Traffic delay and queue length are also key parameters used to assess the effectiveness of roundabout performance. Delay is an important parameter that is used in the performance evaluation of intersections. It is influenced by many variables and hence its determination is somewhat complex. HCM prescribes delay as the primary measure of effectiveness for roundabouts and intersections, with the level of service determined from the delay estimate. Delay in a roundabout can be defined as the time spent on traversing the roundabout in excess of traffic- free flow rate at the roundabout and it is the primary service delivery for the roundabout (Rodegerdts, 2010). The Highway Capacity Manual only includes control delay, which is the delay attributable to the control device. Control delay is the time which a driver spends queuing and then waiting for an acceptable gap in the circulating flow rate while at the front of the queue. Control delay can also be defined as the time a driver takes to decelerate into a queue, stay in the queue, while at the front of the queue wait for an acceptable gap and accelerate out of the queue (Rodegerdts, 2010, HCM, 2000). Queuing occurs when the entry vehicles are waiting for an appropriate and safe gap in the circulating traffic (Sofia et al., 2012). Control delay comprises of both the geometric delay and the stop line delay (Yap et al., 2013, Akçelik, 2005). The mode of operation of a roundabout does not necessarily make the entry vehicle to have total stop before entering the roundabout, but rather yield to the circulating vehicles, look for a safe gap within the circulating vehicle, accept the gap and enter the roundabout. The total stop of a vehicle at entry, unlike at a signalized intersection, is not always necessary which enables it to have a better service delivery and higher entry capacity than a signalized intersection (Kakooza et al., 2005, Sisiopiku and Oh, 2001). Delays still occur at the roundabout, though they might be reduced compared to other forms of at-grade intersections. Collins (2008) states that if the vehicular delay is to be an
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objective of any intersection, then the roundabout should be given a consideration because it will significantly reduce the total vehicle delay. HCM control delay is given as:
𝑑 = 3600 𝑐 + 900𝑇 ([ 𝑣 𝑐− 1] + √( 𝑣 𝑐− 1) 2 + [ 3600 𝑐 ] 𝑣 𝑐 450𝑇 ) + 5 [2.26]
Most stochastic delay model equations are derived from Tanner (1962) average delay equation:
𝑑𝑡 = 1 2 ⁄ ∗ 𝐸(𝑦2) 𝑌 +𝑞𝑒 𝑌𝑒−𝛽2𝑞𝑐 ∗ (𝑒𝛽2𝑞𝑐−𝛽2𝑞𝑐−1) 𝑞𝑐 1−𝑞𝑒 𝑦(1− 𝑒−𝛽2𝑞𝑐 ) [2.27] With, 𝑌 = 𝑒𝑞𝑐 (𝜏− 𝛽1) 𝑞𝑐(1−𝛽1𝑞𝑐) [2.28] 𝐸(𝑦2) = 2𝑌 𝑞𝑐{𝑒 𝑞𝑐(𝜏−𝛽1)− 𝜏 ∗ 𝑞 𝑐(1 − 𝛽1𝑞𝑐) 1 + 𝛽1𝑞𝑐− 𝛽12𝑞𝑐2− 1 2∗𝛽1 2𝑞 𝑐 2 1−𝛽1𝑞𝑐} [2.29]
Where: dt = average delay on approach (s/veh) qae = arrival rate on entering approach (veh/s) qc = arrival rate on circulating flow rate (veh/s)
τ – critical gap (s); β1 – headway (s); β2 – follow-up time (s)
Assuming random arrivals and no queues, Tanner shows that the average delay for isolated minor road vehicles is:
𝑑𝑚 = 𝑒𝜆(𝛼−∆) 𝜑∗𝑣𝑐 − 𝛼 − 1 𝜆+ 𝜆Δ2−2Δ+2Δ𝜑 2(𝜆∆+ 𝜑) [2.30]
Now, if 𝜑 =1 and the minimum gap in the circulating traffic is set to zero, then:
𝑑𝑚= 𝑒𝜆(𝛼−∆) 𝜑∗𝑣𝑐 − 𝛼 − 1 𝜆− 𝛽22𝑣𝑐 2 [2.31] Where: 𝜑 = 0.75(1 − Δ ∗ 𝑣𝑐) and = 𝜑∗𝑣𝑐 1−Δ∗𝑣𝑐 ; ∆ = 2s
Troutbeck (1989) recommends that when estimating average delay (dt), a steady state model equation 2.32 be used:
𝑑𝑡 = 𝑑𝑚∗ {1 + 𝑒𝑥
28 Where;
x denotes the degree of saturation
e is a form factor which can be set to 1 or 0, if no other value is available
Akcelik and Chung (1994b) follows up Troutbeck’s model equation with an allowance for variation over time, as seen in the equation shown below:
𝑑 = 𝑑𝑚+ 900𝑇 ([ 𝑣 𝑐− 1] + √( 𝑣 𝑐− 1) 2 + [8𝑘] 𝑣 𝑐 𝑐∗𝑇 ) [2.33] Where; 𝑘 = 𝑐∗𝑑𝑚 3600; 𝑑𝑚 = 𝑒𝜆(𝛼−∆) 𝜑∗𝑣𝑐 − 𝛼 − 1 𝜆+ 𝜆Δ2−2Δ+2Δ𝜑 2(𝜆∆+ 𝜑) 𝜑 = 0.75(1 − Δ ∗ 𝑣𝑐) And 𝜆 = 𝜑∗𝑣𝑐 1−Δ∗𝑣𝑐 ; ∆ = 2s 𝑐 = 3600 𝜑∗𝑣𝑐 ∗ 𝑒−𝜆(𝛼− Δ) 1− 𝑒−𝜆𝛽 ; 𝛽 = 2.819 − 3.94 ∗ 10−4 ∗ 𝑣 𝑐; and 𝛼 = {1.641 − 3.137 ∗ 10−4}𝛽
Figure 2.4 shows the two curves obtained by the two queue length model functions of which the first term is deterministic (Equation 2.34) and the second term is in a steady state condition (Equation 2.35):
𝐿 = (𝜌 − 1)𝑞𝑒𝑡 + 𝐿0 [2.34]
𝐿 = 𝜌 + 𝐶𝜌2(𝜌 − 1) [2.35]
𝐿 = (1−𝜌)𝜌 For C = 1 Where:
Lo denotes initial queue length; and t denotes any time interval;
ρ denotes traffic intensity (x); and qa denotes demand flow rate; qm denotes capacity; C denotes constant to describe arrival and service patterns.
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Stationary theory Transformed curve L
x y
Deterministic theory (for a given t-value)
ρ
s 1 ρnρ
i Degree of saturation ρSource: Kimber and Hollis (1979)
Figure 2.4: Coordinate transformation for average delay estimation
Generally, the basic equation for traffic delay at a roundabout is:
d = d1 + d2 +d3 [2.36]
Where: d1 = The yield line delay or the follow-up time (s)
d2 = Queueing delay (s) and d3 = geometric delay (s)
According to Guo and Wang, delay in a roundabout is of two main categories, namely the control and the geometric delay (Guo and Wang, 2011). The geometric delay is the reduction in speed due to the effect of the roundabout geometry in the course of traversing the roundabout (HCM, 2000). Geometric delay can further be defined as any delay experienced when a vehicle is traversing the roundabout in the absence of any other vehicle at the roundabout if the driver could identify that he is traversing the roundabout in isolation (Akчelik, 2009, Sofia et al., 2012, Kimber et al., 1986). The value is usually small for a small roundabout, but for large diameter roundabouts it could be significant. The value is usually high for a stopping vehicle because of the time it takes to accelerate to the design speed of the roundabout (Rodegerdts, 2010). This delay is always present
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at the roundabout whether there is the presence of a vehicle or not. Geometry is one of the major factors that have an influence on delay (Al-Omari et al., 2004). The geometric delay excludes the queueing time at the roundabout entry, it could be more than the delay at a congestion with the exception of when the traffic approaches capacity (Kimber et al., 1986). All other elements being equal, delay (and queue) would increase under rainy conditions due to more conservative car following and gap acceptance behaviour. While this seems to be apparent in Figures 2.5 and 2.6. The delay (queue)- volume capacity curves rise rapidly as the volume capacity increases at some points as pointed out in the relationship of delay and volume capacity ratio in HCM (2010). The interest in this study is not to build a different relationship for delay and volume capacity ratio but to determine the effect of rainfall and make some modifications if need be.
Figure 2.5: Delay vs. volume capacity ratio Figure 2.6: Queue length vs. volume capacity ratio