Course Outline Introduction to Transportation Highway Users and their Performance Geometric Design Pavement Design Speed Studies - Project Traffic Queuing Intersections Level of Service in Highways and Intersections 2
Previous class Basic Concepts a. Flow Rate b. Spacing c. Headway d. Speed 2 types e. Density Relationships Graphs 3
Relationships q = n t t = n h i i = 1 q = n n i = 1 h i u s = 1 n 1 1 n i= 1 l t n k = = l 1 q = uk ( ) u i q 4
Consider a linear relationship between speed and density: p p y = k u u 1 = j f k u u 1 = u u k q 2 = f j u u k q Lecture # 11 Department of Civil and Environmental Engineering 5
MODELS OF TRAFFIC FLOW Traffic is rarely uniformly distributed ib t d equal time between arriving vehicles or headways? Must make some assumption for arrival patterns (distribution) 6
Poisson Model Approximation of non-uniform flow Where: P ( n ) = ( λt) e n λt n! P(n) = probability of having n vehicles arrive in time t, t = duration of the time interval over which vehicles are counted, λ = average vehicle flow or arrival rate in vehicles per unit time, and e = base of the natural logarithm (e = 2.718). 7
Poisson Distribution Example Assume: mean = variance (sd^2) λ = 360 veh/h = 0.1 veh/s t = 20 sec P( n) = n λt ( λ t ) e n! 8
Poisson Ideas Probability of exactly 4 vehicles arriving P(n=4) Probability bilit of less than 4 vehicles arriving i P(n<4) = P(0) + P(1) + P(2) + P(3) Probability of 4 or more vehicles arriving P(n 4) = 1 P(n<4) = 1 - P(0) + P(1) + P(2) + P(3) Amount of time between arrival of successive vehicles ( 0 ) = P ( h t ) ( 0 λt λt) e λ t qt 3600 P = = e = 0! 9 e
Poisson Model The assumption of Poisson vehicle arrivals also implies a distribution of the time intervals between the arrivals of successive vehicles (time headway). To show this, note that the average arrival rate as: λ = q 3600 Where: λ = average vehicle arrival rate in veh/s, q = flow in veh/h, and 3600 = number of seconds per hour. 10
Poisson Model Substituting into P(n) equation: P ( n) = ( qt 3600) n n!! e -qt 3600 The probability of having no vehicles arrive in a time interval of length t (P(0)) is equivalent to the probability of a vehicle headway, h, being greater than or equal to the time interval t. ( 0 ) =P ( h t ) =e -qt 3600 negative exponential or P simply called exponential distribution 11
Change of mean and distribution shape 0.25 Mean = 0.2 vehicles/minute Probabilit ty of Occur rance 0.20 0.15 0.10 0.05 Mean = 0.5 vehicles/minute 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Arrivals in 15 minutes 12
Change of mean and inter-arrival times 1.0 of Exceda ance Probability 0.9 0.8 07 0.7 0.6 0.5 0.4 0.3 0.2 Mean = 0.2 vehicles/minute Mean = 0.5 vehicles/minute 0.1 0.0 0 2 4 6 8 10 12 14 16 18 20 Time Between Arrivals (minutes) 13
Limitations of the Poisson Model Mean (average number of cars per time period) must be equal to the variance (variance over all time period) Otherwise use an alternative model (negative binomial, etc.). Department of Civil and Environmental Engineering Lecture # 14
Poisson Distribution Let s grab 6 hrs of 5 min aggregated counts from a station in the Portland freeway network (I-5) for one lane Does it match what we expect? mean = 84.57 veh/5 min/ln (1,014 veh/hr/ln) standard deviation = 9.35 veh variance = 87.44 veh^2 Observed Data Poisson Model 7 0.05 6 0.05 0.04 Frequency 5 4 3 2 P(n) in t=5 min 0.04 0.03 0.03 0.02 0.02 0.01 1 0.01 0 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 0.00 0 7 14 21 28 35 42 49 56 63 70 77 N vehicles 84 91 98 105 112 119 How do we know if a distribution is a good representation of reality? Any objective test? 15
Problem 5.8. An observer has determined that the time headways between successive vehicles, on a section of highway, are exponentially distributed and that 60% of the headways ays between ee vehicles es are 13 seconds or greater. If the observer decides to count traffic in 30-second time intervals, estimate the probability of the observer counting exactly four vehicles in an interval. 16
Queuing delays significance Queuing delays can account for up to 90% or more of a driver s total trip travel time. Examples of queuing: Traffic Signals Toll booths Traffic incidents (accidents and vehicle disablements) 17
Queueing Theory Objects passing through point with restriction on maximum rate of passage Input + storage area (queue) + restriction + output Customers, arrivals, arrival process, server, service mechanism, departures, discipline (FIFO) Input Storage Output Restriction 18
Queueing Theory: Study of Congestion Phenomena Applications: Airplane waiting for takeoff, toll gate, wait for elevator, taxi stand, ships at a port, grocery store, telecommunications, circuits Interested in: maximum queue length, typical queueing times. SERVICE LEVEL! Input Storage Output Restriction 19
Queueing Theory: common assumptions Arrival patterns (λ, in vehicles per unit time): equal time intervals (derived from the assumption of uniform, deterministic arrivals) and exponentially distributed time intervals (derived from the assumption of Poisson-distributed arrivals). Departure patterns (μ, in vehicles per unit time), equal time intervals (derived from the assumption of uniform, deterministic arrivals) and exponentially distributed time intervals (derived from the assumption of Poisson-distributed arrivals). 20
Queueing Theory: Some definitions D/D/1 G/G/m Deterministic arrivals General arrivals Deterministic departures General departures 1 channel departures Multi-channel departures Graphical solution easiest SIMULATION M/D/1 Exponential arrivals Deterministic departures 1 channel departures Mathematical solution M/M/1 Exponential arrivals Exponential departures 1 channel departures Mathematical solution 21
Queuing discipline first-in-first-out (FIFO), indicating that the first vehicle to arrive is the first vehicle to depart, and last-in-first-out (LIFO), indicating that the last vehicle to arrive is the first to depart. For virtually all traffic-oriented queues, the FIFO queuing discipline is the more appropriate of the two. 22
D/D/1 Queuing deterministic i ti arrivals and departures with one departure channel (D/D/1 queue) D/D/1 queue lends itself to a graphical or mathematical solution. 23
Queueing Theory : Conservation Principle Customers don t disappear Arrival times of customers completely characterizes arrival process. Time/accumulation axes Uniform arrivals/departures N(x,t) 3 A=l(t) 2 D=m(t) 1 t 1 t 2 t 3 Time, t @ x 24
Queueing Theory: Departure Process Observer records times of departure for corresponding objects to construct D(t). N(x,t) A(t) 4 1 23 t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, t @ x 25
Queueing Theory: Analysis If system empty at t=0: Vertical distance is queue length at time t: Q(t)=A(t)-D(t) For FIFO horizontal distance is waiting time for jth customer. N(x,t) Uniform arrivals/departures 3 A=l(t) 2 Q(t) D=m(t) 1 W j t 1 t 2 t 3 Time, t @ x 26
Queueing Theory: Analysis Horizontal strip of unit height, width W j N(x,t) 4 1 23 W 2 A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, t @ x 27
Queueing Theory: Analysis Add up horizontal strips total delay Total time spent in system by some number of vehicles (horizontal strips) N(x,t) Total Delay=Area A(t) 4 1 23 t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, t @ x 28
Queueing Theory Total delay = W Average time in queue: w = W/n Average number in queue: Q = W/T N(x,t) 4 1 23 A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, t @ x 29
EXAMPLE 5.7 Vehicles arrive at an entrance to a recreational park. There is a single gate (at which all vehicles must stop), where a park attendant distributes a free brochure. The park opens at 8:00 800 A.M.,,at which time vehicles begin to arrive at a rate of 480 veh/h. After 20 minutes, the arrival flow rate declines to 120 veh/h and continues at that level for the remainder of the day. If the time required to distribute the brochure is 15 seconds, and assuming D/D/1 queuing, describe the operational characteristics of the queue. 30
EXAMPLE 5.7 - SOLUTION Begin by putting arrival and departure rates into common units of vehicles per minute. 480 veh/h λ = = 8 veh/min for t 20 min 60 min/h 120 veh/h λ = = 2 veh/min for t > 20 min 60 min/h 60 s/min μ = = 4 veh/min for all t 15 s/veh 31
EXAMPLE 5.7 - SOLUTION Begin by putting arrival and departure rates into common units of vehicles per minute. 8 t for t 20 min 160 ( t 20) for t 20 min + 2 > the number of vehicle departures is: 4 t for all t 32
EXAMPLE 5.7 - SOLUTION A=l(t)=2(t) Equation of line = 160 + 2(t-20) 80 veh A=l(t)=8(t) D=m(t)=4(t) 33
EXAMPLE 5.7 - SOLUTION When the arrival curve is above the departure curve, a queue condition will exist. The point at which the arrival curve meets the departure curve is the moment when the queue dissipates (no more queue exists). The point of queue dissipation can be determined by equating appropriate arrival and departure equations, that is ( t 20) = t 160 + 2 4 Solving for t gives t = 60 minutes. 34
EXAMPLE 5.7 - SOLUTION Thus the queue that began to form at 8:00 A.M. will dissipate 60 minutes later (9:00 A.M.), at which time 240 vehicles will have arrived and departed (4 veh/min 60 min). Individual vehicle delay: Under FIFO queuing discipline, the delay of an individual vehicle is given by the horizontal distance between arrival and departure curves. So, by inspection of Fig. 5.7, the 160 th vehicle to arrive will have the longest delay of 20 minutes (the longest horizontal distance between arrival and departure curves) 35
The total length of the queue is given by the vertical distance between arrival and departure curves at that time. The longest queue (longest vertical distance between arrival and departure curves) will occur at t = 20 minutes and is 80 vehicles long Total vehicle delay, defined as the summation of the delays of each individual vehicle, is given by the total area between arrival and departure curves 36
In this example, the areas between arrival and departure curves can be determined by summing triangular areas, giving total delay, Dt, as 1 D t = + 2 ( 80 20 1 ) 80 40 2 ( ) = 2400 veh - min Because 240 vehicles encounter queuing-delay (as previously determined), the average delay per vehicle is 10 minutes (2400 veh-min/240 veh), and dthe average queue length is 40 vehicles (2400 veh-min/60 min). 37
Problem 5.14. Vehicles begin to arrive at a parking lot at 6:00 A.M. at a rate 8 per minute. Due to an accident on the access highway, no vehicles arrive from 6:20 to 6:30 A.M. From o 6:30 A.M. on, vehicles es arrive at a rate of 2 per minute. The parking lot attendant processes incoming vehicles (collects parking fees) at a rate of 4 per minute throughout the day. Assuming D/D/1 queuing, determine total vehicle delay. 38
EXAMPLE 5.8 After observing arrivals and departures at a highway toll booth over a 60-minute time period, the observer notes that the arrival and departure rates (or service rates) are deterministic but, instead of being uniform, they change over time according to a known function. The arrival rate is given by the function (t) = 2.2 + 0.17t 0.0032t 2 and the departure rate is given by (t) = 1.2 + 0.07t, where t is in minutes after the beginning of the observation period and (t) and (t) are in vehicles per minute. Determine the total vehicle delay at the toll booth and the longest queue assuming D/D/1 queuing. 39
M/D/1 Queuing exponentially distributed times between the arrivals of successive vehicles (Poisson arrivals) deterministic departures, and one departure channel Obvious example Traffic Signals 40
M/D/1 Basic relationship: ρ = λ μ Where: ρ = traffic intensity, and is unit-less, λ = average arrival rate in vehicles per unit time, and μ = average departure rate in vehicles per unit time. 41
M/D/1 assuming that ρ <1, for an M/D/1: average length of queue in vehicles: Q = 2 ρ 2 1 ( ρ ) average waiting time in the queue (for each vehicle): ρ w = 2μ 1 ( ρ) average time spent in the system (summation of average queue waiting time + average departure time (service time)): 2 ρ t = 2μ 1 ρ 42 ( )
M/D/1 NOTE! that t the traffic intensity it is less than one (ρ <1), the D/D/1 queue will predict NO queue formation. Models with random arrivals or departures, such as the M/D/1 queuing model, will predict queue formations! Q = ρ 2 1 2 ( ρ) w = 2 ρ μ ( 1 ρ) t = 2 2μ ρ ( 1 ρ) 43
M/M/1 Queuing exponentially distributed times between the arrivals of successive vehicles (Poisson arrivals) exponentially distributed departure time patterns in addition to exponentially distributed arrival times one departure channel Traffic applications: Toll booth where some arriving drivers have the correct toll and can be processed quickly, and others may not have the correct toll, thus producing a distribution of departures about some mean departure rate. 44
M/M/1 λ = arrival rate μ = departure (service) rate λ μ ρ = ρ < 1.0 Average length of queue Q ρ 2 = 1 ( ρ ) Average time waiting in queue 1 λ w = μ μ λ Average time spent in system 1 t = μ λ 45
M/M/N Queuing Applications: M/M/N queuing is a reasonable assumption at toll booths on turnpikes or at toll bridges where there is often more than one departure channel available (more than one toll booth open). M/M/N queuing is also frequently encountered in non-traffic but transportation applications such as security checks at airports, vessel queueing at ports/airports and so on. Other non-transportation : checkout lines at retail stores, call centers, etc. 46
M/M/N Multi-channel λ ρ = ρ N < 1.0 μ Average length of queue Average time waiting in queue Q Average time spent in system +1 P N 0ρ 1 N! N 1 ρ N = 2 w t = = ( ) ρ + Q 1 λ μ ρ + Q λ 47
M/M/N Probability of having no vehicles = 1 P0 N 1 n N ρ c ρ + n = 0 nc!! N! 1 ρ N c ( ) Probability of having n vehicles n n ρ P = 0 ρ P Pn for n N P = 0 for n n! N! n n N N N Probability of being in a queue P n > N = N + 1 P0 ρ N! N 1 ρ ( N ) 48
Problem 5.35 Vehicles leave an airport parking facility (arrive at parking fee collection booths) at a rate of 500 veh/h (the time between arrivals is exponentially distributed). The parking facility has a policy that the average time a patron spends in a queue while waiting to pay for parking is not to exceed 5 seconds. If the time required to pay for parking is exponentially distributed with a mean of 15 seconds, what is the fewest number of payment processing booths that must be open to keep the average time spent in a queue less than 5 seconds. 49