Capacity at Unsignalized Two-Stage Priority Intersections

Transcription

1 Capaity at Unsignalized Two-Stage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minor-street traffi movements aross major divided four-lane roadways (also other roads with two separate arriageways) at unsignalized intersetions. The enter of the intersetion, orresponding to the width of the median, often provides room for drivers who have rossed the first half of the major road to stop before proeeding aross the seond major traffi stream. This situation, whih is ommon with multilane major streets, is alled two-stage priority. Here the apaity for minor-street through traffi is larger than at intersetions without suh a entral storage spae. The additional apaity being provided by these wider intersetions annot be evaluated by onventional apaity alulation models. An analytial theory is presented for the estimation of apaity under two-stage priority onditions. It is based on an approah by Harders although major improvements were neessary to math the results with realisti onditions. In addition to analytial theory, simulations were performed that enable an analysis under more realisti onditions. The result is a set of equations that ompute the apaity for a minor-street through-traffi movement in the two-stage priority situation. eywords: apaity, unsignalized intersetion, two-stage priority, median entral storage. Author s address: Prof. Dr.Werner Brilon Dr. Ning Wu Ruhr-University Institute for Traffi Engineering D Bohum Germany

2 Werner Brilon and Ning Wu page Capaity at Unsignalized Two-Stage Priority Intersetions Werner Brilon and Ning Wu INTRODUCTION At many unsignalized intersetions, there is a room in the enter of the major street where several minor-street vehiles an be stored between the two diretions of traffi flow on the major street, espeially in the ase of multilane major-street traffi. This storage spae within the intersetion enables the minor-street driver to pass eah of the major-street streams one at a time, whih an ontribute to an inreased apaity. Part q 5 Output line STOP spaes for pass. ars Input line q q 2 Part I STOP q 8 Figure : Minor-street through traffi (Movement 8) rossing the major street in 2 phases (the theory disussed here is also available if major street provides more or fewer than two lanes per diretion).

3 Werner Brilon and Ning Wu page 2 A model is needed that an desribe this behavior and its impliation for intersetion apaity. A model of this type has been developed by Harders (). His onept has been used here as the basis; it is desribed in the following derivations. However, some major amplifiations as well as a orretion and an adjustment for reality were made to ahieve better orrespondene with realisti onditions. For these derivations an intersetion onsisting of two parts is used (Figure ). Between the intersetion Parts I and there is a storage spae for vehiles. This area has to be passed vehiles turning left from the major street (Movement ) and by minor-street through traffi (Movement 8). Also, a minor-street left-turning vehile (Movement 7, not shown) has to pass through this area. It will be seen that Movement 7 an be treated lie Movement 8. Therefore, these derivations onentrate on the minor-street through traffi (Movement 8) rossing both parts of the major street. The enumeration of movements has been hosen in aordane with Chapter of the 994 Highway Capaity Manual (2). It is assumed that the usual rules for unsignalized intersetions from the highway ode are applied by drivers at the intersetions. Thus Movements 2 and 5 (major-street through traffi) have priority over all other movements. Movement vehiles have to obey the priority of Movement 5, whereas Movement 8 has to give the right-of-way to eah of the movements shown in Figure. In these derivations, Movement 5 stands for all major-street traffi streams in Part of the intersetion. These, depending on the layout of the intersetion, ould inlude through traffi (Movement 5), left turning traffi (Movement 4) and right turning traffi (Movement 6). ANALYTICAL CAPACITY MODEL To determine the apaity of the whole intersetion, a onstant queue on the minor-street approah (Movement 8) to Part I is assumed. Let w i be the probability for a queue of i vehiles in the entral storage spae. Then the probabilities w i for all of the possible queue lengths i must sum up to with i, that is,

4 Werner Brilon and Ning Wu page 3 w i = i= () where is the number of spaes in the entral storage area. The entral storage area of the intersetion is onsidered as a losed storage system limited by the input line and output lines (Figure ). The apaity properties of the storage system are restrited depending on the aspets of maximum input and maximum output. Now different states of the system may be distinguished. State Part I of the intersetion determines the input to the storage area. Under state in situations during whih i vehiles in the storage area are less than the maximum possible queue length, that is, i <, a minor-street vehile (Movement 8) an enter the storage spae if the major-street streams (Movements and 2) provide suffiient gaps. In this ase the apaity of Part I (possible input from Movement 8) haraterizes the apaity; that is, = I = (q + q 2 ) (2) where I = (q + q 2 ) is the apaity of Part I in ase of no obstrution in Part, whih is the apaity of an isolated unsignalized ross intersetion for through minor-street traffi with major-street traffi volume q + q 2. The probability for this State is p = - w. Thus, the ontribution of State to the apaity of Part I for Movement 8 is I, = ( - w ). I (3) Of ourse, during State vehiles from Movement an also enter the storage spae. State 2 For State 2 the storage area is assumed to be oupied; that is, vehiles are queuing in the storage spae. In this ase, normally no minor-street vehile (Movement 8) or vehiles from Movement an enter the storage area. If, however, a suffiient gap for the passage of one

5 Werner Brilon and Ning Wu page 4 minor-street vehile an be aommodated in both Parts I and of the intersetion simultaneously, a vehile an also enter the storage area. The apaity for q 8 (possible input from Movement 8) during this stage is 2 = I+ = (q + q 2 + q 5 ) (4) where I+ = (q +q 2 +q 5 ) is the apaity of the whole intersetion (Part I+) as an isolated ross intersetion for through traffi with major-street traffi volume q + q 2 + q 5. Thus, the ontribution of State 2 to the apaity of Part I is I,2 = w. I+ (5) where w is the probability that vehiles are in the storage spae. States and 2 exlude eah other. The apaity of Part I is the total maximum input to the storage area. Here the volume q of Movement has to be inluded to the partial apaities mentioned above. Therefore, the total maximum input to the storage area is Input = I, + I,2 +q = (-w ). I + w. I+ + q (6) State 3 The output of the storage area is now onsidered, onentrating on Part of the intersetion. For i > eah possibility for a departure from the storage area provided by the major-street stream of volume q 5 an be utilized. The apaity (maximum output of the storage area) of Part in this ase is 3 = = (q 5 ) (7) where = (q 5 ) is the apaity of Part in ase of no obstrution by the upstream Part I, whih is the apaity of an isolated unsignalized ross intersetion for through minor-street traffi with major traffi volume q 5.

6 Werner Brilon and Ning Wu page 5 The probability for this state is p 3 = - w. Thus the ontribution of State 3 to the apaity of Part is,3 = ( - w ). (8) where w is the probability that no vehiles are in the storage spae. No vehiles from Movement (volume q ) an diretly (i.e., without being impeded by Movement 5) pass through the storage area in this state. State 4 For i = (i.e., an empty storage area) no vehile an leave the storage area even if the major-street stream of volume q 5 allows a departure. If, however, a suffiient gap is provided in the major-street streams of both parts of the intersetion simultaneously, a minor-street vehile (Movement 8) an pass the whole intersetion without being queued somewhere in the storage area. The possible output of the storage area from Movement 8 vehiles during this state is 4 = I+ = (q + q 2 + q 5 ) (9) Thus, the ontribution of State 4 to the apaity of Part is,4 = w. I+ () Vehiles from Movement an also pass through the storage area during this state. The number of vehiles from Movement that pass through the storage area during this state is,4,q = w. q () Here,,4,q does not mean the apaity for q, but the demand on the apaity. The traffi intensity of q should be less than the apaity of the Part ; that is, q is subjet to the restrition q <. Otherwise, the intersetion is overloaded and due to this nonstationarity no solution an be derived.

7 Werner Brilon and Ning Wu page 6 States 3 and 4 exlude eah other. Therefore, the total maximum output of the storage area is output =,3 +,4 +,4,q = (-w ). + w. I+ + w. q = (-w ). + w. ( I+ + q ) (2) One might argue that the derivations of I,2 and,4 neglet the travel time of the vehiles from Part I to Part. This, however, is justified. The probability that a minor-street vehile will meet a suffiient gap in Part I and at time t I and time t (with t = t I + t and with t = travel time between the stop lines of Part I and ) is independent of the travel time t if t = onstant for all vehiles and if the two arrival proesses in the major-street streams are independent of eah other. Therefore, the result is the same if t has a realisti positive value or if t is assumed to be. During times when the whole intersetion is operating at apaity beause of ontinuity, the maximum input and output of the storage area must be equal. Therefore input = output (f. Equations 6 and 2); that is, (-w ). + w. I+ + q = (-w ). + w. [ I+ + q ] (3) The total apaity T for minor-street through traffi (Movement 8) regarding the whole intersetion is idential to both sides of this equation minus q. In addition, sine negative traffi volumes are not possible T must fulfill the restrition T output q =, +, +,, q q = max (4) For the easiest ase of =, w + w = (5) Together with Equation 3 and the subsequent explanation,

8 Werner Brilon and Ning Wu page 7 T = ( ) q 2 I I+ I 2 I+ + q for = (6) For > some more general derivations are neessary for whih the following simplifying onditions are assumed: Let q, q 2 and q 5 be onstant over time. Then I = (q + q 2 ), = (q 5 ), and I+ = (q + q 2 + q 5 ) are also onstant over time. Divide the ontinuous time sale into intervals of duration t f = follow-up time = average time interval between the departure of two subsequent minor-street vehiles that enter into the same gap of the major-street flow. It is also assumed that the minimum gap between two vehiles of Movement is of the same size as t f. Let a be the probability that a vehile an enter the entral storage area from intersetion Part I during a time interval of duration t f. Let b be the probability that a vehile an pass intersetion Part during a time interval of duration t f. The a and b variables are introdued only for the following derivations. They need not to be evaluated later for the appliation of the theory. Both a and b are used for the fititious ase in whih Parts I and are independent intersetions. The follow-up time t f for Part I and should be of similar duration for this derivation. Treating the proess of the number of vehiles in the storage spae as a stohastial proess with Marow properties yields w ( t ) = w a w a b w b a ( ) + + ( ) (7) = probability that no vehile is queuing in the storage area at time t This is valid beause the ase of an empty queue at time t an be ahieved by the following possibilities: No queue at time t - t f (prob. = w ) and no arrival (prob. = - a) during t f ; or

9 Werner Brilon and Ning Wu page 8 No queue at time t - t f (prob. = w ) and one arrival (prob. = a) and one departure (prob. = b) during t f ; or One vehile queued at time t - t f (prob. = w ) and no arrival (prob. = - a) and one departure (prob. = b) during t f. By similar onsiderations, the expression for the probability of i vehiles queuing in the storage spae at time t is w i (t) = w. i- a. ( - b) + w. i a. b + w. i ( - a). ( - b) + w. i+ ( - a). b (8) Sine is the maximum number of vehiles in the storage spae, w (t)= w. ( - b) + w. a. b + w. - a. ( - b) (9) Beause of the assumed stationarity of the proess, w, w i, and w do not depend on eah other at time t. Equations 7 through 9 form a system of + equations that an be written as ( ) ( ) w a ab + w b ab = (2) [ ] ( ) ( ) ( ) ( ) wi a ab wi a ab + b ab + wi + b ab = (2) ( ) ( ) w a ab w b ab = (22) For abbreviation A = a a b (23) B = b a b (24)

10 Werner Brilon and Ning Wu page 9 The system of Equations 2, 2, and 22 is then () -A w +B w = () A w -(A+B) w +B w 2 = (2) A w -(A+B) w 2 +B w 3 =... (i) A w i- -(A+B) w i +B w i+ = (25)... (-2) A w -3 -(A+B) w -2 +B w - = (-) A w -2 -(A+B) w - +B w = () A w - -B w = From the first Equation, A w = B w w A = B w (26) From the last equation, A w = B w w A = B w (27) Summing all equations () through (i), A wi + B w i + = A w B w i+ = i (28) The sequene of the probabilities, therefore, is forming a geometri series in whih eah subsequent term is resulting from the prior term by a multipliation with the fator y = A/B. A a ab y = = B b ab (29)

11 Werner Brilon and Ning Wu page That is, wi+ = y wi (3) or i w = y w (3) i Aording to Equation, w i (i =,...,) are subjet to the restrition i= i= w w y i i i= = w = y i = Therefore, w = (32) + y + y y 2 The denominator is the sum of a finite geometri series, i= y i = + y y (33) Thus, and with Equations 3 and 29, w = y y (34) + w = y y + + y (35)

12 Werner Brilon and Ning Wu page Let us now reall Equations 3 and 4 and ombine those with Equations 34 and 35: y y = + + y y y + I + + q y y + y + I+ y y + I ( q ) (36) Note that in this equation the apaities I, and I+ as well as are treated to be nown, whereas the variable y has to be obtained from the equation. As a result, I I+ y = q I+ (37) Using this result for y, the total apaity T for the minor-street Movement 8 is alulated using Equation 4 yields T y y ( I + q ) q (38a) + y y = or T y = + y = + y ( q ) [ y ( y ) ( q ) + ( y ) ] y + + y I+ I+ (38b) It should be noted that for the speial ase of =, the algebrai solution of Equation 6 might give some onfirmation for the above derivations. For y = ( i.e., I = - q ) this expression is not defined. By developing the limiting ase for y, T = + [ ( q ) + ] I+ (39) At this point it should be noted that the apaities I+ = (q + q 2 +q 5 ) and = (q 5 ) an be alulated by any useful proedure, for example, by formulas from gap aeptane theory. But

13 Werner Brilon and Ning Wu page 2 solutions from the linear regression method or Kyte s method, desribed otherwise (3), ould also be used. CAPACITY ACCORDING TO GAP ACCEPTANCE THEORY The simplest formula for the apaity of an unsignalized intersetion with one minor-street and one major-street traffi stream is Siegloh s (4) formula. Several authors (3) have shown that this formula produes also realisti results if the basi assumptions for the formula are not fulfilled. Siegloh s formula is as follows: q t ( q) = e (4) t f where (q) t f =apaity for minor-street movement (veh/s), =follow-up time (s), =average gap between two suessive minor-flow vehile entering into the same major-stream gap, t t =t - t f / 2 (s), =ritial gap (s), =average gap between two suessive major-flow vehile that, as a minimum, is aepted by the minor-stream vehiles to ross the intersetion. The different ases of t - and t f - values that must be distinguished are: t - and t f - values for Part I of the intersetion (State and 2), t - and t f - values for Part of the intersetion (State 3 and 4), and t - and t f - values for rossing Part I and of the intersetion simultaneously in the ase of =. It is realisti to assume that a driver who has to ross the whole major street at one time without having a entral storage area needs longer t - and t f - values than in the first and seond ases.

14 Werner Brilon and Ning Wu page 3 It is justified to assume that the t - and t f - values in the firs and seond ases are of the same magnitude and that espeially the t f - values between both ases are nearly idential. This assumption is important for the following derivations. Realisti values for the t - and t f - values an be obtained from Table. The given ritial gaps t and follow-up times t f are of realisti magnitude ompared with the measurement results wored out by the NCHRP-projet 3-46 (5, 6). Here the ritial gap and the follow-up time for the ase without entral reserve ( = ) are larger then for the two-stage priority ase, whih seems to be more realisti. = i.e. no entral reserve ase ) i.e. a entral reserve of variable (with ) width part I part ase a) ase b) t 7. s 6. s 6. s t f 3.8 s 3.8 s 3.8 s Table : Typial t - and t f - Values for Two-Stage Priority Situations Within Multilane Major Streets under US-Conditions Based on Equation 4 with the assumption that all of the t f - values are nearly idential, I+ = I (4) where = /t f is the maximal apaity for the ase of no ross traffi in the major-street streams in vehile per seond. This relation maes it possible to standardize all of the apaity terms by. If is used in units of vehile per seond, the other apaity terms must use this unit. Of ourse, the unit vehile per hour ould be used for all of the apaity terms. Then it is useful to standardize T in Equations 38 and 39:

15 Werner Brilon and Ning Wu page 4 ˆ T T = (veh/s) (42) Then ĉ T (whih has to be obtained from Equations 38 and 39) an be expressed as a funtion of I / =(q +q 2 ) / and ( - q ) / = ((q 5 ) - q ) /. Thus it is possible to indiate the results of these derivations using graphs (Figure 2) T/ (-) (q+q2)/ (-) [(q5)-q]/ (-) Figure 2: Total apaity ˆ = as a result of Equation 42 (in ombination T T with Equation 38) in dependene of I / = (q + q 2 ) / and ( - q ) / = [(q 5 ) - q ] / for =. Use of graphs of this type with suffiient approximation is also justified in irumstanes that differ from the onditions of gap aeptane theory, for example, If apaities I and are alulated from theories other than gap aeptane or even if they should be measured, or If within gap aeptane theory the ritial gaps t are different for eah part of the intersetion.

16 Werner Brilon and Ning Wu page 5 The only neessary ondition for the appliation of these graphs is that the follow-up times t f be of nearly idential magnitude. SIMULATION STUDIES To test the theory leading to Equation 38, the solution has been further investigated using simulations. For this purpose a simulation model was espeially developed (7). The basi struture of the model is losely related to the ideas of KNOSIMO (8). The important features an be haraterized as follows: The headways in the major-street streams are distributed aording to a hyperlang distribution (8, 9). The ritial gaps and the follow-up times are distributed aording to an Erlang distribution with the parameters given by Grossmann (8) whih are also used in KNOSIMO. Both these assumptions together relate the model loser to reality than the theoretial derivations mentioned earlier. On the other hand, the following assumptions are a simplifiation ompared with reality. No delays due to limited aeleration or deeleration of the vehiles are taen into aount. The travel time t between the two parts of the intersetion are not onsidered, that is, t =. (see disussion following Equation 2). Eah minor-street driver has a minimum delay of t f at the first part of the intersetion if no major-street stream vehile is nearby. This simulates the time whih a driver needs to realize the traffi situation on the major-street when he first approahes the intersetion. This time margin is also neessary for the driver to deide if he an enter the intersetion. Suh an orientation time is not applied for vehiles entering the seond part of the intersetion, where a better visibility is assumed. All traffi volumes are ept onstant over time.

17 Werner Brilon and Ning Wu page 6 The program is organized so that a onstant queue in front of the first stop line of Movement 8 is always maintained. Thus, the maximum number of vehiles that an enter the intersetion an be evaluated. This number is the representation of the apaity for Movement 8. A omprehensive set of simulation runs has been performed for different parameters q, q 2, and q 5. Different attempts were made to find an easy-to-use approximate desription of the results, several of these attempts are given elsewhere (5, 7) together with a statistial assessment of their preision. A good ompromise between easy appliation and highest preision seemed to be the following solution. Instead of T a more realisti solution Tr is used, whih is obtained as a good approximation to the simulation results. Tr = α (veh/s) (43) T where Tr = realisti total apaity for Movement 8 (minor-street through traffi), T = result from the theoretial approah obtained from Equation 38 or from α Figure 2, and = adjustment fator for = =. 32 exp( 3. ) for > (44) These solutions for the total apaity Tr of Movement 8 approximate the simulated results with a standard deviation s (between results for T being simulated and those estimated from Equation 42) aording to Table 2. Other solutions with smaller deviations but more ompliated formulas for the alulation of realisti Tr an be obtained from wor by Brilon et al.(5).

18 Werner Brilon and Ning Wu page 7 s q = 5 q = q = 2 α = eq veh/h veh/h veh/h Table 2: Standard Deviation s for Computed T - Values Compared with Simulated Results To onlude the steps neessary to estimate the realisti apaity of an unsignalized intersetion where the minor-street movements have to ross the major street in two stages (f. Equations 43, 38b, 39, and 44): Tr = α T with T + = y + [ y ( y ) ( q ) + ( y ) ] [ ( q ) + ] I+ I+ for for y y = for = α =. 32 exp( 3. ) for > y = I q I+ I+ where I = (q + q 2) = apaity in Part I, = (q 5 ) = apaity in Part, I+ = (q +q 2 +q 5 )

19 Werner Brilon and Ning Wu page 8 = apaity in a ross intersetion for minor-street through traffi with a major-street traffi volume of q +q 2 +q 5 (all apaity terms apply for Movement 8; they are to be alulated by any useful apaity formula, e.g., the Siegloh-formula, Equation 4), q = volume of priority street left turning traffi in Part I, q 2 q 5 = volume of major-street through traffi oming from the left in Part I, and = volume of sum of all major-street flows oming from the right in Part. [Of ourse, here the volumes of all priority movements in Part have to be inluded. major-street right (6, exept if this movement is guided along a triangular island separated from the through traffi), major-street through (5), major-street left (4); numbers of movements aording to HCM 994 (2), Chapter ]. These equations are only valid for q >. Tr is the modified total apaity of the intersetion for minor-street through traffi. To simplify the alulation proedure, graphs for alulating the apaity Tr were produed by Brilon et al. (5). These graphs enable easy appliations of the theory in pratie. CONCLUSION The two-stage priority situation as it exists at many unsignalized intersetions within multilane major streets provides larger apaities ompared to intersetions without entral storage areas. Capaity estimation proedures for this situation have not been available up to now. In this paper, an analytial solution for this problem is provided. In addition, simulation studies lead to a orretion of the theoretial results. These proedures are already inorporated in the proposed Highway Capaity Manual due 998.

20 Werner Brilon and Ning Wu page 9 Nevertheless, an empirial onfirmation of this model approahes would be desirable. Also the question of the validity of the model for larger -values should be disussed. It is questionable whether the theory also applies for a grid of one-way street networs. Also if these questions are addressed in the future, the theory presented here is reommended for use at unsignalized intersetion in pratie. Delay estimations for the two-stage priority situation an be performed using the onept of reserve apaities () or the general delay formula by Kimber and Hollis (). REFERENCES. Harders, J. Die Leistungsfaehigeit niht signalgeregelter staedtisher Verehrsnoten (The apaity of unsignalized urban intersetions). Shriftenreihe Strassenbau und Strassenverehrstehni, Vol. 76, Speial Report 29: Highway Capaity Manual. TRB, National Researh Counil, Washington, D.C., Brilon, W., R. J. Troutbe, and M. Traz. TR Cirular: Review if International Praties Used to Evaluate Unsignalized Intersetions. TRB, National Researh Counil, Washington, D.C., to be published. 4. Siegloh, W. Die Leistungsermittlung an Knotenpunten ohne Lihtsignalsteuerung (Capaity alulations for unsignalized intersetions). Shriftenreihe Strassenbau und Strassenverehrstehni, Vol. 54, Brilon, W., N. Wu, and K. Leme. Capaity at Unsignalized Two-Stage Priority Intersetions. Transportation Researh Reord, No TRB, National Researh Counil, Washington, D.C., yte, M., et al. Woring paper. NCHRP Projet TRB, National Researh Counil, Washington, D.C., Leme, K. Berehnungsverfahren für Knotenpunte ohne Lihtsignalanlagen - Weiterentwilung - (Calulation Methods for Unsignalized Intersetions - Additional Development -). Diplomarbeit, Lehrstuhl für Verehrswesen, Ruhr-Universität Bohum, Aug. 995.

21 Werner Brilon and Ning Wu page 2 8. Grossmann, M. Methoden zur Berehnung und Beurteilung von Leistungsfaehigeit und Verehrsqualitaet an Knotenpunten ohne Lihtsignalanlagen (Methods for alulation and assessment of apaity and traffi quality at intersetions without traffi signals). Ruhr-University Bohum, Germany, Chair of Traffi Engineering, Vol. 9, Dawson, R. F. The Hyperlang Probability Distribution - A Generalized Traffi Headway Model. In: Shriftenreihe Strassenbau- und Strassenverehrstehni, Vol. 86, pp. 3-36, Bonn 969..Brilon, W. Delays at oversaturated unsignalized intersetions based on reserve apaities. Preprint of a presentation at the TRB-onferene, Washington, Januar 995, to be published in a TRB-reord..Kimber, R. M. and E. M. Hollis. Traffi queues and delays at road juntions. TRRL Laboratory Report LR99, 979.