INTRODUCTION Statistical Properties Dr. Gang-Len Chang Professor and Director of Traffic Safety and Operations Lab. University of Maryland-College Park 1      Definition of Time Headway Poisson Arrival and Exponential Distribution General Poisson Properties Poisson Example Applications Other Types of Distributions 2 Time Headway Distribution Distance 1st 2nd 3rd Gap Occupancy   Time T Given a time horizon: T ⇒ n headways, the distribution of such headways depends on traffic conditions Given a fixed time interval ∆t (T = k ⋅ ∆t ), the number of arrivals during each ∆t is a distribution 3 Poisson Arrival  In order to be able to describe traffic as a poisson process, the following assumptions are required:   The traffic stream must be stationary ⇒ λ (mean arrival rate) = constant The probability that m vehicles appear in the interval (t0, t0+∆t) is independent of t0: Pt ,t + ∆t [ M = m] = Pt [ M = m] The traffic stream has no memory: Pt ,t + ∆t [ M = m] is independent of the details of the process up to time t0 The simultaneous appearance of several vehicles at a location can P[ M ( xi , t , ∆t ) > 1] be neglected; i.e., =0 0  0  0 0 0 lim ∆t →∞ ∆t 4 Poisson Arrival  The number of Poisson arrivals occurring in a time interval of t (= n ⋅ ∆t ) is: (λt ) k ⋅ e − λt k = 0, 1, 2, … P[ M (t ) = k ] = k!  The probability that there are at least k number of vehicles arriving during interval t is: (λt ) k ⋅ e − λt P[ M (t ) ≥ k ] = ∑ k! k '= k ∞ ∴ Poisson is only applicable in light traffic conditions 5 Poisson Arrival Inter-arrival Times = headway Let Lk = time for occurrence of the kth arrival, k = 1, 2, 3,… The pdf fLk(x)dx ≡ P[kth arrival occurs in the interval x to x+dx] = P[exactly k-1 arrivals in the interval [0,x] and exactly one arrival in [x,x+dx]] 1 Lk kth Time k  (λx) k −1 ⋅ e −λx   (λ ⋅ dx) ⋅ e −λ ⋅dx  (λx k −1 ) ⋅ e − λx ⋅ x k −1 ⋅ e − λx λ − λ ⋅dx = ⋅ ]= ⋅ [λ ⋅ dx ⋅ e   = ( k − 1 )! 1 ! ( k − 1 )! (k − 1)!     = f L ( x)dx k 6 Poisson Arrival ∴ f L ( x) = k λk ⋅ x k −1 ⋅ e − λx (k − 1)! , x ≥ 0; k = 1, 2, 3,… ⇒ the kth - order interarrival time distribution for a poisson process is a kth - order Erlang pdf set k = 1 (headway) f L ( x) = λ ⋅ e − λx x ≥ 0 (negative exponential distribution) The probability = P(h ≥ x) ∞ = ∫x λ ⋅ e −λx ⋅ dx = e −λx (C.D.F.) 1 7 Poisson Arrival From a Poisson perspective: If “No vehicle arrives during the time length x” ≡ a time headway ≥ x (λx) 0 −λx P [ M = 0] = ⋅ e = e −λx (same as the previous case) x 0! Note: = λx  Headway is a continuous distribution: P ( h ≥ x ) = e (λx) m ⋅ e − x  Arrival rate is a discrete distribution: Px ( M = m) = m! 8 Poisson Arrival  Congested Traffic Conditions platoon Distance 1st Time T   Two types of headways ⇒ between and within platoons during the same period T T = T1+T2 , each period has a different mean headway λ1 and λ2 9 Multiple Independent Poisson Processes Two Poisson processes: λ1 and λ2 The combined process: N(t) = N1(t) + N2(t) is also a poisson process pdf for λ1 ⇒ λ1 ⋅ e − λ x x1 ≥ 0 (time-period) pdf for λ2 ⇒ λ2 ⋅ e − λ x x2 ≥ 0 (time-period) 1 1 2 2 The two are independent: What is the probability that an arrival form process 1 (type 1 arrival) occurs before an arrival from process 2 (type 2 arrival)? 10 Multiple Independent Poisson Processes x1 and x2 are both random variables ∞ P[ x1 < x2 ] = ∫0 ∞ ∫x f x x ( x1 , x2 ) ⋅ dx1 ⋅ dx2 x1 ≥ 0, x2 ≥ 0 1 1 2 f x , x ( x1 , x2 ) = f x ( x1 ) ⋅ f x ( x2 ) = λ1λ2 ⋅ d − λ x e − λ 1 1 1 ∴ 2 1 2 ∞ ∞ P[ x1 < x2 ] = ∫0 dx1 ∫x dx2 ⋅ λ1λ2 ⋅ e − λ1 x1 ⋅e 1 = Similarly, 2 x2 λ1 λ1 + λ2 ∫ ∞ 0 e −u du = P[ x2 < x1 ] = − λ2 x 2 ∞ = ∫ dx1 ⋅ λ1 ⋅ e −λ1x1 (e −λ2 x2 ) 0 λ1 λ1 + λ2 λ2 λ1 + λ2 11 Multiple Independent Poisson Processes For the entire process: T = T1 + T2 (λ1 and λ2) The probability of a time-headway X > x is?  Total number of arrivals during T period = T1λ1 + T2λ2  P(X > x) during T1 period and T2 period = e − λ x and e − λ x 1  2 Total arrivals having their headways > x Total number of arrivals T1λ1 ⋅ e − λ x + T2λ2 ⋅ e − λ = T1λ1 + T2λ2 1 2x (weighted average) k ∑ Ti λi ⋅ e− λ x i Generalization, P[ X > x] = i =1 k ∑ Ti λi i =1 λ: arrival rate 12 Constrained Flow-Platoon  Headway within a platoon are exponentially distributed with a mean arrival rate λ and minimum headway z0 for x < z0 1, (shifted exponential distribution) P[ X > x] = e −λ '( z − z ) , for z ≥ z  0 0   The relation between λ and λ’ The expected value of the shifted distribution must be equal to the actual mean headway 13 Constrained Flow-Platoon  The arrival rate for such a shifted distribution λ’ 1 where z = 1 / λ λ'= z − z0 ∴ λ'= λ 1 − z0 λ ∴ λ’ ⇒ cannot be observed λ ⇒ actually observed ∴ P[ X > x] = e −( λ 1− z0 λ )( z − z0 ) 14 Some Travel Free, Some Are in Platoon  Combination of two poisson processes: P[X > x] = P[X >x | occurs in travel free traffic] + P[X > x | in platoon traffic] = P1 + P2 T1λ1 ⋅ e − λ x P1 = Total number of arrivals ( = T1λ1 + T2 λ2 ) 1 −( P2 =   λ2 )( z − z 0 ) T2λ2 ⋅ e T1λ1 + T2λ2 1− z 0 λ 2 T1: total observed period during which traffic is not moved in platoon T2: total observed period during which vehicles are moved in platoon 15 Problem   left the right Pedestrians approach from size of the crossing in a Poisson manner λ with average arrival rate λ arrivals per minute (Figure). Each pedestrian  then waits until a light is flashed, at which time all waiting pedestrians must cross. We refer to each time the light is flashed as a “dump” and assume that a dump takes zero time (i.e., pedestrians cross instantly). Assume that the left and right arrival processes are independent Automobile Traffic L R λL Pedestrian Traffic λR Pedestrian Traffic Pedestrian Crossing Light 16   We wish to analyze three possible decision rules for operating the light:  Rule A: Dump every T minutes  Rule B: Dump whenever the total number of waiting pedestrians equals N0  Rule C: Dump whenever the first pedestrian to arrive after the precious dump has waited T0 minutes Presumably, implementation of each rule requires a particular type of technology with its attendant costs, and thus it is important to determine the operating characteristics of each in order to understand tradeoffs between performance and cost 17  For each decision rule, determine:  The expected number of pedestrians crossing left to right on any dump  The probability that zero pedestrians crossing left to right on any particular dump  The pdf for the time between dumps  The expected time that a randomly arriving pedestrian must wait until crossing  The expected time that a randomly arriving observer, who is not a pedestrian, will wait until the next dump 18