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83 Chapter Eight SPECIAL CAPACITY AND DELAY ISSUES There are some conditions that have not been taken into accost In the capacity and delay models discussed so far. These conditions include: the existence of ra~sed/staped median or a two- way left-turn lane (TWLTL3 lane on the major street where a two-stage gap acceptance process was observed path some Livers a flared minor street approach where right turn sneakers are present the existence of upstream signals where heavy major street platoons were observed Me existence of pedestrians which may affect Me capacity and delay Me enhance of a sham through and left turn lane on the major sheet where left turn vehicles may cause blockage of the through traffic These conditions need special consideration when applying the capacity end delay models. The purpose of this section is to summarize some of the findings and recommendations regarding these special situations. TWO-STAGE GAP ACCEPTANCE Among the sites studied for this project, there are a significant number of sites where either a two-way left-turn lane (TWLTL) lane or a raised median exist on the major sew The existence of these facilities usually causes some degree of special gap acceptance phenomena, such as a t`vo-stage gap acceptance process. For example, Me existence of a raised or striped median causes a significant proportion of the minor street drivers to cross the first major sheet approach, and then pause in the middle of the road, to wait for another gap in the other approach. When a TWLTE exists on tiLe major street, the minor street left turn vehicle usually merges into the TWLTE first, Men seeks a usable gap on the other approach while slowly moving for some distance along Me TWLTL. Empirical Data The influence of a TWOS and liaised median on the minor street capacity at TWSC intersections has been investigated using the recommended cap acid mode} and procedure. The methodology used is to compare Me capacities calculated from the standard mode} and procedure with the field capacities measured using Equation 105. Figure 49 shows the mode} testing results for sites with either a TWLTE or a raised median The generalized cntical gap and follow-up times were used. Two regression lines were also plowed in the figure. The scattered data points suggest dial the capacity vanes significantly under these circumstances. It is evident that the theoreticalmode} underestimates the capacity for most cases, which means the existence of a TWLTL or raised median usually causes an Increase In the capacity for the minor street. The magnitude of capacity underestimation is slightly larger for sites with a TWLTE than for sites with a raised median. In other words, the increase of capacity due to a TWLTE is larger than due to a raised median. Based on Me regression results, it was found the ratios between the mode! capacity and field capacity for sites pith a TWLTL is about 2.S, and Me ratio is about 2.l for sites with a raised median. This means that with the existence of a TWLTE or raised/striped median, the capacities calculated using the standard procedure need to be increased by multiplying 2.S or 2.1. However, this result is based only on limited empirical data. coo - 800 600- £ ~ 400 200 O 1 1 1,-1 1 1,-1 1 ~7 . ~ 1 . | Raised Med~- | o . ~ . -a _1- , ~ I \ I TWLTLN I 400 600 800 held Capacity, vehlhr 0 TWLTLN Raised Median 1 000 FIgure 49. Model Testing Results for Sites with TWLTL or Raised Median (~nor LT CapaciW) Analytical Theory The basic capacitor model as well as Be capacity estimation pr~e cannot tee applied directly to Opinions in which two-stage gap acceptance exists. Briton, Wu (1995, Working Paper 24) and WE Bnion (TRB 1996) studied
84 the two-stage gap acceptance phenomena from a theoretical point of view as part of this project. A set of capacity models dealing with the two-stage gap acceptance process was developed A summary of the mode! is presented In this section. At many unsignalized intersections there is a space In the center of the major street available where several minor street vehicles can be stored between the traffic flows of the two directions of the major street, especially in the case of multi-lane major traffic. This storage space within the ~ntersechon enables the minor street driver to pass each of the major streams one at a time. This behavior can contribute to increased capacity. A mode} that can account for this type of behavior was developed by Harders (1968~. His concept was used as a basis for the following derivations. '' _ _1-~ ~ However, some major ampllilcatlons as well as a correction and an adjustment for field conditions have been made to better correspond to field conditions. part 11 J N7? . ~ ~ in line _ _ put Ic spaces fc r passenger car ~ r ~,9 1~ ~- Figure 50. Illusl~abon of an ~tersechon wi~ Two~tage Gap Acceptance Process partl These derivations are based on an intersection consisting of two parts, as shown in Figure 50. ~ Figure 50, the minor street through traffic (movement S) crosses the major street In two phases. Between the partial intersections ~ and ~ there is a storage space for k vehicles. This area has to be passed by the left turner Tom the major street (movement I) and the minor through heroic (movement S). The minor left turner (movement 7) also has to pass through this area. Movement 7 can be treated like movement 8. Therefore, these derivations concentrate on He minor through traffic (movement 8) cross~ngbo~parts of He major street. The designation of movements has been chosen in accordance with Chapter 10 of the 1994 HCM. It is assumed Hat the usual rules for unsignalized intersections are applied by drivers at the intersections. Thus movements 2 and 5 (major through traffic) have priorly over all other movements. Movement ~ vehicles have to give way to priority movement 5, whereas movement ~ has to give He right of way to each of the movements shown In Figure 50. ~ these derivations, movement 5 represents ad major traffic streams at part II of the intersection. These, depending on thelayoutof~e intersection, could include through traffic (movement 5), led turners (movement 4) and right turners (movement 6~. Analytical Mode] for the Determination of the Capacity To determine the capacity of the whole intersection, a constant queue on the minor approach (movement S) to part ~ is assumed. If wi is He probability for a queue of i vehicles queuing in the storage space within the central reserve, then the probabilities wi for all of the possible queue lengths i must sunlupto ~ with O ~ i ~ k, as shown belong: Kiwi= f-O where kis He number of spaces inure storage space within the central reserve. Considering central area of the intersection as a closed storage system, which is limited by the input line and output line, die capacity properties of He storage system are restricted by the aspects of maximum input and maximum output. Different states of the system are distinguished below. Starte 1: Part I of He intersection limits the input of vehicles to He storage area. Under state 1, the number i
85 of vehicles In the storage area is less than Me maximum possible queue length k, i.e. i < k . Dur[ng this state a minor street vehicle from movement ~ can enter the storage space if the major streams (volume v, and vp provide sufficient yaps. This case the capacity of Part ~ (possible input from movement S) characterizes the capacity, i.e.: car = eve + V2) (~o where cove + v2) is the capacity of Part ~ in the case of no obstruction by the subsequent Part 0, which is We capacity of an isolated unsignalized cross intersection for through minor traffic with major traffic volume v, ~ v2. The probability for this state ~ is p, = ~ - we . Thus, the contribution of state ~ to the capacity of Part ~ for movement ~ is c,~ =~1 -we +V2) (~D Dunng state I, vehicles from movement ~ can also enter Me storage space. State 2: For this state it is assumed that the storage area is occupied, i.e., k vehicles are queuing in Me storage space. In this state no minor vehicle from movement ~ or vehicles from movement ~ can get into the storage area. If, however, a sufficient gap for Me passage of one minor street vehicle can tee accommodated at both parts ~ and I~ of the intersection simultaneously, then an additional vehicle can get into the storage area. The capacity for vat (possible input from movement S) during this stage is C2 = eve + V2 + us) (~) where cove + v2 + vs) is the capacity of an isolated cross intersection for through traffic with major traffic volume vat + V2 + vs. Thus, the contribution of state 2 to the capacity of part ~ is CI2 = Wit X ~Vt + V2 + V; (~19) where wk is the probability that k vehicles are In the storage space. State ~ and state 2 are mutually exclusive. The capacity of part ~ is Me total maximum input to the storage area. Here the volume vat of movement I, in addition to the partial capacities mentioned above, has to be included. Therefore, Me total maximum input to the storage area is Input = c`, + cry + v' = (1 - wit x c(v,, + v2) + WE X eve + V2 + Vs) + v~ (120) State 3: To consider the output of Me storage area, this analysis concentrates on part ~ of the intersection. For i > 0, each possibility for a departure homage storage area provided by the major stream of volume vs can be used. The capacity (maximum output of Me storage area) of part In this case is C3 = revs) (121) where C(V5) iS the capacity of part II In the case of no obstruction by the upstream Part I, which is the capacity of an isolated unsignalized cross intersection for through minor traffic wad major traffic volume vs. The probability for this state is: pa = ~ - Wo. Thus the contribution of state 3 to Me capacity of part II IS Cars = (1 - wo) X revs) (122) where we is the probability Mat O vehicles are In the storage space. No vehicles Dom movement ~ (volume vie can directly pass through the storage area in this state (i.e., move without being impeded by movement 5~. State 4: For i = 0 (i.e. an empty storage area) no vehicle can depart Me storage area even if the major stream of volume vs would allow a departure. If, however, a sufficient gap is provided In the major streams of bow parts of the intersection simultaneously, a minor street vehicle from movement ~ can pass the whole intersection without teeing queued somewhere in the storage area. This is the same situation as exists with single stage gap acceptance. The possible output of the storage area from movement ~ vehicles during this state is C4 = ~V1 + V2 + VS) (1~) Thus, the contribution of state 4 to Me capacity of part ~ IS
86 C~4 = Wo X LEVI + V2 + VS) (124) Also vehicles from movement ~ can pass through the storage area in this state. The number of vehicles Tom movement ~ which pass through the storage area In this state is Il.4. ~0 ~ ti2s' Here, C~,4,,,~ does not mean Me cap acid for v,~but the demand on the capacity. The traffic demand of vat should be less than the capacil~,~ of the part ~ cove). i.e. vat is subject to the restriction vat < cove). Obverse, the intersection Is overloaded and as a result of this stationary state, no solution can be denved. States 3 and 4 are mutually exclusive. Therefore, the total maximum output of We storage area is 0U~ut = Can ~ Ca,4 tCa,4,v, = (1 - Why X C(VS) ~ We X COY ~ V2 ~ Van ~ Wo X VI(12o = (1 - We) x C(VS) ~ Wo X twit ~ V2 ~ Van ~ Vt] During times when the entire intersection is operating at capacity, due to reasons of continuity, Me maximum input and output of the storage area must be equal. Thus, since Me input equals the output (see Equations 120 and 126): (} - Wk) X OVA + V2) + W' X ~V1 + V2 + VS) + V! (} - Wo) X C(VS) + Wo X [c(v1 + V2 + Vs) + VI] The total capacity CT for minor Trough traffic (movement 8) when considering the whole intersection is identical to both sides of this equation minus vl. The probabilities wO and wit can be denved as: y- 1 Y - 1 y~1 _ yl w = ~yam _ 1 where (128) (129) ~v1 + V2) - act + V2 + Vs) (130) revs) - V1 - dVl + V2 + Vs) Note Mat wO is Me probability (proportion of the analysis time period) of state 4 under capacity conditions (continuous queuing) on the minor approach. This situation is identical to when there is no median storage and single-stage gap acceptance occurs. Thus, the remailing time proportion contnbudon to capacity, I-wO, is due to the existence of a median. ti27) The remaining proof and assumption details are presented in Working Paper 24. Based on these assumptions, the computational steps which are necessary to estimate the capacity of an unsignalized intersection where the minor movements have to cross the major street in two stages are presented In Table 46. Alternatively, graphical methods are presented In the following sections.
87 Table 46. Summary of Steps in the Capacity Calculation atIntersechons win Tw~Stage Gap Acceptance Process . vl= volume of priory sheet leRt~uning traffic et pert I l v2= volume of major street through1raffic comings the left at part I L v'= volume ofthe sum of all major sheet flows coming In the right at part IL | Here the volumes of all prior movements et pert II have to be included. These are: major right (6, except if this movement is guided along a biangular island Ted from We ~roughbaffic), m~yor~rough (S), major left (4); numbed of movements according to HCM 1994, Chapter 10. c(v1 EVE)= capacity at pelt I c(v,)= capacity at part II c(v'+v2+v5)= capacity at a cross intrusion for minor through traffic with a major sheet traffic volume of v,+v2+v5 Note: all capacity terms apply for movement 8 . They are to be calculated by any useful capacity formula,(e.~, Siegloch~s model or Hardcr~s model) where cull + V2) - c(vl + V2 + V5) (131) C(VS) - Al - C(V1 + V2 + Vs) Ct=~{kX[<V5)-VI]+<VI+V2+V5)} for y= 1 (132) Cl= - xtyi-l)x[c~v5)-vl]+ty-l)xctvl+v2+vs)} for y ~ 1 (133) or= total capacity ofthe into for minor through traffic with = ~ = 1 - 0.32-3 for k ~ O (134) Capacity According to Gap Acceptance Theory It should be noted Hat the capacities c(vl + v2 revs) and c(vS) can be calculated by any useful procedure, e.g., by formulae from gap acceptance theory. Solutions from linear regression methods could also be used. For example, SiegIoch's formula is: c(v ~ = ~ em '° f (135) where c(vp) is the capacity for the minor movement, (veh/s), If is Me foDow-up time, see, to is the to - tf/2, see, to is the critical gap, sec. It is reasonable to distinguish between cases wig different to and tf values: Case a. to and If values for part I of the Intersection (state 1 and state 2) Case b. to and If values for part II of the Intersection (state 3 and state 4) Case c. to - and If values for crossing part I and part II of the intersection simultaneously in the case of k= 0. Itis realistic to assume that a driver who has to cross the whole major street at one time without having a central storage area needs longer to and If values than in cases a or b . For a crossing movement g, it can be assumed that the to and If values in cases a and b are of the same magnitude and particularly that the If values between both cases are nearly identical (this assumption is important for He derivations which follows.
88 Realistic values for the to and tf values can be obtained from Table 47. The given cndcal gaps to and follow-up times tf are of realistic magnitude compared with the measurement results obtained In this project (NCHRP 3- 46, 1995, wowing paper 16~. 1h Table 47, the cndcal gap and the foDow-up time for the case without central reserve tic = 0) are larger than for the two-stage pnont~r case. Table 47. Typical Values for Critical Gap and Follow-Up Time for Two Stage Priority Situations win Multi-Lane Major Suet 0~1 of 0& 7.0 3.8 Based on Equation IQ5, assuming that aD of Me If values are nearly identical, c(v~+v2+v') C(V~+V2)XC(V5) (136) cO cO cO where }/tf is We mid capacity when Were is no conflicting traffic. This relationship for cO makes it possible to standardize all of We capacity terms by cO. If cO is defined In units of veh/s, We other capacity terms must have this unit (the unit veh/h could also be used for ad of the capacity terms). It is also useful to standardize CT In Equations 132 and 133: CT = - (13n o (which has to be obtained using Equations 132 and 133) can then be expressed as a function of cove + v2~/cO and [ckVs) -vim ~ /cO. The results ofthese derivations can then be shown graphically as In Figure 5 I. 1~ (a Figure Slat Total Capacity (0T = cllco) Figure Slb. Total Capacitor (Delco) Graphs of this type may be used as sufficient approximations under circumstances that differ Tom the conditions used in capacity models based on gap acceptance theory. For example: Capacities ctv1 +v2) and cove) maybe computed from other theories than gap acceptance, or they could be measured in the field. gap acceptance theory, the critical gaps tc are different for each part of the ~ntersechom The only necessary condition for the application of these graphs is that the follow-tip times If are of nearly identical magnitude. Graphs for Practical Application Alternatively, the its for the peony given in this section are illustrated In Figures 52(a) and 52(b) for k = 1 and 2 respectively. Here We capacities ckv, + v2) and cove) can be introduced independent of We type of formula Tom
89 which they have been de~minei Another advantage of these graphs is that they can tee applied for any value of v, (major left sum). 1 0.e . 0.. . 0.7 ~ 0.e ~- 0.6 ~ 0~ D.S T t t t ..~.. ...~.. !.. ~~t~\ V.~-~~ 1 'I.\ ~ At;; '',"2 Vt\",~;t'~;'~';: ...,.,..~..~..~.~..~.;.i~ ...~....~..~;...~.~ . ~ or. so s ~_ ~ ' 1 1 1 1 1 0 0.1 02 0 ~0~ 0.5 0 ~0.? 0.8 0.9 1 ~v,tv~l" 0.9 0.e 0.7 0.e 0.6 0 0~ 02 0.1 Figure 52a. Capacities (CT) for Movement 8 in Relation to Standardized Values of Capacities and vat. - 1) 1 0.9 0.e O.7 0.e 0.e 0.4 0.e 02 O. o I.. at t' ..~.. .... , . , , .. , ... , .. , . , ., ... , .. , 0 0.1 02 0.3 0.4 0.6 0.. 0.7 0.. 0.0 1 ~V10~" 0.. 0.e 0.7 0.s 0.5 0.4 0.3 02 0.1 Figure 52b. Capacities (CT) for Movement 8 in Relation to Standardized Values of Capacities and v,. - ) For example, for a two-stage priority intersection witch the leaf icvol~es v~=IOO veh/br, v2-600 veh/br and vs=400 veh/hr, assume that possible storage spaces within Me central reserve - 2, see Figure 52~. The capacities for movement ~ crossing the Intersection separately can be calculated Tom Siegioch's formula 03quation 135~. The corresponding values of to and If can be obtained from Table 47. The parameters for entering the application graph (Figure 52) can be calculated as follows: Pa I: Aim = 1 - ~0 t' (138) 1 ~1~),'_ 3.8` = a 3.8 3COO -- 2 ) = 0.119 (veh/sec) Part HI: c(vs) = 1 -I If ~ ( - )(6-3-8) = 3g~r 3600 2 = 0.167 (veh/sec) wit (140) c0 = tf = 31g = 0.263 (veh/sec) (140) 100 v = 1 3600 = 0.028 (v~h/sec) (141) resulting in the parameters for use in Figure 52: . cave v2) = 0.119 = 045 cO 0.263 (142) C(V53 - V1 _ 0.167 - 0 028 = 0.53 (143) With these two parameters, We relative capacity for movement 8 is shown below: CT = - = 0.36 cO (Few 52(b)). (144) Therefore, the absolute capacitor for movement 8 is: CT= CT X CO = 0.36x 0.263 = 0.095 (veh/sec) = 342 (veh/hr) (14~ If gap acceptance theory is applied (Equation 135) to
Do estimate the basic capacity terms c(vl +v2) and c(vs) and if Me tc and If values are known, the capacity for movement 8 can also be shown by the graphs depending directly on v2 and vs. However, one graph would have to be developed for each possible v1 value. An example of this type of graph, using tc and If values Mom Table 47 (night columns), is shown In Figure 53. 1 600 1 400 1 200 800 600 400 200 O W-",.~>',:.,,''\, 1 W~ N ' \.1 .. ~ V ~ Hi. ~ Hi- ~ .`,'''''i''.''t -`'\>'`\i'`\~` \ A, W. It' ~200 400 600 800 1000 12001400 16OO : . .,....Ij... 1 1 2 1 · 1 1 . I _ i v2 [veh/h] Figure 53. Capacities car for Movement 8 Depending on Traffic Volumes V2 and VS win v~=0 The theory described here can also be used to determine the capacity of the minor left turner (movement 7) under two-stage priority conditions. K there is no separate lane Norris movements the central storage area, the so-called mixed lane formula (Equation 10-9 of the HEM, 1994) should be used to calculate ~e total capacity for movements 7 and g . Delay estimates for the two-stage priority situation can also be performed using the delay formulae In chapter seven. Limitations and Conclusions of the Theory The theory which led to Equation 132 should be treated with caret The concept wo~dbe true if capacities c(v1 + v2 + vs) and c(vs) were completely estimated. However, during a specific maximum time, only k vehicles can enter one major stream gap both in part ~ and part II of Me intersection due to the restricted storage space within the median reserve ofthe two-stage situation. This restriction applies especially for c(v~ + v2) and c(vs) (states ~ and 3 above). This restriction does not apply for c(v, + v2 + vs.) since during state 2 and state 4 for the number of minor stream vehicles (>k) departing during one large gap (being provided simultaneously in major streams ~ and 2 as well as 5) is not~mited Each of the conventional formulae for the capacitor c(v) (e.g. Siegloch's model ~ Equation 135) is based on the assumption that, during large major stream gaps, a greater number of minor stream vehicles can be accommodated. This is not true In the two-stage gap- acceptance situation (state ~ and 3) since here the number of minor vehicles perlarge gap is limited to k in both parts of the intersection. To account for this limited validity of Equation 132, different approaches have been tested. The derivation of an analytical formula that accounts for these effects seems to be impossible. Only a partial solution to address realistic conditions was possible 03rilon, Wu, Lemke, 1995). Therefore, some approximations were necessary: a correction term, ~ (Equation 134), was obtained by calibrating the results of the analytical formula against simulation results. The two-stage priority situation, as it exists at many unsignali~d intersections within multi-lane major skeets, provides larger capacities than intersections without central resene areas. Capacity estimation procedures for this situation have not been available until now. The above model (Table 46) provides an analytical solution for this problem. ~ addidon, simulation studies lead to a correction of the theoretical results. Based on these denvations, a set of graphs was evaluated which enable a simple estimation of the capacity at an unsignalized intersection under two-stage priority. These graphs are ready to be used In practice, nevertheless, an emp~ncal confirmation of this model approach is desirable in filture research Meanwhile, since tills is a commonly encountered phenomenon at TWSC intersections with multilane highways, He theory presented here is recommended for use at unsignalized intersections.
EARED MINOR STREET APPROACH The geometric elements near the stop line on the stop- controDed approaches of many intersections may result In a higher capacity than the shared lane capacity formula or an estimation of field capacity may predict. This is because, at such approaches, two vehicles may occupy or depart from the stop line simultaneously as a result of He large curb radius, tapered curb, or a parking prohibition. The magnitude of this effect win depend In part upon He Ming movement volumes and the resultant probability of two vehicles at the stop line, and upon He storage length available to feed the second position at the stop line. Figure 54 shows a situation where the curb line provides space for two vehicles to proceed one beside the over forward to the stop line. In this case, we could define the storage as key. k spaces / T V - ,. or V7 Vp Figure 54. Illustration of an Intersection with Flared Minor Sheet Approach 91 800 700 .~ 400 300 E 2X I]J 100 Go. . ~ ''. ;,cG.-~ I . . . . . . 7 - 0 100 200 300 400 500 600 700 800 True Capac ty, veh/hr Figure SS. Capacity at~tersections win Flared Minor Street Approaches Clearly, He estimated capacity using this method is less than the true capacity at Intersections with flared minor street approaches, i.e. the capacity at intersections where a flared minor street approach exists is increased to some extent Compaq to when Here is no flared approach. This effect challenges the definition of a continuous queue required to observe capacity in the field: does one exist whenever there are queued vehicles behind those at the stop line, or only when both "sewers" at the stop line are occupied? It was decided that the original definition should be maintained (i.e., at least one vehicle always present at the stop line) and that the ability for rift turn vehicles to "sneak' was a geometric factor which should be explicitly recognized in any new procedure. The issue then is what level of accuracy can be provided based on empirical data and new theoretical models. A comparison between observed and field estimated Troutbeck (1995, Phase ~ report, NCHRP 3~6) has capacity for an example site exhibiting this "right turn reasoned that this is really a short-lane effect. If the sneaker" elect is shown In Figure 55. number of cars queued In the short lane is one (that is, Here are two vehicles at the stop line and there is a line of vehicles behind them), then an estimate can be made of the increase in capacity by calculating the number of gaps that are acceptable: that is, the expected number of gaps per hour that are greater than the critical gap. ~ each one of these gaps allows a vehicle to use the short lane, this would be aneshm~ofthe ex~a~ncrease In capacity (absorption capacity3. However, there may be some of these gaps which are not fully used by the short-lane vehicles, and this may be accounted for by a factor related to the availability of the short lane to arriving vehicles. Each data point In Figure 55 represents a 15-minute interval. The depart flow rate was recorded during each 15-minute interval when a continuous queue existed on the minor sweet approach This observed departure flow rate is regarded as the true capacity of He minor street. The minor street approach capacity was also estimated by dinding 3600 by the sum of the service time and move-up time (see chapter four). This method is only valid for a single-lane minor street approach without a flared position. The majority of the data points shown are below He line on which estimated capacity is equal to He true capacity.
92 At most TWSC Intersections that operate at below capacity, the average queue on the cntical approach is usually low, say less than 4. For intersections with an average queue of less than 4, the second position at the stop line may be available upon arrival, or soon thereafter, so Headmost of these gaps In the major stream traffic could be used. This likelihood would increase at a given site with an increase in the length of the "short lane". Given these observations, for most under-capacity cases, a storage length sufficient to accommodate two vehicles may be sufficient to enable the approach to operate and be analyzed as two separate lanes. Briton et al. CHOP 3-46, 1995, Working Paper 24) suggest that~his effect may be modeled using reasoning similar to the two-stage gap acceptance mode! for minor street left or through vehicles when median storage is available. Their initial thinking is outlined below. Harders (1968) derived He following equation to calculate shared lane capacity for the minor street approach: ~ V7 Ve Vg _ = + +_ (~4 Cm C7 C8 Cg where cm is the shared lane capacity, c', c8 and, c9are He capacity ofeachindiv~dualmovement 7, 8, and 9 (led, through, and right turn); and v', v8, and v, are He traffic volumes of each movement 7, 8, and 9. This equation is also used In the HEM. It is valid for each arbitrary traffic system where haBic flows of different capacities are using the same service facilitr. The usual geometric design of an Signalized ~ntersechon, however, provides space for more than one vehicle waiting at the sup line side by side, e.g., vehicles performing more than one movement can use He stop line position at He same time. If k is defined as He number of spaces for passenger cars belonging to one movement that can queue at the stop line without obstructing the access to the stop line for other movements, it is clear that with k > 0, the capacity of the minor street approach is increased compared with the sharedIane oondidom Wig He increase of k, the total capacitor approaches He case that each movement has its own individual lane of infinite length. No special solution for He estimation of capacity in He case for k >0 could be obtained from the literature. Therefore, a new solution is needed Consider a T intersection where only two minor street movements exist (movement 7 and 9~. For movement 9, if less than k vehicles are queuing In movement 7, each right tum movement can proceed to the stop line. Doing this time, movement 9 can depart throve stop line with its capacity car During over times when there is a longer queue Dan k vehicles in movement 7, no turn vehicle outside the storage area of the flare can depart. Thus during these times, He capacity for movement 9 is zero. Therefore, the capacity for movement 9 is C9 = P(n,~k).c, (147) Similarly, He capacity for movement 7 is given by C7 = P(n,~k).c7 (148) This Is not a solution that is exact in a mathematical sense of queuing theory; rather it is a pragmatic approximation. To solve Equations 147 and 148, a set of equations for He queue lengths of bow of He movements should be applied. The result of an M/M/1 queuing mode! was chosen as a usefill approximation for the queue length distribution and is given by Equation 149. p(n~c)=l-px+~ (~149) where n is He queue length and p is the degree of saturation. The solution is still not correct as it neglects the interdependencies between both movements, especially between He queue length distributions. The foldowing issues would skill need to be resolved before a ~eoredcally robust model can be developed: · overcome He problem of dependence between movements · include movement 8 for the case of a 4-leg intersection Until a better model is available, He following simplified
93 solution teased on an extension of the above discussion can be used (refer to Figure 56~: r ~: I ~ Of SEPARATE ~1 UNSHARED Gil ~ 11~1 INFRARED i 1 mar at_ KACTUAL QUEUEk Figure 56. Capacitor Approximation atIdtersections with Flared Minor Street Approach Assuming the shared lane case is the worst case, calculate the total approach capacity clod, then the delay In vehicle hours. This is equivalent to Me average queue on the approach for the shared lane case. . . . Assuming the separate lane case is the best case, calculate the total approach capacity c,`~ = c,+c8+cg, Men the delay in vehicle hours. I-his is equivalent to the average queue on the approach for the separate lane case. Derive a maximum length in vehicles ken of the flared area above which the traffic flows approximately like it would be on two separate lanes. This could be assumed to be equivalent to the average queue In the second bullet point above plus one vehicle rounded up to an Integer number of vehicles (altematively, a more conservative approach would be to calculate and use the 95th percentile queues from the 1994 HCM instead of the average queues in Me steps above). Using the queue storage Ken, at the site, interpolate between the shared lane formula capacity (with 1~0) and the sum of separate lane capacities (wish k detelInined above) - using either a linear Interpolation (conservative) or an interpolation that follows the shape of the curve given in Equation 149 (more lenient Man the linear assumption). Linear interpolation can be shown to yield Me following capacity: Cedus.1 2 (Cscpe~te- Cohered) k8Ct~ / ~ + Cow (~50) 2 CS,P,,TC KaIl31 / \~ + Coed ~ 1~ I/ kit) Because k will usually be an integer of low positive value less than 10 vehicles, the ratio kCb,~/k~ will only be accurate to one decimal place, or 10 percent of the difference between We extreme capacities. Therefore it is recommenced that linear interpolation be used The critical parameter to determine in this procedure is k,,,=.
94 EFFECT OF UPSTREAM SIGNALS The Ace of nearby upstream sigardi7~ intersections usually causes vehicles alTinng at the intersection to be platoons. Major street vehicles arriving at a TWSC intersection in platoons from a single direction may cause an increase of the minor street capacity compared unthif Spy arrived randomly. The more vehicles traveling in platoons, the higher the minor street capacity for a given opposing volume because there is a greater proportion of large gap sizes which more than one minor sweet vehicle can use. As shownin appendix ~ of the Highway Capacity Manual, the effect of upstream signals on both major street approaches is more complex. The effect of signals on capacity In such cases is at least as good as capacity with a random arrival assumption, but depends upon the alternating pattern of platoons Bom ache left and right that is presented to a crossing or left turn vehicle. This effect of upstream signals is discussed from three perspectives in this section. Firstly, the emp~ncal data is analyzed to discern the effect, if any. Secondly, an analytical model calibrated to Australian data is presented which provides insights into Be effect of distance to Be upstream signal, and the expected difference in the capacitor calculation compared path assuming random amvals. Finally, a model which extends ache lime distance diagram mode} in Appendix ~ of the HEM is presented, which accounts for the effect of platoon dispersion, and the effects of signal timing and coordination. Despite the insights gained, it is apparent that the effect is quite complex, with many input parameters needed to describe the signals' effect. As a topic for further research, it is recommended Cat simulation models be used or developed that adequately account for the many simultaneous influences of upstream signals that affect the capacity of a downstream TWSC Intersection The queue blockage effects of downstream signals were not addressed In this study. A queuing model is needed to estimate the length of queues at the approach to the downstream signalized ~ntersechons. Some indication would need to be prodded as to the proportion of lime that the queue would block the TWSC intersection or affect normal operation of the TWSC intersection. These blockage events could be aggregated over the analysis period to calculate a proportionate reduction in capacity for movements which may be blocked at the TWSC m~ion. A fisher capacity reduction effect would be caused by the queue discharge time at the TWSC intersection until operations revered back to "normal". Empirical Data To test the effect of an upstream signalized intersection on the minor street capacity, sites with an upstream signalized n~ionw~e selected and classified according to their distance from the upstream signalized intersection. The headways formed by the major street traffic were calculated based on the passage time of each vehicle. Using the absorption method, the minor street capacity was estimated. The capacity was measured based on specified time intervals. The number of minor street vehicles Hat can use each major street gap was calculated using Equation 106, and the minor street capacity was obtained by aggregating all He individual gap capacities for minor street vehicles based on the time interval. The major street conflicting flow rate was also obtained for each time interval. The next step was to examine whether the distance to the upstream signalized intersection causes a difference In the capacity estimation. Figure 57 illustrates the comparison between the field capacity measured for sites with the upstream signal less than 0.5 miles upstream with the mode} capacity calculated us~ngHarders' mode} (which assumes random arrivals). A critical gap of 6.4 see and a follow-up time of 3.2 see were used in the calculation. No obvious difference could be observed between the field capacity and model capacity for most of the cases, and He field capacities at higher conflicting volume ranges are lower than the mode} capacitor, which seems counter~nWitive. Figure 58 illusory similar information for sites with He distance of upstream signal between 0.5 and I.0 mile. Again, no significant difference could be observed between the field capacity and mode} capacity. It may be Hat for the major street volumes at these sites, the effect would be more strongly observable if the upstream signal distance was significantly less than 0.5 mile. The only information, regarding upstream signals was their distance. No information on their timing, traffic volumes, or coordination was available. Clearly, these parameters would also be relevant inputs to account for the effect of platooning measured from field. .~
0 200 400 600 800 1000 1200 1400 Conflicting Flow, veh~r | ~ Field-Model Figure S7. Testing We Effect of Upshwn Signal on Minor Street Capacity for Sims win Distance Less Than 0.5 Miles ·wu 800 I. 400- 200 r. . ~ 0 200 400 600 800 1000 1200 1400 Conflicting Flow, veh/hr a Field Model Figure 58. Testing me Effect of Upsbeam Signal on Minor Sheet Capacibr for Sites win Distance Between 0.5 and 1.0 Miles Bunched Distribution Mode! Model 1.4 tested In Chapter six incorporated a bunching factor a and an miming headway to to account for the effect of bunching. However, the mode} did not predict field capacities as wed as the Harders or Siegioch models which assumed random arrivals. However, the concept of bunching is a useful descriptor of He effects of upstream circled stances on arrival headways. Sullivan and Troutbeck (1995) developed annexation for the proportion of flee vehicles or as a function of the distance Tom tile signalized intersection end the flow. This equation is given by 95 a = e ~ ~ (l-Ce~BX) (151) where A is 6, C is site dependent but between 0.05 and 0.25, and B is 0.0068. vp is the major road lane flow in veh/s, and x is the distance In meters. This value of a has an ass~nnedtmof2 sec. A plot of the values Tom this equation shows Hat as tell distance, x, increases, it quickly reaches an equilibrium value for He particular flow, e -A4. In fact the distance is less Han 400m or I/4 mile. Figure 59, USA a plot of (} - C ebbs, confirms this conclusion. It is therefore not surprising that He effect of the proximity of signalized intentions cold not be found Dom the limited field data collected for this study, which had few sites with signals within I/4 mile. 0.95 ,, 0.9 t; lL 0.85 0.8 ///Po~rPr~;- 0 100 200 300 400 Distance from the Signals(m) Figure 59. The Factor (l~e~ Against me Distance x to Pepsin Sisals Assuming the poorest quality of progression (implicit in factor C), the difference in capacitor between an unsignalized intersection 50 m (150') from a signalized intersection to one 800 m (2400') Tom a signalized intersection is only a few percent as shown in Figure 60. The result depends on the cntical gap and He follow-up times. Here two sets of values are assumer The first has a cntical gap of 4 see and a foRow-up time of 2 see, the second has a critical gap of 6 see and a follow-up time of 3 sec. The mimm~headway is assumed 2 see for both of
96 the cases. A general observation relative to critical gap tc is that movements with higher values oftc, If are likely to gain the largest potential benefit from the effects of platooning. 1200 1 000 3 Q 800 ·O c' ._ In m 600 400 200 o HE' .W r - . 1 1 1 1 Todd 1 ~ ~ Tanner 1 ~ . ~ r 1 ~ .-. . 0 200 400 600 800 1000120014001600 Conflicting Priority Volume, veh/hr Figure 60. The Effect of Platoon Progression on Minor Stream Capacity Based on this sensitivity test, it was found that the existence of an upshe~n signalized intersection is likely to have some effect on the minor street capacity. However, the overall effect is not pronounced and of second-order magnitude. The magnitude of this effect on capacity is dependent on distance from a signal, as wed as the tc and If of the subject movement. The distance to the signal is Wear to be less than l/4 mile for the effect to be significant. The empirical data was partitioned into i/: mile bins which may have been too big for the effect to be discernable. A fills test of the effect of signals may require that a suitable simulation model be used as a tool. The effect of platoons on capacity should also translate into an effect on delay. Equation 43, for example, presented a theoretical model for delay Mat would incorporate the effect of bunched arrivals (or and tip. It makes sense ~ntu~tive~y~bat~e effect of regular, platooned arrivals should reduce the random delay component of a delay model. Another NCHRP research project currently underway and conducted by Texas A&M University is addressing this issue In the context of signal intersection delay. Extended T~me-Distance Mode! the absence of an available simulation model, the concept presented in Appendix ~ of the HCM is a good starting point for understanding the effect of upstream signals. However, this model has shortcomings which limit its broad application and validity. Firstly, He effect of platoon dispersion is ignored. This means that the platoons occupy less time at the TWSC Intersection than may actually be the case. ~ An, the "within platoon" flows are higher Han actual, resulting In a potential to undereshmate capacity within some platoon situations. As platoons spread, the proportion of gaps within the platoons that are greater than the critical gap may increase. This is equivalent to saying that He flow rate within platoons decreases. Conversely, in the current Appendix ~ method, "outside platoon" flows occupy more time than may actually be the case,whichmayleadto en overestimate of capacity during these penods. As tile unplatooned time period shrinks, vehicles outside platoons follow more closely, so the number of large gaps decreases. This is equivalent to saying that the flow rate in the non-platooned periods increases. These effects mean that at some point downstream of a signal the platooned and non-platooned flows will be equal to each other. After this point, a random arrival assumption is valid. Microscopic models which account for vehicle bunching as presented In He previous section should capture these elects. The second shortcoming of the HEM 1994 method is that the signals on either side of the TWSC intersection are assumed to have the same cycle length, and be coordinated with a constant offset. This simplification means that at a given TWSC intersection location, the same major street arrival pattern is repeated each signal cycle. Therefore, only one cycle needs to be analyzed, and the results extrapolated for the entire analysis period In the real world, common pycle lengths and coordinated signal are more the exception than the rule. Obviously however, the effect of signals is still valid Cabin a certain distance downstream, even for uncoordinated upstream signals. the signals have fixed, but different cycle lengths, then He arrival pattern will repeat within a period equal to their lowest common multiple ye., the offset changes each cycle, but with a repeating sequence. If the signals run isolated and are actual ere may never be a discernable pattern. Nonetheless, each cycle may produce a higher capacity at He TWSC inl~secdon than random arrivals would, so the
97 effect of upstream signals may still be important to account for. Thirty, the 1994 HEM method assumes that a platoon is discharged for the entire green period of the upstream intersection. A better assumption would be to use the ac~alplatooned volume, discharged at the saturation flow rate until the queue at He signalized approach has . dissipated. This would be less than or equal to He green time. Figure 61 proposes some modifications to extend the range of applicability, end the validity of the current 1994 HEM, Appendix I method.
98 J FLOWS PROBABILITY SIGNIFICANT IF D2 OR D5 Fit 61. Effect of Upon Sign~s ~O2 ~G2 i_ ~2 . \V2\ \ r7777 _ ~ \d \ ,\ D2 ,4 t2 7 t5 ~ //'1 ' 1~ Fat] O5 1 G5 1 R5 J' t5 611` I6. 6., Pi ~ 2 pa ~ P4 . . ~ ~ ~ ' .. ... 1500 F1. (500 m) (1/4 mile) D5 l TIME -_ ~' VN ~ VF 5 VN 35 _ _ SIGNAL 2 , - ~ SIGNAL 5 "SPREAD RATIO" = tat The first shortcoming can be remedied by adding a platoon , dispersion sub-model, similar to the non-recursive Q. = F2s-T + (1-F~Q,', (152) aigonthm used byTRANYST-7P signal analysis software (Yu and Van Aerde, 1995~: where I' is the vehicles arriving at the subject location in time slice t, Qt. is the vehicles departing Tom Me upstream signal in time slice t, F = 1 / (1 + a,Bta), T is '8ta' ta is the
99 distance to upstream signal / average speed of platoon, or is Me dispersion factor alpha (constant for a given road), and ,0 is the dispersion factor beta (constant for a given roadway). The aIgori~m is used to disperse a platoon Q departing at the saturation flow rate from We upstream signal. The "spread ratio" at the TWSC location was defined as the ratio between the time the dispersed platoon occupies the TWSC location and the lime that We platoon required to discharge from the upstream signal. Knowing ~is, We platooned end non-platooned~ow rates can tee ascertained. By applying spread (t/t, Figure 61), an average speed, We platoons with profile Q are projected downstream to the TWSC Intersection using Equation 152 to get profile Q'. Then the spread ratio is calculated by comparing the profiles Q and Q'. It should be noted that the TRANSYT algorithm is not recursive. It cannot be used to get a view of platoon dispersion along a road. Strictly spealdng, it also needs to be calibrated for each road link It is anticipate that defaults dispersion factors for Apical arsenal roadways (multilane, single lane, CBD, non-CBD3 could be provided Despite these cautions, the approach appears to be reasonable. With reference to the bunched c~istnbution model discussed in the previous section, Be platooned proportion is essentially the bunched vehicles and the unplatooned proportion represents the free vehicles. At consecutive locations downstream, some of the headways in the platooned zone will start to have some acceptable gaps (or at least they will be greater Wan We minimum headways and will be classified as free). The unplatooned zone will then start to contain some vehicles that are bunched and consequently, after a certain distance downstream (400 m in the bunched distribution examples) the headway distribution from one direction will start to reach equilibnum. In this sense, both models are a~empt~ng to account for the same phenomenon of platoon · e c aspersion. The spread ratio is used to calculate the flow rates within platoons and outside platoons after dispersion, at Be location of the TWSC intersection. The next issue is to obtain the proportions of the analysis period (Pi, Pa, Pa, P4) for each major street flow regime. Solving the second shortcoming of the current 1994 HCM method requires that the "begin platoon" and "end platoon" events from each major street direction be obtained for the whole analysis hour (direction 2 is major left, direction 5 is major nght). By sorting these events chronologically and examining their sequence, the proportion of the hour is calculated during which each of the following four arrival Bow "regimes" exist: . . no platoons platoons bod1 directions platoons from left only platoons Tom night only For coordinated systems with a constant cycle, only one cycle needs to be examined For non-coordinated, fixed but different cycle systems, a period which is a multiple of He two cycle lengths would need to be analyzed (i.e. one ingpa~n). For actuated signals, the entire analysis period should be analyzed since there may not be pattern. For each controlled movement at He TWSC intersection, the capacity is calculated as the weighted average (using these proportions) of the capacities within these flow regimes (as with the current HEM method). A spreadsheet model incorporating the above improvements was developed as part of this project. To test the effects of platoons on the capacity of TWSC intersections, an experiment was run wi~A,the following mputpalan~s as shownin Table 48. The parameters are an indication of the information required as input to the procedure. The number of parameters which have an effect make it more suitable to be implemented using computer software Man by hand. However, the theoretical basis of the methodic simple to explain. This med~od also requires validation with field data, and calibration for use on a particular arterial street.
100 Table 48. kaput Parameters for Experiment to Assess the Effect of Signals on TWSC Capacitor , ,., , A.,., , .. , , ............. ........ ., ,., .,.,.~. .. ,. , ,., ... ,.,, ........... Cycle tsec) | 90 90 Ill Offset~sec) | 0 0 45 Sahllation Flow Rate (vphgpl) ~ 1800 1800 Gr~n/Cycle ~ 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7 Volume/Capacity 1 0 0.25 0.5 0.75 0.9 1.0 0 0.25 0.5 0.75 0.9 1.0 (Proportion of Green Discharging at Sanction Headway) | Percent Flow within Platoons | 80% 80% | Distance from TWSC (~) | Incremerds of 500 up to limit of signal spacing . Spacing Between Sigr~alstmi) | 0.25 0.5 1.0 Progression Speed (mph) | 30 30 "a" Dispersion Factor | 0.15 0.15 ' a" Dispersion Factor | 0.87 | 0.87 It The TWSC intersection was assumed to be a four legged intersection with each turning movement having an exclusive lane (therefore, knowledge of volumes was not required for capacity calculation). The major sheet was ass~edto have two lanes In each direction, troth left turn pockets. No impedance effects were taken into account. Capacity calculations were performed for the minor left (HCM movement #7), minor through All, minor right (9), and major left approaching from the right (43. The above combinations of inputs produced a batch of over 500 separate runs. Numerous output statistics are product for each run including the proportion of We time period within each flow regime, the weighted capacity of each movement, and the percent and absolute difference from capacity assuming random arrivals. Table 49 provides an aggregation of the percent increase In capacity for each movement compared to random arrivals sort by green time as columns, and signal spacing and distance to signal 2 as rows. For the same experiment, Table 50 provides an aggregation of the vehicular increase In capacity for the minor through movement compared to random sort by signal spacing, distance to signal 2 as columns, and green time, total approach vogue, and offset as rows. Inspection of either tables confirms expected trends: closer signal spacing has a greater effect closer distance to a signal has a greater effect the results are very sensitive to a change in Me Offset between the signals capacity decreases wig increasing opposing volumes Me Rest percent change In capacity from random was 84 percent for the minor left ~ movement, 59 percent for the minor through, 23 percent for the minor right turn, and 10 percent for Me major left turn The greatest vehicular change In capacity from random for the minor through movement was 200 vph. It is interesting to note in this case that chang~ng~che offset by one-half Cycle reduced the change to 106 vph. lThe capacity increases become quite modest when the TWSC intersection is 2000 feet or more Tom either signal (less than 10 percent), and with most input combinations (less than 20 percent). The results of this fictitious experiment hint at potentially significant differences of greater magnitude end beyond the distance downstream found using the bunched distribution model. However, they have not been confirmed by field observations. They may be a result of unrealistic input
101 parameters such as only20 percent of flow teeing outside platoons. This, combined with a high proportion of time for this flow regime would make the non platooned periods very productive in terms of providing capacitor. An important observation to be made from this experiment is that most of the chosen buts are important to account for (cycle length, green time, saturation flow, distance, speed, platooned volume, non-platooned volume). Therefore, a realistic procedure should account for Hem Felicity, probably using computer software, rather than producing tables of average percent increases in capacity to apply by hand Fable 49. Percentage Idcrcasc in Capacity As Function of Upstream Signals as Comb to Random Major Street Arrivals ~ ~ 2e2~ ~5~2-2~ ~ 5~ ~ 5~T ~J~5 ~ ~ ' ~ MinLT 68 78 84 MinTH 49 S6 S9 SOD MinllT 22 23 22 MajLT 9 10 9 0.2S MinLT 64 7S 84 MinTH 46 S4 60 1000 MinRT 13 13 12 MALT S S S MinLT 39 42 40 MinTH 28 31 29 SOO MinRT 22 23 22 MajLT 9 9 9 MinLT 33 34 3S MinTH 24 24 25 1000 MinRT 14 13 12 MajLT 6 S S MinLT 31 33 36 MinTH 22 24 26 lSOO MinRT 7 6 S O.S MajLT 3 2 2 MinLT 3S 41 46 MinTH 26 30 34 2000 MinRT 2 2 2 MajLT 1 1 1 MinLT 42 S2 S9 MinTH 32 39 4S 2SOO MinRT 1 1 O MajLT O O O MinLT 31 34 34 MiIITH 23 2S 2S SOO MinRT 22 23 22 MajLT 9 9 9 MinLT 20 19 MinTH 14 14 14 1000 MinRT 13 13 12 MajLT S S S MinLT 10 8 MinTH 7 4 6 1 lSOO MinRT 7 4 S MajLT 3 0 2 MinLT 4 3 3 MinTH 3 2 2 2000 MinRT 2 2 2 MajLT 1 1 1 MinLT 3 1 1 MinTH 2 1 O 2SOO MinRT 1 1 O MaiLT ~ O O O 77 SS 18 7 81 S9 9 30 22 17 7 28 20 34 25 4 49 36 o 66 50 o 27 20 18 7 14 10 9 6 4 4 1 2 1 1 o 1 1 1 o 77 SS 21 9 76 SS 12 5 38 28 21 9 33 23 12 5 34 24 2 43 32 2 5S 42 o o 31 23 21 9 18 13 12 S 8 2 3 2 2 1 o o Notes: (1) D2 - Di~to Uh;beam Signal on~e I~ Side of ~e Minor S~ Appn~acb, (ft); (2) Green Time - Green Time for ~e Major Sbeet ofthe Upsbeam SiF-l': see, (3) Ihe Numbers Shown in~e Table are Capac~ Idcreases in Perce~age Comparetto Random Major S1reet Arrivals.
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104 Microscopic Simulation The spreadsheet mode! discussed above is a simple macroscopic deterministic model. However, the interplay between the venous input parameters is Rely to be best captured by use of a microscopic simulation model. To evaluate tibe behavior of a permit leR turn model, Velan and Van Aerde (1996) have used microscopic simulation to determine the effects of various degrees of platooned flow from an upstream signal on the major left tum. The degree of platoon~ng is expressed In terms of the green/cycle ratio through which the same opposing flow is filtered. The results of one experiment is shown in Figure 62. For example, with an opposing flow of 900 veh/br, the capacibr increased from approximately 400 vet shout platooning (g/c=~.O) to approximately 800 veh~r with strong platooning (glc=0.5) of the opposing flow. Since in this test, 100 percent Lithe opposing flow was assumed to be platooned, the unplatooned periods yield most of the capacity as We green/cycle parameter is reduced. In reality, cap acid increases of this magnitude are less likely. On most signalized arsenals, there are numerous intermediate driveways, as well as mining movements at the upstream signal, and laggards dropping out of the platoons. They ad contribute to some non-zero proportion of unplatooned flow. 1800 , : 1500 s ~1200 . c, ~ 900~ g 600 . Q 300- 1 1 1 1 1 1 ~1 1 ~ 1 , ~ _ ~ . . ~ . ~ l stew ~ 1 1 l l l l l l ~1 . . . . o 0 300 600 900 1200 1500 1800 2100 Opposing Flow, veh/hr . . 1 1 l _ gIc=0.3_ gic=o.s 1 gic=0.7_ 9/C=0.~_~ g/c=1.0 Figure 62. impact of me Level of Platooning in We Opposing Plow on the Minor Street Capacity The above discussion indicates that it is Important In many urban situations to include the effect of upstream signals Intersections when determining the capacil~,r of adjacent TWSC Intersections. This was confined by a user survey during Phase ~ of Me project where users aIrnost n~mmously requested Mat a new pro - =e acknowledge the effect of upstream signals. Therefore, it is recommended Mat a procedure be incorporated directly Into the bow of the HCM chapter for unsignalized intersections, rather Han the current Appendix I. THE EFFECT OF PEDESTRIANS ON CAPACITY AND DELAY Capacity and delay of vehicular tragic at unsignali~d intersections may also be affected by pedestrians. Where a cross walk is provided, pedestrians In some states have the highest priority over any other traffic movements. Since most of the sites collected during the project are either located In suburban area or rural areas, very low pedestrian volumes were observed. Therefore, the effect of pedestrians could not be obtained using the collected data. At many IWSC intersections with low pedestrian volumes, whatever methodology is Implemented is not expected to result in a large decrease in capacity. However, as the level of pedestrians increases to levels common in downtown situations, some decrease in capacity should be accounted for. Conversely, the capacity of pedestrian movements In situations requiring gap acceptance is not addressed In the current HCM. Many of the required input parameters which are needed such as pedestrian bunching, critical gaps, and crossing speeds should probably be under the preview of tile Pedestrian chapter of the HCM. So should simpler gap acceptance processes such as bock crossings be covered in the Pedestrian chapter. It is possible ~at, given such information, the Unsignalized intersection chapter could expand upon the midblock crossing methodology to account for the more complex ~nte~ctions with vanoush,r ranked boning movements at an unsignalized intersection. Therefore, the cap acid of pedestrian movements are not addressed here, although it is recognized as an important issue. Such a procedure would provide guidance for signal warrants, and the need for median refuges for pedestrians to cross in two stages. In the absence of empirical data, a critical gap could be derived based on perception, safety margin, and walldng times. Further studies will be necessary to accost for the effect of pedestrians on capacity and delay of vehicular traffic at unsignalized intersections. In the interim, two theoretical procedures were developed to account for pedestnans. The first procedure takes a simple view of the time that
105 cross~ngpe~ians "block" lower ranked movements. In the second procedure, pedestrians crossing each leg of Me intersection are assigned a "rally,', as for vehicular tuning movements, and~eated in much the same way as vehicles, but with specific pedestrian charactenstics. A theoretical adjustment to vehicular capacity is made by including conflicting, higher pr~onty pedestrian streams into the conflicting volume, and including their impedance due to their higher pnority. These procedures indicate a range of possibilities for Including pedestrians into a TWSC capacity model. The Highway Capacity Committee will need to consider at what level of detail pedestrians should be incorporated, in light of pedestrians' actual behavior. This will probably require validation TV emp~ncal observations. Blockage Time Method Pedestrian impedance is not Me same as vehicle impedance since the pedestrians. can all cross He road at the same time. This means that they do not consume intersection capacity unless there are so many pedestrians that they form a congruous stream. Perhaps, therefore, they should not be included as an Impedance In the sense of vehicular impedance. Impedance is essentially a way to account for the time that the higher ranked traffic streams wig be using the intersection. For the case of pedestrians, the additional time that they will use the intersection (and cars cannot) win be a constant time based on walking speed. Consider a major street left mining vehicle exiting the Intersection onto a minor street leg which pedestrians need to cross. The few issue is to establish the average time between the left Owners being able to depart (ignoring pedestrians for now3. This is a "block" time and TV ndicate the times that the rank ~ major sheet through and right vehicles are using the intersection (The concept of a block time is also used for impedance calculations in the HCM). The block time is: e v/C_1 h= (153) Vp where vp is the opposing traffic to major left turn movement and to is the critical gap for major left turn movement. Then, during this block time only, the probability that no pedestrians will arrive should be established. The probability that zero pedestrians wiD arrive during tb, is ~ a t o f t h e h e a d w a y b e t w e e n p e d e s ~ i ~ s b e i n g g r e a t e r ~ a n Ah: -vend (154) This is then the proportion of lime a pedestrian is expected to tee present when a left turner is present. If the walk time is t,,~,, then the product pO x tw should be subtracted from the time available to all the lower ranks (~an rank I) each time a left turn occurs. The Impedance is then: Po tw (155) to This impedance should be used as a factor that is multiplied with the other impedance factors. The vehicle impedance factors would be calculated the same way as In the current HEM procedure. Pedestrians Treated as Distinct Movements While recognizing some peculanties associated with pedestrians, this method takes a more uniform approach to both vehicular and pedestrian movements at an intersection. Unless otherwise noted, the terms "pedestrian" or "volume of pedestrians" in the following discussion are defined as the number of groups of pedestrians accepting the same gap in the opposing tragic stream Defining the average size of groups for venous pedestrian flows is a separate task not addressed here. . Firstly, the venous pedestrian movements must be ranked between (or equal to) the appropriate vehicular movements. Once Cat has been done, they can be included in the cat HEM procedure in a manner quite consistent wig how vehicles are treated: They are part of the cornicing volume, since they define "begin gap" or "end gap" events. This will usually be a small elect for normal pedestrian volumes. Pedestnansafebrisimpliedby the act of assigning them a rank or priority. Theyimpede lower ranked movements. However, tile duration of the impedance due each pedestrian crossing should only be for blocking one travel lane, since vehicles performing a given honing
106 movement tend to turn in front of or behind pedestnans once their target lane is clear. This rule win prevent an overestimate of Impedance due to pedestnans crossing wide streets. With reference to the HEM chapter, pedestrian impedance should also be adjusted for the codependence effect between impedances of higlaer ranked movements as is done for vehicles. It is useful to define which leg of the intersection is being crossed. Let subscnpt: S denote crossing of We subject minor street approach O denote crossing of the opposite minor street approach ~ denote crossing of We major street to the leg of the subject minor sweet approach R denote crossing of the major street to the right of the subject minor street approach Ranking Pedestrian Movements. A decision also needs to be made regarding the appropriate ranldng of pedestnans between the vehicular movements. This may be a policy issue which vanes by jurisdiction. For example, both AASHTO and He MUTCD imply Hat pedestrians must use acceptable gaps In major street (rank 1) traffic Stearns and that pedestrians have pnonty over all minor sweet traffic at a TWSC intersection. Refer to Figure 63. Cross Intersection Rank 1: 2, 3, 5, 6 1,l, ~ R. S. O 9, 12 a) ~8 11 4 7 10 Figure 63. Rink of Movements T-lotersection t tl 5 _ _ _ _ ! ~ Id_ 2 ~ a i,' I' Rank 1: 2, 3, s WAS a) 3: 7 In the HCM, major lens 1,4 and minor rights 9,12 ad have rank 2. An issue for major street crossing pedestrian types ~ and R is whether they have higher or equal rank Han major leR turns 1 and 4. If so, then they belong to a new rank"1.5". This is probably an unsafe assumption. If not, then they need a new rank "2.5". However, they still rank higher than minor rights 9,12 which means that these rights new a lower rank This would suggest the following rank sequence: · rank 1; (1,4); (L,R); (9,12); rank 3; rank 4 where ranks 1,3 and 4 are as in the HCM For pedestrians O and S crossing the minor legs, they rank higher than all minor approach movements. They may also rank higher than the major left turns 1,4 and major right turns 3, 6 while crossing the exit legs of the minor street approaches, although pedestrians often yield to them. If this yielding to major lefts is assumed (a safer assumudon). then the entire ranking seau~ce for all pedestrians becomes quite simple for a cross intersection: . rack 1; (1,4); (all pedestrians L,R,O,S); (9,12); rank 3; rank 4 for a T intersection the ranking sequence is: · rank 1; 4; (all pedestrians L, R. S); 9; rank 3 In general, the ring can be summanzed as: major sweet vehicles, pedestrians; miller street vehicles This ranking, together wad the recommendation that pedestrian groups crossing any leg only be assumed to block the leg for a subject vehicle movement for the time required to cross one lane is conservative in terms of die magnitude of impedance it will produce for vehicles. This is appropriate given the lack of emp~ncal calibration or validation data. Conflicting volume. Following He sequence of the HEM Methodology section in Chapter 10, it is questionable whether or how to include passenger car equivalents as a function of approach grade for pedestrians in Table 10-! of He HCM. This would be difficult since pedestrian group sizes vary as a function of pedestrian flow. Once group size was [mown, a pedestrian group on a flat grade could be assigned a weight based on crossing time for one travel lane. To get p.c.e.'s for non-zero grades, crossing dine could vary based on variation in walking speeds on different grades.
107 The diagrams and conflicting tragic equations ~ Figure 10-3 of the HEM would be modified to include the pedestrian group volumes VL, VR, VS, and VO (as shown In Figure 63), where V is the number of pedes~ians/group size. For planning applications, the number of pedestrians could be defined as it is In the chapter on signalized Intersections: low= 50/hr moderate= 200~r high= 400/hr As is done wad major sweet nght turning vehicles which In some cases have a weight of 0.5, each of We four pedestrian movements could be included in conflicting traffic equations, with weights of I.0 for higher ranked pedestnan movements and weights of O for lower ranked movements, except for a proportion of lower ranked pedestrians in "non-compliance" with their ranking rules. In practice, pedestrian volumes would usually comprise only a small proportion of total conflicting volume. Pedestrian Critical Gap and Crossing Time. Although a critical gap for pedestrians is not needed if calculation of pedestrian capacity is not a goal of Me Chapter Ten procedure, a discussion of it is worthwhile to raise some interesting issues. Conceptually the critical gap could be defined as: [c fr + t~ + fir (~156) where . . range of 4 to 6 seconds. It is probably a function of group size. The 4 second value is used by Me MUTCD as a minimum "walk" indication. The safety margin is salmon, but perhaps in the order of 2 seconds. Although these first two terms is not needed unless a calculation of pedes~iancapaci~is Arm, the Bird term wig n=l, Me time one travel lane is occupied, is needed to calculate the impedance due to pe~rians. Wing speed is a function of age and flow and has been documented to be between 2.5 to 6.0 feet~second. The MUTCD recommends 4 feet/second, except for elderly or child pedestrians where 3 feet/second may tee more appropnate. The HCM chapter on pedestrian capacity provides speed flow profiles for pedestrians and uses a crossing speed of 4.5 feet/second. tr is the perception and reaction time t, is the safety margin tw= w . n / v + (N-~)tf is the walk time for group w is the travel lane wide n is the number of lanes crossed at a time (this could be a two stage process if a median is present). This is related to which lanes are being used by the vehicle defining the "end gap" event. v is the waking speed N is the number of rows In pedestrian group if is the follow-up time for consecutive rows of pedestrians Perception reaction time could be assumed to be in the Using the above equation, and typical parameter values, crossing a 5 lane road may require 21 seconds, whereas a 3 lane crossing may require 15 seconds. Whether these are strictly critical gaps is debatable. The decision process as to how a pedestrian decides to cross a road is unclear. On a multilane road the decision may be quite complex, involving decisions about which lanes opposing tragic is in, and whether to make a two-stage crossing (note: two- stage gap acceptance by pedestrians could be analyzed In a similar way to the method presented earlier for automobiles). The blockage time for crossing one 12 feet urge Ravel lane will be in the range of 2.5 to 4 seconds for one row of pedestrians. Follow up time for pedestrian groups may be assumed to be He time between connive rows of pedestrians wit}lin a group, in the order of 2 seconds per row. The relatively simple expression for wale time to in Equation 156 was adapted from He ITE Traffic Engineering Handbook (Pline, 1992). It should be noted that adaptation of other expressions may be more appropriate at high volume pedestrian crossings with two- way platoons of pedestnans, such as have obtained Tom shockwave analysis at signalized crosswalks by Viricler, et al. (1995~. However, at high pedestrian volumes, He pedestrian signal warrant may in any case be met. Impedance byPedesf1ians. Generally, pedestrian groups would impede He lower ranked minor street vehicles while crossing one travel lane. The equivalent vol,~me/capaci~ ratio of pedestrians required to calculate impedance would be: Y/C = Np-~600 t~57y
108 Then, the impedance factor 1= 1 - V/C (158) Table 51 shows approximate impedance factors that may be appropriate for planning applications. The assumptions would need to be verified with empirical data. As with vehicular impedances, the statistical dependence between queues of different streams could be accounted for using the HEM equation 10-6 and Figure 10-6. Field observations have shown Mat such a blockage effect is usually very small, because the major sweet usually provides enough space for the blocked rank ~ vehicle to sneak around. Models could be developed Dom a theoretical point of view when Me major street width does not allow a through vehicle to sneak. At a minimum, incorporating this effect requires Me following information: . Table 51. Sensi~vi~ Test For Iinpedance Factor Due To Pedestrians (s=4fps;w=128) l ..~:::::.~ .~ ~ . .~ : -~::::~. ~.:.:~ .~ ~. ........................................ . ~. ~..... . . Pa - r Low 50 1 1 ~ 0.96 Moderate 200 1.2 1 0.86 High 400 1.S 1 0.77 This section has raised the need to adequately account for the effect of pedestnans on capacity at TWSC intersections. It is apparent that there are many issues regarding bow methodology, and parameter values that are inadequately addressed In the literature. However, it is imperative to account for the effect of pedestrians in high pedestrian use areas, at least at a planning level of analysis, as well as Dom a policy perspective. DELAY TO MAJOR STREET THROUGH VEHICLES TraDic engineers are also interested In knowing He effect of a shared lane on the major sheet approach where left tum vehicles may block rank ~ through or right mining vehicles. If no exclusive left turn pocket is provided on the major street, a delayed left turn vehicle may block the rank ~ vehicles behind it and cause them some delay. The current HEM procedure does not account for this effect and assumes no delay would be experienced by major street rank ~ vehicles. This effect not only delays rank ~ vehicles. While the delayed rank ~ vehicles are discharging from the queue formed behind a major left turning vehicle, they impede lower ranked movements with which t hey conflict. However, He current 1994 HEM does provide an impedance for major street left turn vehicles In a shared lane. This section proposes Cat it also be used to estimate delay to rank ~ vehicles. the proportion of rank ~ vehicles being blocked (the entire shared lane would be I/n) the average delay to He major sheet left Ming vehicles which are blocking Trough vehicles In the simplest procedure, the proportion of major rank ~ vehicles not being blocked (i.e. In a queue free state) is given byp*0 in equation 10-10 of the 1994 HCM (p*0j should be substituted for the major left turn factor poj in equation 10-3 of the 1994 HCM when calculating He capacity of lower ranked movements which conflict). Therefore, the proportion of rank ~ vehicles being blocked is I-p*O~ . Note that on a multilane road, only the major street volumes in the lane which may be blocked should be user! in the calculation as Vi, and Viz. On multilane roads if it's assumed that blocked rank ~ vehicles do not bypass the blockage by moving across into other through lanes (a reasonable assumption under conditions of high major skeet flows) Men Vi, = Vim. The average delay to rank ~ vehicles on this approach is given by: d,,,,,, I = (1 p 0) d~49 ( N ) V,l + Y,3 (1 - p Of) x dame Kit No 1 (159) N=1 Because ofdle unique characteristics associated with each site, the decision on whether or not to account for this effect should be left to He analyst. Geometnc design features such as an adjacent exclusive right turn lane, a large curb radius, or a wide shared left and through travel lane may enable rank 1 vehicles to bypass He blockage caused by major left tuning vehicles. Also, convicting traffic vogues in such adjacent bypass lanes must provide sufficient gaps to accept bypassing vehicles. It is interesting to note that when investigating factors affecting He critical gap for He major sweet left tom (see Wowing Paper 16, NC~P 346, 1995), it was found that
109 Me critical gap for the left turn was affected by the designation of an exclusive versus shared lane as follows: tc = 3.92 + 0.59 x.Exclusive (1603 where Exclusive = 0 or ~ wad a shared or exclusive left turn lane, respectively. This effect may be as a result of the "pressure" that a major street left turner feels while blocking through traffic. CONCLUSIONS This section has introduced both emp~ncal evidence and suggest theoretical models to adjust the basic capacity or delay equations to account for some common occurrences at TWSC intersect `~ two-stage gap acceptance; flared minor street approaches; effects of upstream signals; effects of pedestrians; and delay to major street rank ~ vehicles. Singly or In combination, these effects can cause significant adjustments to the basic capacity and delay models. The user survey conducted earlier in this study indicated that methods to account for all these effects are desirable. If one of these effects were to be included In Me recommended computational procedure, then it may be Important to include all of them, if the conditions are present at a given site, since they may be either additive (two-stage, flared approaches and signals increase capacity), negative (pedestrians and shared major left ~ through lanes decrease capacity) or canceling In Weir overall effect. Some effects may require significant judgement or data to be provided by the analyst. For example, the proportion of platooned and non-platooned flows. Other physical inputs such as flared, median, or shared lane geometries may be easier to obtain or judge.
110
