim intrval t = avrag arrival rat (vh/unit tim) n = # o vhicls arriving in a spciic tim intrval t = slctd tim intrval (duration o ach counting priod (unit tim)) Intrprtation: I you wantd to now th probability o a crtain numbr o vhicls arriving during a crtain lngth o tim thn t would b th lngth o tim and n would b th numbr o vhicls. Th low rat would b givn as λ. Ma sur that you spciy λ and t using th sam tim units. Considr a highway with an hourly low rat o 120 vhicls pr hour, during which th analyst is intrstd in obtaining th distribution o 1-minut volum counts. What is th distribution? What is th lilihood o sing 3 vhicls arriv? What is th cumulativ lilihood o sing wr than 4 vhicls arriv? What is th rquncy that you would xpct to s ach count or a 60 minut priod? Stup = (120 vh/hr) / ( sc/hr) = vh/s t = vh/sc 60 sc = 2 vh OR = (120 vh/hr) / (60 min/hr) = 2 vh/min t = 2 vh/min 1 min = 2 vh What is th probability o btwn 0 and 3 cars arriving (in 1-min intrval)? n 4 n 0 n 1) n 2 n ) 3!

4 Exampls o using oisson Distribution On an intrsction approach with a lt turn volum o 120 vph. (oisson), what is th probability o sipping th grn phas or th lt turn traic? Th intrsction is controlld by am actuatd signal with an avrag cycl lngth o 90 sconds. m = Avrag numbr o lt turn vhicls pr Cycl numbr o cycls pr hour = /90 = 40 m = 120 vph / 40 = 3 vhicls/cycl robability o x = 0 0) = = 4.9 % (1.96 cycls/hours) mxm x) x! Exampls o using oisson Distribution An intrsction is controlld by a ixd tim signal having a cycl lngth o 55 sconds. From th northbound, thr is a prmittd lt turn movmnt o 175 vph. I two vhicls can turn ach cycl without causing dlay, on what prcnt o th cycls will dlay occur? m = Avrag numbr o lt turn vhicls pr Cycl numbr o cycls pr hour = /55 = m = 175 vph / = 2.67 LT vhicls/cycl robability o x > 2 x! (x>2) = 1 [0) + 1) + (2)] = 1 [ ] = 49.9% [ Cycls/hour] mxm x) n 3 What is th probability o mor than 3 cars arriving (in 1-min intrval)? n 3 1 n i or (14.3%) n i 4

5 oisson distributd vhicl arrivals implis a distribution o th tim btwn vhicl arrivals (i.., tim hadway) Dscribd by ngativ xponntial distribution. To dmonstrat this, lt th avrag arrival rat,, b in units o vhicls pr scond, so that q vh h sc h vh sc h t) qt Substituting into oisson quation yilds n qt qt n) n! (Eq. 5.25) robability o no vhicls arriving in tim t is h t). Driv Ngativ xponntial distribution rom oisson, 0) h t) qt qt 1 1 Not: 0 x 1 0! 1 Assum vhicl arrivals ar oisson distributd with an hourly traic low o 120 vh/h. Dtrmin th probability that th hadway btwn succssiv vhicls will b lss than 8 sconds h<8). Dtrmin th probability that th hadway btwn succssiv vhicls will b btwn 8 and 11 sconds 8<h<11). 5

By dinition, h t 1 h t h 8 1 h 8 h 8 1 1 qt 120(8) 1 0.766 0.

6 By dinition, h t 1 h t h 8 1 h 8 h qt 120(8) h 11 h 11 h (11) For q = 120 vh/hr

Continuous probability distributions

Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd