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245 A P P E N D I X C Freeway Off-Ramp â Queue Spillback Analysis 1. Introduction The HCM (Chapter 14) provides three LOS checks for diverge segments, and failure (LOS F) may occur in any of the following two cases: â¢ the total demand flow rate on the approaching upstream freeway segment exceeds the capacity of the upstream freeway segment; â¢ the off-ramp demand exceeds the capacity of the off-ramp. The HCM methodology also provides a LOS evaluation based on the density of the ramp influence area (Exhibit 14-3), but it only yields a LOS range of A through E; failure due to excessive density is not considered in the methodology. The first case of LOS F is addressed by the Oversaturated Segment Evaluation procedure (HCM Chapter 10) and is not the focus of this methodology. The Queue Spillback Analysis, described in this document targets the second case of LOS F, when the off-ramp demand exceeds the capacity of the off-ramp. The methodology of this appendix also addresses cases of spillback due to insufficient capacity at the ramp terminal downstream of an off-ramp. The methodology described in Appendix A - Off-ramp Spillback Check presents the necessary steps to determine whether spillback from an off-ramp is expected to occur, based on a standard 15-min period analysis. This appendix provides the methodology for evaluating operations when spillback occurs. The approach is based on the Freeway Facilities Oversaturated Segment Evaluation (HCM Chapter 25), where performance measures are computed at the 15-s time step level. Section 2 introduces the basic link-node structure that is applied to model off-ramp segments in the methodology. Section 3 presents the concept of spillback regimes as a function of the off-ramp queue. Section 4 presents a glossary with the definition of all parameters used in the procedure. Section 5 presents the methodology to evaluate the impacts of an off-ramp queue spillback, and discusses each step and the respective calculations. 2. Evaluation of operations along off-ramp segments To evaluate the interaction between the freeway mainline and the downstream off-ramp terminal, the link-node approach used by the HCM Chapter 25 to evaluate oversaturated freeway facilities is expanded, with additional links and nodes to represent the off-ramp segment. As shown in Figure C-1, the mainline node for the off-ramp (Node 3) is connected to the off-ramp segment, which has a three-node structure: â¢ Ramp node 3.1: interface between the diverge segment (exit lanes) and the upstream end of the ramp proper. The volume that flows through this node is equivalent to the amount of vehicles that are able to leave the freeway; â¢ Ramp node 3.2: interface between the ramp proper and the arterial intersection approach. The volume that flows through this node is equivalent to the amount of vehicles that are able to leave the ramp proper and enter the intersection;
246 â¢ Ramp node 3.3: the last node in the off-ramp represents the discharge capacity of the arterial intersection approach. The volume that flows through this node is equivalent to the amount of vehicles that are able to enter the intersection; Figure C-1 â Expanded link-node structure to evaluate the off-ramp segment The geometry of an off-ramp is seldom a homogenous road segment, and additional lanes are frequently added closer to the arterial intersection approach. Figure C-2 illustrates a sample off-ramp, considering its entire length from the deceleration lane to the stop bar at the downstream signalized intersection. The ramp proper is the uniform ramp segment with a downstream boundary defined by the point where additional lanes are provided. When modeling the off-ramp geometry, the method considers the channelization at the approach as imbalances in the turning movements may cause queues on a subset of lanes. Figure C-2 shows a typical queue formation resulting from a left-turn movement that operates with insufficient capacity. In this scenario, the approaching left-turn vehicles are positioned in the leftmost lane and spillback may occur even if not all lanes of the approach are oversaturated. Figure C-2 â Sample geometry of an off-ramp considering the arterial intersection with heavy demanded left-turn
247 The type of ramp terminal is an important input into the analysis. Signalized intersections operate in cyclical patterns, and therefore those have fluctuating queue lengths. For certain demand scenarios, this can result in queues backing up into the freeway and then discharging multiples times within a 15-min time period. Stop-controlled intersections and downstream merge segments (in the case of a freeway-to-freeway connection), on the other hand, have a more uniform discharging rate. For cases other than signalized intersections, off-ramp queues are assumed to develop or discharge linearly based on the relationship between demand and capacity. 3. Evaluation of operations on the freeway mainline: spillback regimes The impact of queue spillback on the freeway mainline varies as a function of the queue length and the lanes blocked. Four spillback regimes are defined (Elefteriadou et. al, 2016) Regime 1 The queue ends within the deceleration lane and does not spill back into the mainline freeway (Figure C- 3 (a)). During undersaturated conditions, the deceleration lane serves as a transition zone between speeds on the mainline (typically 55 â 75 mi/h) and advisory speeds posted along the off-ramp (typically 20 â 50 mi/h). When queues begin to form on the deceleration lane, the available deceleration distance is reduced and speeds along the rightmost lane are affected. Regime 2 The queue of vehicles extends upstream beyond the deceleration lane, but sufficient lateral clearance on the right-hand shoulder allows for additional queue storage. In this case there is no transition zone within the deceleration lane and drivers decelerate and join the back of the queue more abruptly, resulting in turbulence and reduced speeds in the rightmost lane (Figure C-3 (b)). If no lateral clearance exists immediately upstream of the deceleration lane, Regime 2 conditions are not possible. In some cases, this regime does not occur even if storage is available; this depends on local driver behavior and is site-specific. Regime 3 The queue extends to the rightmost lane of the freeway mainline (Figure C-3 (c)). This may occur either when there is no shoulder available for additional queue storage, or when drivers choose to queue in the rightmost lane once the deceleration lane is entirely occupied. Non-exiting vehicles on the rightmost lane are delayed or change lanes, which causes increased turbulence and reduced speeds in both rightmost lanes. Regime 4 The queue blocks the rightmost lane, and drivers occasionally or often use the adjacent freeway mainline lane next to the rightmost freeway mainline lane to force their way into the queue, blocking thus an additional lane (Figure C-3(d)). During this regime, speed and capacity are significantly reduced. The effects of spillback vary from site to site and from time period to time period due to driver behavior and site geometry. Data collection at locations around the US has shown that at some sites drivers block the adjacent lane, while at other sites they do not, regardless of the queue spillback length at the site.
248 Figure C-3 âOff-ramp spillback regimes 4. Glossary of variable definitions This glossary defines internal variables used in the methodology for off-ramp queue spillback evaluation. The structure of the variables is similar to the one used in HCM Chapter 25 â Freeway Facilities Supplemental. Facility variables â¢ QIA(i, p): Length of the queue influence area (ft) for segment i during time period p, measured from the back of the queue. Segment variables â¢ KBBL(i,j): background density (pc/mi/ln) at the blocked lanes in segment i, when queue spillback occurs at a downstream segment j â¢ KBUB(i,j): background density (pc/mi/ln) at the unblocked lanes in segment i, when queue spillback occurs at a downstream segment j â¢ LCR(i,t,p): rate of lane change maneuvers in the queue influence area upstream of a queue from an off-ramp, for segment i during time period p and time step t. â¢ LD(i,p): available deceleration lane length (ft) for segment i during time period p. This variable is used to calculate performance measures for ramp segments (Chapter 14 - LD.) â¢ MQ1(i,t,p): mainline queue length of off-ramp unserved vehicles in the rightmost mainline lane, for segment i during time period p in time period t. â¢ MQ2(i,t,p): mainline queue length of off-ramp unserved vehicles in the rightmost mainline lane, for segment i during time period p in time period t. If Regime 4 is not expected to occur, this parameter value is set to zero. â¢ NQ(i): number of blocked lanes if the off-ramp queue backs up into the freeway mainline. This parameter is a function of the prevailing spillback regime at segment i as provided by the analyst. The value for this parameter is an input and can be either 1 (Regime 3 - one blocked lane) or 2 (Regime 4 â two blocked lanes); â¢ OFRFUP(i,t,p): flow that can exit at the closest off-ramp downstream of i during time step t in time period p. â¢ OFRLQ(i,t,p): queue length of off-ramp unserved vehicles for diverge segment i during time period p in time period t.
249 â¢ OFRUV(i,t,p): number of off-ramp unserved vehicles for segment i during time period p in time period t. â¢ SBKQ (i,t,p): spillback queue density for segment i during time period p in time period t. â¢ SBLC(i,t,p): number of lane change maneuvers within the Queue Influence Area at node i, during time step t in time period p. â¢ SBLQ(i,t,p): queue length within segment i during time period p in time period t, caused by a downstream off-ramp bottleneck. â¢ SBQS(i,p): total available off-ramp queue storage (ft) for a diverge segment i during time period p, if the subject segment has an off-ramp bottleneck. It is calculated as a function of the available storage lengths in the deceleration lane, shoulder and prevailing spillback regime. â¢ SCEQ(i,N,NQ): equivalent capacity of the unblocked portion of a segment i with N total lanes and NQ blocked lanes. â¢ SL(i,p): available shoulder length (ft) for segment i during time period p. If the value of SL is greater than zero, any off-ramp queues that exceed the deceleration lane will occupy the shoulder before blocking mainline lanes. â¢ TIA(i,p): total influence area (ft) of a queue from an off-ramp bottleneck on segment i, during time period p in time period t. It is calculated as the sum of parameters QIA(i,t,p) and MQ(i,t,p). Node variables â¢ CAFBL(i,t,p): capacity adjustment when one or more lanes of segment i are entirely blocked during time period p in time period t. This is used to calculate friction effects that cause through vehicles to slow down due to the presence of a queue in the rightmost lanes. â¢ CAFUP(i,t,p): capacity adjustment factor of node i during time step t in time period. This capacity adjustment factor affects approaching vehicles within the queue influence area (QIA) upstream of an off-ramp queue. This factor accounts for the turbulence caused by intense lane changing within the queue influence area as vehicles adjust their position when there is a downstream off-ramp queue. â¢ MFBL(i,t,p): mainline flow rate that can cross the blocked portion of node i during time step t in time period p. â¢ MFUB(i,t,p): mainline flow rate that can cross the unblocked portion of node i during time step t in time period p. â¢ MIBL(i,t,p): maximum flow desiring to enter the blocked portion of node i during time step t in time period p. â¢ MIUB(i,t,p): maximum flow desiring to enter the unblocked portion of node i during time step t in time period p. â¢ MO2BL(i,t,p): maximum number of passenger cars that can enter the blocked portion of segment i, during time step t and time period p, due to the presence of a queue in the downstream ramp segment. â¢ MO2UB(i,t,p): maximum number of passenger cars that can enter the unblocked portion of segment i, during time step t and time period p, due to the presence of a queue in the downstream ramp segment. â¢ NEXTOFR(i): index of the nearest downstream diverge segment relative to subject node i. â¢ OFRDIST(i): distance (ft) from node i to the start of the deceleration lane at the nearest downstream off-ramp. â¢ OFRPCT(i,j): percent of the off-ramp demand at segment j over the mainline entering volume at segment i. Ramp variables â¢ RC(i,p): capacity of the ramp proper (pc/h) during time period p in time period t. Capacity values for the ramp proper are provided in HCM Exhibit 14-12.
250 â¢ RF(i,t,p,k): flow (pc/ts) that can enter the ramp proper at segment i during time period p in time period t and level k. â¢ RI(i,t,p,k): maximum flow (pc/ts) desiring to enter the off-ramp on segment i during time period p in time period t and level k, including queues accumulated from previous time periods. â¢ RKB(i,t,p,k): ramp proper queue density (pc/mi/ln) for segment i during time period p in time period t and level k. â¢ RL(i): length of ramp proper (ft) for segment i. â¢ RN(i): number of ramp lanes for segment i. â¢ RNV(i,t,p,k): maximum number of passenger cars within the ramp of segment i at the end of time step t during time period p and level k. The number of vehicles is initially based on the calculations of Chapters 12, 13, and 14, but, as queues grow and dissipate, inputâoutput analysis updates these values during each time step. â¢ RSTG(i,t,p,k): maximum number of passenger cars that can enter the ramp level k of segment i, during time step t and time period p, due to the presence of a queue in the downstream ramp segment. â¢ RUV(i,t,p,k): number of unserved vehicles at the entrance of the ramp proper of segment i at the end of time step t during time period p and level k. Any values of RUV greater than zero indicate the occurrence of queue spillback from an off-ramp. Intersection (ramp terminal) variables â¢ ID (i,t,p,k): discharge capacity (pc/ts) for intersection movement k in segment i during time period p in time period t. â¢ IF(i,t,p): flow (pc/ts) that can enter the intersection on segment i, level k, during time period p in time period t. â¢ II(i,t,p,k): maximum flow (veh/ts) desiring to enter the intersection on segment i, level k, during time period p in time period t, including queues accumulated from previous time periods. â¢ IL(i,k): storage length of movements at intersection of segment i, for level k (ft) â¢ INV(i,t,p,k): number of vehicles at the intersection of segment i, for level k at the end of time step t during time period p â¢ IO(i,t,p): flow (pc/ts) that can be discharged from the intersection on segment i, level k, during time period p in time period t. â¢ ISTG(i,k): total available storage length at intersection of segment i, for level k (ft) â¢ IUV (i,t,p,k): number of unserved vehicles at the entrance of the intersection of segment i, for level k, at the end of time step t during time period p 5. Evaluation of operations on the freeway mainline: step-by-step methodology description The methodology for evaluating off-ramp queue spillback is integrated to the core methodology for Freeway Facilities Oversaturated Segment Evaluation (HCM Chapter 25). Figure C-4 through Figure C-7 show the core methodology, highlighting additions and changes to address off-ramp queue spillback.
251 Source: adapted from HCM 6th Edition Exhibit 25-3 Figure C-4 â Freeway facilities oversaturated segment evaluation procedure, adapted for off-ramp queue spillback evaluation
252 Source: adapted from HCM 6th Edition Exhibit 25-3 Figure C-5 â Freeway facilities oversaturated segment evaluation procedure, adapted for off-ramp queue spillback evaluation - continued
253 Source: adapted from HCM 6th Edition Exhibit 25-3 Figure C-6 â Freeway facilities oversaturated segment evaluation procedure, adapted for off-ramp queue spillback evaluation - continued
254 Source: adapted from HCM 6th Edition Exhibit 25-3 Figure C-7 â Freeway facilities oversaturated segment evaluation procedure, adapted for off-ramp queue spillback evaluation - continued
255 Step 1 - Calculate background density for unblocked lanes on each segment in the case of queue spillback The first step in the Oversaturated Segment Evaluation procedure computes a background density (KB), for each segment at the start of each time period, defined as the expected density when there is no queueing on the segment. It is used as a reference to estimate how many vehicles occupy a given segment at undersaturated conditions, creating an initial reference point for oversaturated analyses. When Regime 3 or Regime 4 occur, there is blockage of one or more freeway lanes in the affected segments, and the through vehicles aim to move to the unblocked lanes. The capacity of the unblocked lanes must be calculated at the initialization step, to be used as a reference value. For a segment i with N lanes, a subset NQ of lanes will be blocked when spillback occurs (NQ = 1 for Regime 3 and NQ = 2 for Regime 4). Therefore, the capacity of the unblocked lanes will be equivalent to a similar segment with (N - NQ) lanes, adjusted for the impact of the blockage using a capacity adjustment factor CAFBL. The values of CAFBL are equal to the Incident Capacity Adjustment Factors of Chapter 11, Freeway Reliability Analysis (Exhibit 11-23), as there are currently no data available to accurately assess the impacts of blockage due to spillback. These values may be conservative, as during incidents capacities may be further reduced due to the presence of police vehicles. Table C-1 presents the recommended values for CAFBL. Table C-1. Capacity adjustment factors for lane blockage (CAFBL) as a function of the number of directional lanes and the number of blocked lanes Directional Lanes 1 Blocked Lane 2 Blocked Lanes 2 0.70 N/A 3 0.74 0.51 4 0.77 0.50 5 0.81 0.67 6 0.85 0.75 7 0.88 0.8 8 0.89 0.84 The equivalent capacity SCEQ of segment i, with N lanes and NQ blocked lanes, is estimated as: ðð¶ð¸ð ð,ð,ðð ðð¶ ð,ð â ðð ð¶ð´ð¹ (Equation C- 1) Figure C-8 presents an example of a basic 4-lane directional segment operating in Regime 4 (2 blocked lanes). The capacity of the unblocked lanes will be equivalent to the capacity of a 2-lane basic segment with a capacity adjustment factor CAFBL = 0.50 (4 directional lanes with 2 blocked lanes). Figure C-8 â Equivalent segment capacity for unblocked lanes when lane blockage occurs For the segment of Figure C-8, capacity at ideal conditions is:
256 ð = 2,400 ðð/â/ðð (Capacity per lane) or ðð¶ = 9,600 ðð/â (Segment capacity) When Regime 4 occurs (2 blocked lanes), the equivalent capacity SCEQ is obtained as the equivalent capacity of a 2-lane segment multiplied by a corresponding CAFBL of 0.5 (Table C-1): ðð¶ð¸ð = 2 ð¥ 2,400 ð¥ 0.5 = 2,400 ðð/â Next, the unblocked background density KBUB is calculated. This parameter estimates the background density of the uncongested portion of a given segment operating under a two-pipe regime due to a queue spillback from a downstream off-ramp. To estimate this value, the method first determines the ratio of the Expected Demand (ED) that will move to the uncongested side of the segment. When queue spillback occurs in a diverge segment j, the parameter OFRPCT(i,j) is defined as the percent of the off-ramp demand over the mainline entering volume vf: ðð¹ð ðð¶ð ð, ð = ð£ ðð£ (ð) (Equation C-2) For any segment i, upstream of segment j and affected by the off-ramp spillback from segment j, the ratio of vehicles traveling towards the off-ramp at segment i is given by OFRPCT(j), while the ratio of vehicles traveling through in the unblocked lanes is given by (1- OFRPCT(j)). Therefore, the unblocked background density KBUB at any segment i upstream of an off-ramp spillback in a segment j is given by: ð¾ðµððµ(ð, ð) = ð¾ðµ[ð¸ð·(ð) Ã (1 â ðð¹ð ðð¶ð(ð)), ðð¶ð¸ð(ð)] (Equation C-3) Where: KBUB(i,j)I = background density at the unblocked lanes in segment i, when queue spillback occurs at the downstream segment ED(i) = expected demand at segment i , as defined in HCM Chapter 25 OFRPCT(i) = rate of off-ramp flow and mainline flow at segment i KB (v,c)I = density at a segment with demand flow rate v and capacity c, as provided by HCM Chapters 12 (basic), 13 (weaving) and 14 (merge and diverge) Step 2 - Initialize the freeway facility These calculations are performed at the start of the analysis, to prepare the flow calculations for the first time step and specify return points, such as background density (KB), for later time steps. This subsection presents the additional parameters required for queue spillback analysis. Number of mainline blocked lanes The number of mainline blocked lanes is stored in the parameter NQ(i) and is determined by the prevalent queue spillback regime as provided by the analyst. If the back of an off-ramp queue is calculated to reach the freeway mainline, two possible spillback regimes may occur: â¢ Regime 3: blockage of one lane in the freeway mainline â Set NQ(i) = 1 â¢ Regime 4: blockage of two lanes in the freeway mainline â Set NQ(i) = 2
257 The analyst should select one of these two regimes based on prevailing driver behavior at the site and in the vicinity of the site. Shoulder length The available shoulder length must be input by the analyst for queue spillback analysis, and is stored under the parameter SL(i) for oversaturated calculations. Deceleration lane length The deceleration lane length is provided by the analyst for the analyses of diverge segments, and is stored under the parameter LD(i) for oversaturated calculations. Spillback queue storage length The maximum storage length for off-ramp queues on segment i is computed as a function of the segment length L(i), the deceleration lane length LD(i) and the number of queued lanes NQ(i). Figure C-9 provides guidance on measuring each of the components required for Regimes 3 and 4. Figure C-9 â Maximum off-ramp queue storage length at diverge segments with occurrence of (a) Regime 3 queue spillback and (b) Regime 4 queue spillback, when no shoulder is available Figure C-10 illustrates queue length measurements for special cases of queue spillback, when a shoulder is present but its storage length is not sufficient to accommodate the unserved vehicles. Regime 3A (Figure C-10a) occurs when there is blockage of one mainline lane in addition to the shoulder. Regime 4A (Figure C-10b) occurs when there is blockage of two mainline lanes in addition to the shoulder. Figure C-10 â Maximum off-ramp queue storage length at diverge segments with occurrence of (a) Regime 3A queue spillback and (b) Regime 4A queue spillback, when shoulder is available
258 Step 2A - Model off-ramp geometry The three-level node structure for the off-ramp shown in Figure C-1 must be modeled to reflect the geometric characteristics of the site, as illustrated in Figure C-2. This is accomplished by setting a âbranchâ structure, where a node can connect to multiple links downstream. If a node is connected to more than one downstream link, the flow through the node will be constrained by the downstream link with the highest queue storage ratio. The ramp structure must be modeled from the downstream end towards the upstream end: â¢ For the most downstream location provide one node for each lane group or movement at the approach; â¢ For the next upstream change in alignment provide one node for each ramp proper lane connecting to a distinct lane group downstream. The data structure used in the methodology computations should be adjusted according to this âbranchâ structure. Most parameters in the Oversaturated Segment Evaluation methodology are computed as a 3- dimensional array (i, t, p), where i is the segment index in the freeway facility and t refers to a specific time step within a given time period p. In the case of two-lane ramps that need to be evaluated independently, a new dimension k will be added to the ramp parameter arrays to account for the specific lane under analysis. Lanes are numbered right from left; therefore, k=1 stands for the rightmost lane and k=2 for the leftmost lane of the ramp. Example 1 â In this example, there is only one lane connecting the freeway exit to the entry leg of the downstream roundabout. Therefore, only one node is required at each location (single branch structure, with k=1 in all nodes), as shown in Figure C-11. Figure C-11 â Node structure for Example 1 Example 2 â A single-lane ramp connects with a stop-controlled T-intersection ramp terminal (Figure C-12). The intersection node is comprised of two branches (k=2), while the ramp proper has only one lane (k=1). Each movement of the intersection (LT and RT) is represented by a node, and when there is a queue on either one of the movements, the one with the longest length will constrain the flow of vehicles from the ramp proper.
259 Figure C-12 â Node structure for Example 2 Example 3 â A two-lane ramp connects with a signalized intersection ramp terminal (Figure C-13). Both the intersection and the ramp proper nodes comprise of two branches each (k=2). At the downstream end, one node is defined for each lane group at the intersection (LT and RT). According to the ramp geometry, left-turning drivers will be positioned along ramp lane 2, while right-turning drivers will be located along ramp lane 1. Therefore, two nodes are also defined at the upstream location. If the queue storage ratio for any of the ramp lanes reaches 1, vehicle flow in the respective upstream node will be constrained, resulting in queue spillback on the freeway mainline. Figure C-13 â Node structure for Example 3 Step 2B - Determine spillback regime for each diverge segment Field observations have shown that locations that experience recurring queue spillback always have the same type of spillback regime when the queue extends beyond the deceleration lane (Regime 3 or 4). Regime 4 occurs often at ramp junctions with a lane drop. At these locations, the exiting traffic can access
260 the off-ramp with a single lane change. Therefore, drivers are more likely to wait until they are closer to the exit to change lanes, blocking the adjacent through lane. However, not all lane drop exits experience a Regime 4 queue spillback. Regime 4 occurs more frequently in locations with more aggressive driver behavior. Local information and driver behavior should be taken into consideration in determining the prevailing regime at a given site. For operational analyses of existing locations, it is recommended that the analyst provides the expected spillback regime based on observed field conditions. For planning level purposes where no field data is available, Table C-2 provides the expected queue spillback regime as a function of the number of exiting lanes and driver aggressiveness. Table C-2. Default spillback regimes as a function of ramp geometry and driver aggressiveness Ramp geometry Driver aggressiveness Low Medium High Diverge Regime 3 Regime 3 Regime 3 Lane drop Regime 3 Regime 4 Regime 4 Step 2C - Determine queue influence area (QIA) The Highway Capacity Manual (Chapter 14) provides the following definition for the ramp influence area for off-ramps operating under steady conditions: âFor right-hand off-ramps, the ramp influence area includes the deceleration lane(s) and Lanes 1 and 2 of the freeway for a distance of 1,500 ft upstream of the diverge point.â When there is queue spillback in one or more freeway lanes, drivers would react to the presence of the queue further upstream resulting in increasing lane changes and additional turbulence upstream of the ramp influence area (Figure C-14). In this step the methodology estimates the length of the Queue Influence Area (QIA), measured upstream from the back of queue. Figure C-14 â Queue Influence Area with increased turbulence Field data (video observations and loop detector data) were used to estimate the length of the Queue Influence Area. This measurement process is illustrated in Figure C-15. For a given off-ramp bottleneck, the distance between the exit and the upstream loop detector is known and fixed. The back of queue length Q(t) at time t was measured by video observations of congested diverge segments. Speed measurements from the loop detector were obtained with resolutions ranging between 20s and 60s (depending on the source) and analyzed to determine the onset of congestion. Oversaturated conditions were determined to occur when a speed drop greater than 20% occurred in at least one lane, and sustained for at least 15 min.
261 Figure C-15 â Measurement of queue influence area length based on queue lengths For the timestamp tbd when congestion begins in at least one lane, the back of queue length is also known from video observations. The distance between the detector and the back of queue Q(tbd) at the congestion initiation time is defined as the Queue Influence Area. The speed prior to congestion, sbd, was also measured (Figure C-16), and used in further calculations. Figure C-16 â Sample measurement of queue lengths and speeds at the time of breakdown
262 This process was performed for all data obtained. It was observed that different locations operate at significantly different speeds prior to congestion, therefore the Queue Influence Area measurements were normalized to estimate the reaction headway, defined as the travel time between detector location (where breakdown occurred) and the back of the queue: â = 3600 Ã ðð¼ð´5280 Ã ð (Equation C-4) Where: hR = reaction headway (s) QIAbd = measured Queue Influence Area at time of breakdown (ft) sbd = measured speed at the beginning of congestion at the upstream detector (mi/h) After all queue spillback observations were analyzed, the measured values of the reaction headway can be found in the histogram of Figure C-17, with a median value of 10.95 s. Figure C-17 â Frequency distribution of measured headways Using this median value of 10.95s, the length of the QIA is estimated as a function of the segment free- flow speed (FFS), as shown in Table C-3. The exact location of the QIA varies as a function of the queue length. The QIA values are shorter than the ramp influence distance of 1,500 ft. However, the two concepts are very different and are used differently in analyzing ramp operations: the ramp influence area is used to analyze undersaturated conditions, while the QIA is used to analyze oversaturated conditions. Since drivers can only detect a downstream queue visually, they have shorter times to react when compared to the presence of undersaturated off-ramps, where signing and navigation information is provided in advance and allows drivers to adjust their position earlier. Table C-3. Queue influence area as function of the segment FFS Segment FFS (mi/h) Queue Influence Area (ft) 50 810 55 900 60 980 65 1060 70 1140 75 1220
263 When Regimes 3 or 4 occur and lane blockage is present in the mainline, the estimated QIA is added to the queue length to determine the extent of spillback effects. If an upstream node is located within the combined length of the queue and QIA, capacity adjustment factors must be applied to account for the spillback effects. Step 2D - Determine ramp proper capacity and speed The first off-ramp parameter to be determined is its capacity (RC), defined as function of the ramp free- flow speed and is obtained from HCM Exhibit 14-12, replicated below in Figure 18. Source: HCM 6th Edition Exhibit 14-12 Figure C-18 â Capacity of ramp proper for off- ramps The RC is compared to the off-ramp demand, and if the demand-to-capacity ratio is greater than 1.0 then spillback is expected to occur. Determining the speed-flow relationship at the ramp proper is also critical for the analysis. Speed data along off-ramps are scarce, but a few field observations at off-ramp speed-flow curves (Figure C-19) have shown that speeds have little variation as a function of demand. Figure C-19 â Sample speed-flow curves for: (a) I-694 EB to Silver Lake Rd. and (b) I-94 EB to Brooklyn Blvd. Minneapolis/MN Determining the speed-flow relationship at the ramp proper is also critical for the analysis. Ramp speeds can be obtained through the following equation: ð = 1 â 0.109 Ã ð£1000 Ã ð (Equation C-5) where ð = ramp speed (mi/h);
264 ð£ = ramp demand flow rate (pc/h) ð = ramp free-flow speed (mi/h) The speed-flow relationship for ramps is linear and speed decreases with higher ramp flows, as presented in Figure C-20. The maximum allowed values of vR are bounded by ramp capacity, consistent with guidance provided by Chapter 14 â Merge and Diverge segments (Exhibit 14-12). Figure C-20 â Speed-flow curves for freeway ramps The ramp density at capacity (RKC) is not necessarily equal to 45 pc/mi/ln as assumed for freeway mainline lanes. This parameter is required to evaluate the queue density at the ramp proper when operating in oversaturated conditions. The ramp density at capacity can be found by dividing the capacity by speed. Table C-4 summarizes the values of RKC as a function of the Ramp FFS. Table C-4. Ramp density at capacity, as a function of Ramp FFS Ramp FFS (mi/h) Capacity (pc/h/ln) RKC (pc/mi/ln) 50 2200 44.0 45 2100 46.7 40 2100 52.5 35 2000 57.1 30 2000 66.7 25 1900 76.0 20 1900 95.0 15 1800 120.0 Step 2E - Determine intersection storage capacity The storage capacity at the intersection, ISTG, is obtained as the sum of the available storage of every lane group, multiplied by the number of lanes. If the off-ramp has multiple branches at the intersection (k > 1), then the available storage capacity must be computed for each branch k individually. This distinction is necessary to evaluate cases with unbalanced demands at the intersection, when the queues developed in
265 one oversaturated movement may extend upstream and block the throughput of all movements at the off- ramp. ISTG is estimated as: ð¼ðððº(ð,ð, ð) = ð ð¥ ð¿ ð¥ ð¿ (Equation C-6) Where: Nm = number of lanes serving movement m at the intersection Lm = storage length for movement m at the intersection (ft) N = number of movements at the approach Lh = average vehicle spacing in stationary queue (ft/veh) (HCM Equation 31-155) Step 2F - Determine initial number of vehicles at the off-ramp The computation of the number of vehicles in the facility at every time step is critical for deriving performance measures of oversaturated freeway facilities. Similar to the Freeway Facilities Oversaturated Segment Evaluation methodology, the estimation of the number of vehicles in the ramp during oversaturated conditions requires a reference value for undersaturated conditions to be computed during the initialization steps. First, the initial number of vehicles on the ramp during undersaturated conditions is determined as an initial reference point. Given the ramp speed-flow relationship from Figure C-20, the density at an off-ramp segment can be obtained by dividing the off-ramp flow rate (vR) by its free-flow speed (SR). Then, the total number of vehicles is obtained by multiplying the ramp density by the ramp length (RL) and number of lanes (RN), as follows ð ðð(ð, 0,0, ð) = ð£ ,ð ð¥ ð ð¿(ð, ð)ð¥ ð ð(ð, ð) (Equation C-7) Where: RNV(i,0,0,k)) = number of vehicles in the ramp proper at the initialization step IN(i,k) = off-ramp demand at the first time period in the analysis (pc/h) Qk = off-ramp free-flow speed (mi/h) The initial number of vehicles in the intersection approach are also determined as an initial reference point, as follows: ð¼ðð(ð, 0,ð, ð) = ð¼ð(ð, ð) ð¥ ð (Equation C-8) Where: INV(i,t,p,k) = number of vehicles at the intersection of segment i, for level k at the end of time step t during time period p IN(i,k) = number of lanes serving the subject approach k Qk = back-of-queue length for the subject approach k (veh) The back-of-queue length Qk is estimated using equations corresponding to the intersection type at the ramp terminal (Table C-5). Table C-5. Reference HCM equations for back-of-queue length estimation Intersection type Reference Equation
266 Signalized 31-149 TWSC 20-68 AWSC 21-33 Roundabout 22-20 At signalized intersections, due to their cyclic nature, queues form and discharge at different times for different movements. Therefore, a reference point within the cycle must be selected as a starting point in the methodology. The methodology assumes pretimed signalization, or converts actuated control to the equivalent pretimed pattern. Typical signalized intersections at ramp terminals have the off-ramp approach as the minor movement, with a start of green on the right side of the barrier (Figure C-21). It is recommended setting a reference point at the onset of green for phases 3 and 7, as the back-of-queue lengths at this time can be easily estimated using the methodology of Section 4, HCM Chapter 31. Figure C-21 â Selection of a cycle reference point to determine the initial number of vehicles within the approach Step 2G - Determine the capacity of the downstream terminal The methodology to evaluate the capacity of the terminal is specific to each intersection type and relies mostly on the respective HCM chapters (19 through 23). Signalized Intersections For a signalized intersection approach, the capacity for each movement at each time step is a function of the signal phase sequence and the capacities of the individual movements at the intersection. Figure C-22 illustrates a sample signalized intersection approach from an off-ramp, with two lane groups: left-turn (Phase 3) and right turn (Phase 8).
267 Figure C-22 â Sample signalized intersection approach from an off-ramp Input Parameters The required parameters to evaluate the capacity of a ramp terminal capacity are generally the same required for standard signalized intersection analyses, as listed in Exhibit 19-11. Arrival type: Chapter 19 of the HCM (Exhibit 19-14) provides guidelines for selecting the appropriate Arrival Types based on the characteristics of arterial operations, such as quality of progression and coordination. For an off-ramp approach to the intersection, vehicles arrivals can be considered random. Therefore, Arrival Type 3 (random arrivals) is recommended to analyze the off-ramp approach at a signalized ramp terminal. Phase duration and effective green time: The duration of each phase at the signal can be fixed (pre-timed control), or variable (semi-actuated or actuated control). For the former case, phase duration is known. For the latter, an average phase duration is estimated as described in Section 2 of HCM Chapter 31 â Signalized Intersections Supplemental. The effective green time g for each phase can then be computed according to HCM Equation 19-3: ð = ð· â ð â ð (Equation C-9) Where: g = effective green time (s) Dp = phase duration (s) l1 = start-up lost time = 2.0 (s) l2 = clearance lost time = Y + Rc â e (s) Y = yellow clearance interval Rc = red clearance interval e = extension of effective green = 2.0 (s) Converting approach capacity from time periods to time steps The standard signalized intersection analysis is performed in 15-min periods, while the queue spillback evaluation requires a 15-second approach compatible with the Freeway Facilities oversaturated
268 methodology. Therefore, an adjustment is necessary to calculate the capacities of each movement in 15- second intervals. The cycle length C can be divided into n time steps, with a duration of 15 s each (Figure 23). If an integer number of time steps is not obtained, the difference is included in the first time step of the next cycle. Then, green times for each time step from 1 to n are computed. This procedure must be repeated for every time step within the 15 minutes time period, resulting in a total of 900/15 = 60 time steps. Figure C-23 âConversion of green times to time steps The capacity ID for each approach and for each time step, is then obtained by multiplying its respective green time by its capacity, as shown: ð¼ð·(ð, ð¡,ð, ð) = ð ð ðºð(ð, ð¡, ð,ð) Ã ð (Equation C-10) Where: Nk = number of lanes serving movement k sk = saturation flow rate for movement k (veh/h/ln) GT(i,t,p,m)= green time for each movement m (s) The green time parameter GT(i,t,p,m) measures the available green time for a given intersection movement m, downstream of a freeway segment i, in time step t and time period p. It can range from 0 (when the movement has red through the entire time step length) to 15 (movement has green through the entire time step length). The heavy vehicle factor fHV needs to be applied to the equation for intersection discharge to make the units used in intersection capacity (veh/h) consistent with the flow rates used in uninterrupted flow methods (pc/h).
269 Merge segments (freeway-to-freeway connectors) When two freeway facilities are connected through a ramp junction, the merge segment at the downstream facility becomes the ramp terminal. In this case, the capacity of the ramp terminal is equal to the merge capacity at the downstream freeway. If the downstream segment operates at undersaturated conditions, there is no data available in the literature to estimate the capacity of the merge segment. In this case, the merge capacity is considered unrestricted (for a computational engine, a very high capacity value such as 9999 pc/h can be assumed) and the only constraint at the freeway-to-freeway interaction will be the capacity of the ramp roadway, provided by HCM Exhibit 14-12. If the downstream segment operates at oversaturated conditions, the merge capacity is constrained by the congested conditions in the mainline. The Freeway Facility Oversaturated Segment Evaluation procedure computes, for every time step, the parameter ONRO as the maximum number of vehicles that can merge into the freeway in a given time period. Therefore, in these conditions the merge capacity can be obtained by analyzing the downstream freeway facility and aggregating the yielded values of ONRO to an hourly flow rate. Step 2H - Determine reference index for next downstream off-ramp This step is essential for building the computational engine for this procedure, but it is not important for understanding the overall methodology. The Freeway Systems methodology uses the parameter OFRF(i,t,p) to store the off-ramp flow rate at diverge segment i. When a segment upstream of an off-ramp is evaluated for queue spillback, the off-ramp flow rate must be referenced in order to estimate the incoming flows for the blocked and non-blocked lanes. Therefore, a new variable NEXTOFR(i), is introduced to reference the index of the closest diverge segment downstream of segment i. This is illustrated in Figure C- 24, where the node (i+2) represents a diverge segment with an off-ramp flow vR. When the queue extends upstream to node i, the approaching flow vf is segregated into two groups: the exiting vehicles that will join the back of the queue, and the through vehicles that will use the non-blocked lanes. Figure C-24 â Illustration of mainline flow rate split into blocked and unblocked lanes For nodes i and i+1, the closest downstream off-ramp is located at node (i+2), therefore the following parameter is computed: ðð¸ðððð¹ð (ð) = ð 2 The use of the parameter NEXTOFR facilitates referencing diverge segments downstream of a given segment i and will be used for the spillback analysis procedure described over the next section.
270 Step 9A - Perform spillback analysis This is a new step in the Freeway Facilities Analysis method (Figure C-4). In this step, spillback effects in a diverge segment are determined after the off-ramp flow OFRF is determined (steps 7/8). Determine ramp input, RI The ramp input, RI, represents demand, and it is the number of vehicles that wish to travel through the ramp proper node during a given time step. It takes into account the off-ramp demand, OFRF (as defined in the Freeway Facilities Oversaturated methodology) and the number of off-ramp unserved vehicles from the previous time step, RUV. The OFRF parameter already takes into consideration any bottleneck segments upstream of the diverge that may meter the off-ramp demand (HCM Equations 25-23 through 25-25). The RI is calculated as: ð ð¼(ð, ð¡,ð) = ðð¹ð ð¹(ð, ð¡,ð) + ð ðð(ð, ð¡ â 1,ð) (Equation C-11) Where: OFRF(i,t,p) = flow that can exit at the off-ramp i during time step t in time period p RUV(i,t,p,k) = number of unserved vehicles at the off-ramp exit at segment i, during time step t in time period p Calculate flow to the off-ramp and number of unserved vehicles The ramp maximum flow RF represents capacity, i.e., the number of vehicles that are able to enter the ramp proper by crossing the boundary node between the diverge segment and the ramp proper. It is calculated as the minimum of three variables: RI, RC and RSTG. ð ð¹(ð, ð¡,ð) = min(ð ð¼(ð, ð¡, ð, ð),ð ð¶(ð, ð),ð ðððº(ð, ð¡,ð, ð)) (Equation C-12) The parameters RI and RC have been previously defined. The parameter RSTG represents the maximum number of vehicles that can enter the ramp due to a queue inside the ramp proper. The calculations follow the same approach taken by the Mainline Output 2 (MO2) parameter (Equation 25-11). It starts by calculating the maximum number of vehicles allowed on the ramp at a given ramp queue density RKQ: ð ð¾ð(ð, ð¡,ð, ð) = ð¾ð½â [(ð¾ð½ â ð ð¾ð¶) ð¥ ð ð¹(ð, ð¡ â 1,ð)] / ð ð¶(ð, ð¡,ð) (Equation C-13) The calculation of RKQ takes an approach similar to the calculation of the mainline queue density KQ (Equation 25-10), with the following remarks on the inputs: â¢ The jam density parameter KJ uses the same value adopted for the mainline calculations â¢ The ramp density at capacity RKC is determined based on the ramp FFS (Table C-4) â¢ The parameters SF (segment flow) and SC (segment capacity) from Equation 25-10 are replaced with RF (ramp flow, previously defined) and RC (ramp capacity, previously defined) The maximum ramp storage constraint RSTG is then calculated using an approach similar to the Mainline Output 2 (MO2) parameter from the Oversaturated segment evaluation procedure. This constraint limits the number of vehicles able to enter the off-ramp due to the presence of a queue within the ramp proper. RSTG is calculated as:
271 ð ðððº(ð, ð¡, ð, ð) = ð ð¹(ð, ð¡ â 1,ð, ð) + ð ð¾ð(ð, ð¡,ð, ð)ð¥[ð ð¿(ð) ð¥ ð ð(ð)]â ð ðð(ð, ð¡ â1,ð, ð) (Equation C-14) Next, the number of unserved vehicles at the ramp entrance RUV is calculated. For each time step, the number of unserved vehicles is computed as the value from the previous time step, plus the difference between demand (RI) and throughput (RF) at the ramp node. RUV is calculated as: ð ðð(ð, ð¡,ð, ð) = ð ðð(ð, ð¡ â 1,ð ,ð) + ð ð¼(ð, ð¡,ð, ð) â ð ð¹(ð, ð¡,ð, ð) (Equation C-15) Where: K = number of different branches at the intersection If there are multiple branches k at the ramp proper (two lane ramps), RI and RF are compared for each branch k to obtain RUV for each branch k. The total number of unserved vehicles at the ramp RUV(i,t,p) is then obtained as the sum of RUV for each lane: ð ðð(ð, ð¡,ð) = ð ðð(ð, ð¡,ð, ð) (Equation C-16) Calculate approach input, II The intersection approach input II is the number of vehicles that wish to travel through the intersection node during a given time step, i.e., its demand. It takes into account the off-ramp flow RF and the number of unserved vehicles on the approach from the previous time step IUV. II is calculated as: ð¼ð¼(ð, ð¡, ð, ð) = ð ð¹(ð, ð¡,ð, ð) + ð¼ðð(ð, ð¡,ð, ð) (Equation C-17) Calculate maximum ramp output The maximum allowable ramp output (RO) is calculated as a function of the available storage space within the intersection approach, minus the number of vehicles present at the previous time step and the number of vehicles discharged (IDC) in the present time period. RO is estimated as: ð ð(ð, ð¡,ð, ð) = ð¼ðððº(ð, ð) â ð¼ð(ð, ð¡ â 1,ð, ð) + ð¼ð·ð¶(ð, ð¡,ð, ð) (Equation C-18) Calculate intersection approach flow and number of unserved vehicles The intersection flow IF represents the number of vehicles that are able to cross the boundary node between the ramp proper and the intersection (i.e., its capacity). It is computed as the minimum value between the number of vehicles that wish to enter the intersection and the maximum number of vehicles allowed to enter the intersection due to the available queue storage in the intersection: ð¼ð¹(ð, ð¡,ð, ð) = min (ð¼ð¼(ð, ð¡,ð, ð),ð ð(ð, ð¡,ð, ð)) (Equation C-19) If the number of vehicles trying to enter the intersection exceeds the amount of vehicles allowed to enter the intersection, then the number of total unserved vehicles must be computed and considered in the intersection input II during the next time period:
272 ð¼ðð(ð, ð¡,ð, ð) = ð¼ðð(ð, ð¡ â 1, ð, ð) + ð¼ð¼(ð, ð¡,ð, ð) â ð ð(ð, ð¡,ð, ð) (Equation C- 20) Update number of vehicles at the ramp terminal intersection The number of vehicles at the intersection, INV, is updated every time step based on the NV from the previous time step, plus the number of vehicles that enter the intersection approach minus the number of vehicles that are discharged. The maximum allowable total number of vehicles is function of the available storage at the intersection, ISTG. NV is calculated as: ð¼ðð(ð, ð¡, ð) = ð¼ðð(ð, ð¡ â 1,ð) + ð¼ð¹(ð, ð¡,ð, ð) â ð¼ð·ð¶(ð, ð¡, ð, ð) (Equation C- 21) Calculate number of unserved vehicles at the off-ramp The number of unserved vehicles, OFRUV, at the entrance of the ramp proper is updated every time step as the difference between the number of vehicles that wish to enter the ramp proper (RI) and the flow through the ramp node (RF): ðð¹ð ðð(ð, ð¡,ð) = ð ð¼(ð, ð¡,ð) â ð ð¹(ð, ð¡,ð) (Equation C- 22) Calculate intersection approach output The intersection flow, IO, represents the actual number of vehicles discharging from the intersection approach. It is computed as the minimum value between the intersection discharge capacity and the sum of number of vehicles present in the intersection and the intersection input demand: ð¼ð(ð, ð¡,ð, ð) = min (ð¼ð·ð¶(ð, ð¡, ð, ð), ð¼ð(ð, ð¡ â 1,ð, ð) + ð¼ð¼(ð, ð¡, ð, ð)) (Equation C- 23) Update number of vehicles at the ramp roadway The number of vehicles at the ramp proper, RNV, at the end of each time step is calculated based on the number of vehicles from the previous time step plus the number of vehicles that entered the ramp minus the number of vehicles that left the ramp: ð ðð(ð, ð¡,ð, ð) = ð ðð(ð, ð¡ â 1, ð, ð) + ð ð¹(ð, ð¡,ð, ð) â ð¼ð¹(ð, ð¡,ð, ð) (Equation C- 24) Determine the back-of-queue length and spillback regime Field observations have shown that off-ramp queues blocking mainline lanes are typically not stationary. These queues usually consist of a platoon of closely-spaced vehicles moving at very low speeds (< 15mph). The spacing between vehicles is also longer than the average vehicle spacing in stationary queues, represented in the HCM by Lh (Equation 31-155). Therefore, the density of the spillback queue follows the queue density at the ramp (RKQ, as previously defined), which allows the estimation of the queue length OFRLQ. This parameter estimates the total queue length upstream of the off-ramp if all unserved vehicles formed a single queue: ðð¹ð ð¿ð(ð, ð¡, ð) = ðð¹ð ðð(ð, ð¡, ð)ð ð¾ð (ð, ð¡,ð) (Equation C- 25)
273 Next, the mainline queue length, SBLQ, is compared to the available spillback queue storage for the prevalent spillback regime for the given time step, as follows: If OFRLQ = 0 â Regime 0 If 0 < OFRLQ â¤ LD â Regime 1 If SBLQ > LD : If SL(i,p) > 0: If OFRLQ < (LD + SL) â Regime 2 Else: Regime 3 / 4 Finally, the queue length in the mainline lanes MQ1 (lane 1) and MQ2 (lane 2) are obtained as a function of the expected spillback regime. The total queue length OFRLQ minus the available storage lengths at the deceleration lane and shoulder computes the queue length that the associated blockage. If the site experiences Regime 3: ðð1(ð, ð¡,ð, ð) = ðð¹ð ð¿ð(ð, ð¡,ð, ð) â ð¿ð·(ð)â ðð¿(ð) ðð2(ð, ð¡,ð, ð) = 0 (Equation C- 26) If the site experiences Regime 4: ðð1(ð, ð¡,ð) = ðð2(ð, ð¡,ð) = [ðð¹ð ð¿ð(ð, ð¡,ð) â ð¿ð·(ð)â ðð¿(ð) ] / 2 (Equation C- 27) Check for impacts on upstream nodes The freeway nodes upstream of a congested off-ramp may be affected by spillback as queues grow. When this occurs, the methodology calculates the length of the queue in the upstream segment. The length of the queue within the subject segment will then be used to evaluate whether the capacity of any upstream node is affected by the queue. For upstream segments that may be affected by spillback, the queue length within the segment (measured from its downstream end) must be computed and stored in the parameter SBLQ. This check is performed for every node upstream of a congested off-ramp (Figure C-25).
274 Figure C-25 â Procedure for evaluating the impact of queue spillback on upstream nodes and determination of the queue length within upstream segments When queue spillback occurs in a downstream off-ramp, the length of the mainline queue measured from the start of the deceleration lane is known from the previous step. If a given segment has any queues blocking one or more lanes, three possible scenarios may occur at the node (Figure C-26): (a) Lane blockage: Queues extend through the entire segment and reach the upstream node, causing the subject node to operate in a two-pipe regime. The blocked lanes operate in a congested regime, with their capacity constrained by the off-ramp capacity. The unblocked lanes, on the other hand, operate at uncongested conditions with a small reduction in capacity due to the friction of through vehicles passing along congested lanes. For the through lanes, an adjustment factor CAFBL is applied. This condition occurs when the Spillback Queue length SBLQ(i) is equal or greater than the Segment Length L(i). (b) Increased turbulence: Queues extend partially through the segment and the upstream node is located within the Queue Influence Area (QIA). This region is characterized by intense turbulence as vehicles quickly perform lane changes to adjust their position reacting to the queue ahead, and all lanes in node i have their capacity reduced by an adjustment factor CAFUP. This condition occurs when the sum of the Spillback Queue length SBLQ(i) and the Queue Influence Area QIA(i) is equal or greater than the segment length L(i). (c) No effect: Queues extend partially through the segment but the upstream node is located within the Queue Influence Area (QIA). For this condition, no capacity adjustment factors are applied to the node i. This condition occurs when the sum of the Spillback Queue length SBLQ(i) and the Queue Influence Area QIA(i) is smaller than the segment length L(i).
275 Figure C-26 â Illustration of different impacts of an off-ramp queue at node i: (a) lane blockage, (b) increased turbulence and (c) no effect Calculate capacity adjustment factors Based on how upstream nodes are affected as described under Step 6B (Lane Blockage, Increased Turbulence or No Effect), the corresponding impacts on capacity are computed in this step. This section describes the calculations of capacity adjustments depending on how upstream nodes are affected. Lane blockage adjustment factor When one or more lanes are blocked, the subject node is analyzed as a two-pipe operation, with a congested flow in one or more lanes of the ramp side and uncongested flow in the remaining lanes. The capacity of these lanes is equal to the number of queued vehicles discharged at the downstream segment. The flow rate attempting to cross the node through the congested lanes is equal to the off-ramp flow rate (OFRF) at the closest downstream off-ramp Increased turbulence adjustment factor When a node falls under the Increased Turbulence case (Figure C-26b), all lanes are affected by the turbulence caused by the intense lane changing. In this case, an adjustment factor CAFUP is applied uniformly to the node capacity: ð¶ð´ð¹ðð(ð, ð¡,ð) = 1 â ð¼ Ã ð¿ð¶ð (ð, ð¡,ð) (Equation C- 28) The calibration adjustments Î± and Î² were calibrated to match field conditions. Based on observed field data, the recommended values for the parameters are Î± = 0.52 and Î² = 0.81. The parameter LCR estimates the rate of lane change maneuvers performed by vehicles within the Queue Influence Area trying to adjust their position when spillback occurs. Vehicles traveling towards the exit ramp will move to the shoulder lane attempting to join the back of the queue, while vehicles traveling through will move to the median lanes in order to avoid the queue. Therefore, the lane change rate LCR is computed as:
276 ð¿ð¶ð (ð, ð¡,ð) = ððµð¿ð¶(ð, ð¡, ð)ðð¹(ð, ð¡,ð) (Equation C- 29) The parameter SBLC represents the number of lane change maneuvers performed in the queue influence area. In order to compute SBLC for a given node, the number of vehicles driving toward the off-ramp must be estimated for each freeway lane. For each lane i, the parameter pi represents the percent of the off-ramp demand vR traveling on the subject lane. In order to estimate the values of pi as a function of the distance from the off-ramp to the subject node, the following steps and assumptions are used: a) Within the influence area (1,500 ft from the exit point), the off-ramp demand flow rate vR is entirely positioned in the two rightmost lanes, according to the guidance provided in HCM Chapter 14. Therefore, the sum of the off-ramp flow rate percentages in the ramp influence area p1,R and p2,R is equal to 1. The methodology to estimate lane-by-lane flow distribution in freeway segments allows the estimation of the Lane Flow Ratio (LFR) for lanes 1 and 2. The proportion between p1,R and p2,R can then be estimated as follows: ð , = , ð , = (Equation C- 30) b) According to the guidance provided in HCM Chapter 14, the influence of ramps rarely extend beyond 8,000 ft. Therefore, for any nodes located beyond 8,000 from the off-ramp, the distribution of pi is taken as equal among all N freeway lanes: ð = 1ð (Equation C- 31) c) At intermediate distances from the off-ramp dOFR ranging between 1,500 ft and 8,000 ft, the distribution values of pi can be obtained through linear interpolation between the cases previously described. Figure C-27 â Distribution of pi as function of distance from the off-ramp exit, for a 3-lane segment The value of pi as function of the distance from off-ramp exit dOFR can then be obtained through the following equation:
277 ð = ð , + 1ð â ð ,ð Ã (ð â 1,500)6500 (Equation C- 32) As the lane-by-lane distribution of the off-ramp flow is known, the number of lane change maneuvers, SBLC, can then be estimated. For Regime 3 cases (one blocked lane), the number of lane changes is obtained as follows: ððµð¿ð¶(ð, ð¡,ð) = (ðð¹1(ð, ð¡,ð) â ð£ , ) + (ð â 1) Ã ð£ , (Equation C- 33) The equation adds the number of through vehicles in lane 1 that move to lane 2 to avoid the queue and the number of exiting vehicles in the remaining lanes that adjust their position to join the back of the queue, multiplied by the necessary number of lane changes. Figure C-28 illustrates an example of the proposed equation applied to a 4-lane segment. Figure C-28 â Illustration of lane change maneuvers within the queue influence area in a 4-lane segment with Regime 3 For Regime 4 cases, the following equation is applied to obtain SBLC: ððµð¿ð¶(ð, ð¡,ð) = 2 Ã (ðð¹1(ð, ð¡, ð) â ð£ , ) + ðð¹2(ð, ð¡,ð) â ð£ , + (ð â 2) Ã ð£ , (Equation C- 34) Figure C-29 illustrates an example of the proposed equation applied to a 4-lane segment. Figure C-29 â Illustration of lane change maneuvers within the Queue Influence Area in a 4-lane segment with Regime 4 Step 9 - Calculate mainline input
278 The Oversaturated Segment Evaluation procedure computes the Mainline Input (MI) for each node, in every time step. It is defined as the maximum flow desiring to enter the subject node during the current time step. An adjustment is necessary when the subject node is operating in a two-pipe regime, as the blocked and unblocked portions will be subject to different input demands. Since exiting and through drivers segregate when approaching a queue, the mainline input demand in the blocked side consists of the off-ramp demand, while the remaining demand will move to the unblocked portion. When node i operates in a two-pipe regime, the Mainline Input (MI) parameter is split into two components: MIUB, representing the mainline input in the unblocked lanes, and MIBL, representing the mainline input joining the back of the queue. These parameters are computed as follows: ðð¼ðµð¿(ð, ð¡,ð) = ðð¹ð ð¹(ðð¸ðððð¹ð (ð), ð¡,ð) (Equation C- 35) ðð¼ððµ(ð, ð¡, ð) = ðð¼(ð, ð¡,ð) â ðð¼ðµð¿(ð, ð¡,ð) (Equation C- 36) Step 12 - Calculate on-ramp maximum output If there is a merge segment upstream of an off-ramp bottleneck, the capacity of on-ramp output may be affected due to the blockage caused by the spillback queue. The Oversaturated Segment evaluation procedure calculates the on-ramp maximum output through HCM Equation 25-18, based on a series of potential constraints that include ramp metering, the on-ramp capacity, the capacity of the merge, or the presence of downstream queues. At high demands on both the freeway and the on-ramp, zipper merge (one- to-one) is expected to occur. Therefore, a new capacity constraint is added to Equation 25-18, included in the equation below in bold font and illustrated in Figure C-30: ððð ð(ð, ð¡,ð) = ððð ( â©âªâª âªâ¨ âªâªâª â§ ð ð(ð, ð¡, ð)ððð ð¶(ð, ð¡, ð) ððð¥ ( â©âªâª â¨âª âªâ§ððð ðð¹(ð + 1, ð¡ â 1, ð) + ððð ð¹(ð, ð¡ â 1, ð)ðð3(ð, ð¡ â 1, ð) + ððð ð¹(ð, ð¡ â 1, ð) â ðð¼(ð, ð¡, ð)ððð ðð¶(ð, ð¡, ð)ðð¹(ð + 1, ð¡ â 1, ð) + ððð ð¹(ð, ð¡ â 1, ð)ðð3(ð, ð¡ â 1, ð) + ððð ð¹(ð, ð¡ â 1, ð) /2ð(ð, ð)ð¹ð(ð¶ðð¹ðµð¬ð¿ð»(ð), ð,ð))ð Ã ðµð¸(ð¶ðð¹ðµð¬ð¿ð»(ð)) (Equation C- 37)
279 Figure C-30 â Impact of a queue spillback on the discharge capacity of an upstream on-ramp If one or more lanes are blocked due to a downstream off-ramp bottleneck, the throughput in Lane 1 will be equal to the maximum exit throughput in the congested off-ramp if the site operates in Regime 3, or 50% of the maximum exit throughput in the off-ramp, if it operates in Regime 4. It is assumed that the on-ramp and the flow arriving from the upstream on Lane 1 contribute equally to the downstream Lane 1 flow, and thus the on-ramp maximum output, in this case, is assumed to be half of the downstream throughput in Lane 1. Step 21 - Calculate mainline output (2) The Oversaturated Segment Evaluation methodology calculates the maximum number of vehicles, MO, that can exit a node, constrained by a downstream bottleneck or by merging on-ramp traffic. Among the potential constraints to calculate MO, the Mainline Output 2 accounts for the growth of queues on a downstream segments, eventually limiting the maximum number of vehicles that can enter it. When there is a queue in a downstream segment caused by a downstream off-ramp bottleneck, the segment is expected to operate under two distinct densities (Figure C-31). Therefore, the total number of vehicles in the downstream segment takes into account two different density values: the ramp queue density (RKB), prevailing at the queued area in red, and the background density (KB), prevailing in the remaining area of the segment (blue). Figure C-31 â Illustration of different density values within one diverge segment If there are no spillback effects, the segment operates with a uniform density. In this case, the constraints for the unblocked and blocked portions (MO2UB and MO2BL, respectively) are calculated proportionately to the number of unblocked and blocked lanes:
280 ðð2ððµ(ð, ð¡,ð) = ðð2(ð, ð¡,ð) Ã (1 âðð(ð))ð(ð) (Equation C- 38) ðð2ðµð¿(ð, ð¡,ð) = ðð2(ð, ð¡,ð) Ã ðð(ð)ð(ð) (Equation C- 39) If node i operates under Increased Turbulence (node is in the Queue Influence Area), the unblocked portion of segment i will operate similar to a regular segment. Therefore, the component MO2UB is equal to MO2 but proportional to the number of lanes in the unblocked portion: ðð2ððµ(ð, ð¡,ð) = ðð2(ð, ð¡,ð) Ã (1 âðð(ð))ð(ð) (Equation C- 40) For the blocked portion of segment i, the parameter is calculated as equal to MO2 proportional to the number of lanes in the blocked portion plus an additional number of vehicles due to the presence of a partial queue. This additional number of vehicles is obtained by the bold terms in the following equation, which takes into account the difference between the queue spillback density (RKQ) and the segment queue density (KQ), multiplied by the queue length: ðð2ðµð¿(ð, ð¡,ð) = ðð2(ð, ð¡,ð) Ã ðð(ð)ð(ð) + ðºð©ð³ð¸(ð, ð â ð,ð)ðððð Ã ðµð¸(ð, ð â ð,ð)Ã [ð¹ð²ð¸(ð¶ðð¹ðµð¬ð¿ð»(ð), ð â ð,ð) âð²ð¸(ð â ð, ð â ð,ð)] (Equation C- 41) If node i experiences lane blockage, the values of queue density must be computed for both the unblocked (KQUB) and blocked (KQBL) portions of segment i. For the unblocked portion, the queue density KQUB is calculated similarly to Equation 25-10, but the inputs for segment flow (SF) and segment capacity (SC) are replaced by their equivalent parameters SFUB and SCEQ: ð¾ðððµ(ð, ð¡,ð) = ð¾ð½ â [(ð¾ð½ â ð¾ð¶)] Ã ðð¹ððµ(ð, ð¡ â 1,ð)]/ðð¶ð¸ð(ð,ð) (Equation C- 42) The queue density for the blocked portion is computed as equal to the ramp queue density: ð¾ððµð¿(ð, ð¡,ð) = ð ð¾ð(ðð¹ð ðð¸ðð(ð), ð¡ â 1,ð) (Equation C- 43) With the queue density values for both the blocked and unblocked portions known, the MO2 components MO2BL and MO2UB can be computed: ðð2ððµ(ð, ð¡,ð) = ðð¹ððµ(ð, ð¡ â 1,ð) â ððð ð¹(ð, ð¡, ð) + [ð¾ðððµ(ð, ð¡,ð) Ã ð¿(ð) Ã(ð(ð,ð) â ðð(ð,ð))] â ððððµ(ð, ð¡ â 1,ð) (Equation C- 44) ðð2ðµð¿(ð, ð¡,ð) = ðð¹ðµð¿(ð, ð¡ â 1,ð) â ððð ð¹(ð, ð¡, ð) + [ð¾ððµð¿(ð, ð¡,ð) Ã ð¿(ð) Ãðð(ð,ð)] â ðððµð¿(ð, ð¡ â 1,ð) (Equation C- 45)
281 Step 22 - Calculate mainline flow The Oversaturated Segment Evaluation procedure computes the Mainline Flow through a subject node as the minimum of several variables, as presented in HCM Equation 25-16. If the node experiences spillback, the calculation of Mainline Flow must consider the flow through both the blocked and the unblocked portions of the node. Therefore, the Mainline Flow (MF) parameter is split into two components in an approach similar to the Mainline Input: the component MFUB represents flow across the node in the unblocked lanes, while the component MFBL represents the flow across the node in the blocked lanes. For both components, the resulting flow is computed as the minimum value between input and the maximum allowed flow. For MFUB, the maximum allowed flow is equal to the capacity of unblocked lanes in the downstream segment, represented by the parameter SCEQ as computed in the initialization step: ðð¹ððµ(ð) = min (ðð¼ððµ(ð, ð¡,ð), ðð¶ð¸ð(ð, ð¡,ð),ðð2ððµ(ð, ð¡,ð)) (Equation C- 46) For MFBL, the maximum allowed flow is equal to the flow allowed to enter the nearest downstream off- ramp RF, as presented in the following equation: ðð¹ðµð¿(ð) = ððð (ðð¼ðµð¿(ð, ð¡,ð),ð ð¹(ðð¸ðððð¹ð (ð, ð¡, ð),ðð2ðµð¿(ð, ð¡,ð)) (Equation C- 47) Next, the Mainline Flow MF through node i is computed as the sum of the blocked and unblocked portions, as follows: ðð¹(ð, ð¡,ð) = ðð¹ððµ(ð, ð¡,ð) + ðð¹ðµð¿(ð, ð¡,ð) (Equation C- 48) Step 23 - Update number of vehicles in the blocked portion of the segment The number of vehicles in the blocked portion NVBL during increased turbulence is updated based on the number of vehicles in the previous time step and considers the number of vehicles that are able to leave the current and upstream segment : ðððµð¿(ð, ð¡,ð) = ðððµð¿(ð, ð¡ â 1,ð) + ðð¹ðµð¿(ð â 1, ð¡,ð) + ððð ð¹(ð â 1, ð¡,ð)âðð¹ðµð¿(ð, ð¡,ð) â ðð¹ð ð¹(ð, ð¡,ð) (Equation C- 49) Step 24 - Additional number of vehicles ÎNV due to the off-ramp queue The additional number of vehicles in the segment due to an off-ramp queue is obtained by the following equation: âðð(ð, ð¡,ð) = ðð¹ð ð¿ð(ð, ð¡,ð) Ã ð ð¾ð(ð, ð¡,ð) â ð¾ðµ(ð,ð)5280 (Equation C- 50) Step 30 - Calculate segment performance measures The aggregated segment flow for a 15-min time period is obtained as the sum of flows for every time step (HCM Equation 25-30): ðð¹(ð,ð) = ðð ðð¹(ð, ð¡,ð) (Equation C- 51)
282 Similarly, the aggregated off-ramp ramp is aggregated at a 15-min time period: ðð¹ð ð¹(ð,ð) = ðð ðð¹ð ð¹(ð, ð¡, ð) (Equation C- 52) The additional density in the queued lanes is obtained by aggregating the additional number of vehicles ÎNV(i,t,p) in the off-ramp queue: âð¾(ð,ð) = 1ð Ã ð âðð(ð, ð¡,ð) (Equation C- 53) Similar to the mainline, the flow in the ramp roadway is also aggregated: ð ð¹(ð,ð, ð) = ðð ð ð¹(ð, ð¡, ð, ð) (Equation C- 54) The aggregated density at the ramp is calculated as the average of the number of vehicles inside the ramp along the time period: ð ð¾(ð,ð, ð) = 1ð ð ðð(ð, ð¡,ð, ð) (Equation C- 55) Finally, the speed at the ramp for a time period p is obtained by dividing the total ramp flow in the time period by its average density: ðð (ð,ð, ð) = ð ð¹(ð,ð, ð)ð ð¾(ð,ð, ð) (Equation C- 56)
283 Case Study: Evaluating Queue Spillback on a Freeway-to-Freeway connector This case study illustrates the application of the off-ramp spillback methodology to evaluate a network comprised of two freeway facilities (I-75 SB to SR-826 SB, Miami, Florida), as shown in Figure C-32. Due to congested conditions at the downstream merge segment (SR-826), spillback is expected to affect the operations of the upstream freeway facility (I-75). Vehicles traveling from node A to D are likely to have their travel time severely affected if spillback occurs. Figure C-32 â Study site for freeway-to-freeway queue spillback check, Miami, FL This freeway-to-freeway connector is modeled as two separate freeway facilities. The upstream freeway (Facility 1: I-75) is modeled as a diverge section that is connected to the downstream freeway (Facility 2: SR-826). The systemâs detailed geometry is shown in Figure C-33.
284 Figure C-33 â Individual freeway facilities: (a) I-75 SB and (b) SR-826 SB Input data Traffic demands for the freeway facilities and ramps are provided in Table C-6 in 15-minute time periods. Table C-6. Traffic Demands for the Subject Freeway Facilities Time Period Freeway Facility 1 (I-75 SB) Freeway Facility 2 (SR-826 SB) Mainline demand flow rate (veh/h) Diverge demand flow rate (veh/h) Mainline demand flow rate (veh/h) Merge demand (veh/h) 1 5400 1400 4000 1400 2 6200 3000 5700 3000 3 6000 3400 5600 3400 4 4500 800 4500 800 Additional input parameters are as follows: â¢ Urban area with level terrain; â¢ Grade: 0%; â¢ Regime 4 is expected; â¢ Base FFS: 65 mi/h (I-75) and 67.1 mi/h (SR-826); â¢ Ramp FFS: 55 mi/h; â¢ Ramp side: right for both facilities; â¢ Lane width: 12 ft; â¢ Right side clearance: 10 ft; â¢ Traffic composition: 12% trucks on both freeway and ramps; â¢ Ramp length: 3588 ft; â¢ Acceleration lane length: 1500 ft; â¢ No shoulder available;
285 â¢ Deceleration lane length: 700 ft; â¢ Number of ramp lanes: 2; and â¢ Familiar facility users. Performance measures - individual facilities The performance measures for both freeway facilities, if analyzed independently, are presented in Table C-7 and Table C-8. Facility 1 (I-75) is undersaturated, while Facility 2 (SR-826) experiences congestion in time periods 2 and 3. Ignoring the interactions between these two facilities would lead to inaccurate estimations of performance for the upstream facility. The merge segment (segment 2) in the SR-826 facility operates at LOS F, and the on-ramp capacity may be affected leading to queue formation and potential spillback. Table C-7. Performance measures for I-75 (Freeway facility 1) Time period Segment 1 Segment 2 Segment 3 Segment 4 Basic Basic Diverge Basic 1 C C B B 2 C C C A 3 C C C A 4 B B A B Table C-8. Performance measures for SR-826 (Freeway facility 2) Time period Segment 1 Segment 2 Segment 3 Segment 4 Segment 5 Basic Merge Basic Diverge Basic 1 B C C B C 2 C F E F E 3 C F F F E 4 C C C C C Spillback check The analysis of SR-826 using the Freeway Facilities Oversaturated Segment Evaluation provides the expected on-ramp queue for every time period. The first check compares the off-ramp demand to the ramp roadway capacity, as shown in Table C-9. The ramp queue starts to develop during time period 2. At the end of this time period, a ramp queue length of 1188 ft is expected, yielding a queue storage ratio of 0.33. Therefore, spillback is not expected during time period 2. During time period 3 a ramp queue length of 5160 ft is expected with a queue storage ratio of 1.41. Therefore, spillback will occur during time period 3.
286 Table C-9. Estimation of queue length and storage ratio at the SR-826 on-ramp Time period Total number of queued vehicles Number of queued vehicles in each lane Average vehicle spacing (ft) Queue length (ft) Ramp length (ft) Queue storage ratio Spillback occurs? [A] [B] = [A]/2 [C] [D] = [B]*[C] [E] [F] = [D]/[E] 1 0 0 - 0 3588 0.00 No 2 38.3 19.15 62 1188 0.33 No 3 159.1 79.55 65 5160 1.44 Yes 4 0 0 - 0 0.00 Yes Spillback analysis Since spillback is expected to occur, the methodology of this chapter (Figure C-4 through Figure C-7) is applied to evaluate its impacts on I-75 SB. The application of the methodology for each time period is presented below. Time period 1 No oversaturated conditions occur, therefore no additional calculations are needed for this time period. Time period 2 During time period 2, the downstream merge segment operates at LOS F and the on-ramp capacity is expected to be reduced. Step 1 - Calculate background density for unblocked lanes on each segment in the case of queue spillback The diverge segment at I-75 has 5 lanes and Regime 4 (two blocked lanes) is expected. Therefore, when queue spillback occurs this segment operates with two blocked lanes (lanes 1 and 2) and three unblocked lanes (lanes 3 through 5). The capacity per lane at the diverge segment SC(3) is 2,350 pc/h/ln or 11750 pc/h. For the time step level analysis, the capacity is converted to 48.95 passenger cars per time step (ts), Therefore, the capacity for the unblocked portion of the segment is given by (Equation C- 1): ðð¶ð¸ð(ð,ð,ðð) = ðð¶(ð,ð â ðð) Ã ð¶ð´ð¹ The capacity adjustment factor CAFBL is obtained from Table C-1. For a segment with 5 directional lanes and 2 blocked lanes, an adjustment factor CAFBL = 0.67 is applied. Therefore, the equivalent capacity of the unblocked portion is given by: ðð¶ð¸ð(3, 5, 3) = 48.95 Ã 0.67 = 38.8 pc/ts or 7872 pc/h
287 The unblocked background density KBUB is calculated next. For time period 2, an expected demand of 4165.8 pc/h for the mainline is used in the calculations. The KBUB parameter of the unblocked lanes is computed as the density of a 3-lane basic segment with a capacity SCEQ = 7872 pc/h: ð¾ðµððµ(3, 5, 3) = 30.4 pc/h/mi Step 2 - Initialize the freeway facility When spillback occurs, the subject freeway facility is analyzed as a link-node structure similar to the oversaturated procedure for freeway facilities. The facility structure is also expanded to consider the ramp segments. Figure C-34 illustrates the structure for the current analysis. Node 4.1 represents the interface between the diverge segment and the ramp roadway, while node 4.2 represents the interface between the ramp roadway and the merge at the downstream facility. Figure C-34 â Link-node structure for spillback analysis â I-75 SB Step 2C - Determine queue influence area (QIA) The queue influence area is obtained as function of the segment FFS, as shown in Table C-3. Therefore, for a FFS = 65mi/h, the QIA length is equal to 1060 ft. Step 2F - Determine initial number of vehicles at the off-ramp The ramp speed at the expected demand is obtained as: ð = 1 â 0.109 Ã ð£1000 Ã ð ð = 1 â 0.109 Ã 16791000 Ã 55 = 44.9 mi/h Next, the ramp background density is obtained: ð ð¾ðµ = ð£ð = 167944.9 = 37.4 ðð/ðð/ðð The initial number of vehicles in the ramp is then computed as:
288 ð ðð(3,0,2,1) = 37.4 Ã 35885280 Ã 2 = 50.8 ðð Step 2G - Determine the capacity of the downstream terminal The capacity of the merge is obtained by analyzing the downstream freeway facility using the oversaturated segment evaluation procedure and aggregating the parameter ONRO for an hourly flow rate. During this time period, the merge capacity is constant at 13.4 pc/ts or 3217 pc/h, while the ramp demand is 14 pc/ts or 3369 pc/h. Given the demand and capacity at the merge, the queue in the ramp roadway increases by 0.6 pc for every time step. Figure C-35 illustrates the ramp queue and the total number of vehicles in the ramp, considering an initial number of 50.8 pc in the ramp at the start of the time period as previously computed. Figure C-35 â Queued vehicles and total number of vehicles in the ramp â time period 2 Step 9A - Perform spillback analysis The flow RF that can travel across the ramp node 4.1 and enter the ramp roadway is obtained as the minimum of demand (RI), the ramp roadway capacity (RC) and the constrained capacity due to a downstream queue in the ramp (RSTG), as shown in (Equation C-12): ð ð¹(ð, ð¡, ð) = min(ð ð¼(ð, ð¡,ð, ð),ð ð¶(ð, ð),ð ðððº(ð, ð¡,ð, ð)) The capacity of the ramp roadway for a 2-lane ramp with FFS = 55mph, is equal to 4,400 pc/h or 18.3 pc/ts. Therefore, the capacity of the ramp roadway is not a constraint to ramp flow. The other potential capacity constraint RSTG is calculated through (Equation C-14): ð ðððº(ð, ð¡,ð, ð) = ð ð¹(ð, ð¡ â 1, ð, ð) + ð ð¾ð(ð, ð¡,ð, ð)ð¥[ð ð¿(ð) ð¥ ð ð(ð)]â ð ðð(ð, ð¡ â 1,ð, ð) The constraint RSTG is dependent on the number of vehicles in the ramp RNV, which increases progressively as the queue grows along the ramp. Figure C-36 compares the decreasing value of RSTG with the ramp input RI during time period 2. At the end of the time period, the capacity is still greater than demand, therefore no spillback occurs at the end of this time period as predicted by the queue spillback check previously described.
289 Figure C-36 â Ramp capacity and ramp inputs â time period 2 Since spillback does not occur, no additional calculations for the mainline are required. Step 30 - Calculate segment performance measures Since spillback does not occur during this time period, the performance measures for the mainline do not need to be recalculated. Since the ramp experiences queueing, the ramp speed in this time period is calculated using (Equation C- 54) through (Equation C- 56): ð ð¹(ð, ð, ð) = 4 Ã ð ð¹(ð, ð¡,ð, ð) = 1679.5 pc/h/ln ð ð¾(ð, ð, ð) = 160 Ã ð ðð(ð, ð¡,ð, ð) = 71.6 pc/mi/ln ðð (ð,ð, ð) = ð ð¹(ð,ð, ð)ð ð¾(ð, ð, ð) = 1679.571.6 = 31.9mi/h Time period 3 The same steps are repeated for time period 3. The ramp analysis is summarized in Figure C-37. For this time period, the ramp demand is 15.4 pc/ts, while the merge capacity is 13.9 pc/ts. Since demand is greater than capacity, the number of vehicles increases gradually, causing the capacity constraint RSTG to decrease each time step. At time step 14, the value of RSTG becomes equal to the merge capacity (13.9 pc/ts), which implies that the ramp has reached jam density and the maximum flow that can enter the ramp is equal to the flow that departs the ramp. Therefore, queue spillback into the mainline starts at time step 15.
290 Figure C-37 â Ramp capacities and ramp inputs â time period 3 After the onset of queue spillback, the number of unserved vehicles at the exit is computed every time step through the parameter OFRUV(i,t,p). Then, the expected length of the mainline queue OFRLQ(i,t,p) is computed based on the number of unserved vehicles and the ramp queue density RKQ, as shown in (Equation C- 25): ðð¹ð ð¿ð(ð, ð¡,ð) = ðð¹ð ðð(ð, ð¡,ð)ð ð¾ð (ð, ð¡,ð) The ramp queue density RKQ is obtained using (Equation C-13): RKQ(i, t, p, k) = KJâ [(KJ â RKC) x RF(i, t â 1, p)] / RC(i, t, p) ð ð¾ð(ð, ð¡,ð, ð) = 190â [(190 â 46.5) Ã 13.87)] / 18.33 = 81.4 pc/mi/ln Figure C-38 illustrates the expected spillback queue length during time period 3. Figure C-38 â Spillback queue length â segment 3 (diverge) â I-75 SB
291 The parameter OFRLQ represents the length of the queue if all unserved vehicles were queued in a single line. Given the segment geometry (Figure C-39), the operating regimes and flow modes can be obtained as a function of OFRLQ: â¢ 0 < OFRLQ â¤ 1,400 ft: Regime 1 â¢ 1400 ft < OFRLQ â¤ 3000 ft: Regime 4, with increased turbulence â¢ 3000 ft < OFRLQ: Regime 4, with lane blockage (queue extends upstream beyond the diverge) Figure C-39 â Available queue storage â segment 3 (diverge) â I-75 SB As previously shown in Figure C-38, the maximum queue length OFRLQ at time period 3 is equal to 4696 ft. Since queues develop along lanes 1 and 2, at the end of time period 3 the back of queue will be located 848ft upstream of the boundary of segments 2 and 3. The length of the queue influence area (QIA) is 1060 ft, and when it is added to the back of the queue it does not reach the upstream node of segment 2. Therefore, segment 2 capacity is not affected by the turbulence area upstream of the queue. Figure C-40 â Back of queue length, including QIA, at the end of time period 3 Step 30 - Calculate segment performance measures The ramp speed is computed similarly to time period 2: ð ð¹(ð,ð, ð) = 4 Ã ð ð¹(ð, ð¡,ð, ð) = 1707 pc/h/ln
292 ð ð¾(ð,ð, ð) = 160 Ã ð ðð(ð, ð¡, ð, ð) = 108.4 pc/mi/ln ðð (ð, ð, ð) = ð ð¹(ð,ð, ð)ð ð¾(ð,ð, ð) = 1707108.4 = 21.5 mi/h For the freeway facility, performance measures are computed for the blocked and unblocked portions of each segment. Segment 3 (diverge) â blocked portion Similar to the ramp, the flow through the blocked portion is aggregated for this time period: ðð¹ðµð¿(ð,ð) = 4 Ã ð¡ðððð ðð¹ðµð¿(ð, ð¡,ð) = 3030 pc/h The average density is obtained as the sum of two separate components. The average number of vehicles in the blocked portion of the segment is computed as: ð¾ðµð¿(ð,ð) = 160 Ã ðð(ð, ð¡,ð) = 51 pc/mi/ln The increase in density due to the lane blockage ÎK is obtained as: âð¾(ð,ð) = 1ð Ã ð âðð(ð, ð¡,ð) = 20.1 pc/mi/ln The total density is then computed as: ð¾(ð,ð) = ð¾ðµð¿(ð,ð) + âð¾(ð,ð) = 70.1 pc/mi/ln Finally, the speed in the blocked lanes is obtained through the fundamental equation: ððµð¿(ð,ð) = ðð¹ðµð¿(ð,ð)ð(ð,ð) Ã ð¾(ð,ð) = 30302 Ã 70.1 = 21.2 ðð/â The same process is repeated for the unblocked portion of the segment, except the ÎK component is omitted as no queues occur in these lanes: ðððµ(ð,ð) = 56.1 ðð/â Time period 4 During time period 4, the congestion at the downstream facility (SR-826) dissipates, which allows the ramp to discharge at the ramp roadway capacity (4,400 pc/h, or 18.33 pc/ts). Given the low ramp demand during this time period, the queue can be cleared quickly (9 time steps, or 135s). After the 10th time step, the freeway facility returns to undersaturated conditions.