**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

*Capacity and Level of Service at Unsignalized Intersections: Final Report Volume 1 - Two-Way-Stop-Controlled Intersections*. Washington, DC: The National Academies Press. doi: 10.17226/6340.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

*Capacity and Level of Service at Unsignalized Intersections: Final Report Volume 1 - Two-Way-Stop-Controlled Intersections*. Washington, DC: The National Academies Press. doi: 10.17226/6340.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

*Capacity and Level of Service at Unsignalized Intersections: Final Report Volume 1 - Two-Way-Stop-Controlled Intersections*. Washington, DC: The National Academies Press. doi: 10.17226/6340.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

**Suggested Citation:**"Chapter Two. Available Methodologies." Transportation Research Board. 1996.

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s Chapter Two AVAILABLE METHODOLOGIES Capacity analysis at TWSC intersections depends upon a clear description and understanding of the interaction of drivers on the minor or STOP-controlled approach with drivers or vehicles on the major street. Both gap acceptance and emp~ncal models have been developed as a means to describe this interaction. Critical gap and follow-up time are the two major parameters used in venous gap acceptance capacity models. Most delay models were developed based on the degree of saturation, or vol~ne/capacity ratio. This chapter documents the theoretical background of critical gap and follow-up time estimation, and various capacity and delay models Cat have been developed for analyzing TWSC intersections. CRITICAL GAP AND FOLLOW-UP TIME Definitions Gap acceptance theory relies upon three basic elements, including the size and distribution (availability) of gaps In the major traffic swam, the usefulness of these gaps to Me minor stream Divers, and the relative Lions of the various traffic sins atthe~ntersection. The first element to consider is the proportion of gaps of a particular size in He major traffic skeam offered to the driver entering Tom the minor stream, as wed as the pattern of ~nter-a~Tival times of vehicles. The second element to consider is the extent to which drivers find the gaps of a particular size useful when attempting to enter the intersection. It is generally assumed In gap acceptance theory that drivers are both consistent and homogeneous. The critical gap Its) is defined as the minimum-length time internal that allows intersection entry to one minor stream vehicle. Thus, Me Diverts critical gap is the minimum gap in the opposing traffic stream that would be found acceptable. A particular Diver would therefore reject any gaps less than this cntical gap and would accept any gaps greater than or equal to this critical gap. The follow-up time (t) is the time span between the departure of one vehicle form the Tninor stream and Me departure of the next, under a condition of continuous queuing. Put another way, If is the headway that would define the saturation flow rate for the lane if there were no conflicting vehicles on movements that have the pnonty of entering the intersection. Another parameter used while measuring follow-up time and field capacitor is called move-up time (to,,) Move-up time is defined as the time span between the departure of one vehicle Tom the minor steam to Me arrival at the stop line of Me next vehicle under a condition of continuous queuing. Move-up time is only a portion of the follow-up time. Critical gap and follow-up time are the two major parameters for various gap acceptance capacity models. The values ofthese two parameters significantly affect the final capacity result; therefore, it is important to correctly measure these two parameters based on certain traffic and intersection conditions. The 1985 HCM adopts values from German studies, however, some inconsistency has been found when the values are applied to traffic conditions in the United States. This study has produced a comprehensive estimation of cntical gap and follow-up time measurements based on U.S. conditions. Several previous studies addressed critical gap and follow- up time measurement procedures. However, the majority of He studies only discussed the estimation methodology from a theoretical point of view. Few studies have addressed in detail how to measure these parameters in the field. For example, most of the studies reviewed in the literature search address the estimation process In relation to available data regarding accepted and the rejected gaps, however, none of the these studies has explained how to define gap events when a minor steam driver faces different conflicting streams. Without this discussion, the procedure remains at a theoretical level and does not provide enough guidance to be used in practical traffic engineering applications. Critical Gap Estimation Procedures Inreality,thecridcal gap is not a constant value. Instead it is a variable vnth different values for different drivers and for each individual driver over time. Consequently, the critical gaps Hat drivers use for Heir decision-making process at unsignalized intersections have a stochastic distribution, characterized by: a minimum value as the lower threshold, which is greater Man zero, an expectation of average critical gap (or mean

6 critical gap), which is often denoted as "the critical gap" In theoretical models, a standard deviation, and a skewness factor, which is expected to be positive, that assumes and accounts for a longer tad! on the right side of the distribution. This distribution and its parameters cannot be directly estimated because Me cntical gap cannot be observed. Only the rejected and accepted gaps can be measured. To estimate the critical gap, procedures must be used to estimate the distribution or its parameters as closely as possible using the accepted and rejected gap data. Such an estimation procedure should be consistent. If the minor street Livers within a specified composition of traffic streams have a given distribution of cntical gaps, We procedure should be able to reproduce this distribution closely. The procedure should reproduce the average critical gap reliably, without being dependent on other parameters such as: . . . traffic volumes on the major or minor street, delay experienced by Me drivers, and over external influences OD]Y if the results from a procedure are consistent can it be used to study the influence of external parameters on the critical gap. Othe~w~se the influences identified through emp~ncal studies might result from the inconsistency of the estimation procedure, in which case they are not really related to He extemal parameters being investigated. Tests of consistency are reported In chapter three. Several previous studies have addressed critical gap estimation methods and procedures. Miller (1972) has documented some of the early procedures, and additional procedures have been developed more recently. The most commonly used procedures include the following: maximum likelihood procedure (e.g., Troutbeck, 1992) SiegIoch (1973) Meshwork (1970) Raff (1950) Harders (1976) Hewitt (1992) Roget mode] (e.g., Cassidy, 1994) SiegIoch's method is quite simple and reliable for estimating critical gap (tc) and follow-up time (t) from saturated conditions (i.e. continuous queueing on the minor street). His procedure includes He following steps (see Figure I): . . . . Observe a traffic situation during times when Here is, Handout interruption, at least one vehicle queueing in the minor street. Record die number of vehicles, "i", entering each main stream gap of duration "I". For each of the gaps accepted by "i" drivers, compute He average of He accepted gaps (shown as x's in Figure I). Find the linear regression of these averages (average gap as a function of i). The increase of this regression line Mom i to i+! iStf. The intersection of He regression line wig He honzontal axis gives- to = tc - tf /2.

1 9t 8t an a) 7 cay ·_ a) 6 > a as of 5 4 3 2 1 O - I I I~ 1 1 1 1 1 1 11 1 1 0 5 10 15 Gap Size, see 20 25 30 Eo Average Values - Regression Line t=ftn) Figure 1. Siegloch's Method of Estimating Critical Gap and Follow-up Time SiegIoch's method is easy to apply and reliable, since the memos used to estimate cntical gap and follow-up time is exactly compatible with the derivation of the corresponding SiegIoch capacity formula. However, the method is only suitable for use with data dunng oversaturated conditions. Traffic operations in undersaturated conditions can also provide information about to and If, however, the SiegIoch procedure cannot be applied to undersaturated conditions. It is more complicated to estimate the critical gap to from traffic observations of undersaturated conditions because a critical gap cannot be directly measured; however, it can be assumed that the critical gap for a driver on the minor street is greater than the Diverts maximum rejected gap and smaDer~an~e accepts gap. This is hue if Me driver behaves consistently. A series of accepted gaps ta (gaps in the priority stream accepted by minor street vehicles) can be descnbed by an emp~ncal statistical distribution Unction (see Figure 2~; however, We distribution function of critical gaps to must be to the left of the ta distnbution. O - >0.5 . ~ E O 3 4 5 6 7 Crap, sea Figure 2. Cumulative Distribution Functions of Accepted Gaps F,(t) and Critical Gaps Fcff) By assuming exponentially distributed priority stream gaps and a normal distributions for ta and to, Ashworth (196S,1970) found that the average critical gap to can be estimated Tom {a (mean of the accepted gaps ta in

8 seconds) using the following equation: t = ~ - V X S2 (1) where Vp is the priority traffic volume, veh/sec, and sa2 is Me variance of the distribution of tat sec2. Hewitt (1985,1992) also developed a procedure for estimating to under more realistic conditions than those assumed by Ashworth Troutbeck (1992) describes a maximum likelihood estimation procedure for cntical gaps. This procedure can be used to estimate the critical gap under traffic conditions that are not necessarily oversaturated. The details of the maximum likelihood procedure are discussed below. The maximum likelihood method of estimating the critical gap dishibudon is based on the fact that a driver's critical gap is greater then his largess rejected gap and smeller then his accepted gap. The first step is to assume a probabilistic distribution for the critical gaps. For most cases this can be assumed to be log-nor~nal. This distribution is skewed to the right and has non-negative values, as would be expected In these circumstances. The distribution is reasonably general and is acceptable for most studies. The foHow~ng notations are used for subsequent equations: Yi is the logarithm of the gap accepted by the ith driver Yi is ~ if no gap was accepted xi is the logarithm of We largest gap rejected by the ith Liver xi is zero if no gap was rejected ~ is the mean ofthe distribution of the logarithms of the individual drivels critical gaps 02 iS Me variance of the distribution of the logan~ns of Me individual Diverts cntical gaps fit ~ is the probability density function for the normal distribution Fit ~ is the cumulative distribution function for the normal distribution The maximum likelihood of a sample of n drivers having accepted and largest rejected gaps of (ye x) is i.! II [Fluff) - F(X`~] n The logarithm, L, of Me likelihood is Men n L = ~ ~ [~f) ~ AXED] f-1 (2) (3) The maximum likelihood estimators, ~ and 02 Mat maximize ~ are the solutions to the two equations: BL = o aL = o ~2 That is, they are solutions of: (4) (I Iffy`) FOXY) - BL ~B~ BM (O = , = 0 8p f-1 [if) ~ [(X`) AL _: ~2 It can then be shown Mat This then leads to the two equations Mat must be solved iteratively using numerical methods: at) FOXY) - n ~`s2 ~2 = 0 (7) f-1 [if) ~ F(X`) annex' = -~X) arty = X 2p J(X) (8) (9) `) it) O f-1 Fief) ~ F(X`) (10)

9 n (X' ~ it) ,tt,X`) ~ (A' ~ 4) ](Yt) ~ pity`) ~ jinx) = 0 (11) where fix), ffy3,F(x) and F(5r~) are also functions of ,0 and ~2 A computer program was developed to solve these equations. The mean cntical gap to and the variance s2 can then be computed by: ~ = en ~ °.so2 S2 = [C2 (e°2 -I) It is this mean cntical gap that has been used In venous gap acceptance capacity and delay models. These cntical gap estimation procedures, along with the Ram Harders, and loft mode} procedures, were evaluated regarding their consistency to produce estimates of critical gap. The results of these tests are reported In chapter three. t~2' (13) where Obtaining Follow-up Time Unlike the cntical gap, Me follow-up time can be directly observed. For such an evaluation, the times between vehicles from the minor street entering the same gap of the priority stream should be measured, during periods of continuous queueing. For practical purposes, at least nf observations should be used to get an estimate of sufficient reliability(S%probability that We estimate is in a range of rf around Me Rue estimated. Assuming Of= 0.4 x If (Herders, 1976), the following equation can be obtained from sampling theory: n =a x ~ f S of (14) of is the necessary number of observations rf is the relative error = ef / If of is the absolute error At is Me standard deviation of Me statistical distribution of Me If at is a function of S. and is given as follows: If S = 90°/0, at= 0.4; If S = 95%, at= 0.6; If S = 99%, al= ~

10 CAPACITY MODELS Two general types of capacity analysis models have been used for TWSC intersections, gap acceptance models and emp~ncal models. Stance unsignal~zed intersections give no positive indication or control to the driver regarding when he or she can enter the intersection, the driver alone must decide when it is safe to enter the intersection by looking for a safe opportunity, gap or headway in the major stream traffic. This is the basis of Me gap acceptance process that is used in most analysis models of capacity and level of service at unsignalized intersections. At TWSC intersections, a driver must also respect the priority of other drivers. Other vehicles may have priority over the Diver byingto enter the traffic stream and the driver must yield to these drivers. Various models have been developed based on the gap acceptance theory and different assumptions of Me gap acceptance process. Models developed based on gap acceptance theory are closely related to queuing theory. Emp~ncal models were developed using regression techniques. This section documents these available models as well as Me related theories. Gap Acceptance Models This sub-section presents the basic premise of two stream capacity models based on gap acceptance. Then the hierarchy of the different traDic streams at a TWSC intersection and therefore Me definition of conflichug volumes for a given movement are discussed. Capacity with Two Traffic Streams. To understand traffic operations at a TWSC intersection, it is useful to concentrate on the simplest case first (Figure 3~. AD gap acceptance methods for TWSC intersections are derived from a simple queuing mode! in which the crossing of two traffic streams is considered. A pnonbr traffic stream (major stream) of volume vp (veh~r) and a non-priority traffic stream (minor stream) of volume vn (vestry are used in this queuing model. Vehicles from Me major stream can cross the conflict area without any delay. Vehicles Dom He minor stream are only allowed to enter the conflict area if the next vehicle from the major stream is still at least to seconds away (tc is the critical gap). Otherwise, they must wait. Moreover, vehicles from the minor stream can only enter the ~ntersechon at least If seconds after the departure of the previous minor stream vehicle (tf is the follow- up timed. I TV Noe~pnor~ ~/ 1 , ~ v P Priory dram Figure 3. Illustration of the Basic Queueing Theory The mathematical derivation of the cap acid en for the minor stream is as follows. Let gets be the n~nberofminor stream vehicles that can enter into a major stream gap of duration I. The elected number offer t-gapsperhow~s3600 x up x fell, where fits is the statistical density function of the gaps m the major stream, an~vp is (he To We olthe major steam in vps. Therefore, We capacity provided by t-gaps each hour is 3600 x vp x fit) x g(t). To determine the total capacity, expressed in veh/s, the following must be Antedated over the whole range of major stream gaps: ~0 an = Vp Jo At).git) dt t ~ O (1~ where en is the maximum traffic volume that can depart Tom the stop line in the minor steam in

11 veh/sec, vp is the major steam volume in veh/sec, fate is the density function for the distribution of gaps in the major steam, and gets is the number of minor stream vehicles that can enter into a major stream gap of size t. Two types of headway distributions for Me major stream have been used In gap acceptance models: Me negative exponential distribution, and Cowan's M3 (Cowan, 1975) distnbution. The probability density function of Me negative exponential distribution is given in Equation 16: J(t) = Ae~'` (16) where A is ~earT~val rate or traffic volume on the major stream, veh/sec. Cowan's M3 distribution modifies the negative exponential distribution by introducing a "bunching" factor a. The resulting probability density function is given in Equation 17: J(fl= 1 -a O Ale -l(t - 'A t> t m t = t m t < t m (1~ where tm is Me minimum inter-vehicle Tacking headway, a is the proportion of free vehicles Raveling wad headways greater Man tm seconds, and is a decay constant given in Equation I8: A = ~ '' V (~) Based on the gap acceptance model, the capacity of the simple two-stream situation (Figure 3) can be evaluated using elementary probability theory methods if the following assumptions are used: constant to and if values (driver population is homogenous and consistent), negative exponential distribution for priority stream headways, and · constant traffic flows for each traffic stream. For first assumption, two different formulations for the term gate must be distinguished. This is the reason for two different families of capacity equations. The first fancily assumes a stepwise constant function for gets, resulting In the following integer values for gate: g(t) = ~ n.pn~t) (~19) n ~ O where putt) is the probability that n minor stream vehicles enter a gap in the major stream of duration t. Pn (t)= ~ ~ for arc+ (n-l) x tf ~ t C tc + n x If The second family of capacity equations assumes a continuous linear function for gate, which may result in non-integer values for gate. This approach was first used by Siegioch (1973) and later by McDonald and Armitage (1978~. O for to< g(t)= ~ t° for trio (21) tf where to = tc ~ 2 Once again it should be emphasized that, in both Equations 19 and 21, to and If are assumed to have constant values for all drivers. Combining Equations 15 and 19 results in Me capacity equation used by Drew (1968) and by Harders (1968~. These authors denved the equation in Me following form:

12 or v e-vp.tc ~-e P r (22) Vpe~~P(tC - tr) n e vp i, 1 (23) Combining Equations 15 and 21 results In SiegIoch's (1973) formula: en = -.e P ° (24) J These formulae result In a relationship of capacity to conflicting flow as illustrated In Figure 4. Both approaches for gets produce useful capacity formulae where the resulting differences are small and can usually be ignored for practical applications. 1200 _ 1 000 ' .=' 600 400 200 O . ~ v . =~ == _= 200 400 600 800 1000 1200 1400 1600 Conflicting Priority Volume, veh~hr Slegloch --- Harders Figure 4. Relationship Between Capacitor (C=) and Priority Street Volume (vp) for the Two Streams (for this example, tc = 6 see, and h= 3 see) More general solutions have been obtained by replacing We exponential headway distnbution for Me pnor'~,r stream with a more realistic one (e. g., a dichotomised distribution). If the stepw~se constant function for gate is used (Equation 19), this more general equation is av e~A("~'~ - = P At (25) 1-e~ r This equation is illustrated In Figure 6. This is also similar to equations reported by Tanner (1967), Troutbeck (1986), Cowan (1987), and others. If ~ is set to 1 and tm to 0, Men Harders' equation (Equation 22) is obtained. If Me Linear relationship for gets according to Equation 21 is used, Men the associated cap acid equation is av e~A('°~'~ c = -P n Atf or (2o (1-Vptm) e ° is (27) of This was proposed by Jacobs (1979) and has also been referred to as the Troutbeck modification to the Siegioch equation. Changing the a term has a pronounced effect on capacity, as shown In Figure 5. 1200 _ 1000 c) ~ 800 ·= 400 is' 2CiO 0 `i I 1 ~- r~ -Tanned Fourth ~ 1 1 a= 1 -~=Q~ ~- ~-I= ~ ~ ~- 0 200 400 600 800 1000 1200 1400 1600 Conflicting Priority Volume, veh/hr Figure 5. The Effect of Changing cc in Equation 26

13 Equations 25 and 26 require two additional parameters, ~ and tm, as defined in Equation 17. Sullivan and Troutbeck (1994) have developed a computer program to calibrate these two terms based on field measured headway date. The program has been modified dunug this research project to calibrate a and tm for each site based on Me traffic flow charactenstics. Unfortunately, the three assumptions discussed earlier are idealized, and do not reflect actual operations. To address more realistic conditions, researchers have tried dropping one or more of these assumptions. SiegIoch (1973) studied different types of gap distributions for the priority stream (Figure 6) using analytical methods. S~m~lar studies have also been perfonned by Troutbeck (1986), Catchpole and Plank (1986~. Grossmann (1991) investigated these effects using simulations. These studies showed ~at: . Capacity decreases if the constant to and if values are replaced by realistic distributions (Grossmann, 1988~. Drivers may be inconsistent ~ i.e. one driver can have different cntical gaps at different times). A ~iverm~ghtreject a gap one time and accept it at other times. This effect results In Increased capacity. The use of consistent and homogeneous drivers in the analysis produces much the same result as the use of inconsistent and heterogenous drivers. If the exponential distribution of major stream gaps is replaced by more realistic headway distributions, capacity increases by about We same order of magnitude as the decreasing capacity effect of using a distribution for to and if values (Grossmann 1991 and Troutbeck 1986). The elect of a driver's valving behavior in relation to opposing flow was also investigated during this research. For example, drivers may adjust their critical gap and foDow-up time values according to Me magnitude of the opposing flow. Drivers may tend to accept smaller gaps when the opposing flow is high and they experience higher delays. Therefore, Me critical gap and follow-up time may no longer be constant values, but flow-dependent values. Also, a minor stream vehicle may enter a small gap and force Me major stream vehicle to be delayed. This situationis referred toas"limitedprior~ty,'.Figure 7 shows Me hypothetical capacity results using flow dependent cntical gap and follow-up time values and considering limited priority situations. In this figure, the bunched headway distnbution for Me major stream was used, and Me following linear functions were assumed for the critical gap and foDow-up time: to= 5.15 - 6.3 x vp, and If= 3.0 - 3.0 x vp. 1200 1000 ~ 800- .~ 2t 600 .~`,', 400- . m 200 O ~ / I . _ ~ __ =_ _ 200 400 600 800 1000 1200 1400 1600 Conflicting Priority Volume, veh/hr 1 1 1 1 1 Jacobs(Siegloc}^routbeclc) | Figure 6. Comparison of Capacities for Different Types of Headway Distributions in the Major Street Traffic Flow (for this example, tc = 6 see, tf = 3 see, and t, ~ = 2 see) 1 1 1 1 1 1 1 Flow Dada _~ 0 200 400 600 800 1000 1200 1400 1600 18 30 Major Stream Flow, veh/hr - Figure 7. Capacities with Flow Dependent tc and tf and Limited Priority

14 When to and tf are made flow dependent and/or "limited pnority?' is assumed capacity is increased at the lower flows and reduced at the higher flows compared to the case assuming a constant critical gap, follow-up time, and random headway distribution. The Emoted priority case produces results that are close to being linear relationship, providing good reason to believe that the approaches used by Kimber (1980) and Kyte (1991) are reasonable. in summary, many unsignalized intersections have complicated driver behavior patterns, and there is often lithe to be gained Dom using a distribution for the variables to and tf or complicated headway distributions. Moreover, using simulation techniques, Grossmann showed that these effects compensate for each other, so that the simple capacity Equations 22, 23, and 24 give realistic results In practice. Hierarchy of Traffic Streams. Unlike We simplest case (depicted in Figure 3) from which the basic gap acceptance capacity formulae are derived, a hierarchy of traffic streams exists at all TWSC intersections. These different levels of priority are established by traffic ndes as follows: . Rank ~ steam has absolute pnorits,r and does not need to yield right of way to another stream, Rank 2 stream must yield only to a rank stream, Rank 3 stream must yield to a rank 2 stream and In turn to a rank ~ stream, and Rank 4 stream must yield to a rank 3 stream and in turn to rank 2 and rank ~ streams (minor sheet left turn movement at a 4-leg intersection). These trailic streams are illustrated In Figure 8. The figure also illustrates that the major street left turn movement has to yield to the through traffic on the major road. The left turning traffic from the minor road has to yield to all other streams, including We queuing tragic In the rank 2 stream. Rank 1: 2.3,~6 2: 1,4,E,12 3: ~11 4: 7,10 a) b) CroN; intcrsoct;on T-lat Lion 1 1~1~ Rank 1: ~.5 2: 4,. 3: 7 Figure 8. Traffic Streams and Their Level of Ranking (note: the numbers adjacent to the arrows indicate the enumeration of streams given in the 1994 HCM) No rigorous analytical solution is known for the derivation of We capacity of rank 3 movements such as the minor street left turn movement at a T- junction (movement 7 in Figure S). In this case, models using gap acceptance theory typically use impedance factors as an approximation. Impedance is a concept introduced by Harders (1968), which has since been reconsidered wig some significant refinements by Briton and Grossmann (199 I). For each movement the probability that no vehicle is queuing et the entry is given bypO. This probability of a queue-tree state is given with sufficient accuracy by Equation 28 (BnIon, 1988~: Pot = 1 - x (28) where Po2 Is the probability of a queue-free state for a rank 2 movement, x is the degree of saturation, v/en, v is We traffic volume of a rank 2 stream, and en is the capacity of the rank 2 stream. As a practical approximation of impedance to rank 3 vehicles by opposing rank 2 vehicles, the following considerations apply. Only during the portion pO 2 of the total time, can vehicles of rank 3 enter the intersection due to highway code regulations. Therefore, for rank 3 movements, the basic value en for the potential capacity must be reduced to pO 2 X On to get the real potential capacity ce cc,3 Po,2 Cn,3 (29)

The foDow~ng examples refer to turning movements numbered in Figure 8. For a Tjunction, this means C,7 = po4 X C,,,7 (30) For a crossjunction, this means c,, = Px x cat CC.II Px Call with PA = PHI X po'4 01) (32) (33 The subscripts refer to the index of the movements according to Figure 8. Next, the values of poll and Poll can also be calculated according to Equation 33. For rank 4 movements (minor street left turn at a 4- leg intersections, the statistical dependence between the Poi values in rank 2 and rank 3 movements cannot be calculated from analytical relations and must be emp~ncally derived. They have been evaluated in numerous simulations by Grossmann (1991; cf. Brilon and Grossmann 1991~. Figure 9 shows adjustments to account for the statistical dependence between queues in steams of ranks 2 and 3. To calculate the maximum capacity for the rank 4 movements (numbers 7 and 10 in Figure S), the auxiliary factors pzg and Pelt should be calculated first: Pa,! = 0~65py~ PYJ + 0.6 ~(34) e 15 09 1 1 1 1 1 1 1 1 1~ of ===== 1 1 ~1 _0.7 --_-- :~ ~ 0O2 __ - -- LLII O.! O _~ ~ ~ ~ ~ ~ __~_ O 0.1 02 03 0.4 05 0.6 0.7 0.8 0 ~1 Pyti Figure 9. Reduction Factor to Account for Me S~6shcal Dependence Between Streams of Ranks 2 and 3 (E3rilon, Grossman, 1991) where i equals g or 11 (~e movement numbers according to Figure 8) and Pyi ~Px POi (the product Pyi is used as Me entry value on the horizontal axis of Figure g). The maximum cap acid of the minor street lefiturn movement at a 4-leg intersection is calculated as C~7 = (pan x Po,~2) Cn,7 (35) c. lo = (Pal x pO.9) c,,.~O (36) Conflicting Traffic Volume. For each traffic stream, the maxi n potential capacitor en should be calculated using the method shown In chapter To using We sum of all conflicting traffic volumes with higher rank Man the rank of the traffic stream in question. To aid in correct calculations, Table 1 can tee used This table basically corresponds to the German guidelines from bow 1972 and 1991 as well as to the 1985 and 1994 HEM.

16 Table 1. Evaluation of Conflicting Volume vp , 1 1 ~c; ~ ~ 2 ik5~'.}~ ~ s ~ c-- '1 ~ . Major Street LeR Turn 1 v, + vie. 4 v2+v,~. Minor Street Right Tum 9 vie + 0.S Vet) 12 v,2, + 0.S vie, Minor Street Through Movement 8 v2 + 0.S v,~, + v, + via + v' + V4 1 1 V2 + Vie + V5 + O.S Vie) + V! + V4 Minor Sheet Left Stun 7 v2 + 0.S Vet) + V5 + Vi + V4 + Vim + Vit5) 10 V5 + O.S Vat) + V2 + V! + V4 + V94 - + Vl5) , *The subscripts refer to Me traffic streams denoted in Figure 2.8. Notes: 1). If there is a nght-tum lane, v, or v6 should not be considered. 2). If there is more than one lane on the major road, v2 and v, are considered as traffic volumes on the right lane. 3~. If right-turning traffic from the major road is separated by a biangular island and has to comply with a YIELD or STOP sign, v6 and v, need not be considered. 4). If right-tu~ng traffic from the minor road is separated by a triangular island and teas to comply with a YIELD or STOP sigh V9 and via need not be considered. S). If movements 11 and 12 are controlled by a STOP sign, vat, and v'2 should be halved Iraqis equation. Similarly, i;fmovements 8 and 9 are controlled by a STOP sigh v, and V9 should be halved. 6). If the minor approach area is wide, V9 and v,2 can be omitted or Weir values halved Note Bathe assumed "weights" (~.0 or 0.5) of the conflicting volumes In Table ~ for the major street nit and left turns were revised later In this report. Several authors have objected that traffic streams of rank; 2 and 3 are included in these calculations In two places. These streams are, on the one hand, added to the main street volumes (Table I). On the over hand, they are introduced into the Impedance factors poby which the basic capacities on of rank 3 and rank 4 streams are diminished to calculate the actual capacities cc. Briton (198S, cf. Figures 7 and 8) has discussed arguments Mat support this double introduction. Two discrete effects of this approach are: Impedance effect - During times of queuing In rack 2 steams (e.g., the major sweet left turn), the rank 3 vehicles (e.g. the minor street left turn at a Tjunction) are prohibited Dom entering the intersection by traffic regulations and the highway code. Since the portion of time provided for rank 3 vehicles is Po2, the basic capacity calculated In chapter two for rank 3 streams has to be diminished by the factor Po2 for the corresponding rank 2 steams (Equation 29~. Conflicting volume effect - Even if no rank 2 vehicle is queuing, these vehicles influence rank 3 operations, because a rank 2 vehicle approaching the intersection within a time of less Wan to prevents a rank 3 vehicle from entering the intersection. Grossmann (1991) has shown that, of the possibilities considered, Me memos described is realistic enough to represent the range of traffic volumes that occur in practical applications. ~ other words, the impedance effect of hither ranked traffic streams accounts for the unusable time to the subject movement caused by the queuing of these higher ranked streams. The second effect of these higher ranked streams on the uncontrolled approaches before they enter a queue is that they form part of the traffic stream through which the subject movement must accept a gap. Therefore, higher rank movements may define the beginning or end of a gap. The weights assigned to conflicting traffic movements in Table ~ also have implications for how they help to define the availability of gaps. By implication, conflicting movements with weights of O or ~ should either be fully excluded or included in deriving the critical gap parameter for the subject movement, respectively. A weight of 0.5 for

17 example may be interpreted to mean that about half of these vehicles are perceived by the subject vehicle as being in conflict and therefore preventing a gap from being accepted Empirical Capacity Models The capacity of Me simple two-stream problem can also be evaluated based on an emp~ncal basis without any queueing theory. This is Me type of model used in the United Kingdom (Kimber, 1977) and by Kyte (1991). . .. ^. ~ The fundamental notion of this solution is as follows: consider Me simple intersection with one priorly traffic stream and one non-pnonty traffic stream during times of a steady queue (i.e., at least one vehicle queuing on the minor street). During these penods, the volume of traffic departing from Me stop line is,the capacity. This capacity should depend on the priority traffic volume vp during the same period. To derive this relationship, observations ot~traffic operations of the intersection have to be made during periods of oversaturation on Me minor steam approach. The total time of observation is then divided into periods of constant duration (e.g., ~ minute). During these I-minute intervals, the number of vehicles in both the priority flow and the entering minor street traffic are counted. These counts are transformed into traffic volumes. Normally, these data points are scattered over a wide range. Then, these data points may be represented by a linear regression line. Half of the variation In data points may result from using one- minute data. The other half is caused by variations In driver behavior or the geometric design of intersections. Under practical conditions, evaluation intervals of more than ~ minute (e.g., 5 minutes) cannot be used, because longer intervals result In a large reduction of the sample size because there are normally only a few oversaturation periods of such length at typical unsignalized intersections. This regression method would produce linear relations for en of Me form: en = b - c X up where b, c are constants of regression. (37) Instead of a linear fi~nchon, other types of regression could be used as well, for example: en = A em A r ~=,~ (38) Here, the regression parameters A and B could be evaluated using linear regression after a logarithmic transformation of Equation 38. This type of equation is of He same form as SiegIoch's capacity formula Using this analogy, it can be shown that A = 3600/tf. If the empirical regression theory is applied to the complete hierarchy of traffic movements at an intersection, Equation 37 can be amplified into a multinomial linear regression equation of He type On = d - ~ gf Vp, f (39) where d is a constant, i is the subscript for movements wig priority over the subject movement, gi is a constant for movement i, vpi is the traffic volume (velour) for movement i. The constants b and gi are evaluated by multiple linear regression using the available data. This is the type of formula Hat is used by Kimber (1977~. In general, some interrelations between traffic movements of different ranks of priority should be taken into account as well. Therefore, the regression equation should normally be of the type: en = A - EBf Vp, f f f ~ _~Ci;vpiVp; (40) where A is a constant, Bi is a constant for minor movement i, i is a subscript for minor movement, j is a subscript for movements with priority over movement i, and i ~ j, and rank (i) crank 0~. The sums should be formed over each possible combination of i and j according to He rank of the subject movement under consideration. The Bird term wi~the product of priority volumes takes into account the impedance effects between movements

18 of different ranks of priority. Although this general equation may represent an adequate solution, no investigations of such comprehensive regression analysis are known. In addition to the influence of priority stream traffic volumes on the minor street capacity, Me influence of the geometric layout of the intersection should also be included. To do this, the constant values could be related to road widths or visibility or even over characteristic values of the intersection layout by using another set of I~near regression analysis. investigation of TWSC intersection capacity has a number of advantages and disadvantages: The advantages include: . . . . Using a similar approach, Kyte (1991) also developed empirical models using We regression technique: Cn = ~ Af V' t4~' In addition, Kyte proposed another method for We direct estimation of ~ntersechon capacity In We field. The method is based on the fact that the capacity of a single-channel queueing system is We inverse of the average service time. The service time at an unsignalized Intersection is He time that a vehicle spends in the first position of a queue. This capacity estimation method is discussed In detail in chapter four. The estimation of delays and queue length using the empincal approach is again derived using queueing Peony. Here, however, these equations use the maxims entry Bow as an input, and so delay is not calculated Tom to and If values. In practice, these empincal regression delay equations are always combined wad the Kimber-Hollis capacity. Using the empirical regression technique for the - There is no need to establish a dleoretical model. Reported emp~ncal capacities are used. The influence of geometric design elements can be taken into account. The effects of prionty reversal and forced priority are taken into account automatically. There is no need to descnbe driver behavior in detail. The disadvantages include: Transferability to other countries or other times (driver behavior might change over time) is quite limited. For application under different conditions, a large sample must always be evaluated. No real understanding of traffic operations atone intersection is achieved byte user. Generally, He derivations are based on driver behavior under oversaturated conditions. Each situation to be described with the capacity formulae must first be observed. On the one hand, this requires a large effort for data collection. On the other hand, many of He desired situations are found infrequently, since these oversaturated intersections are often changed to signal control or modified using over measures before data can be collected. . . .

19 DELAY MODELS In the calculation of the capacity of a two stream TWSC intersection, it is assigned that He headways In the major stream have a specified distribution. This distribution can be changed to another distribution and a new equation for capacity can be developed. The capacity estimates are Independent of the order in which the major stream headways arrive. The Intersection would have the same capacity if the gaps offered to the minor stream drivers in one hour, were ordered so that the smallest gap was presented first then the next largest, the next largest and so on. However, the same ability to re- order gaps is not permissible for delay estimates. Using these usual gap acceptance assumptions, and assumptions about the amval patterns of the minor stream drivers and the order of the arrival of the major steam headways, the delays can be estimated. Delays are a function of all of these stream and driver attnbutes. There are generaDytwo~nds of delay models: steady state solutions and time dependent solutions. This section documents these delay estimation models, including the definition of delay, the underlying theory, and their formulae. Definitions of Delay In queuing theory, delays are estimated using a vertical stacking model. Vehicles are assumed to all stop at the stop line and are stacked vertically. Accelerations and decelerations are instantaneous. This delay, ceded q~eueing delay, includes some accelerations, such as In the Diverts move-up process, but not ad of them. The remainder of the delay is geometric delay. Total delay is the sum of the queueing and the geometric delays. Kimber et al. (1986) discuss the role of queuing and geometric delay and introduce the term "pure geometric delay", which is the delay caused by the geometry of the intersection when the driver is certain that no other vehicles are approaching. If the driver needs to check for He presence of other vehicles then his negotiation speed is lower. In the analysis presented here, He geometric delay includes three components: · the "pure geometric delay"; · the extra delay associated with He presence of other vehicles; and · the delay from stopping or slowing doom. Some of the delay calculated as geometric delay has already been included In the queuing delay. Figure 10 is an example showing the factors associated with the geometric delay. Here, die minor street is illustrated as yield controlled. Consider a gap with a duration of exactly the critical acceptance gap, le. Under gap acceptance rules, only one vehicle can enter in this gap and it may enter at only one instant. It is assumed for this discussion that this instant is to after the passage of the last major stream vehicle (and to - to before the next major stream ones. If the entry queue is empty, then a minor stream alTiving at this instant would not be delayed It is assumed that the minor stream vehicle would arrive at the stop line at t, traveling at its approach speed. If it arrived either shortly after or shortly before t, then it would have been delayed. Figure 10 illustrates the idealized trajectory of this undelayed vehicle based on these gap acceptance assumptions. More realistically, an entering driver would need to slow down to negotiate the intersection and would incur a delay of dock dI,eg and daccell before being able to resume the departure speed. The vehicle would cross the stop line at fir s after He last major stream vehicle, and would then have a geometric delay of the sum ofthese three lesser delays, as shown In Figure 10. dgeom = ddeCat + dun +d~t (42) where dgeom is the geometric delay, d, is He delay that occurs when decelerating Dom He approach speed to the speed at which He driver is able to negotiate the intersection, dog is the delay Hat occurs when traveling through the conflict point at the negotiation speed, and daeCe,~ is the delay that occurs when accelerating Dom the negotiation speed to the departure speed. The negotiation speed is dependent upon the maneuver type and can also be dependent on the cntical gap. The negotiation speed is the minimum speed at which He vehicle would Gavel when driving Trough the intersection when there is a long gap in the major streams. This driving behavior is assigned to be exhibited when a gap is equal to the critical gap and the driver has arrived at the specified time shown In Figure 10. At a TWSC intersection, He negotiation speed is zero and the minor stream vehicles always stop. The delay dreg

20 would also be zero andante ddoo ~1 and d, would be longer than shown in Figure 10. Similarly Me negotiation speed may not be constant. For a night ton movement at a TWSC intersection, the negotiation speed might start from zero, Men increase to 10 mph before Me vehicle rounds the corner. The vehicle would then accelerate to a higher speed. There are other cases of geometric delay resulting from various arrival patterns. This has been more fully documented in the NCHRP 3~6 Working Paper 19 (NCHRP 346, 1995~. In this study, queueing delay was measured Tom Me field videos. Major Stream Vehicles - Distance Trajectory Trajectoty ~ I4 dale W/~/:se~abon / Figure 10. Geometric Delay for an Undelayed Vehicle Arriving in a Gap of Tc Underlying Theory This section is a review of several of the melons available to estimate delays. For this discussion, it is assumed that Here are only two streams - one stream has priority and the other steam has to yield right of way to the priority steam. The priority traffic stream (major stream) has a volume of vp (veh/hr). The non-priority traffic stream (minor stream) has a volume of vn (veh/hr3. Steady-State Delays Based on the Gap Acceptance Assumptions. A general form of the equation for the average delay per vehicle is d = Did,, (~1 + Y £ ~ (43) where y and ~ are constants, x is the degree of saturation (vn / en), en is He capacity of the minor stream to enter the intersection, given a major stream flow of vp, and DO is Adams' delay (Adams, 1936~. Arlamq' dela~r is the average delay to minor steam vehicles when minor stream flow is very low. It is also the minimum average delay experienced by minor steam vehicles. Troutheck (1990) gives equations for fly, £, and Dmjr. based on formulations by Cowan (1987~. If minor stream vehicles are assumed to arrive at random, then y is equal to 0. On the other hand, if there is platoon~ng In the minor stream, Den ~ is greater than 0. These equations include the gap acceptance parameters: the critical gap and the- follow-up time. For random minor stream arrivals, £ iS given by eVP if - V [f - ~ + V (e VP " _ By) v (e VP tr ly) Note Hat ~ is approximately equal to I.0. Dan, depends on the platoon~ng characteristics in He major stream. If the platoon size distribution is geometnc, then Troutbeck (19X6) found Hat sac - t~ 1 At2 _ 2t + 2t a D . = e _ t _ _ + m (45) mm avp c A 2(tmA + a) where this the ass~nedmin~mu~n headway between major stream vehicles and a is the proportion of tree vehicles, or He bunching factor. The blanching factor ~ is in effect a platoon size parameter. The more vehicles that travel In bunches, He longer the platoons. The average platoon size is 1/~ although He variance of the platoon size distribution can vary independen~dy of lo. Differences in He platoon size distribution affect He average delay per vehicle, as shown in Figure Il. Here, He critical gap was 4 see, He follow-up time was 2 see, and He priority steam flow was 1000 veer. To emphasize the point, tibe average delay for a displaced exponential sonority stream was 4,120 seconds, when He minor stream flow was 400 Whir. This is much greater Man He values

21 for Be Tanner and exponential headway examples, which were around ~ I.5 seconds for the same major stream Bow. The two curves for Tanner headways have a different ordenng of the headways. The curve with the "geometric bunches" is where the headways are assumed to be independent of each other. The curve path the Borel- Tanner bunches has the same distribution of headways but there are more longer platoons or bunches. The vertical difference between these curves is a direct result of this ordering. The average delay is also dependent on the average platoon size, as shown In Figure 12. The differences In delays are dramatically different when Be platoon size is changed. 50 , Displaced 1 Legate /B~ = / 40- - / }Bengal / Bomb // / _~ Tanncr Headway <10~. ~ ~ ~ ~ ~ O- . . . . . . . . . . . . . . . O 200 400 600 800 1000 Major Stream Flow, vehlhr Figure 11. Average Steady-State Delay Calculated Using Different Headway Distributions so 40 30 In Cal 20 'a 1 0 o 2 0 200 400 600 800 1000 Major Stream Flow, vehlbr Figure 12. Average Steady-State Delay Based on Geometric Platoon Size Distribution and Different Mean Tanners (1962) mode} has a different equation for Adams' delay, because the platoon size distribution In the major stream has a Borel-Tanner distribution. This equation is D = ev' ('e - ·,J ~ ~ ~ opt" ~ 2ta'Vp A) (46) Another solution for average delay has been given by Harders (1968~. It is not based on rigorous queuing theory. However, as a first approximation, the following equation for the average delay to non-pnonty vehicles is quite useful. d= - e -(V' ·c ~ very + t' n _ v (47) 3600 M/G/! Queueing System. Queuing theory can also be used to approximate He gap acceptance process by assuming the simple two-stream system (Figure 3) can be represented by a M/G/l queue. The sewer is the first queueing position on the minor street. The input into the system is formed by the vehicles approaching from He minor street which are assumed to arrive at random (i.e. exponentially distnbuted) arrival headways ("My. The time spent in the first position of the queue is the service time. This service time is controlled by He priority steam, with an unlalown service time distribution. The "G" represents a general service time. Finally, the " I" in MIG/l stands for a single service channel, for example, one lane in the minor street. For He MIG/l queueing system, in general, the PolIaczek- Khintchine formula is valid for the average delay of customers in He queue x W(1 + C2) D = ~ (48) g 2 (1 - X) where W is the average service time (the average time a minor street vehicle spends in the first position of He queue near the intersection), and Cw is the coefficient of variation of service times,

22 Cw = ~ (49) and Vary is the variance of service times. The total average delay of minor street vehicles is then d=Dq+ W. In general, Me average service time for a s~ngle-channe} queuing system is the reciprocal of the capacity. If the capacitor is derived from Equations 25 and following, and includes We service fine W In the total delay, We following results are obtained: Cn ~ x C] t50' where ~1 + C2 2 Up to this point, the denvabons are generally valid. The real problem now is to evaluate C. Only the extremes can be defined, which are: Regular service: Each vehicle spends the same dme~ntihe first position. This gives Vary = 0, C = 0, and C=0.5 This is the solution for the M/DA queue. Random service: The times vehicles spend in the first position are exponentially distributed. This gives Vary = E~,C=1 and, C=l.O. This gives the solution for the ~1 queue. Unfortunately, neither of these simple solutions applies exactly to He unsignali~d intersection problem However, as an approximation, some authors recommend He application of Equation 50 with C = 1. Equation 43 can be further transformed to d = Den, (1 + y) (1 + ~ + y 1 - x) (51) where £ Andy are documented ~ Troutbeck (1990). This is similar to the PoDaczek-Khintchine fonnula (Equation 48~. The randomness constant C is given by (~+£~/~+~) md the hum 1~ x 0~) c" be considered to be an equivalent capacity or service rate. Both terms are afimchonofthe critical gap parameters to and If and of the headway distributions. However, C, it, and £ values are not available for all conditions. As a generalproperty of the M/G/1 system, the probability pa of the empty queue is given by Pa = ~ - X ~s2' This formula is sufficiently realistic for practical use at unsignalized Intersections. MAG2/] Queueing System. Other researchers have found Bathe service lime distribution in the queuing system is better described by two types of service times, each of which has a specific dis~ibudon: We is the service time for vehicles entering He emptr system (no vehicle is queueing on the vehicle's arrival) W2 is Be service time for vehicles joining He queue when other vehicles are already queuing Again, id bode cases the service time is the time the vehicle spends waiting in tiLe first position at Be stop line. The Initial concepts for this solution were Produced by Denser (1962, 19643 and in a comparable way by Tanner (1962) as well as by Yea and Weesalml (1964~. The average time that a driver spends in He queue of such a system is given by Yeo's (1962) formula: Cn ~E(W~) - E(W2) ~ E(W2~ (53) g 2 v y where Dq is the average delay of vehicles in the queue at higher positions Mange first, E(W,) is the expectation of We, E(W~23 is the expectation of (We x Wit, E(W2) is He expectabonofW2, and E(W22) is He expectation of (W2 X W21. Note also He following definitions:

23 · v = y+z y = 1 ~n E(W2) Z = Cn E(W1) cn= capacity The probabili~ pO of the empty queue is Po = YV (54) The application of ~is formula shows that ~e di~erences with Equation 52 are quite small ~ < 0.03~. Including the service time in the total delay results ~n ~e following equation (see Brilon, 19gg): E(W) Cn ty X E(W~ ~ Z X E(W251 (55) ~ = ~ - x v 2 v x y Formulae for the expectations of W. and W2 respectively have been developed by Kremser (1962~: E(W') = ~ (eV9tc _ ~ _ y t) + [i (56) E(W~2) = 2 (eV'c - ~ - vpt~-~ t' - ~ ~ ~ {c ~Sn E(W2) =-tl - e ~ ~ E(W22) = 2e 2 (e ~P`c - vptc) (1 -e ~~* ~ - vp [` e ~~,'r `59' ~p However, Kremser (1964) showed that the validib~r of these eq~ions is restricted to the special case of tc equal to tf, which is unrealistic for TWSC unsignal~zed ~ntersections. Daganzo (1977) gave an ~mproved solution for E(W2) and E(W2), which was extended by Poeschl (1983~. These new formulae were able to overcome Kremser's (1964) res~ictions. However, Kremser's first approach, Equation 56, g~ves qu~te reliable approx~mate results for tc and tf values ~at apply to realistic unsignalized ~ntersections. The foDow3ng comments can also be made about the newer equations. The formulae are so complicated that they are far from being suitable for practice. Computer programs must be wr~tten to provide numerical solutions. These formulae are only valid under the three assumptions presented earlier. That means that, for practical purposes, the equations can only be regarded as approximations and only apply for undersaturated or steady-state conditions. Time-Dependent Solution. Each of the solut~ons g~ven by conventional queueing theory above are steady-state solutions. These are the solutions that can be expected for non-t~me~ependent traffic volumes aDer an infinitely long t~me, and they are only applicable when the degree of saturation x is less ~an I. In practical tetms, Morse (1962) found that the results of steady state queueing theory are only usefill approximations if the observation dme, T, is considerably greater than the expression on the right side of Equation 60. T > ~ ~ ~ (60) This equation can only be applied if cn and vn are nearly constant during time interval T. If T is not greater that the right side of the equation, then time~ependent solutions should be used. Mathematical solutions for ~e time-dependent problem have been developed by Newell (1982) but may need to be simplified for practic~ng analysts. There is, however, a heur~stic approximate solution for ~e case of ~e peak hour efEect given by Kimber and Hollis (1979) which is based on ~e ideas of Whiting, who never published his work. During the peak period itself, haDic volumes are greater ~an those before and ailer that peno~ They may even exceed capacity. For this situation, the average delay during ~e peak penod can be estimated as: d = D~ + E + ~ (61) D~ = 2(~2 + G - F) (62)

24 F = - : n n en)] (63) G = tic- - (Cn - An) E| (64) CnO (CnO - VnO) (65) h = en ~ CnO + Vn° (6o h Y = - en (6 where en is the capacity of the intersection entry during the peak period of duration T and cnO is We capacity of the intersection entry before and after the peak period, vn is the minor street volume during the peak period of duration T. and v,,, is the minor street volume before and after Me peak period. Note: each of these terms is given in veh/s; each delay term is in seconds). C is again similar to Me factor C mentioned for the M/G/l system, where C = ~ for unsignalized intersections and C = 0.5 for signalized intersections. (Refer to Kimber and Hollis, 1979) This formula has proven to be quite useful in estimating delays. A simpler equation can be obtained by using a co-ordinate transformation memos. This is a more approximate method. The steady-state solution is acceptable for sites with a low degree of saturation and the deterministic solution is satisfactory for sites with a high degree of saturation say, greater than 3 or 4. The co-ordinate transformation method is one approach to estimating delays that falls between these two extremes. The equations do not have a theoretical basis. The steady-state solution for the average delay to the entering vehicle is given in Equation 6X: S { 1 - X5 ~(68) where the terms Dn,,r`, A, and ~ were defined previously. The deterministic equation for delay, dd is d, = D . + to + (Xa - 1) cnT O , overwise Ad > 1 (69) where Lo is the initial queue, T is the time the system is operating In seconds, and on is the entry capacity. These equations are diagrammatically illustrated in Figure 13. For a given average delay, the co-ord~nate hansfonnation method gives a new degree of saturation, xt, which is related to the steady-state degree of saturation, X5, and the deterministic degree of saturation, Xd, such that x, - x, = 1 - xs = a (70) (D In 0.5 O o , 7 -- 1 . . · . 0.5 ~1.5 2 Degree of Sabura~don, x figure 13. The Coordinate Transforrnabon Technique Rearranging Equations 68 and 69 gives two equations for X5 and xd as a function of the delays dd and d5 which then gives xt as: 2(d~ - Dig ~ =;Vn do ~ Din - rid x, = T do - Dot cDm,n (71) Rearranging Equation 71 and cropping the t subscripts

25 gives: where and D, = 1 (]A2 + B - A) A = T(1 - x) B = 4D~ T(1 -XXI +-Y) + TASK fir) -(1 -S (-+D~,1n (72) - _ - Dmm(2-~) (73) 1~1 (74) Equation 74 ensures that the transformed equation watt asymptote to the Feministic equation and gives a family of relationships for different degrees of saturation and periods of operation. A simpler equation was developed by Akcelik and Troutbeck (1991~. The approach here is to re-arrange Equation 68 to give: = ~ + T Ax_) ~ ~ (78) The average delay predicted by Equation 7g is dependent on the initial queue length, the time of operation, He degree of saturation and He steady state equation coefficients. This equation is an approximation and can then be used to estimate He average delay under over- saturated conditions and for different initial queues. The delay equation for unsignal~zed intersections in the HCM 1994 has He above formulation. Reserve Capacity. ~ He 19gS HCM, reserve capacity was used as the measure of electiveness. This decision was based on He fact that average delays are closely related to reserve capacity. This relationship is shown In Figure 14. Based on this relationship, a good approximation of He average delay can also be obtained from reserve capacities. For instance, Equation SO with C equal to 1 gives the equation: d = 1 ~ 1 ~ Cn ~ 1 -x J (79) a = ~X D~(Y + £X,) (75y 5 mm and this reduces to Thisis approximately equalto: d = ~= R- (so) D~,(Y + Art) a ~ d5 - D=,..... (76) Kthis Is used In Equation 70, then rearranged, the resulting equation for the non-steady-state delay is: ~ - D ~ ~ 2 1~+ (X-~)T t]~_~(X-~ -2~ (~+y)~ (77) A similar equation for He M/M/1 queueing system can be obtained if £ iS set to 1, y is set to zero, and Drain is set to I/cn. The result is: where R is the reserve capacity, en - vn. . According to Figure 14, as a practical guide, a reserve capacity R> 100 pcu~ generally ensures an average delay of less than 45 seconds, which is He LOS F threshold in ~e1994HCMforunsignalizedintersections.

o 50 . \\ Va-W - he (me ~ Va -12SO. 1000, 750, 500 ) ~30- '/ ~ _ ~ 20 ~ , Van lSOO crew O 100 200 300 400 Reserve Capacity R. vehthr 1 Figure 14. Average Delay d in Relation to Reserve Capacity R Briton (1995) has developed equations for average delay using the reserve capacity concept for use when Me intersection is oversaturated. Using a revised deterministic equation to account for We effect of the initial queue before the peak and assuming the equilibrium queue length after the peak is the same as the equilibrium queue leng~before We peek period: d = -B +;B2 +b (81) where B= ~ (bR--) (82) b = |-R ~°~ 2 (1 R1)) c 1: (83) and Ris the reserve capacitor and No is We lethal queue. In addition, Rf is an arbitrary point with suggested values as follows: · For To ~ h, Rf= -IOOvph · For T~ 0.5h, Rf= -200 vph · For To 0.5 h, Rf= -300 vph or -400 vph Also, T Is We duration of the peak period In hours, c is the capacitor of We system cluing the peak penod in veh/s (c is also considered to be the capacity before and after We peak), and R. is the reserve capacity before and after We peak period in vps. Kyte (1991) has proposed an empirical approach directly relating delay to reserve capacity: d = ~ e-~(Ca-V) (84) where ,B's are the regression coefficients, en is the minor steam capac~W, veh/hr, and vn is the minor stream volume, veh/hr. Effect of Capacity Estimation on Delay Estimation. The MIM/l queueing mode} can be used to demonstrate how changes In the capacity estimate can influence the delay es imate,using the approximate Briton (1995) relationship for the delay in oversalted conditions given as Equation g ~ and following If it is assumed that T is equal to ~ hour,NOis 0, end R. is halftime capacity, then, for the same via, Me delay changes considerably as qm changes. Shown In Figure 15, this illustrates the importance of using the correct capacity estimate. 50 40 30 In Cat 20 10 o o Capacity, vehlhr 200600 1000 1400 11300 ~I ~I ~ 500 1000 1500 2000 Minor Sbeam Flow. veh~r Figure 15. Average Delay Against He Minor Stream Flow and for Different Capacities For an example of how important the estimate of capacity is to Me delay estimate, assume Mat Me minor stream flow is 800 vehJhr in Figure 15. The delay would be estimated to be a function of the capacity, as shown in Table 2. No matter how sophisticated the delay methodology used, the quality of the forecast is dependent on obtaining a good Mate of capacity. If the capacity estimate is inaccurate,

27 It is evident that the average delay during near-saturated and oversaturated conditions is strongly dependent on the peak duration T. Any increase in the peak period results in a significant increase in the average delay. SUMMARY This chapter documents the theoretical background of critical gap estimation procedures, capacitor and delay models for TWSC intersections. SiegIoch's method for estimating critical gap and follow-up time is easy to use, however, it requires continuous queued condition on Me minor street approach. The maximum likelihood method can be used under conditions that are not necessarily oversaturated, and it provides both Me mean cni~cal gap and standard deviation. 250 5: a) `~ 200 - ~ 150 a: 100 50 o . ~ ~ Z then Me delay estimate wall also be inaccurate, no matter which delay mode! is used. Table 2. Example of Average Delay and Capacity for vn = 800 veh/hr l 1800 3.6 l 1600 4.S l 1400 6.0 l 1200 8.9 l 1000 17.3 Elector the Duration ofthe Peak Period on the Estimate of Delays. The over term Mat has a significant influence on Me estimate of delay is Me duration of the peak period (or Me analysis We period). This sensitivity to T is shown In Figure 16. 300- 1 1 . 1 1 1 1 1 1 12` it Anal lane Penod T -+ l l l l J~ 1 ~ F 4 = ~ , r 0 200 400 600 800 1000 1200 1400 Minor Stream Flow, vehthr Figure 16. Example of We Effect of Changing T in Me Delay Model Although more generalized capacitor models have been developed by considenng non-random major steam headway and using vaned cntical gap and follow-up time values, capacity models proposed by Harders and Siegioch can generally give reasonable results, when random major stream headway and constant critical gap and follow-up time are used. Various delay models were developed based on Me degree of saturation (either volume/capacity ratio or reserve capacity. The accuracy of delay estimation primarily depends on accurate capacity estimation. Time~ependent delay models should be used under oversaturated conditions.

28 .,