Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland, Collg Park
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
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) K-th : T(intrval) nk vhicls (.g. 3) Distribution is rfrrd to arriving vhicls pr intrval: n, n,, nk 3
If k80 intrvals (0 sconds pr intrval) Av. vhicls pr intrval /80m No. of Vh/T Obsrvd Frquncy Obs Vhicls 0 94 0 63 63 4 3 6 >3 0 0 80 Poisson distribution: Light traffic conditions If T is 30 scs, Thn: m m 3 Comput pr 30 sconds 4
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
Tst th distribution No. of Vh/T Obs Total Prob Th Fr Frquncy 0 94 0 P(0) 80 97 63 63 P() 80 59.9 4 P() 80 8.5 3 6 P(3) 80 3.8 >3 6 0? 0.8 80 80.0 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
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
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 '! ) ( ] ) ( [
Poisson Arrival Intr-arrival Tims hadway Lt Lk tim for occurrnc of th k-th 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
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
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!
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 <<
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
Ngativ Binominal Distribution Traffic counts with high varianc tnd ovr both a pak priod and an off-pak 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
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
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
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
Ngativ ponntial frquncy curv Bar indicat obsrvd data takn on sampl siz of 609 8
Statistical distributions of traffic charactristics 9
Dashd curv applis only to probability scal 0
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
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)
Eampl of fhiftd ponntial fittd to frway data 3
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
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
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
7
Distribution Modls for Spds Normal distributions of spds Lognormal modl of spds Gap accptanc distribution modl 8
Cumulativ (normal) distributions of spds of four locations 9
Sam data as abov figur but with ach distribution normalizd 30
Lognormal plot of frway spot spds 3
Comparison of obsrvd and thortical distributions of rjctd gaps 3
Lag and gap distribution for through movmnts 33
Distribution of accptd and rjctd lags and gaps at intrsction lft turns 34
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
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 (tim-priod) 0 (tim-priod) Th two ar indpndnt: What is th probability that an arrival form procss (typ arrival) occurs bfor an arrival from procss (typ arrival)? 36
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 + + 0 u du ] [ + < P
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 + + 0 u du ] [ + < P
Multipl Indpndnt Poisson Procsss For th ntir procss: T T + T ( and ) Th probability of a tim-hadway 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 Constraind Flow-Platoon 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
4 Constraind Flow-Platoon 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
Constraind Flow-Platoon 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
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