Traffic Flow Analysis (2)


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1 Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland, Collg Park
2 Tim Hadway Distribution Distanc st nd 3 rd Gap Occupancy Tim T Givn a tim horizon: T n hadways, th distribution of such hadways dpnds on traffic conditions Givn a fid tim intrval t ( T k t), th numbr of arrivals during ach t is a distribution
3 Traffic counts Numbr of vhicls pr intrval st : T(intrval) n vhicls (.g. 3) T 0 scs nd : T(intrval) n vhicls (.g. ) 3 rd : T(intrval) n3 vhicls (.g. 0) Kth : T(intrval) nk vhicls (.g. 3) Distribution is rfrrd to arriving vhicls pr intrval: n, n,, nk 3
4 If k80 intrvals (0 sconds pr intrval) Av. vhicls pr intrval /80m No. of Vh/T Obsrvd Frquncy Obs Vhicls > Poisson distribution: Light traffic conditions If T is 30 scs, Thn: m m 3 Comput pr 30 sconds 4
5 Distributions for Traffic Analysis Poisson Distribution: light traffic conditions Total occurrncs m ( av. valu) Total obsrvations of tim σ / m m P( ) m /! 0,,, m t t: slctd tim intrval m P(0).g. P( ) P( ) m m p( m)! p( m) ( )! m P ( ) P( ) m Svral poisson distributions with man valus: m, m, m3, Thn m N m i i Limitations: only for discrt random vnts 5
6 Tst th distribution No. of Vh/T Obs Total Prob Th Fr Frquncy P(0) P() P() P(3) >3 6 0? P ( ) m! P(0) m 0 0! 0 Av. vhicls pr intrval /80m Th probability of having X vhicls arriving at th counting lin during th intrval of 0 sconds 6
7 Th probabilitis that 0,, cars arriv at ach T(0 scs) intrval Can b prssd as: Pn ( ) i 0 i m i! m Pn ( ) i 0 i m i! m For th cas of X or mor Pn ( ) i 0 i m i! m P i y ( ) y i i m i! m 7
8 Poisson Arrival Th numbr of Poisson arrivals occurring in a tim intrval of is: k 0,,, Th probability that thr ar at last k numbr of vhicls arriving during intrval t is: Poisson is only applicabl in light traffic conditions 8 ( t) n t! ) ( ] ) ( [ k t k t M P k t k k t k k t k t M P '! ) ( ] ) ( [
9 Poisson Arrival Intrarrival Tims hadway Lt Lk tim for occurrnc of th kth arrival, k,, 3, Th pdf f Lk ()d P[kth arrival occurs in th intrval to +d] P[actly k arrivals in th intrval [0,] and actly on arrival in [,+d]] L k k th Tim ( ) ( k )! ( d)! k d k ( ) ( k )! d [ d ] k k ( k )! f k ( L ) d 9
10 Poisson Arrival f L k k k ( ) ( k )!, 0; k,, 3, th kth  ordr intrarrival tim distribution for a poisson procss is a kth  ordr Erlang pdf st k (hadway) f ( ) 0 (ngativ ponntial distribution) L Th probability P( h ) d (C.D.F.) 0
11 Poisson Arrival From a Poisson prspctiv: If No vhicl arrivs during th tim lngth a tim hadway P [ M Not: ( ) 0] 0! 0 (sam as th prvious cas) Hadway is a continuous distribution: Arrival rat is a discrt distribution: P( h ) P ( M m) m ( ) m!
12 Distributions for Traffic Analysis Binomial Distribution For congstd traffic flow  P( ) c P p n n ( ) varianc man P is th probability that on car arrivs Man valu: m np Varianc: s np( 0,,,, n p) P is unknown from th fild, but can b stimatd from th man and varianc of obsrvd vhicls pr intrval <<
13 Binominal distribution m and s can b computd from th fild data (no. Vhicls pr intrval) ^ p ^ n ( m s ) m m/ p m ( m s ) 3
14 Ngativ Binominal Distribution Traffic counts with high varianc tnd ovr both a pak priod and an offpak priod (.g., a short counting intrval for traffic ovr a cycl, or downstram from a traffic signal P( ) c + k k P k q k 0,,, p ˆ m s m k ˆ s qˆ ( pˆ m ) p (0) k p + k p( ) q p( ) 4
15 Summary Th Poisson distribution rprsnts th random occurrnc of discrt vnts. Poisson distribution fits th vnts of man qual to varianc, spcially undr light traffic. Binomial distribution can b fittd to congstd traffic conditions whr th varianc/man ratio substantially lss than on. Ngativ binomial distribution can b fittd to traffic whr thr is a cyclic variation in th flow and man flow is changing during th counting priod. 5
16 Continuous Distributions Intrval (btwn arriving vhicls) Distribution Ngativ Eponntial Distribution Lt V: hourly volum, V/3600 (cars/sc) P( ) V t ( ) 3600 P(0) Vt / 3600! Vt / 3600 If thr is no vhicl arriv in a particular intrval of lngth t, thr will b a hadway of at last t sc. P(0) th probability of a hadway t sc 6
17 7 H hadway Man hadway T 3600/V T t t h P / ) ( T t t h P / ) ( < Varianc of hadways T 3600 / ) ( Vt t h P
18 Ngativ ponntial frquncy curv Bar indicat obsrvd data takn on sampl siz of 609 8
19 Statistical distributions of traffic charactristics 9
20 Dashd curv applis only to probability scal 0
21 Shiftd Eponntial Distribution P( h t) ( tτ ) /( T τ ) P( h < t) ( tτ ) /( T τ ) P( t) and 0, at t< τ (min. hadway) P( t) p[ ( t τ ) /( τ )] T τ T
22 Shiftd ponntial distribution to rprsnt th probability of hadways lss thn t with a prohibition of hadways lss than τ. (Avrag of obsrvd hadways is T)
23 Eampl of fhiftd ponntial fittd to frway data 3
24 Erlang Distribution P( h t) k t 0 for k for k for k 3 ~ k T / S kt ( ) T i kt / T i! Rducd to th ponntial distribution kt kt / T P( h t) + T kt kt P( h t) + ( ) + ( ) T T! kt / T k: a paramtr dtrmining th shap of th distribution T: man intrval, S : varianc * k, th data appar to b random * k incras, th dgr of nonrandomnss appars to incras 4
25 Composit Hadway Modl Constraind flows (Av. hadway T) Unconstraind fr flows (Av. hadway T) 5 + < ) p( ) p( ) ( ) ( τ τ α α T t T t t h P
26 Slction of Hadway Distribution Gnralizd Poisson distribution (Dns Traffic) or P( ) P( ) k ( + i) j k k i j j! k+ i ( ) ( k + i )! km +/ ( k ) 0,,, 0,,, k, k 3, P (0) + 3 P() +! 3! P(0) + +! 4 5 P() + + 3! 4! 5! 3 6
27 7
28 Distribution Modls for Spds Normal distributions of spds Lognormal modl of spds Gap accptanc distribution modl 8
29 Cumulativ (normal) distributions of spds of four locations 9
30 Sam data as abov figur but with ach distribution normalizd 30
31 Lognormal plot of frway spot spds 3
32 Comparison of obsrvd and thortical distributions of rjctd gaps 3
33 Lag and gap distribution for through movmnts 33
34 Distribution of accptd and rjctd lags and gaps at intrsction lft turns 34
35 Poisson Arrival Congstd Traffic Conditions platoon Distanc st Tim T Two typs of hadways btwn and within platoons during th sam priod T T T +T, ach priod has a diffrnt man hadway and 35
36 Multipl Indpndnt Poisson Procsss Two Poisson procsss: and Th combind procss: N(t) N (t) + N (t) is also a poisson procss pdf for pdf for 0 (timpriod) 0 (timpriod) Th two ar indpndnt: What is th probability that an arrival form procss (typ arrival) occurs bfor an arrival from procss (typ arrival)? 36
37 Multipl Indpndnt Poisson Procsss and ar both random variabls 0, 0 Similarly, 37 < 0 ), ( ] [ d d f P, ) ( ) ( ), ( d f f f [ ] P < 0 d d 0 ) ( d u du ] [ + < P
38 Multipl Indpndnt Poisson Procsss and ar both random variabls 0, 0 Similarly, 38 < 0 ), ( ] [ d d f P, ) ( ) ( ), ( d f f f [ ] P < 0 d d 0 ) ( d u du ] [ + < P
39 Multipl Indpndnt Poisson Procsss For th ntir procss: T T + T ( and ) Th probability of a timhadway X > is? Total numbr of arrivals during T priod T + T P(X > ) during T priod and T priod and Total arrivals having thir hadways > Total numbr of arrivals T + T (wightd avrag) Gnralization, T P[ X + T k i i > ] k T i i T i i i : arrival rat 39
40 40 Constraind FlowPlatoon Hadway within a platoon ar ponntially distributd with a man arrival rat and minimum hadway z 0, for < z0 (shiftd ponntial distribution) P[ X > ] '( zz0 ), for z z0 Th rlation btwn and Th pctd valu of th shiftd distribution must b qual to th actual man hadway
41 4 Constraind FlowPlatoon Hadway within a platoon ar ponntially distributd with a man arrival rat and minimum hadway z 0, for < z0 (shiftd ponntial distribution) P[ X > ] '( zz0 ), for z z0 Th rlation btwn and Th pctd valu of th shiftd distribution must b qual to th actual man hadway
42 Constraind FlowPlatoon Th arrival rat for such a shiftd distribution ' z z 0 ' z whr 0 / cannot b obsrvd actually obsrvd z P[ X > ] ( )( zz0 ) z 0 4
43 Som Travl Fr, Som Ar in Platoon Combination of two poisson procsss: P[X > ] P[X > occurs in travl fr traffic] + P[X > in platoon traffic] P P P + P T Total numbr of arrivals ( T + T ( )( z z z 0 T T + T 0 ) T : total obsrvd priod during which traffic is not movd in platoon T : total obsrvd priod during which vhicls ar movd in platoon ) 43
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