16 Performance Measures
Performance measure statistics are only calculated for the simulation duration time, not during the warm-up time. Therefore, the warm-up time should be sufficient to fill the network with vehicles so that performance measure calculations do not include simulation time when parts of the network are empty. Performance measures are calculated on either a point (detector) basis or a link basis. Whereas all vehicles that cross a detector within the simulation duration time are factored into the performance measure(s) calculation, for link-based measures, only vehicles that both enter and exit a link within the simulation duration time are considered. Thus, in addition to any difference due to point- versus space-based measures, the sets of vehicles included in the respective calculations will typically not be the same. For example, for a detector placed near the end of a long link, some vehicles may enter the link during the warm-up time and cross the detector during the simulation duration time. These vehicles would not be factored into the link performance calculation because they only exited, not both entered and exited, the link during the simulation duration time.
16.1 Flow Rate
Point (detector) measurement
\(q_p=\frac{n_p}{t}\)
where:
\(q_p\) = traffic flow at a point in vehicles per unit time,
\(n_p\) = number of vehicles passing a designated roadway point during time t, and
\(t\) = duration of time interval.
Link measurement
\(q_d=\frac{n_d}{t}\)
where:
\(q_d\) = traffic flow over a specific distance in vehicles per unit time,
\(n_d\) = number of vehicles traversing a specific distance during during time t, and
other terms as defined previously.
16.2 Average Speed
Time-mean (Spot) Speed (Detector Measure)
\(\bar{u}_t=\frac{1}{n_p} \sum_{i=1}^{n_p}\ u_{i}\)
where:
\(\bar{u}_t\) = time-mean speed in unit distance per unit time, typically mi/h,
\(u_{i}\) = point speed measurement for vehicle i, and
other terms as previously defined.
Space-Mean Speed (Link Measure)
\(\bar{u}_s=\frac{1}{\frac{1}{n_d}\sum_{i=1}^{n_d}\ \frac{1}{l/t_i}}\)
where:
\(\bar{u}_s\) = space-mean speed in unit distance per unit time, typically mi/h,
\(l\) = length of roadway used over which measurements are made in unit distance, typically miles,
\(t_i\) = travel time for vehicle i over distance l in unit time, typically hours, and
other terms as previously defined.
16.3 Density
Density Calculation (Detector Measure)
Detector occupancy can be used to approximate density as follows.
\(PropOccupancy =\frac{T_o}{T}\)
where:
\(PropOccupancy\) = proportion of time that a vehicle occupies the location of the point detector,
\(T_o\) = total time the point detector is occupied by a vehicle, and
\(T\) = total observation time.
Relationship between Occupancy and Density
\(k_{est}=PropOccupancy \times \frac{5280}{L_e}\)
where:
\(k_{est}\) = estimated density, in units of veh/mi,
\(L_e\) = effective vehicle length (average vehicle length + detector length), in units of feet, and
\(5280=\) feet per mile.
Density Calculation (Link Measure)
\(k=\frac{q_d}{\bar{u}_s}\)
where:
\(k\) = density in units of vehicles per unit distance, typically veh/mi/ln, and
other terms as defined previously.
16.4 Percent Followers
Percent Followers (Detector Measure)
Percent Followers is the percent of vehicles designated as followers within simulation
\(PF=\frac{n_{follow}}{n_p} \times 100\)
where:
\(PF\) = percent followers at a designated roadway point during the measurement time period, and
\(n_{follow}\) = the number of follower vehicles passing a designated roadway point with a headway \(\le\) 2.5 seconds (see Washburn et al. (2018)).
16.5 Follower Density
Follower Density (Detector Measure)
Follower Density is the density times the proportion of followers
\(FD=\frac{PF}{100} \times \frac{q_p}{\bar{u}_t}\)
where:
\(FD\) = follower density at a designated roadway point during the measurement time period, in units of followers/mi/ln, and other terms as defined previously.
16.6 Acceleration Noise
Acceleration Noise measures the standard deviation associated with successive deceleration and acceleration in an uninterrupted-flow environment. It provides an indication of the smoothness of traffic flow.
\(\sigma=\left\{ \left(\frac{1}{T}\right)\int_{0}^T\ [a(t)]^{2}dt \right\} ^{1/2}\), Assuming mean (\(\mu\))=0
\(\sigma=\left\{ \left(\frac{1}{T}\right)\sum_{i=0}^T\ [a(t)]^{2}Δt\right\}^{1/2}\), (In simulation)
\(\sigma=\left\{ \left(\frac{1}{T}\right)\int_{0}^T\ [a(t)-\mu]^{2}dt\right\}^{1/2}\), Assuming mean (\(\mu\)) \(\neq0\)
where:
\(\sigma=\) standard deviation,
\(a(t)=\)acceleration value at time t,
\(\mu=\) mean, and
\(T=\) total time in motion for the trip segment.
16.7 Travel Time
Detector or link measurement
The travel time of a vehicle is the total time of movement between two points, including any time of delay that may have occurred from traffic lights, stop signs, etc. to reach a destination.
\(\bar{t}=\frac{1}{n_d}\sum_{i=1}^{n_d}\ t_{i}\)
where:
\(\bar{t}=\) average travel time (units of time), and
other terms as previously defined.
16.8 Running Time
The running time of a vehicle is the time spent in motion as it travels between specific stations on a given traffic network. Running time will not include any delays that vehicle may encounter from traffic control devices such as stop signs, signals, etc. In order to determine the running time of a given network, the travel time of vehicles that were not included in a traffic-controlled queue anywhere along the length of the roadway are used.
16.9 Delay
Individual Vehicle Delay (Link Measure)
\(Delay_{ij} = Link Travel Time_{ij} – \frac{LinkLength_{j}}{Desired Speed_{i}}\)
where:
\(Delay_{ij}=\) difference between actual travel time and travel time at desired speed for vehicle i on link j (s)
\(LinkLength_{j}=\) length of link j (ft)
\(Desired Speed_{i}=\) Desired Speed for vehicle i in subject link (ft/s)
\(Link Travel Time_{ij} = Link Exit Time_{ij} - Link Entrance Time_{ij} =\) time for vehicle i to travel the length of link j (s)
where:
\(Link Entrance Time =\) time when vehicle i entered link j (s), and
\(Link Exit Time =\) time when vehicle i exited link j (s).
Total Vehicle Delay (Link Measure)
\(TotalDelay_{j}=\sum_{i=1}^N\ Delay_{ij}\)
where:
\(Total Delay_{j}=\) summation of individual vehicle delays, for vehicles \(1-N\), that traveled full length of link j (veh-s)
Average Vehicle Delay (Link Measure)
\(AverageDelay_{j}=\frac{TotalDelay_{j}}{TotalVehicles_{j}}\)
where:
\(AverageDelay_{j}=\) the average delay experienced by all vehicles traversing link j during the simulation period (i.e., after warm-up time) (s/veh), and
\(TotalVehicles_{j}=\) the number of vehicles traversing the full length of link j during the simulation period (i.e., after warm-up time) (veh)
Note about Control Delay: Control delay is the portion of the total delay attributed to the influence of a traffic control (signal, stop/yield sign). Care must be taken when selecting the links from which to extract delay to use for control delay purposes, as control delay consists of deceleration delay and acceleration delay, in addition to stop delay.
16.10 Queue Length
Queue Length (Link Measure)
\(Avg. Queue Length_{j} = \frac{Total Delay_{j}}{Simulation Duration}\)
where:
\(Avg. Queue Length_{j}=\) average queue length for link j (veh),
\(Total Delay_{j}=\) total delay of vehicles on link j (veh-s), and
\(Simulation Duration=\) simulation duration (not including warmup time) (s).
16.11 Saturation Flow Rate
Saturation Flow Rate (Detector Measure)
\(h_{sat}=\frac{T_{i}-T_{4}}{i-4}\)
\(SLT=T_{4} - 4h_{sat}\)
\(s=\frac{3600}{h_{sat}}\)
where:
\(SLT=\) start up lost time,
\(T_{i}=\) the time it takes for the front axle of vehicle i in the queue to cross the stop bar post green signal indication on the intersection, i ranges from 6–10 vehicles, depending upon the observed queue length,
\(s=\) saturation flow rate (veh/h),
\(h_{sat}=\) saturation headway (s/veh), and
\(3600=\) number of seconds per hour
16.12 Stop Rate
Stop Rate is the average number of stops per hour. (Link Measure)
\(Stop Rate \text{(stops/h)} = \frac{Total Stops}{Simulation Duration \text{ (h)}}\)
where:
\(Stop Rate_{j}=\) the average number of stops per hour on link j
\(Total Stops_{j}=\) the total number of vehicle stops on link j, and
\(Simulation Duration=\) simulation duration (not including warmup time) (s)
Notes about Total Stops measure: If a vehicle stops one or more times on the link, it only gets counted as one stop. In the case of cycle failures, this number will likely not reflect the actual number of stops. To be counted as a stop, SwashSim only considers the ‘StoppedInQueue’ status, not ‘SlowingInQueue’ or ‘QueueDischarge’.
16.13 Vehicle-Miles-Traveled (VMT)
\(VMT=n_d \times l\)
where:
\(VMT\) = vehicle-miles of travel, in units of veh-mi, and
other terms as defined previously.
16.14 Vehicle-Hours-Traveled (VHT)
\(VHT=n_d \times TT_{avg}\)
where:
\(VHT\) = vehicle-hours of travel, in units of veh-h,
\(TT_{avg}\) = average travel time (equivalent to \(\bar{t}\) in the average travel time calculation above, in units of h, and
other terms as defined previously.
Note that space-mean speed can also be calculated as
\(\bar{u}_s=\frac{VMT}{VHT}\)