The Analysis and Design of

Transcription

1 The Analysis and Design of Freeway Entrance Ramp Control Systems H. NATHAN YAGODA, Comptran Systems Corporation, Wayne, N.J.; and LOUIS J. PIGNATARO, Polytechnic Institte of Brooklyn This paper describes some of the reslts obtained from an analytic stdy ndertaken as part of the Glf Freeway Srveillance and Control Project in which the on-line, dynamic control of individal entrance ramps is investigated. For this stdy 2 alternate control philosophies are considered. The first philosophy concerns the control of ramps on which a string of one or more vehicles is released when a sitable merge opportnity arises and when all previosly released vehicles have sccessflly merged into the freeway stream. The second control philosophy concerns ramps on which the reqirement that the ramp be cleared of all previosly released vehicles is relaxed to increase the merge capacity of that ramp. In this second case, the controller may release an additional vehicle whenever the expected delay associated with attempting a merge into a detected gap is less than the expected delay associated with waiting for the ramp to clear. Here the controller is a seqential decision-maker that evalates the expected delay associated with all previosly released vehicles that have not yet merged and reflects this information into the control process. Fixedtime and demand-capacity metering controllers are special cases of the control system analysis presented. SEVERAL FREEWAY CONTROL projects are presently nder way in the United States, and each is attempting to improve the operational characteristics of a freeway by se of an entrance ramp control system. However, the operational modes for the respective control systems and the rationale that nderlies individal control system designs appear to be qite different. As a reslt of this diversity, there appears at first to be no clear-ct design philosophy with which to approach a new access ramp control system design problem. Becase of this deficiency, a stdy grop at the Polytechnic Institte of Brooklyn ndertook the development of a more nified design theory for dynamic control systems for freeway entrance ramps. The stdy, initiated in Agst 1967, was part of the Glf Freeway Srveillance and Control Project condcted by Texas A&M University in Hoston, Texas, for the U.S. Brea of Pblic Roads. The portion of that stdy described in this paper concerns the design of a dynamic control algorithm for an arbitrary entrance ramp configration, sbject to the assmption that control is exercised by se of a green-amber-red traffic signal located at a fixed position on the ramp. Control systems designs that tilize other controller configrations, sch as speed signs or mltiple signal stations, are not inclded in this presentation. For the type of systems considered, 2 identifiable primary fnctions are (a)to improve the merge service offered to vehicles that enter the freeway and (b) to improve the operation of the freeway. The first fnction is performed by any system that better coordinates the arrival of the merge vehicle at the merge zone with the availability of a high-qality merge opportnity, and the second fnction is performed by any system that improves the freeway operational characteristics sch as volme, speed, density, Paper sponsored by Committee on Freeway Operations and presented at the 49th Annal Meeting. 56

2 accident rate, and occpacy. These 2 fnctions are not always compatible, bt there is evidence to indicate that compatible soltions do exist. In particlar, observation of the Glf Freeway reveals that operation nder conditions of ramp control (6) yields the following: - 1. The rsh period volme increased by abot 1 percent; 2. The speed on the test section increased by abot 3 percent; 3. The average travel time over a 5-mile section decreased from 16 to 11 mintes; and 4. The nmber of accidents dring the 2-hor morning peak decreased from 145 to 75 per year for the 3 inbond lanes over the 6.5-mile controlled section. DYNAMIC RAMP CONTROL To nderstand the natre of the control system design problem for a freeway entrance ramp reqired that initial consideration be focsed on the Gap Acceptance Control System. In this system any intervehiclar headway in the otside freeway lane in excess of T 1 seconds is defined as a gap into which a vehicle that seeks to enter the freeway may be placed. To achieve the placement of entering-vehicles into detected gaps, however, the gap detection process mst be carried on sfficiently pstream from the merge point to allow both for synchronization after sitable travel time of vehicles on the ramp and for srvival of detected :I \ THRESHOLD Figre 1. Gap detection rate as a fnction of threshold limit and lane volme. TL THRESHOLD Figre 2. Gap acceptance by driver as a fnction of gap size. Q_ " Figre 3. Gaps accepted and not accepted by drivers for moving merge. gaps dring the passage from the detection point to the merge point. In addition, the gap threshold T 1 mst be sfficiently large so that an acceptable portion of the drivers execte a synchronized moving merge. Within this framework the design problem is then to select locations for the traffic control signal and the gap detector eqipment and to specify the gap threshold Tl' Location of the control signal is restricted by the need for sfficient room to accelerate prior to the merge manever and by the desire to provide sfficient room for the qee that develops behind the signal. Coordinated with this is the need to place the gap detector as close to the merge point as possible to obtain the best possible gap srvival conditions. The choice at any particlar ramp is ths limited. The specification of T 1 to provide an acceptable rate of gap acceptance by merging drivers, however, is arbitrary and a large r ange for choice exists. When T 1 is chosen to be small, ther e are many, gaps and the smaller of these are likely to be rejected by the drivers. For larger vales of T 1 there are fewer gaps, bt the smaller of the gaps are deleted and the likelihood that a driver will reject the offered gap is redced. If the design concept is to be properly established, first consideration mst be given to the rate at which gaps (i.e., headways in excess of T 1 ) appear in a traffic stream of volme q. Figre 1 shows that the gap detection rate, µ(t q), is a fnction of both the threshold limit T 1 and the lane volme q. Then, it is necessary to note that for very large gaps the probability that a driver accepts the gap for a moving merge is approximately 1, and for seqentially smaller gaps the probability of acceptance by the driver is redced toward a finite limit, K, between and 1 (Fig. 2). This lower limit corresponds to the probability that a driver can force entrance into a stream at a point at which no headway was detected. Based on these 2 concepts, sbdivision of all gaps into 2 grops is then possible; the 2 grops inclde those that are accepted for a moving merge and those that are not accepted for a moving 57

3 58 >.(l} if 3 > r-+----t-, e:oo 9 ro.oo Time Figre 4. Mean of time-dependent Poisson process sed to characterize the ramp arrival process. 5!. :S OO 6 7= 8 9 K> t TIME Figre 5. Time -dependent freeway volme. merge (Fig. 3). Corresponding to this division, µa (T 1, q) is the rate at which a ccepted gaps are detected, and µr(t 1, q) is the rate at which r ejected gaps are detected; the sm of these qantities, µ(t 1, q) is the rate at which all gaps in excess of T 1 are detected, that is, Single-Vehicle-Release-After Completion Mode For any specified vale of gap threshold, T 1 = T, the rate at which gaps are detected is simply µ(t, q). Ths a driver who randomly arrives at the controller sffers an expected delay of 1/ µ(t, q) seconds before a gap is detected. Then, the driver mst proceed down the ramp for which the expected travel time is denoted by tr. In addition, the probability that the offered gap is nacceptable is eqal to the ratio of nacceptable gaps to total gaps, µr(t, q)/ µ(t, q). Hence, an extra expected delay associated with the rejection of an offered gap is (1) 1 µs(q) where µ 8 (q) is the occrrence rate of gaps that are acceptable for merge by drivers who have previosly rejected the gap offered to them and are stopped in the merge zone. These qantities can be sed to evalate the total expected service time for a driver who arrives at the ramp control signal for merge service onto the freeway as 1 µr{t, q) t = +t + e µ1t, q) r µ(t, q) 1. µs(q) (2) As a reslt, when operation is restricted to a one-at-a-time mode and release is predicated on the completion of the merge manever by all previosly released vehicles, a single-vehicle-release-after-completion system exists for which the ramp service rate µ' is a fnction of the threshold limit T and the volme q in the otside freeway lane. This service rate is 1 µs(q) µ(t' q) µ' = te = µr(t, q) + µ 8 (q)[l +tr µ(t, q)] (3) when te is large compared with 1/ q so that the time interval between the instants at which seqential gaps are soght is large compared with average headway in the stream. (Note that the condition probability of a gap at the instant t = T given a gap at the instant t = approaches the nconditioned probability of a gap at the instant t = T when T is made very large. For an Erlang headway process on the freeway, this is eqivalent to te» 1/ q.) The Design of a Controller for the Single-Vehicle-Release Mode One sefl and fairly common model for an rban freeway employs stochastic processes with slowly varying parameters to accont for both ramp arrivals and highway flows. In particlar, the peak-period ramp-arrival process is often described as a time-dependent Poisson process (!)with A.(t) as shown in Figre 4. In addition, the

4 intervehiclar spacings for vehicles on the highway are described as independent samples from an Erlang distribtion ' ch. 9): 59 a a-1 -aqt f(t) _ (aq).t e - (a - 1)1 (4) where a is an integer and q varies slowly as a fnction of time corresponding to average volme (Fig. 5). Based on these descriptions and sbject to the assmption that the process parameters vary slowly, one controller design philosophy is to maximize the service rate µ' sbject to a limitation of downstream freeway capacity. The service rate µ.' can be rewritten with the argments omitted as ' 1 1 µ. = tr + (µ.r + µ.s)/ µµ.s = tr + 1/ µ.h From this form it is seen that, for a given tr, the maxima of µ' correspond to the maxima ofµ.". Hence, setting dµ /dt = located the vales of T corresponding to the relative minima of µ. '. In particlar, (5) redces to (6) becase (µ,r + µ. 8 )2 is positive and finite, and µ 8 I. From Eq. 6, (d/dt) [ln (µ.) - ln ( + µ. 8 )J = O (7) Becase T corresponds to the threshold that is set on the minimm spacing between vehicles on the freeway into which a moving merge will be attempted, all spacings in excess of this threshold are gaps. For any given threshold limit, the rate at which gaps appear in a stream with volme eqal to q vehicles per second is - r= (aq)a ta-1 e-aqt dt µ. - q JT (a - 1}! a-1 i -aqt (agt) =qe.i 1 1 i=o When the probability that a presented gap of t is accepted by a driver for a moving merge is described c; ch. 9) by -Kt Pa (t) = 1 - e (8) (9) then the rate of sccessfl moving merges is = r= ( 1 - -Kt) ( aq) t e dt [ a a-1 -aqt] µ,a q JT e (a - 1)! = q. e -aqt L: (aq.) 1 _ (aq) a e -(aq+k)t :t (aq -i: )TJl (1) a-1 a a 1 1 i ==O i. (aq + K) i=o 1 '

5 6 and the rate of nsccessfl moving merges is a a-1 p _ (ag) e- (aq+k)t '"" [(ag + K)T] µ'r - q a n! (aq + K) n=o (11) When the indicated derivatives are evalated and employed in Eq. 6, then algebraic maniplations yield µ, = q e(aq+k)t {I:" ( (aq + K)a (aq)n - (aq + K)11 (ag)aj Tn} (12) S (aq + K)a n=o n! as the eqation from which the optimm threshold settings are obtained, sbject to the constraint T ". Becase this eqation is of the form µ 8 = f(t) where (13) and becase the sm Ths, f(t) > for T >. where Cn > for every n a-1 E Cn Tn > for T > O n=o f addition, the derivative of the expression f(t) is f'(t) = (d/dt) A e-bt a-1 E Cn Tn J [ n::o (14) (15) a-1 n] f'(t) == -KA(aq + K)a e-bt E (ag) [ n. n= (17) fspection of this qantity reveals that f'(t) is negative for all positive T. Therefore, f(t) is monotonic decreasing for T ;;;,. On this basis, the conclsion is that a niqe optimm soltion for the controller threshold T exists. That vale is when 1 [ a a] µ 8 " q a (aq + K) - (aq) (aq + K) Otherwise, the vale is the positive nmber Topb obtained as the soltion to Eq. 12. After Topt is evalated as described, the control policy is specified next. In particlar, when the sm of the volme on Lile freeway and the demand 11 Lhe ramp does not exceed the volme limitation for the freeway, a threshold of T opt is sed for the ramp. This ensres the highest pos sible service rate for demand on the ramp and reslts in minimm expected delay and minimm expected qee length. When the sm of the freeway volme and ramp demand exceeds tile volme limitation, a threshold of Topt is not acceptable. Instead a th1 eshold Tc mst be employed sch that the swn of tlie freeway volme and the served portion of the demand is eqal to or less fan the volme limitation. (18)

6 At this point, the heretofore ndefined qantities of freeway volme, ramp service, and volme limitation mst be considered more exactly. For this prpose it is noted that experience and traffic flow theory (3) indicate that the short-term prodction of any given point on a freeway cannot exceed some pper bond Q, approximately 2, vehicles per hor (vph), withot significant risk of breakdown in the flow of traffic. Ths the TA minte rnning average of flow past a critical point mst be limited to approximately QTA vehicles, depending on the facility. When this short-term average volme is below the specified capacity limit, additional vehicles from the ramp can be admitted into the stream, provided that the rnning average of the sm remains below the critical vale. Ths the average ramp service is limited at most to the difference between the limiting and actal average volmes. Many soltions are possible to the controller design problem that satisfy this restriction. One sch possibility allows the threshold to be set at T opt provided that the r estriction is met, and inhibits all merging otherwise. By this process all available roadway capacity is sed as qickly as possible, and additional waiting vehicles are inserted into gaps as additional room for single vehicles arises. This soltion, when associated with the limit condition in which Topt eqals, becomes the capacity adjsted metering system (4, 5). A second approach to the-controller design problem involves the gradal adjstment of the threshold as a fnction of freeway volme. This techniqe has the sal advantages associated with smooth variation in controller policy; however, it also has the added limitation, extra delay, associated with smoothing. In particlar, the first policy carries the risk of inserting an acceptable averaged nmber of vehicles into the stream too qickly ths casing breakdown, and the second policy incldes the risk of adjsting too slowly ths overloading the stream. The Mltiple-Vehicle-Per-Gap Merge Mode In the generalized release-after-completion merge mode, a string of vehicles is released to attempt to merge into a sitable single gap after all previosly released vehicles have completed the merge operation. To design a controller for this action reqires only the specification of the gap threshold Tn that corresponds to the smallest gap into which a string of n vehicles may attempt a merge. When this is done, the seqence of nmbers (Tn} determines the nmber of vehicles that may attempt to merge into any given gap. The sbseqent analysis and controller design is simplified by the assmption that the gap threshold is chosen so that the probability that the last vehicle in a released string of n fails to merge is sbstantially larger than the probability that 2 or more vehicles in the string balk. Ths, the expected service time ten for a string of n vehicles that is released to attempt to merge into a gap in excess of Tn bt less than Tn+l is ten = the expected travel time for the first vehicle to reach the merge zone + the sm of the expected intervehiclar headways between vehicles in the string + the expected extra delay de to gap rejection by the nth vehicle =tr+ (n - 1) h +(Prob n are released and the nth balks) (1/µs) ( ) ( I ) = tr + n - 1 h + 1 µs [ JJRn(T n) - µrn(tn+l)] [µ(tn) - µ(t 11+1)J where tr = the expected travel time for 1 vehicle on the ramp, h = the expected intervehiclar headway, µ 8 = the rate at which vehicles stopped at the merge zone execte merges into the freeway stream, µ(t) = the rate at which gaps in excess of the threshold setting of T appear in the stream, and µr (T) = the merge rejection rate for the nth vehicle in a string when the release n threshold is set at T. 61 (19)

7 62 Next, it is noted that the rate at which merge opportnities appear in the stream, µtotal' is co IJTotal = L n (the rate of merge opportnities for strings of exactly n vehicles) n=l co = L n[µ(tn) - µ(tn+1)j = L µ(tn) n=l n=l (2) Based on this, the expected service time associated with the merge opportnities offered by the stream Te is T e = r; [(merge opportnity rate for strings of n vehicles) (total expected delay] n=l for the merge of a string of n vehicles)/ (total merger opportnity rate) f: µ(tn) - µ(tn+l) {tr+ (n _ l)h + (l/µ ) [µrn(tn) - µrn(tn+l)j + 1 } n=l µtotal S µ(tn) - µ(tn+l) µ(t1) = 1 -{(tr - h)µ(tj + hµtotal + (1/ µs) I: [µrn(tn) - IJRn (Tn+l)] + 1} µtotal n=l co L [µrn(tn) - l"rn(tn+l)] + [ 1 +(tr -h) (T1)] I'S n=l = h l"sijtotal (21) Finally, µrn (T)>> µrn(tn+l) is implied by the assmption that the probability of gap rejection by the last released vehicle in a string is mch greater than the probability of a balk by any other vehicle in that string. Ths, co L: µrn(tn) + µs [1 +(tr - h) µ(t)j n=l Te= h J.lsllTotal and the merge capacity limit for the stream is (22) where 1 µstream = - = Te 1 µsµtotal h + -co =--- (h + t ') L µrn(tn) + µs[ 1 +(tr - h)µ(t1)] n=l t' =------µ_s_µ_t_o_ta_l co L µrn(tn) + IJS[l +(tr - h)µ(t1)] n=l (23) (24) Hence, the merge service capacity is maximized when t' is minimized. A controller design based on this is considered next.

8 The Design of a Controller for String-Release Mode The extension of controller action to inclde the release of strings of vehicles for insertion into adeqately large gaps offers several advantages at the expense of separate, individalized merge service. In particlar, the safety level associated with the singlevehicle-release-after-merge mode is exchanged for a higher merge capacity when demand indicates that this is necessary. The reslt is a control law that offers individal service when demand is low and that allows for extended merge capability when demand is high, provided that adeqate freeway capacity exists. Inspection of the expression for the capacity of the ramp controller, J.LStream reveals this qantity depends on the set of threshold settings (Tnl. In general, this expression need not be nimodal, and the coordinates of the local maxima are not always identified by a seqential search. There is a good basis, however, for setting the thresholds by a seqential optimization. In particlar, becase there is no information inclded in the model to indicate the actal demand for service, an increase in the ramp capacity by the increase of the 2-vehicle merge rate at the expense of the 1-vehicle merge rate is detrimental to system performance when the actal demand reqires only 1-vehicle merges. This sitation is contrasted by the following: When the vale of T 1 is selected to maximize the 1-vehicle merge process and then T 2 is selected to maximize the 2-vehicle merge process given T, 1 the resltant system always provides the highest 1-vehicle merge rate and yields extra capacity throgh 2-vehicle merges as reqired by demand. Althogh this controller may not provide as large a total ramp capacity as is available by the simltaneos selection of T 1 and T, 2 only the actal demand will determine nder which controller the ramp provides better service. When the intervehiclar headways on the freeway are described as independent sam - ples from the Erlang distribtion 63 a a-1 f(t) = (aq) t -aqt (a - l)! e (25) and the probability that the last vehicle in a string of n vehicles accepts an offered gap of t seconds or less is described by the cmlative Erlang sn-1 Pan(t) = 1 - e-snbnt L: m=o (26) the vales of J.LRn(Tn), µ(tn), and µ 8 can be evalated. Here and = - T a-1 (aqt }t Tn l=o µ(tn) = q f(t)dt = qe aq n L: t (27) a+m-1)! ] aq + snbn) a +m Likewise, when P s(t) is the standing merge headway acceptance probability,

9 64 and r-1 m """"" (a + m - 1) 1 (rc/ V ) µs = q.!i:o m! (a - l )! (1 + re aq) a + m (29) when Ps{t) is cmlative Erlang type r with mean eqal to l/c. The expression for ramp capacity for this case has been evalated for some examples sbject to the philosophy of seqential optimization. Capacity verss threshold is shown for otside lane volmes between 8 and 2, vehicles per hor in Figres 6 throgh 13 to indicate the system sensitivity to adjstment of the threshold T 1 Figres 14 throgh 21 show capacity verss string length for similar volmes. In all cases the lower order thresholds are set at their optimm vale before the threshold of interest is varied. In addition, it is assmed that the gap acceptance probability that corresponds to the end vehicle in a string is type 3 Erlang with a mean that varies linearly with string length. Finally, it is noted that when the maximm string length is restricted to 1, and the gap acceptance probability for single vehicles is Erlang type 1, the string merge capacity simplifies into the 1-vehicle merge capacity previosly developed. CAPACITY AND SENSITIVITY OF A STRING-RELEASE-AFTER-COMPLETION MERGE-CONTROLLER-AN EXAMPLE At this point the capacity of a ramp control system that releases a string of vehicles to attempt a merge into the freeway stream is considered. For this operational mode the nmber of vehicles in the string is dependent on the size of the detected gap, and release is conditional on the completion of sccessfl merges by all previosly released vehicles. Hence, the ramp is clear before an additional string is released. Several cases considered for the controller design presented in the previos section inclded the following: 1. Two alternate length entrance ramps were measred in "seconds of time between the instant the control signal is trned to green and the instant the first merge-vehicle arrives at the merge zone"; ths tr= 5 seconds and tr= 1 seconds are sed to investigate the effect of ramp length. 2. For alternate gap acceptance criteria were employed to describe drivers that balk at the moving-merge attempt. Here 1/c eqal to 1.5, 3., 4.5, and 6. seconds are respectively sed for the mean gap sizes that drivers who stop at the merge zone find acceptable for a standing merge. 3. Five freeway volmes that correspond to q eqal to 8, 1,1, 1,4, 1,7, and 2, vph were investigated to stdy the effects of variation in volme. In addition, the following parameter vales were employed as reasonable or typical or both: 1. The standing-merge gap acceptance probability is described as Erlang type 3, ths r = 3; 2. The moving-merge gap acceptance probability for the nth vehicle in a string is Erlang type 3, ths sn = 3 for all n; 3. The mean acceptable gap Ior Ute nth vehicle in a string is l/bn = 3n , ths the first, second, third, and nth vehicles accept a gap of 1. 5, 4. 5, 7. 5, and 3n seconds respectively on one-half of the merge attempts; and 4. The freeway intervehiclar headway process is described by an Erlang type a distribtion with mean eqal to 1/q (where q is the volme), and here, a is the nearest integer vale of the expression a =. 92 e3. 6q and is based on the experimental reslts obtained on the Glf Freeway ().

10 :c > I 5 ffi : : " 2VPH,;_ a.: > I t- If. :'.l ffi : -' : t- z 3 Q" 2VPH 4 CONTROLLER GAP-THRESHOLD LIMIT-SECONDS 3 CO NTROLLER GAP-THRESHOLD LIMIT -SECONDS Figre 6. Single-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 1.5 seconds. Figre 7. Single-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 3. seconds. 3 3 <?SO :i: a.: > ' lll I : t- 8 Q BOOVPH C"llOOVPH O=rqOOVPH Q:l7VPl"t Q: 2 VPH > I t ;;' I ffi " or '-' j &! 1- z =14VPH 17 VPH 4 6 l Q 2 "" CONTROLLER GAP-THRESHOLD LIMIT- SECONDS CONTROLLER GAP-THRESHOLD LIMIT - SECONDS Fi9re 8. Single-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 4.5 seconds. Figre 9. Single-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 6. seconds.

11 66 5 : >'. i. : 8VPH O JIOOVPH "._ cr "".l. i5 _, er 3 14VPH Q 17\IPH 2VPH CONTRDl.LER GAP-THRESHOLD LIM IT-SECONDS :C :,: I,_ 5 :3 It w er w "2._ 8 VPH Q: I IOOVPH Q:l4VPH " w _, _, Q s 17DVPH cr 1. ii Oti 2: VPh --'---'--'---'"-L---'----'--'-- 'L-.J I 5 6 CONTROLLER GAP-THRESHOL D LIMIT - SECON DS Figre 1. Single-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 1.5 seconds. Figre 11. Single-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 3. seconds. 4 > I.,._ '5 aoo w cr w._ too cr w ::l. 1 Q.1:14VPH CONTROLLER GAP- THRESHOLD LIMIT- SECONDS 2 :C.: > I,_ s J5._ :i w cr._ w 1 _,. 5 : z ') s. 14 VPH Qs.17VPH O!e OOVPH I CONTROLLER GAP -THRESHOLD LI MIT- SECONDS Figre 12. Single-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 4.5 seconds. Figre 13. Single vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge of 1.5 seconds, and mean acceptable gap for standing merge of 6. seconds.

12 O=BOOVPH 4 Q HOO VP BOOVPH. >- Q 14\/PH'!:i--!-& '3 3 " : Q. 2 " 1 2VPH I 3 4 MAXIMUM STRING LENGTH -VEHICLES 4 & 6 MAXIMUM STRING LENGTH-VEHICLES Figre 14. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 1.5 seconds. Figre 15. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 3. seconds. Before the reslts of the nmerical analysis are discssed, 2 points are worth noting 1. When the time reqired by a vehicle to reach the merge zone is tr seconds (after the signal is changed to green), then the maximm ramp volme in the single-vehiclerelease-after-completion mode is 1/tr. This qantity eqals 72 and 36 vph when the respective vales of tr are 5 and 1 seconds. When strings of n vehicles each are released in the release-after-completion mode and the assmption is sed that intervehiclar headway is n eqal to 3 seconds, the corresponding merge rate is 1/(tr + 3n - 3 ) '3 11 ioo " '" 1- " Q 2VP!i i1r 3 " i 1 = I MAXIMUM STRING LENGTH - VEHICLES 4 MAXIMUM STRING LENGTH - VEH ICLES Figre 16. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 4.5 seconds. Figre 17. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 1 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 6. seconds.

13 68 6 & W >- I- <! *JOO w a:! zoo :I! 1 4 Q a 8 VPH >- Q "IJOOVPH I- l4vph Q 17VP Q 2VPH " " 1!13 a: w!12 11! x i MAXIMUM STRING LENGTH - VEHICLES MAXIMUM STRING LENGTH- VEHICLES Figre 18. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap fo; moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 1.5 seconds. Figre 19. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 3. seconds. 5, oo t: if w " JOO Q Q 8Vl'H lloovph 14VPH Q 17VPH ::> Q 2VPH 2 1 >- 1f 4 F s a 8 VPH ll OOVPH ;:; 14 VPH JOO " 17VPH w a: w 2VPH 2 a: ::> "! x MAXIM! IM STRING LENGTH- VEHICLES Figre 2. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 4.5 seconds. 6 MAXIMUM STRING LENGTH-VEHICLES Figre 21. Mltiple-vehicle-release-after-completion mode with expected ramp travel time of 5 seconds, mean acceptable gap for moving merge for nth vehicle of 3n seconds, and mean acceptable gap for standing merge of 6. seconds.

14 As n approaches infinity, this qantity approaches 1,2 vph for all vales of tr. This limit corresponds to the ncontrolled capacity limit of the entrance ramp. 2. The capacity of an entrance ramp in the release-after-completion mode depends on both the expected length and the expected freqency of the strings of vehicles that may be released to attempt a moving-merge into the freeway stream. For sch operation, an increase in the freeway volme always reslts in a decrease in the nmber of vehicles in a released string. The expected nmber of released strings (i.e., the expected freqency), however, may either increase or decrease with an increase in freeway volme becase this qantity depends on both the nmber and the sitability of freeway gaps. In particlar, for low freeway volmes almost all gaps are sitable for a merge attempt by 1 vehicle, and an increase in the nmber of gaps implies an increase in the allowable nmber of released strings; however, for high freeway volmes only a portion of the gaps are sitable for merge attempts, and an increase in the nmber of gaps implies both that the average gap becomes smaller and that the nmber of gaps sitable for merge attempts decreases. The se of parameter vales previosly listed in the models that were developed in the prior section makes it possible to nmerically evalate representative entrance ramp capacities. The reslts of sch nmerical evalation with a string-release-after-completion merge controller are presented graphically for ease of interpretation. In Figres 6throgh 13, the ramp capacity is shown as a fnction of threshold setting for the single-vehiclerelease-after-completion mode, with volme as a parameter. These crves show that a threshold of approximately 1 second yields a generally near-maximm capacity for all cases. However, the sensitivity of capacity to tj eshold setting varies sbstantially. The short ramps (i.e., tr = 5 seconds) tend to have higher capacities and greater sensitivity the tr = 1 second cases have lower rates and are less sensitive to variations in threshold. Likewise, there is a fnctional dependence of ramp capacity on the natre of the merge zone. For short ramps, the ease of exection of standing merges sbstantially affects the ramp capacity, btonlongrampsthis qantity is less significant. More exactly, the nmerically evalated reslts indicate the following: 1. As the threshold is increased the occrrence of balks becomes less significant, and the capacities are asymptotically identical (Le., independent of c for a given q and tr). 2. The qality of service is improved at the expense of added delay when the threshold is increased, and the expected service time for a given threshold Te is simply the reciprocal of the associated ramp capacity. 3. For all cases investigated, the largest ramp capacity for a single-vehiclerelease-after-completion controller corresponds to a short ramp with an excellent merge zone (i.e., tr = 5 and 1/c = 1.5 seconds). Here, a ramp capacity of approximately 45 vph is indicated for a heavy freeway flow condition (i.e., q of 1, 7 to 2, vph), bt actal service mst be sitably restricted below this vale to prevent a breakdown of the stream. 4. The dijferences between the ramp capacities shown by the crves in Figres 6 throgh 13 and the maximm capacities of 36 and 72 vph that correspond to tr = 1 and 5 seconds i espectively a-re attribted to the combined effects of limiting by the threshold setting and by gap rejection (i.e., by both moving and stopped vehicles). The maximm ramp capacity available in the string-release-after-completion mode has been evalated by a seqential search, sbject to the restriction that a second vehicle is not released nless a gap of 3 seconds or larger is detected {i.e., becase the headway h = 3 seconds). Similarly, the release of a third vehicle reqired a gap of at least 6 seconds, the release of a forth vehicle reqired a minimm gap of 9 seconds, and so on. The resltant ramp capacities are shown as a fnction of maximm string lengths in Figres 14 throgh 21 with freeway volme as a parameter and for varios vales of tr and c. These crves show that the largest benefit of mltiple vehicle release occrs at low freeway volmes; at high volmes, little extra capacity is gained. Hence, the largest volme available for the cases investigated is approximately 48 vph when tr = 5 seconds, 1/ c = 1.5 seconds, and q = 8 vph. This is sbstantially below the limit of 1,2 vph that corresponds to q =. However, when the case of tr = 1 seconds and 1/c = 3 seconds is considered as an example, it is observed that, when the freeway volme gradally drops from 2, to 8 vph, the single-vehicle-merge ramp 69

15 7 capacity remains almost constant at approximately 25 vph and the string-release mode yields an atomatic increase in capacity p to 35 vph. Finally, it is noted that, when a ramp capacity of 4 vph is reqired for this ramp, a i edction in tr from 1 seconds to 5 seconds yields a string-release capacity in excess of 4 vph. However, when a ramp capacity of 5 vph is needed, se of the string-release- after-completion mode is nacceptable and release-before-completion mst be implemented. This mode of operation is considered in the next section. One final point remaining with regard to the nmerical reslts is that the parameter vales were chosen on the basis of reasonableness in order to yield typical reslts for presentation. The models developed are qite general and may be applied to alternate sitations by the proper selection of parameter vales. THE RELEASE-BEFORE-COMPLETION MODE Althogh the release-after-completion mode exhibits many desirable attribtes, sch as high safety level and high qality of merge service, the qestion of whether to release an additional vehicle while previosly released ve11icles r 1 on the ramp reqires consideration. In particlar, when a possible merge opportnity arises for a vehicle awaiting service behind the ramp control signal the expected service delay for that vehicle may be significantly redced when that vehicle is released withot awaiting merge-completion for the previosly released vehicle. However, when this operational mode is to be employed, the qestion of what vale of ga:p threshold to se mst be reexamined. When a possible merge opportnity is detected in the freeway traffic stream before all p1 eviosly released vehicles have merged, the qestion of whetl1er to release the next qeed vehicle can be converted into a qestio of which action minimizes the expected delay. Here the e:icpected delay sbject to release of the vehicle mst be compared with the expected delay sbject to nonrelease to determine which decision to implement. Tile resltant analysis yields a controller that is a seqential decision-maker in which both the target gap sizes and present state of vehicles on the ramp affect the decision process. Analysis of the operation of an entrance ramp in the release-before-completion mode can take several forms. One possibility is to assme that the actal ramp service rate is restricted to a vale that does not case a breakdown in the flow of vehicles on the freeway. Then, an investigation of expected delay for the next vehicle awaiting service simplifies to a qestion of which decision yields the earliest expected merge instant. For this case, the following points are noted: 1. There are n nmerged vehicles on the ramp; 2. The last previos release of a vehicle occrred at t = t 2 ; 3. The target gap sizes are Tv r 2,., Tn respectively; 4. A target gap of n+l is detected at T = td; 5. The expected delay between the completion of standing merges is 1/JJ 8 ; 6. The probability that the next vehicle can complete a moving merge into the pron+ 1 spective target is 1T Pa (Ti); and i=l 7. The expected delay before a sccessfl merge, given the merge zone is cleared, is tr + 1/ " (i.e., µ"is the merge rate when tr = ). Based on these definitions, the expected additional time reqired to complete the merge of the (n + l)th vehicle (i.e., beyond the time reqired to complete the merge of the nth vehicle and provided that no balk has occrred) is (3) when the vehicle is released to attempt entry into the r n+ 1 gap, and

16 71 (31) when release is inhibited ntil the ramp is clear. Comparison of these 2 qantities reveals that when the gap r n+ 1 satisfies the ineqality n Pa(rn+l) [(td - t)µ 8-1] 1T Pa(ri) - [µstr + (µ 8 /µ") - 1] i=l (32) the (n + l)th vehicle sffers less delay in the merge process by not waiting for the ramp to clear. Ths for the operational condition in which the fnction f( ) on the right side of the ineqality is negative, the ineqality is always satisfied, and any vale of T"n+l is acceptable for a merge attempt. Similarly, when f( ) exceeds nity, no gap is acceptable becase Pa(rn+l) is a gap-acceptance pi obability and cannot exceed nity. Finally, when (.) is between and 1, a threshold Tn+l exists sch that -1 Tn+l =Pa [f(.)j (33) where P;\.) is the inverse of the gap acceptance probability fnction and P; 1 [Pa(t)J =t. For a ramp that has experienced a stoppage and offers zero probability of a moving merge for the (n + l)th vehicle, the expected delays associated with the release or the nonrelease of that vehicle at td are respectively and (34) (35) Comparison of these qantities indicates that, for any ramp on which the expected travel time tr exceeds the difference between the expected delay 1/ µg (associated with a standing merge from the merge zone) and the travel-free service time 1/µ" (associated with the sccessfl clear-ramp, single-vehicle-merge rate), the release of the next vehicle into any size gap is warranted. Setting the pi obability of a moving merge for the (n + l)th vehicle in the previos case eqal to reinforces this conclsion; this is exactly the same criterion as that for release eme1 ges. The Design of a Controller for the Release-Before-Completion Mode Under those operational conditions in which an increase becomes necessary or desirable in the prodction of a particlar entrance ramp above the level achieved in the release-after-completion mode, it can be achieved by operation in the release-beforecompletion mode. To do so, however, involves a trade-off between the qality of mergeservice offered and the merge prodction. For a long ramp (i.e., tr is large) with a merge zone that is nobstrcted so that the expected standing merge delay is comparable to (thogh more than) the expected delay in the attempt of a moving merge, analysis has shown that the release-before-completion mode offers an improved merge rate for the ramp, provided that the freeway stream does not break down. This property allows for a variation of the controller threshold (or merge-rate limit or both) to increase the ramp prodction as reqired within the limits necessary to preserve the stability of the st1 eam. Sch operation has been inplemented on the Glf Freeway when qee lengths beyond the available storage capacity have made action necessary. In addition, ramp operation with a fixed threshold in the release-before-completion mode has been more the rle than the exception on that Freeway. By contrast, when a ramp is short (i.e., tr is small) and the merge zone is marginal so that standing merge opportnities are rare compared with moving merge opportnities,

17 72 operation in the release-before-completion mode does not always yield an increase in ramp merge capacity. In particlar, analysis reveals that when a stoppage of a released vehicle warrants that the release-before-completion operation b inhibited ntil the ramp is cleared. The existence of this restriction in the releasebefore-completion operation, however, is temporary and in no way negates the overall advantage of increased ramp capacity that reslts, provided that the controller selects only adeqately large gaps that occr within an adeqately short period after the last previos release. SUMMARY Under either control philosophy described, the threshold that yields the maximm ramp capacity mst be identified first and, based on that vale, a threshold s etting is established that both provides an appropriate level of merge service and preserves the stability of the freeway stream against breakdown in flow cased by excess density. By this process, merge capacity will often be redced below its maximm vale to increase the level of service offered to drivers. Ths, the controller may attempt to operate in a mode in which a single vehicle is released whenever the ramp is clear, the demand is low, and the probability of a sccessfl moving merge is adeqately high. When it is necessary to increase the available ramp merge capacity, the operational mode is modified so as to redce the level of service by the relaxation of control limits ntil adeqate mer ge capacity is achieved. In this way the controller is continosly adjstable to both actal demand and actal fr eeway volme. Based on analysis o( the reslts obtained by se of the control modes -developed in this paper, we believe a sond analytic theory exists for the design of freeway entrance ramp control systems. In particlar, the relationship between merge capacity and system parameters is explicitly developed for several control modes. These reslts make possible the evalation of both the capacity limitations inherent in the design of a particlar ramp control system and the control system parameters necessary to provide a specified merge capacity for the ramp. To aid in the design process, several typical ramp control configrations were analyzed in detail. These inclded the following conditions: (a) placement of a traffic control signal at between 5 and 1 seconds of tr avel time from the merge zone, (b) mean gap acceptance limits for vehicles stopped in merge zone of between 1.5 and 6. s econds, and (c) freeway volmes in the otside lane of between 8 and 2, vph. As a r eslt of this analysis, the following observations are made: 1. The gap rejection phenomenon was fond to increase the minimm expected ramp service time, in the single-vehicle-release-after-completion mode, to a level between 1.25 and 3. times the expected vehiclar travel time on that ramp. For example, for a long ramp with an expected travel time of 1 seconds, the maximm ramp capacity is redced to between 5 and 8 percent of the 36-vph limit that the 1-second travel time imposes on this mode; 3,6 seconds per hor divided by a minimm of 1 seconds per vehicle yields a maximm flow in the single-vehicl e-release-after - completion mode of 36 vph. Here, the larger capacity corresponds to a volme of 8 vph in the otside freeway lane; the lower capacity occrred with a freeway volme of 2 In both cases, the assmption is that a relatively good merge zone exists becase the mean acceptable gap for a standing merge from the merge zone was defined to eqal 1.5 seconds. For a second example, with an expected travel time of 5 seconds and a short merge zone, the reslts indicate a redction in maximm ramp capacity to between 35 and 65 percent of the limit of 72 vph that the 5-second travel time imposes on this mode. Again, capacity depended on freeway volme. 2. The release of more than 1 vehicle per string can increase the ramp capacity withot forsaking the release-after-completion operational mode. For sch operation, mltiple-vehicle-release opportnities are rare when high freeway volmes exist and (36)

18 little is gained in these instances. However, when freeway volmes a.re low (e.g., 8 vph per lane), improvements of approximately 5 percent ove1 single-vehicle operation are possible. 3. Operation in the string-release-after-completion mode appears to be limited at most to between 15 and 4 percent of the ncontrolled ramp capacity, with the higher prodction generally corresponding to the lower freeway volmes (i.e., 8 vpb per lane). To provide ramp capacities in excess of this limit reqires that release-beforecompletion operation be employed. 4. When the diuerence between the expected service time reqired to <;:omplete a standing merge from the merge zone is less than the total expected service time reqired to complete a moving merge from behind the ramp control signal (i.e., inclding the expected travel time), then the release-before-completion mode exhibits a maximm ramp capacity that eqals the ncontrolled ramp capacity. However, the actal service provided by this controller mst be constrained by consideration of freeway stream stability and actal demand characteristics. For example, metering at any fixed rate elow 1, 2 vph wold be possible for several of the cases investigated in this report, provided that stream stability was not a problem. REFERENCES 1. Highway Capacity Manal. HRB Spec. Rept. 87, Drew, D. R. Traffic Flow Theory and Control. McGraw-Hill, New York, Haight, F. A. Mathematical Models for Traffic Flow. Academic Press, New York, Wattleworth, J. A. Peak-Period Control of Freeway Systems: Some Theoretical Considerations. Northwestern Univ., Evanston, Ill., PhD dissertation, Hodgkins, E. A. Effects of Volme Controls on Freeway Traffic Flow-A Theoretical Analysis. Pblic Roads, Feb Bhr, J. H. Freeway Ramp Control. Civil Engineering, ASCE, Jne