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APPENDIX D DEVELOPMENT OF UNIFORM DELAY EQUATIONS FOR TRAFFIC-ACTUATED CONTROL An analytical model for the estimation of phase times, cycle lengths and volume/capacity ratios was described in detail In Appendix C. This information is used by the HCM Chapter 9 delay model to assess the level of service ~OS) for each movement. The computational structure of the Appendix C timing estimation model introduces an improved method for delay estimation at a traffic-actuated intersection. Potential improvements to the HCM Chapter 9 delay model are presented In this appendix. The traditional delay formulation used by virtually all analytical models is based on two components, or terms, which are added together to produce the computed delay per vehicle: I. The un~forrn delay term, which determines the delay that would occur if all vehicles arrived in a completely uniform manner. This term is computed as the area under the queue accumu- lation polygon (QAP). 2. The, incremental delay term, which adds a correction factor to compensate for randomness in the amval patterns and the occasional oversaturation that may result. Since the QAP's must be developed in detail by the signal timing estimation model described in Appendix C, the value of the uniform delay term (i.e., the area contained within the QAP) may be computed by a simple extension to the signal timing estimation procedure. The detailed development of the equations for computing the QAP areas is presented here. There are nine distinctive shapes that may be taken by the QAP, each of which is associated with a specific phasing alternative. Each of these cases requires a different computational formulation, or set of equations. Each case is developed as a separate figure in this report. Each figure shows the shape of the QAP and presents the derivation of an equation that determines the uniform delay by computing the area under the associated polygon. A summary of the phasing alternatives illustrated in Figures D-] through D-9 is presented in Table D-~. The derivations are presented in a format similar to the format used in the supplemental worksheet now contained in HCM Chapter 9 for uniform delay computations with compound left turn protec- tion. In all cases, the uniform delays may be determined using only those variables (volumes, satura- tion flow rates and signal timing) that are used by the existing HCM Chapter 9 methodology. Each of the figures conforms to a common terminology with respect to its labeling. Intervals are illustrated along the horizontal axis as follows: Appendix D: Page
r indicates the effective red time g indicates the elective green time gq indicates the portion of the permitted green time blocked by a queue of opposing vehicles gu indicates the portion of the permitted green time not blocked by a queue of opposing vehicles indicates the portion of the protected green time required to service the queue of vehi- cles accumulated on the previous phases indicates the extension to the protected green time that occurs while the controller waits for a gap in the amving traffic long enough to terminate the phase gf indicates the portion of the green time in which a through vehicle in a shared lane would not be blocked by a led turn vehicle waiting for the opposed movement to clear. This condition occurs ondy at the beginning of the permitted green when one or more through vehicles are at the front of the queue. Points in the cycle at which the queue size is important to the computations are also identified as fol lows: Qr indicates the queue size at the end of the elective red Qq indicates the queue size at the end of the interval gq Qp indicates the queue size at the end of the permitted green period Q'p indicates the queue size at the end of the perrnined Been period, adjusted for sneakers Qg,, indicates the queue size at the beginning ofthe protected green (green arrow) period Qf indicates the queue size at the end of the interval gf The flow rates that determine the slopes ofthe various sections of the QAPs are indicated es follows: qa indicates the arrival flow rate' which is assumed constant over the cycle s indicates the saturation flow rate for protected movements sp indicates the saturation flow rate for permitted movements. Appendix D: Page 2
The slope ofthe lines representing the departure of vehicles Tom the queue generally represents the net departure rate, i.e., the saturation flow rate minus the amval rate. Note that, in cases where protected phases exist, these are arranged to be the last to occur. The length of these phases will be determined by their detector actuations. The actual length will be the sum of the time required to service the queue that exists at the beginning of the phase plus the extension time. Table D-~. Summary of phasing alternatives for the computation of uniform delay Figure l Description of Phasing Alternative 1 D-1 | Single protected base l D-2 | Permitted left fur s from an exclusive lane. l D-3 | Permitted left tur i from a shared lane (gq > g] Dot | Permitted leR tur i from a shared lane (gq ~ g) D-5 | Compound left to n protection: HCM Chapter 9 Case 1 D-6 | Compound left to n protection: HCM Chapter 9 Case 2 D-7 | Compound left tu n protection: HCM Chapter 9 Case 3 D-8 | Compound left tu I protection: HCM Chapter 9 Case 4 . . D-9 | Compound left tu ~ Appendix D: Page 3
- ~ d ._ U) - C, ._ o lo Qr = r qa Qq Qr Qp = Qp/ = 0 1 qa~ / r A Single Protected Phase Qr . .. .. .. l i s-q \QP= Qp =0 Time (seconds) 0.5 ~ r Qr + Qr2 / (s - qa) ~ forisolated operation D1 - qa C D _ 0 5 [ r Qr + Qr2 / (s - qa) ~ (1 - P) f for coordinated 1 - qa C 1 - (g / C) phases 2 & 6 only Figure D-1. Uniform delay computation for a single protected phase Appendix D: Page 4
Permitted Left Turns from an Exclusive Lane , ~I ~ I ~I ._ cn - ._ ;> o Qr r qa Qq Qr gq qa Condition 1: QST < gu Qp = Qp/ = 0 D. = Condition 2 QST ~ gu Q = Q ~ P q Qpt = 0 D1 = Qq . _.._. ._ .._ Qr / l / qa ~ ~/;1 -/W . 1 _ I ~, P qa ~Qp=QP' .Qp' Time (seconds) O.5[(r+ gq)Qq+Qq /(sp qa)] qa C gu (sp qa) 05 [ (r + gq)Qq + gu (Qq + Qp)] qa C QST= Qq S. ~ q Figure D-2. Uniform delay computation for permitted left turns from an exclusive lane Appendix D: Page 5
Permitted Left Turns from a Shared Lane (:q > ~f) - ._ U) - C~ ._ ~C o o z Qr r qa Qr 76. r e ~ ' Time (seconds) Qf = Qr gf (S qa) Qq = Qf + (gq gf) qa Q = Qf + (gq gf) (qa 1 + PL (EL2 Condition 1: QST < gu QST = Qq S. ~ q ~pp > 1 if n0pp = 1 Qp = Qpl = 0 D = 0 5 [r Qr gf (Qr Qf) + (gq gf) (Qf + Qq) + Qq / (Sp ~ qa): qa C Condition 2: QST ~ gu Qp = Qq ~ gu (Sp qa) Qpl = 0 [ Qr gf (Qr Qf) (gq gf)(Qf Qq ) gu (Qq QP )] qa C Figure D-3. Uniform delay computation for permitted left turns from a shared lane (gq ' gr) Appendix D: Page 6
Permitted Left Turns from a Shared Lane (:q < gf) o~ ._ u, ._ o o z Qr = r qa Qf = Qr gf (S ~ qa) Qq = Qr ~ gq (S qa) Condition 1: QST < gu Qp = Qp/ = 0 Qr ~/7 S-qa WF;7~ r \: Qq l;;''-''-''-'' \e Qf 1 ' :~a Q gqll I Qp=Q/\ ~ Qp' Time (seconds) QST= Qq S. ~ q 0 5 ~ r Q + gf (Qr + Qf) + Qf2 / (Sp - qa): Condition 2: QST > gu Qp = Qf gu (Sp qa) Qp' = 0 [ Qr gf (Qr Qf) gu (Qf + QP )3 D1 = qa C Figure D-4. Uniform delay computation for permitted left turns from a shared lane (gq < gf3 Appendix D: Page 7
Compound Left Turn Protection (Case 1) ~1 c: i ._ u, - c~ ._ ~: o o z Qr r qa Qq = gq (qa sq) Sq = 0 s s = q E L2 Condition 1: QST ~ gu Qp = Qp/ = 0 Qr / i\ 1 \ 1 I ~ S ~ qa Qq ~-- .. qa~ S~: qa i (~: QP/ V i 1 1 l Time (seconds) ~nOPP > I (where s'=s/0.95) -ifnOpp=1 QST = Qq S. ~ q D _ 05 [ r Qr + Qr / (s ~ qa) + gq Qq + Qq2 / (sp - qa)] 1 qa C Condition 2 QST > gu Q = Qq ~ gu (Sp Qp~ = 0 D, = qa) O.S ~ r Qr + Qr / (S ~ qa) + gq Qq + gu (Qq QP) ~ qa C Figure D-S. Uniform delay computation for compound left turn protection (Case 1) Appendix D: Page 8
Compound Left Turn Protection (Case 2) 0D ._ v~ - ._ o o z Qr r qa Qq = Qp 1/ + gq (qa sq) Qp'/ = Qr g (s qa) Sq = 0 s (where s'=s/0.95) - ifn0pp= S - q E L2 Condition 1: QST ~ gu Qr ~ \x,: Sp- q / QP i I ~ QP/ I~ QP=Q~ ~S)P/' r \ ~S ~ qa Time (seconds) ~nOpp > 1 1 QST = Qq S. ~ q Qp = Qp/ = 0 D1 = [ r g (Qr QPN) gq (QP/I + Qq) + Qq / (sp ~ qa) ] qa C Condition 2 QST ~ gu QP = Qq gu (sp qa) Qp/ = 0 D = 0 5 [ r Qr + g (Qr + Qp//) + gq (Qp// + Qq) + gu (Qq Qp) ] qa C Figure D-6. Uniform delay computation for compound left turn protection (Case 2) Appendix D: Page 9
Compound Left Turn Protection (Case 3) :S ._ v, - v ._ o o z Qr = Qp / + r qa Qq = gq (qa sq = 0 s = q Qp = Qfq Qp / = 0Qp ~ s a j~Sp- qa~ qa- S. q V 1 Qr .. .. .. ~- Qp ', r ~ S ~ qa T~me (seconds) q) sl EL2 gu (Sp qa) if nOpp > 1 (where s' = s / 0.95) if nOpp where Sa = Min (Sneakersmax ~ Qp) D _ 05 [ r (Qp' + Qr) + Qr / (s ~ qa) + gq Qq + gu (Qq + Qp)] 1 qa C Figure D-7. Uniform delay computation for compounc! left turn protection (Case 3) Appendix D: Page 10
Compound Left Turn Protection (Case 4) :~ 5 ._ U) C, ._ 5: o o z Qr r qa Qq = Qr + gq (qa sq) sq = 0 s = q s/ EL2 Condition 1: QST < gu Qp = Qp/ = 0 Qq q S. L~ / Qr ~ ~1 Sp-qa I N: I Qp- Qp~ Time (seconds) QST= Qq S. ~ q if nOpp > 1 (where s' = s / 0.95) if nOPP 1 = 0.5 [ r Qr + gq (Qr + Qq) + Qq / (sp qa) ] qa C Condition 2 QST ~ gu Qp = Qq ~ gu (sp qa) Qp/ = 0 D = 0 5 ~ r Qr gq (Qr Qq) + gu (Qq + Qp) ] Figure D-8. Uniform delay computation for compound left turn protection (Case 4) Appendix D: Page 11
Compound Left Turn Protection (Case 5) :- - ~: ._ u, c~ ·_ ~: c~ o o z Qr r qa Qq = Qr + gq (qa sq) sq = 0 Sq = Qq qa- Sq ~W 1 W: Sp - qa ~ ....... i ~S~qa Qr - .. .. .. .. ,~ / q~7/ Qp Q~ Time (seconds) ~nO" >! Es (where s'=s/0.95) ifnOpp=T L2 Qp = Qq gu (Sp qa) Qp / = 0Qp ~ Sa where Sa = Min (Sneakers~:, Qp) D = 0 5 [ r Qr gq (Qr Qq) gu (Qq Qp) Qp~ / (s qa) ] qa C Figure D-9. Uniform delay computation for compound left turn protection (Case 5) Appendix D: Page 12
