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

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