# Step-by-Step Signalized Intersection Delay and Level of Service Analysis Let's solve the problem step by step. **Note:** Since the original question doesn't provide the full table with flows and saturation flow rates, I'll explain the process and use variables where data is missing. You can plug in your actual numbers for a complete answer. ## Given Data - **Number of Signal Phases:** 3 - **Cycle Length, \( C \):** 60 s - **Total Lost Time per Cycle, \( L \):** 12 s - **Critical v/c Ratio:** .85 - **Saturation Flow Rates (\( s \)) and Flows (\( v \)):** *Assumed as variables* - **Movements for Each Stage:** *Assumed given in your table* We will calculate for: - (a) Westbound left lane group - (b) Westbound approach (sum of all westbound movements) - (c) Whole intersection (sum of all critical movements) --- ## Step 1: Calculate Effective Green Time (\( g \)) \[ g = C - L = 60\ \text{s} - 12\ \text{s} = 48\ \text{s} \] This 48 seconds is shared among all phases. --- ## Step 2: Calculate Flow Ratios (\( y \)) For each critical lane group: \[ y_i = \frac{v_i}{s_i} \] Where: - \( v_i \): Flow (vehicles per hour) for lane group \( i \) - \( s_i \): Saturation flow rate (vehicles per hour) for lane group \( i \) **Sum the critical flow ratios for all phases:** \[ Y = \sum y_i \quad \text{(sum of each phase's critical movement(s))} \] --- ## Step 3: Allocate Effective Green (\( g_i \)) to Each Phase \[ g_i = \frac{y_i}{Y} \times g \] --- ## Step 4: Calculate Degree of Saturation (\( x \)) \[ x_i = \frac{v_i}{s_i \cdot (g_i/C)} \] If you use the target critical \( v/c \) ratio of .85, compare this calculated value to .85. --- ## Step 5: Calculate Average Delay per Vehicle (\( d \)) — HCM Uniform Delay Formula For each lane group or approach, use the HCM formula: \[ d = \frac{.5 C (1 - g/C)^2}{1 - \min(x,1) \cdot (g/C)} \] Where: - \( C \): Cycle time (s) - \( g \): Effective green for the movement (s) - \( x \): Degree of saturation *For total delay, may sum uniform and incremental delay, but if arrival is uniform and \( x < 1 \), the uniform delay is the main component.* --- ## Step 6: Level of Service (LOS) Classification | Delay (seconds/veh) | LOS | |---------------------|-----| | - 10 | A | | 10 - 20 | B | | 20 - 35 | C | | 35 - 55 | D | | 55 - 80 | E | | >80 | F | --- ## Example Calculation ### (a) **Westbound Left Lane Group** Assume: - \( v_{WL} = 200 \) veh/hr - \( s_{WL} = 180 \) veh/hr #### 1. Flow Ratio: \[ y_{WL} = \frac{200}{180} = .111 \] #### 2. Assume Sum of all critical flow ratios (\( Y \)): Suppose: - \( Y = .90 \) #### 3. Effective Green for Lane Group: \[ g_{WL} = \frac{.111}{.90} \times 48 = 5.92\ \text{s} \] #### 4. Degree of Saturation: \[ x_{WL} = \frac{200}{180 \times (5.92/60)} = \frac{200}{177.6} = 1.126 \] But since \( x > 1 \), use \( x = 1 \) for calculation. #### 5. Uniform Delay: \[ d_{WL} = \frac{.5 \times 60 \times (1 - 5.92/60)^2}{1 - 1 \times (5.92/60)} \] Calculate: - \( g/C = 5.92/60 = .0987 \) - \( 1 - g/C = .9013 \) - Numerator: \( .5 \times 60 \times (.9013)^2 = 30 \times .8123 = 24.37 \) - Denominator: \( 1 - 1 \times .0987 = .9013 \) \[ d_{WL} = \frac{24.37}{.9013} = 27.04\ \text{s/veh} \] #### 6. LOS: - \( d_{WL} = 27.04\, \text{s} \) → **LOS C** --- ### (b) **Westbound Approach** Sum all westbound flows (\( v_{WB} \)), sum corresponding critical flow ratios, repeat the process above. --- ### (c) **Whole Intersection** Sum all critical movement flows and saturation flows, get total \( Y \), and compute average delay over all vehicles. --- ## **Summary Table** | Movement | Delay (s/veh) | LOS | |----------------------------------|--------------|-------| | (a) Westbound Left Lane Group | 27. | C | | (b) Westbound Approach | (calculate) | (LOS) | | (c) Whole Intersection | (calculate) | (LOS) | --- ## **Final Answers** - **(a) Westbound left lane group delay:** 27. s/veh (**LOS C**) - **(b) Westbound approach delay:** *(calculate as above, using sum of flows)* - **(c) Whole intersection delay:** *(calculate as above, using sum of critical flows)* --- > *Plug in your actual flow and saturation flow values for the full answer. The steps above show you how to do the calculation for each component.* --- ## **References** - Highway Capacity Manual (HCM), 6th Edition, Chapter 19: Signalized Intersections --- **If you provide actual flow values and saturation flows for each movement, I can give you the precise numerical answers for (b) and (c).**