Simulating Colliion Probabilitie of Landing Airplane at Non-towered Airport John F. Shortle Richard Xie C. H. Chen George L. Donohue Stem Engineering & Operation Reearch Dept. George Maon Univerit Fairfa, VA 22030 Abtract Making greater ue of maller airport i one wa that ha been propoed to increae the capacit of the National Airpace Stem. A major difficult i that man mall airport do not have control tower, and thu capacit i everel limited during poor viibilit. We conider a propoed tem in which airplane elf-eparate, o the are able to land at higher capacitie without a control tower. Before uch a tem i implemented, it mut firt be hown to be afe. Safet i a difficult metric to meaure and predict, becaue accident are o rare. Even computer imulation can be low becaue of the long time to oberve accident. One methodolog that ha been ucceful in aeing aviation afet through imulation i TOPAZ (Traffic Organier and Perturbation AnalZer). In thi paper, we appl the methodolog to ae the afet of the propoed non-towered tem. In particular, we etimate the probabilit of colliion on the runwa in poor viibilit. 1. Introduction The National Airpace Stem (NAS) i predominatel a hub-and-poke network, with about 60 hub airport and a maimum capacit of about 40 million operation per ear. Current forecat predict that demand for thee airport will oon eceed capacit (e.g., ATAG (2000), Donohue and Shaver (2000), EUROCONTROL (1999,2000)). Cloel coupled with the iue of capacit i the iue of afet. Fling airplane cloer together ha the potential to increae the likelihood of an accident. Thu, an increae in capacit mut be accompanied b a demontration that uch an increae will be afe. A capacit increae, the relative afet mut decreae to achieve the ame abolute accident rate over time. In fact, the commercial accident rate ha remained relativel table over the lat two decade (Barnett and Higgin (1989), Machol (1995)), but the abolute number of accident ha increaed due to more operation. Safet i a ver difficult metric to predict and meaure. The main reaon i the carcit of data. Van E (2001) etimate the rate of plane-to-plane colliion for commercial aviation between 1980 and 1999 to be about 0.3 colliion per million flight. Thu, to determine the probabilit of an accident, a long obervation period i required. Evaluating a new tem deign i even more difficult, ince it i impractical to implement change and then count accident. 1
One wa to predict afet i through computer imulation. However, even thi can be difficult. For eample, uppoing colliion occur at a rate of one in 10 7 flight, imulating one hundred flight per econd till take over 24 hour to generate a ingle imulated colliion, and over a month to generate a tatiticall ignificant ample. Thu, alternate method, beide traight-forward imulation, are generall required. One methodolog that ha been ucceful i the TOPAZ (Traffic Organier and Perturbation AnalZer) modeling methodolog, developed at the National Aeropace Laborator NLR, the Netherland (Blom, Bakker, et al. (2001)). The methodolog provide a two-tep framework for aeing afet. The firt tep qualitativel aee afet b identifing haard relevant to the cenario in quetion. The econd tep quantitativel etimate afet through imulation. In the econd tep, analtical model, a well a conditioning technique, are ued to improve the efficienc of imulation. In thi paper, we appl the TOPAZ methodolog to predict colliion probabilitie of landing airplane at mall airport that do not have a control tower. Making greater ue of maller airport i one wa that ha been propoed to increae the capacit of the NAS. In fact, it ha been etimated that 98% of the people in the U.S. live within a 30-minute drive of a public airport (Holme (2002)). Providing greater acceibilit to thee airport could potentiall divert paenger awa from major hub, increaing overall tem capacit. Without a control tower, capacitie in Intrument Meteorological Condition (IMC) can be a low a three landing per hour, depending on the proimit of the airport to nearb radar coverage. We conider a propoed tem where a nearb, upporting controller i reponible for the initial eparation of airplane entering the airpace near the airport (that i, leaving radar coverage). But, the pilot are reponible for elf-eparation after that. We pecificall invetigate the probabilit of a colliion on the runwa in thi tem. Thi anali i a tep in determining the potential capacit at uch non-towered airport. The paper i organied a follow: Section 2 review ome eiting model for etimating colliion rik. In particular, we review the Reich colliion model, which ha been ued etenivel in the TOPAZ modeling methodolog. Section 3 review ome of the quantitative technique which have been ued in the TOPAZ methodolog. Section 4 preent a propoed concept of operation for landing airplane at a non-towered airport. Section 5 preent the afet model and numerical reult. 2. Colliion Prediction Model Thi ection provide background on quantitative model that have been ued to etimate colliion and conflict probabilitie. We retrict our dicuion to method which can be implemented on a computer. For eample, we do not dicu qualitative method, uch a haard aement method or human factor modeling, or real-time method involving flight imulator and human in the loop. Thi ection particularl focue on the Reich colliion model, ince it i a component of our imulation anali. 2
Man of the model in thi ection have been developed to evaluate en-route afet. Thee model tpicall aume that airplane fl in traight line, in level flight, with no controller intervention. Although thee aumption are retrictive, ome of the model can be combined with imulation to addre more comple problem, a we how in Section 3. Interection Model The implet cla of model are interection-tpe model. In thee model, one aume that plane fl along pre-determined, croing route (Figure 2.1), generall at contant velocitie. Under thee aumption, the probabilit of a colliion at the point P can be computed from the arrival rate of airplane along each path, their velocitie, and the airplane geometrie. For eample, Siddiqee (1973) derive the colliion probabilit at P auming airplane are uniforml ditributed along each path. Geiinger (1985) generalie Siddiqee model to three dimenion, allowing for climbing and decending plane. P Figure 2.1. Airplane fling on croing route The flight path can alo be more complicated. For eample, Barnett (2000) ue a grid a an approimation to free flight. Airplane are allowed to fl approimate traight-line path in an direction, where the true traight path are approimated b path on a grid. Uing aumption imilar to thoe above, Barnett etimate the probabilitie of colliion at the interection point of the grid. Probabilitic Conflict Model A P Uncertaint in plane poition B Region of Conflict (e.g., 5 nmi radiu) Figure 2.2. Uncertaint in airplane poition 3
A imilar cla of model are geometric conflict model (e.g., Paielli and Erberger (1997, 1999), Irving (2002)). Thee model allow for uncertaint in each airplane location (thu, the plane can deviate from the flight path). Figure 2.2 how the baic model. The location of plane A and B are uncertain, and are modeled uing bivariate normal random variable. If plane A ever get within a 5 nmi radiu of plane B, then there i a conflict. Determining the probabilit of a conflict require geometricall decribing poible location for plane A and B that would ield a conflict, and then integrating thee ditribution over the conflict region. A with the interection-tpe model, thee model generall aume level flight with contant velocitie (ee Paielli and Erberger (1999) for a generaliation to non-level flight). The Reich Colliion Model The Reich model wa originall developed to etimate colliion rik for oceanic travel over the North Atlantic and to determine the appropriate pacing of flight path (Reich (1966)). In uch flight, airplane are not under direct radar control, and onl periodicall report their poition to controller. Thu, over time, light error in navigation ma caue an airplane to drift off coure. If thee error go undetected, an airplane ma tra onto a neighboring flight path and collide with another airplane. We decribe thi model in more detail, ince we ue it in out imulation anali. Firt, we tart with ome notation. Let r 1( t ) and v ( t) 1 be the poition and velocit vector of one airplane at time t; let r 2 ( t ) and v ( t) 2 be imilarl defined for a econd airplane. To implif notation, we drop the ubcript t, keeping in mind that thee and related quantitie have an implicit dependence on time. Let r ( r, r, r ) r1 r2 and v ( v, v, v ) v1 v2 be the relative poition and velocit vector of the two airplane. Now, r trace out a path in time (Figure 2.3). If r get too mall, there i a colliion between the airplane. Mathematicall, if r (0,0,0), the two airplane imultaneoul occup an identical point in pace. 2 2 2 r Face F Figure 2.3. A geometric repreentation of the Reich colliion model. The Reich model aume that each airplane i haped like a bo with dimenion (along-track length), (acro-track width), and (vertical height). Under thee aumption, two airplane are touching when, for eample, one airplane i in front of the 4
other b a ditance, or when one airplane i behind the other b a ditance (that i, whenever r ). More generall, a colliion occur along an direction whenever r pae through the 2 2 2 bo, centered at the origin in Figure 2.3. Such an event i called an incroing. 1 The probabilit of an incroing depend on the joint probabilit denit function (PDF) f ( r, v) f ( r, r. r, v, v, v ). The Reich model make the following aumption on thi PDF: 1. The ditribution i independent in the,, and dimenion. That i, f(r, r, r, v, v, v ) = f (r, v ) f (r, v ) f (r, v ), where f, f, and f are marginal ditribution of f ( r, v). 2. The ditribution i contant in the relative poition over the dimenion of the aircraft. In other word, f (r, v ) = f (0, v ) when r, and imilarl for the other dimenion. Intuitivel, the ditribution hould not change much over a ditance a mall a the dimenion of an airplane. 2 3. The ditribution i independent in the poition and velocit component. That i, f (r, v ) = f r, (r ) f v, (v ), and imilarl for the other dimenion. (Thi aumption i not in Reich original aumption; however, Eq. 2.1 below, which i frequentl quoted from Reich paper (e.g., Haelrigg and Buch (1986), Bakker and Blom (1993) require thi aumption). Aumption 1, 2, and 3 impl: f(r, r, r, v, v, v ) = f r, (0) f r, (0) f r, (0) f v, (v ) f v, (v ) f v, (v ) whenever r i on or within the boundar of the bo in Figure 2.3. In addition, Reich (1966) aume: 4. Plane travel along parallel track without making turn. Thu, the orientation of the colliion bo doe not change. 5. All plane have the ame geometric hape. 6. There i no evaive maneuvering b the pilot or intervention b the controller. Under the above aumption, the total incroing rate through all ide of the bo i: 3 ( t ) f (0,0,0) A E v, (2.1) r where the ubcript i denote the three dimenion,, and, f r i the marginal denit of f ( r, v), and A i i the area of the face perpendicular to dimenion i. Alo, f r and E vi are implicit function of time. For an intuitive derivation of Eq. 2.1, ee Appendi A. Since Eq. 2.1. give the incroing rate at time t, the total epected number of incroing over the time interval [a, b] i: i 1 i i 1 Mathematicall, there ma be multiple incroing. In realit, thi would correpond to onl one colliion. Thu, the probabilit of an incroing i an upper bound on the probabilit of a colliion. 2 However, if one change the interpretation of,, and to repreent a conflict bo (intead of a colliion bo) with dimenion of everal nautical mile, then thi aumption ma not be valid. 5
b E ( t) dt. (2.2) a If there are more than two airplane, then Eq. 2.2 mut be evaluated for ever poible airplane pair and then ummed to get the total number of epected incroing among all airplane (ee Eq. 5.2 in Section 5). An advantage of the Reich colliion model i that it account for all poible direction of the aircraft. Some of the other model, b dicounting the vertical dimenion, onl account for colliion through the four ide of the bo. Generalied Reich Model Some of the aumption in the Reich model are quite retrictive in particular aumption 1 and 3, which tate that all component of ( r, v) are mutuall independent. In particular, velocit in one direction uuall depend on velocit in another direction (for eample, the acent rate generall depend on the along-track rate). Removing aumption 1 and 3, Bakker and Blom (1993) derived a generalied Reich colliion model. In particular, the incroing rate through a ingle face (face F in Figure 2.3) and it oppoing face i: ( t) 0 v 0 v f ( v, r f ( v, r, r, r ) dv dr dr, r, r ) dv dr dr (2.3) where f ( v, r, r, r ) i the marginal ditribution of f ( r, v). Thi i a generaliation of Eq. 7.1 in the appendi. The total incroing rate through all face i ( t) ( t) ( t) ( t), (2.4) where (t) and (t) are defined imilarl to Eq. 2.3. Although Eq. 2.3 i difficult to evaluate numericall, Blom and Bakker (2002) how that if f ( r, v) i a miture of Gauian ditribution, then evaluation of the integral i much eaier. Blom and Bakker alo argue that aumption 4 can be removed b auming that =. In other word, if the length of a plane i approimatel the ame a it wing pan, the bounding bo doe not change when the plane turn. That i, the colliion bo in Figure 2.3 keep a fied orientation regardle of the orientation of the two airplane. 3. Simulation Method The previou ection dicued everal analtical model to etimate colliion rik. A central problem with thee model i that the onl appl to imple cenario. While the model work reaonabl well for en-route problem, the do not directl appl in more complicated etting J for eample, near an airport where flight path are not traight and where controller intervene when airplane deviate off coure. The analtical model alo do not account for equipment reliabilit or factor related to controller workload. One wa to model thee compleitie i through imulation. In thi ection, we dicu a imulation method ued in everal TOPAZ model (for intance, Blom, Klomptra, and 6
Bakker (2001)). The method ue dnamicall colored petri net (DCPN ) a a framework for imulation. Since colliion are o rare, the method alo ue the generalied Reich colliion model to improve the efficienc of imulation. Figure 3.1 how the baic idea. Intead of returning the number of colliion, the imulation return the probabilit denit f ( r, v) (the ditribution for the relative poition and velocit of an airplane pair). Uing thi, the method compute the probabilit of a colliion uing the generalied Reich model a dicued in Section 2 (Eq. 2.3 and 2.4; alo ee Blom and Bakker (2002)). Simulation f ( r, v) Generalied Reich Model Colliion Probabilit Figure 3.1. Baic mechanim for combining imulation with an analtical model. DCPN can imulate a wide variet of tem dnamic including flight dnamic and controller-pilot interaction. To illutrate the ue of DCPN in the contet of aviation afet, we give a imple eample (Figure 3.2). Appendi B give the DCPN ued to imulate the mall airport application dicued in the net ection. For further detail on DCPN, ee Everd, et al (1997). For an eample of DCPN applied to imultaneou approache of two airplane on two converging runwa, ee Blom, Klomptra, and Bakker (2001). Figure 3.2 how two eparate (but not independent) petri net. The right petri net model whether or not a runwa enor i working. The token (red dot) indicate the current tate. The tranition (gra boe) repreent event which trigger the token to move from one place to another. In thi petri net, we uppoe that time between tranition are independent random variable. For eample, if the follow an eponential ditribution, then the right petri net i a continuou time Markov chain. We can alo ue uch petri net to model human cognitive tate - for eample, whether a pilot i relaed, bu, or frantic (thi i imilar to the human cognition model in Hollnagel (1993)). For an eample of petri net applied in thi contet, ee Blom, Daam, and Nhui (2001). G 1 G 2 G 3 Runwa Senor Working Final Approach T 1 On Runwa G 4 T 3 Eit Stem T 2 Mied Approach Not Working Figure 3.2. Simple petri net with poible mied approach. The left petri net i more mathematicall comple. The token correpond to airplane and the place (open circle) correpond to phae of flight. In the figure, there i one airplane on the runwa and one airplane in the final approach. Aociated with each 7
token i a i-dimenional vector (not drawn) giving the poition and velocit of that airplane. Each place ha i differential equation which govern the evolution of thi vector when the token i in that place. For eample, G 1 repreent the et of differential equation that govern the airplane evolution when the airplane i in the final approach. The differential equation can be tochatic, allowing for random perturbation due to wind or pilot error. For eample, two of the differential equation repreented b G1 could be: dp vdt dv av b( p ) (3.1) 0 d Here, p and v are the poition and velocit of the airplane along the direction; dw repreent Brownian motion; 0, a, b, and are contant. The equation repreent a pilot who i tring to keep the airplane centered along the runwa line at 0, in the preence of wind. The firt two term repreent the pilot control and the lat term repreent wind. G 1 would alo have four other equation correponding to the and direction. For an introduction to tochatic differential equation, ee Okendal (1992). We can alo link the two petri net in Figure 3.2. For eample, to model the functionalit of the runwa enor, we define the tranition T 1 in the left petri net to trigger when: The airplane in final approach ha jut croed the runwa threhold (baed on it poition vector), and - The runwa i not occupied or - The runwa i occupied, but the runwa enor i not working. Thu, the tate of the right petri net affect the trigger event of the left petri net. Thi logic can alo be drawn uing tandard petri net notation. However, we do not do thi to avoid clutter. We can alo create a econd tpe of link between the two petri net: making the differential equation on the left a function of the petri net tate on the right. An eample would be uing the right petri net to model the tate of an Intrument Landing Stem (ILS). When the ILS i not working, pilot deviate more from the glide path. Thu, two et of differential equation are needed to model the airplane evolution, depending on whether or not the ILS i working. Appendi B give the baic petri net tructure for the mall airport application decribed in the net ection. 4. Application: Non-towered Airport Thi ection decribe a cenario involving a mall airport with no control tower. The goal i to increae the capacit of uch airport in IMC. Currentl, procedural eparation rule dictate that onl one airplane i allowed into the airpace near the airport at one time (in IMC). Thi guarantee ± that two airplane are not imultaneoul fling near th airport outide of radar coverage. However, thi can ield capacitie a low a three operation per hour depending on nearb terrain and proimit to radar coverage. One olution that ha been propoed i to equip airplane with a elf-eparation capabilit. Nearb controller have reponibilit for the initial eparation of airplane 8
into the local airpace, but after that, airplane mut eparate themelve. Referring to Figure 4.1, we conider the following concept of operation: Since the local airport doe not have a control tower and i outide radar coverage, en-route arrival-departure traffic i controlled b a upporting air traffic controller (ATCo) at a TRACON or ARTCC. The upporting ATCo meter airplane into the local airpace through one of two approach leg. The two approach path combine to form a T ±. The ATCo i reponible for the initial eparation of the airplane. Once an airplane enter the airpace, the pilot i reponible for maintaining eparation with other airplane. At thi point, the airplane i outide radar control of the ATCo. A mall terminal enor located on the ground at the airport provide radar-like coverage for the local airpace. The enor fuion tem tranmit airplane poition to all airplane in the local airpace via a ground-to-air data link. A Cockpit Dipla of Traffic Information (CDTI) dipla the location of thee airplane to the pilot. Pilot ue the dipla to maintain eparation. The airport i equipped with an intrument landing tem (ILS) to aid landing. A a deterrent to a colliion on the runwa, an infrared enor on the ground detect the preence of an airplane on the runwa. If the runwa i occupied, the landing pilot receive a warning in the cockpit. Figure 4.1. Airplane landing at mall, non-towered airport. 9
Since thi i a propoed concept, imulation i required to ae the tem afet. Three location where plane-to-plane colliion are mot likel to occur are (Figure 4.1): 1. Interection of the T ±. The controller ha not properl eparated incomin airplane. 2. Final approach. A fater airplane overtake and collide with a lower airplane. 3. Runwa. One airplane fail to eit the runwa before the approaching airplane land. In thi paper, we concentrate onl on the lat colliion tpe, ince there are fewer degree of freedom on the runwa, o thi tpe of colliion ma be mot likel. 5. Anali and Reult Thi ection decribe the anali to etimate the probabilit of colliion on the runwa. A preliminar tep in afet aement i an identification of haard (alo ee Blom, Bakker, et al. (2001)). Since the focu of thi paper i on imulation, we do not elaborate on thi tep, but give ome eample for illutration. Haard which could lead to a colliion on the runwa are: 1. An airplane land and become diabled, o it cannot eit the runwa (blown tire, partial crah landing, etc.). 2. An airplane land, and the pilot become dioriented; intead of eiting to the tai-wa, the pilot ta on the runwa while going to the gate. 3. The pilot top on the runwa and doe not immediatel pull off. 4. The runwa enor fail. 5. The communication link between thi enor and other airborne airplane fail. 6. The pilot fail to notice a warning from the runwa enor. 7. The pilot notice the warning, but chooe to ignore it. 8. The pilot i ditracted and doe not ee another airplane on the runwa. Thi could be due to poor viibilit or becaue the pilot i concentrating on landing hi or her own airplane. 9. The runwa i lick, o a landing airplane cannot decelerate a quickl a normal. And there are man other haard. Appendi B give the petri net tructure which model the cenario decribed previoul. The petri net incorporate all of the haard lited above. For eample, haard 4 and 5 are modeled b the two tate Runwa Senor ± petri net. Stochatic differentia equation govern the airplane evolution for each phae of flight. In general, thee are econd order repone model where the pilot trie to maintain a target peed (different for each airplane) along a contant heading or glide lope, ubject to random perturbation due to wind or pilot control. We now dicu the imulation logic at the landing and the relationhip between the petri net in Appendi B. If airplane i reache point A in Figure 5.1 and airplane j i diabled on the runwa, then airplane i will land if the following happen (otherwie i will fl a mied approach): Pilot i doe not receive a warning from the runwa enor becaue either: o The enor i not working, or o The enor i working, but pilot i fail to notice the warning 10
and Pilot i fail to viuall ee airplane j while between point A and B. A Viibilit Ceiling Airplane j Airplane i Figure 5.1. Airplane landing on runwa. B Runwa Further aumption we make in the model are: An airplane which fail to eit the runwa remain on the runwa for a random amount of time (following an eponential ditribution). At touchdown, the pilot immediatel ee a diabled airplane on the runwa. At thi point, the pilot decelerate the plane at the maimum poible rate. The runwa ma be wet or dr. If the runwa i wet, the maimum deceleration rate i le than the normal deceleration rate. In addition, we make the following aumption regarding weather, approach path, and airplane tpe: Weather condition are IMC. The viibilit ceiling i 250 feet. The runwa viual range i? mile. Pilot fl a 3 degree approach path. All airplane have the ame flight characteritic and onboard equipment. The airport ha a ingle runwa with a eparate tai-wa. The runwa i 5,000 feet in length. Following the method from Blom, Klomptra, et. al (2001). We define to be a time uch that there i ero probabilit of a colliion between airplane i and j prior to time. (Without lo of generalit, we aume that airplane are indeed in order of their arrival, and we onl conider poible colliion when i < j.) Since we are onl intereted in colliion on the runwa, we can define: = time airplane i land (at B in Figure 5.1) while airplane j i on the runwa. We define if j eit the runwa before i land, or i flie a mied approach, or i land before j. Alo, define 1 if airplane i collide with airplane j B 0 otherwie (Again, we onl conider cae where i < j, o B = 0 for i j ). Since, there i ero probabilit that airplane i collide with airplane j prior to, the total probabilit that airplane i collide with airplane j i: Pr( B 1) Pr( 1 ) Pr( ). (5.1) B 11
The total epected number of colliion E(N) i then E ( N) Pr( 1). (5.2) i j To compute the colliion rate, we compute the right-hand ide of Eq. 5.1. Firt, to evaluate Pr( ), we run the imulation (petri net in Appendi B) and count the number of occurrence that (that i, the number of time that ome airplane i land while another airplane j i diabled on the runwa). To evaluate Pr( B 1 ), we firt oberve B Pr( B 1 ) ( t dt. 3 (5.3) ) To compute thi, we run the imulation with parameter adjuted to achieve an artificiall high number of intance where. In other word, we uppoe that there i a ver high probabilit that (a) airplane become diabled on the runwa, and (b) that pilot and enor tem fail to oberve the diabled airplane. Each time that, the imulation collect data on f t ( r, v), the relative poition and velocit of the landing airplane with repect to the diabled airplane. The ubcript t denote the dependence of thi denit on time, where t repreent the time after. (In other word, t = 0 refer to the time when airplane i land, while j i till on the runwa). Uing the generalied Reich model, we compute ( t ) uing Eq. 2.3 and 2.4, a decribed in Section 2. Then, we integrate thi (a in Eq. 5.3) up to ome maimum value. Prob ( Colliion on Runwa ) 1.00E-06 8.00E-07 6.00E-07 4.00E-07 2.00E-07 0.00E+00 0 3 6 9 12 15 Landing Per Hour Figure 5.2. Colliion probabilitie on the runwa Figure 5.2 how the reult of the preceding anali and imulation. The figure how the colliion probabilit on the runwa a a function of the arrival rate. The hape of the curve i eplained from the term in Eq. 5.3. Firt, Pr( B 1 ) i relativel contant a a function of the arrival rate. In other word, given airplane i land while 3 We aume here that there i a one-to-one correpondence between incroing and colliion. Thi i reaonable, ince airplane j i tationar. Since airplane i i alwa moving forward, there i no poibilit of more than one incroing. 12
airplane j i diabled on the runwa, the probabilit of a colliion doe not depend on the relative pacing of arriving airplane. Pr( ), on the other hand, doe depend on the arrival rate. A the arrival rate goe toward infinit, the probabilit a following airplane arrive (get to point A in Figure 5.1) while another airplane i diabled on the runwa goe to one. Thu, Pr( ) depend onl the probabilit of the enor failing and the pilot failing to notice the diabled airplane. Since thee are all contant value, Pr( ) approache thi net probabilit. Thi give the leveling affect in Figure 5.2. Since thi i a propoed tem, little data are available to etimate parameter in the model. Below are ample value we ued a etimate for haard in the model. A: Landing airplane doe not eit runwa (Haard 1-3) Pr(A) = 2.5 10-3 B: Runwa enor i not working (Haard 4-5) Pr(B) = 10-4 C: Pilot fail to notice warning from enor (Haard 6-7) Pr(C) = 3 10-2 D: Pilot fail to ee airplane on runwa (Haard 8) Pr(D) = 3 10-2 E: Runwa i lick (Haard 9) Pr(E) = 10-2 Figure 5.3 how enitivit anali for ome of thee parameter. The figure how that the colliion probabilit i more enitive to the probabilit that an airplane fail to eit the runwa (Pr(A)) than to the probabilit the runwa enor fail (Pr(B)). Of the above parameter, Pr(A) and Pr(D) are the mot ignificant. Factor Change in P(colliion) 12 10 8 6 4 2 0 0.1 1 10 Scaling Factor for Individual Parameter Pr(A) Pr(B) Figure 5.3. Senitivit of colliion probabilit to model parameter 6. Concluion In thi paper, we applied the TOPAZ modeling methodolog to etimate colliion probabilitie of airplane landing at airport without control tower. Etimating afet metric i a ver difficult problem. One reaon i that traight-forward imulation i generall too low. Thi paper ued two technique to increae the efficienc of imulation: (1) Conditioning on haardou event (o the probabilit of a imulated colliion i higher) and (2) uing analtical model to enhance the reult of imulation. Performance uing thee technique wa good, ince we were able to get tatiticall ignificant reult without eceive computer time (everal hour on a PC). One 13
potential difficult i whether or not it i alwa accurate to fit a miture of Gauian ditribution to the imulation output a input to the generalied Reich model. A econd difficult with afet modeling i that it i impoible to account for all haard that might lead to a colliion. For eample, thi paper did not conider haard related to improper initial eparation due to controller error. Thi omiion biae the colliion probabilit etimate to be lower than actual. On the other hand, ome parameter etimate are likel conervative, biaing the etimate in the other direction. Thu, concluion baed on the abolute reult are tentative. Since thi paper conider a propoed tem, the pirit i to reveal general trend and to identif critical parameter that have the highet impact on afet. In particular, the anali revealed a leveling off of the runwa colliion rik a a function of arrival rate, contrar to an epected quadratic or eponential growth. Appendi A. Derivation of Reich Model Thi ection give an informal derivation of Eq. 2.1, from the Reich colliion model. We firt look at incroing through a ingle face of the bo for eample, face F in Figure 2.3. An incroing occur on thi face whenever: r i on the face F (that i, r, r and r ), and v i pointing to the left (that i, v < 0). The above two event are independent (b aumption 3). For the firt event, the probabilit denit that r i on face F i: fr (, r, r ) dr dr fr (0,0,0) dr dr A fr (0,0,0), where A i the area of face F; the econd term follow from aumption 2. The econd event occur whenever v < 0. Thu, the average rate that r move through face F (given that i r i on face F) i: 0 v f ( v dv. v, ) Combining the previou equation give the average incroing rate through face F: 0 A f 0,0,0) v f ( v dv. r ( v, ) Similarl, the incroing rate through face F and it oppoite face (which we call (t)) i: ( r v, r t ) A f (0,0,0) v f ( v ) dv A f (0,0,0) E v. (7.1) Eq. 2.1 follow b umming Eq. 7.1 (and imilar equation) over three dimenion. 14
Appendi B. Petri Net Diagram of Non-Towered Airport Acknowledgement The author wih to thank Henk Blom and Bert Bakker for their helpful and generou intruction in appling the TOPAZ methodolog. Reference Air Tranport Action Group. 2000. European Traffic Forecat 1985-2015. http://www.atag.org/etf/inde.htm. Acceed Jul. 22, 2002. 15
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