Queuing Models for Analyzing the Impact of Trajectory Uncertainties on the NAS Flow Efficiency

Transcription

1 Queung Models for Analyzng the Impact of Traectory Uncertantes on the NAS Flow Effcency Prasent Sengupta *, Monsh D. Tandale *, Jnwhan Km * and P. K. Menon Optmal Synthess Inc., Los Altos, CA The applcaton of Queung Theory to quantfy the relatonshps between traectory uncertantes due to avaton operatons, precson of navgaton and control, and traffc flow effcency n the Natonal Arspace System s dscussed. Ths work bulds on a prevous research effort on Markovan queung network models of the NAS. Two approaches for ncorporatng the traectory uncertantes n the queung networks are dscussed. In the frst approach, an approxmate queung method that models the servce dstrbutons wth the frst and the second moments of the traffc flow parameters s dscussed. Ths approach employs the Queung Network Analyzer formulaton dscussed n the lterature. The second approach advances a methodology for ntroducng the traectory uncertantes through a set of uncertanty servce nodes n a Markovan model of the NAS. The modelng approach s dscussed for multple spatal dscretzatons. Sample results for the U.S. Natonal Arspace System are gven. U I. Introducton nderstandng the relatonshps between traectory uncertantes due to avaton operatons, precson of navgaton and control, and the traffc flow effcency s central to the desgn of Next Generaton Ar Transportaton Systems (NextGen). In all that follows, the traffc flow effcency s defned as the degree to whch the arcraft s delayed due to congeston effects as compared to an unmpeded flght along the same route. Congeston arses when the demand exceeds capacty n a regon of the arspace, whch requres that the upstream arcraft enterng the arspace must be delayed untl the capacty becomes avalable. An mportant element of the NextGen concept beng developed by NASA and the Jont Plannng and Development Offce (JPDO) s the Traectory-Based Operatons (TBO). Traectory-Based Operatons s a paradgm shft from the current clearance based approach and employs four-dmensonal (4D) traectores as the bass for managng the ATM system, addng tme as the fourth dmenson to lattude, longtude and alttude. If the predcted 4D traectory of every arcraft s avalable, the flghts can be scheduled to have conflct-free traectores. However, arcraft may not be able to precsely execute the planned 4D traectory due to the uncertantes arsng from avaton operatons and precson of navgaton and control. Fgure llustrates the varous uncertanty sources contrbutng to arcraft traectory uncertantes. Due to the dscrepancy between the planned and executed 4D traectores, congeston may be observed resultng n flght delays and reduced flow effcency. The obectve of the present research s to employ results from Queung Theory to quantfy the relatonshp between traectory uncertantes and traffc flow effcency n the Natonal Arspace System. A prevous paper ntroduced the concepts pursued n ths research and presented prelmnary results based on Markovan network analyss of the NAS. That paper also presented the lterature survey outlnng prevous work performed n ths area. Markovan network analyss s prevalent n lterature due to the mathematcal tractablty and presence of closedform analytcal solutons. However Markovan analyss assumes degree-of-freedom exponental dstrbutons to descrbe the arrval and servce processes. In an exponental dstrbuton the varance s equal to the square of the mean and cannot be specfed ndependently. Thus, there s no drect approach for analyzng the effect of changng * Research Scentst, 95 Frst Street Sute 40, Senor Member AIAA. Chef Scentst and Presdent, 95 Frst Street Sute 40, Assocate Fellow AIAA.

2 the varance of the traectory uncertanty on the traffc flow effcency n the Markovan network analyss. To overcome ths dffculty, ths paper proposes to employ approxmate queung network analyss methods that model the arrval and servce processes wth the frst and the second moments. In partcular, ths research effort mplements network flow analyss equatons developed by Whtt for the Queung Network Analyzer (QNA) 3, an approach orgnally developed at Bell Laboratores to calculate approxmate congeston measures for a network of queues. Fgure. Varous Uncertanty Sources Contrbutng to Traectory Uncertantes n Avaton Operatons The paper s organzed as follows. Secton II descrbes the spectrum of stochastc network models avalable for the study of Ar Transportaton. Secton III descrbes the structure of the Arspace-Level Queung Network model of the NAS and the estmaton of model parameters from NAS traffc data. The Queung Network Analyzer (QNA) 3 formulaton that provdes approxmate solutons for queung networks wth G/G/s queues s presented n Secton IV followed by the prelmnary results for the arspace-level queung model. Secton.Fnally, conclusons are presented n Secton VII. II. Spectrum of Stochastc Ar Transportaton Network Models Fgure shows the spectrum of stochastc ar transportaton network models beng consdered n the present research effort. Fgure. Spectrum of Stochastc Ar Transportaton Network Models

3 On upper end of the spectrum n terms of accuracy and fdelty le the Monte-Carlo smulatons of the Natonal Arspace System (NAS) usng hgh fdelty arspace smulatons such as FACET 4 (Future ATM Concepts Evaluaton Tool) and ACES 5 (Arspace Concept Evaluaton System). However, Monte-Carlo smulatons usng hgh fdelty arspace smulatons are computatonally expensve and produce non-analytc results. On the other hand, queung network models can provde much faster solutons for the stochastc arspace dynamcs at a lower level of accuracy. Queung network models create an abstracton of the NAS, wth regons of the arspace beng modeled as nodes n the network. Accordng to the queung termnology, the nodes (regons of arspace) are enttes that provde a servce to the customer (arcraft), wth the servce beng the operaton of the arcraft n that arspace. The nodes are nterconnected by a system of routes whch specfy the flows between the nodes. Fgure 3 llustrates a queung network. Each node s assgned a number of servers whch specfy the maxmum number of customers that can be smultaneously served by the node. Ths s a measure of the capacty of the arspace. For example, the Montor Alert Parameter (MAP) can be loosely consdered as a measure of the capacty of an Ar Traffc Control Sector. The nodes provde servce to the customer as long as the number of actve customers beng served s less than the node capacty. If the number of customers exceeds the node capacty, the customers wat n the queue, whch captures the effect of delay due to congeston. The arrval and servce processes at the nodes are descrbed by dstrbutons whch quantfy the stochastc nature of the arrval and servce. Fgure 3. Queung Network Wthn the doman of the queung theory, models of varous fdeltes can be constructed. Markovan queung models have the least fdelty as the nput and servce dstrbutons for each node are constraned to be exponental. The soluton for a queung network wth Markovan nodes s drect because of the specfc propertes of the exponental dstrbuton. Closed form analytcal solutons are avalable for the congeston analyss at each node. Snce the varance of the dstrbutons cannot be specfed ndependent of the means n the Markovan model, artraffc uncertanty models that prmarly affect the varance of the servce process cannot be readly ncorporated. Coxan Models provde the hghest fdelty among the queung models as the Coxan dstrbuton can approxmate any gven arrval or servce dstrbuton wth arbtrarly hgh accuracy. However the soluton methodology nvolves explct state enumeraton 6 and numercal propagaton of the state equatons (forward Chapman-Kolmogorov Equatons) 6,7 whch can be become unmanageable due to the large number of states n the system. Approxmate soluton methods provde hgher fdelty than the Markovan models wthout the computatonal expense assocated wth the Coxan models. Approxmate solutons characterze the arrval and servce dstrbutons wth the frst and the second moments. The propagaton of the frst moments (mean) though the network can be acheved usng the flow balance equatons, smlar to the Markovan model. In addton to the mean, the approxmate soluton methodology provdes lnear equatons for the propagaton of the second moment or varances through the network. Once the means and the varances of the flow are avalable at the nput to every node, the queung metrcs at the node such as the mean number n system, wat tme n queue, and queue length can be calculated usng approxmate formulae. Note that the soluton methodology assumes that the nodes are stochastcally ndependent once the approxmate flow parameters are obtaned. The approxmate soluton methodology provdes a convenent way to ncorporate the uncertanty models n the queung network. The change n means and varances of the servce processes due to the uncertantes can be provded to the model, and the flow balance and the varance propagaton equatons then capture the effect of the uncertantes over the entre network. 3

4 III. Arspace-Level Queung Network Model of the NAS The Arspace-level queung model shown n Fgure 4, models the en route flght segment as flows between pars of arports. The model consdered n ths report contans 40 maor arports n the contnental US. It models all other arports as Arport X. The 40 arports consdered are: PHX, ATL, CLT, BOS, ORD, MDW, DTW, CLE, PIT, DEN, DFW, IAH, IND, CVG, MCO, TPA, MCI, STL, LAX, LAS, SAN, MEM, BNA, FLL, MIA, MSP, LGA, TEB, EWR, JFK, SFO, SJC, SLC, SEA, PDX, BWI, DCA, RDU, PHL, IAD. These and are shown by green dots n Fgure 4. The blue lnes show all the lnks between the arport pars consdered whle the red lnes hghlght the lnks between Chcago- O-Hare (ORD) and all other arports consdered n the present model. Note that the lnks are abstract representatons of the connectons between the arports and do not depct the actual flght route of the arcraft as a straght lne segment. Fgure 4. Arspace-Level Queung Network Model of the NAS Fgure 5 llustrates the flght segments consdered for modelng the queung network between a par of arports n the present study. The queung models for the flght segments are formulated as follows:. Tax-Out: The tax segment s modeled as a G / G / s queue. The servce tme for the tax segment s the unmpeded tax tme determned from the Bureau of Transportaton Statstcs (BTS) 8 data. The number of parallel servers s s computed by matchng the traffc ntensty segment of the tax segment wth the takeoff runway segment, yeldng: µ. s = µ. s tax tax runway runway () Both arrvng and departng arcraft share the same tax space, and the tax servce s treated as a prorty queung servce wth arrvng (landng) arcraft gven hgher prorty than departng (take-off) arcraft.. Takeoff / Runway: The departure runways for all arports are modeled as sngle server queues wth servce rates obtaned from the Arcraft Departure Rates (ADR) publshed n the BTS 8 data. Smlar to the tax queues, the takeoff and landng segments are currently modeled as a server wth shared servce for arrvng and departng arcraft queues. 4

5 Fgure 5. Flght Segments between One Arport Par n the Arspace-level Queung Model 3. Clmb: The clmb segment s modeled as a G / G / s queue wth number of parallel servers s s obtaned by matchng the traffc ntensty segment of the clmb segment wth the take off runway segment. The servce tme n clmb s derved usng the BADA database 9. Clmb uncertantes are ncorporated nto the model accordng to the procedure outlned n References and. 4. En route: The en route segment between each arport par s modeled as a G / G / queue. Snce the avalable en route arspace s much larger than the demand under normal operatng condtons, the en route segment s modeled wth nfnte capacty. The nomnal en route flght tme s between each arport par s obtaned by extractng data from a FACET smulatons of arcraft flght data from TRX fles. Fgure 6 shows the nomnal servce tme dstrbuton for the en route LAX-JFK node of the Arspace-level queung model. En route uncertantes are ncorporated nto the model accordng to the procedure outlned n References and. Fgure 6. Nomnal servce tme dstrbuton for the LAX-JFK node of the Arspace-level queung model 5

6 5. The Descent, Landng Runway and Tax-In models are developed smlar to the Clmb, Take-Off Runway and Tax-Out models respectvely. Fgure 7. Incorporatng an Arport Model nto the Arspace Model Fgure 7 shows how the dfferent models descrbed above are ncorporated nto a network of queung nodes. The fgure depcts the connectvty of the nodes for the th arport and s representatve for all 40 arports n the NAS. The departng flows from arport gates are drected towards the Tax node, then to the Runway node, followed by the Clmb node. After undergong servce at the Clmb node, the flows are splt between the all arports n the NAS, based on pre-calculated flow fractons. Flows from other arports to the th arport are represented by the outputs of the (,), (,) (n,)th lnks. The aggregate of these flows are drected to the Descent node, followed by the Runway and Tax nodes, respectvely. It should be noted that the departure flow from the Clmb node to Arport X s treated as a sequence of departures from the network at the Clmb node, and the arrval flow from Arport X to the Clmb node s treated as a sequence of external arrvals to the network at the Descent node. Smlarly, the arrvals from gates to the Tax node are treated as external arrvals to the network, and departures from the Tax node to the gates are treated as departures from the network. Although FACET models arcraft traectores after arcraft clmb has been completed, t s assumed for the purpose of modelng that these flows are dentcal to the gate departure flow. The arrvng flow nto the arports can then be calculated based on the connectvty of the arports to each other. Therefore, the competng departng flow and arrval flows, for the Tax and Runway nodes can be calculated, and the theory of prorty queues 0 can be used to obtan queung parameters for the arrvng and departng flows through these nodes. The arrvng and departng flows can also be used to construct the flow fracton from the Tax node to the Runway node, and from the Runway node to the Tax and Clmb nodes, respectvely. It may be shown from the above, that for n modeled arports n the system, a total of 4 queung nodes exst, and 4 flows can be determned through these nodes. The connectvty matrx for the system ncludng the arport models s of dmenson 4 4. The followng secton presents approxmate soluton methodology for queung networks wth G/G/s queues. IV. Approxmate Soluton for Queung Networks wth G/G/s queues The Queung Network Analyzer (QNA) formulaton 3 bulds on the well-known Markovan model for M/M/s (Markovan arrval, Markovan servce process wth s parallel servers). In a Markovan queung model, the mean cannot be dfferentated from the varance and only Markovan nputs are allowed for the arrval and servce dstrbutons. However, n the QNA model one can defne frst and second moments of arbtrary arrval and servce 6

7 dstrbutons, whch may better approxmate the real processes. Note that whle Markovan analyss s an exact soluton to an approxmate model that does not ft the nter-arrval and servce tme dstrbutons exactly, the QNA s an approxmate soluton to a hgher-fdelty model that matches the nter-arrval and servce tme dstrbutons more closely. A. Flow Balance Equatons The mean traffc flows nto queung network nodes (.e. the nternal arrval rates) are estmated by solvng the flow balance equatons. The expected flow rate through each node ( λ ) s gven by 7 n λ = p + λ q 0 () where n : number of nodes q : proporton of those customers completng servce at node that go to next node p 0 : external arrval process to node The QNA model bulds on the flow balance calculaton by calculatng the varances for these nternal arrval rates. Ths process s outlned n the next secton. = B. Flow Varablty Equatons The heart of the QNA approxmaton 3 s the system of equatons yeldng the varablty parameters for the nternal flows,.e., the squared coeffcents of varaton or the coeffcent of varaton, defned as the rato of the standard devaton to the mean. The coeffcent of varaton where and a and c a = a n + c b, a = b are constants dependng on the nput data a = + w The varables n these equatons are: m : number of servers at node = λ µ m, where ρ / p ca s obtaned usng the followng expresson: n n {( p c ) + [( ) ( ) ]} 0 0 p q + q ρ x b λ s the arrval rate and λ = λ q, the arrval rate from node to node. = [ ν + ( ν )( ρ )] (3) ν (4) = w p q (5) µ the servce rate of th node = λ / λ, proporton of arrvals to that came from, 0 c 0 : varablty parameter of the external arrval process to node Here x, ν and nternal arrval process and wth w depend on the basc data determned prevously, e.g., on ca whch s beng solved for. The varables x, ν and ( max[ c,0.] ) ) ρ, m and c 0, but not on the w n ths verson of QNA are 0.5 x = + m (6) w s ν = 0 (7) [ ( ) ( )] + 4 ν = ρ (8) n = p = 0 ν (9) Here p s defned as n equaton (4) and c s s the squared coeffcent of varaton for the servce process. The key approxmaton equatons (3) through (9) whch yeld the varablty parameters for the nternal flows are all

8 based on the basc method dscussed n reference 3, asymptotc methods and the statonary nterval method. Here the equatons for the basc operatons: superposton (merge), splttng and departure wll be gven n the followng. Superposton: The summaton n (0) s ust the matrx element Splttng : Departure Process: c d ν = c (0) λ k k λ 8 p and the c s the external arrval process to that node. c = p c + p () ρ ( )( c ) + ( c ) = + ρ a s () m C. Queung Analyss of Each Node After the varance propagaton throughout the network s completed, the frst two moments for the arrval process to each node are avalable and hence each node can be analyzed n solaton. The M/M/s queung model s appled to each node to determne the nodal queung metrcs such as the number of arcraft n the system L, the number n the queue L, the total average tme spent by an arcraft n the system W and the tme spent n the queue q q W. The nodal queung metrcs were calculated usng the followng QNA approxmatons for the general arrval process, general servce processes and multple-server queues (G/G/s). c c W a + s qqna = W q( M / M / s) (3) where W qqna and W q (M/M/s) are the wat tmes for the QNA and Markovan models and squared coeffcents of varaton for the arrval and servce processes respectvely. c a and cs are the The other QNA system parameters, W QNA (the total tme n the system), L qqna (the queue length) and L QNA (the number n the system) follow from equaton (3). The wat tme n the system s gven by W QNA = W qqna + (4) µ The number n the queue L qqna s gven by Lttle s formula 4 The number n the system s gven by L L QNA = λ (5) qqna W qqna = λ W + ρm (6) qqna Further detals on the QNA methodology can be found n Reference 3. V. Modfed Markovan Network Soluton wth NAS Uncertantes As mentoned earler, the basc Markovan formulaton does not facltate the addton of NAS uncertantes that modfy the varance of the nomnal dstrbuton. To overcome ths drawback a modfed soluton methodology was proposed pror to the mplementaton of the QNA formulaton. Ths nvolves the use of an addtonal node n seres wth the servce parameter equal to the recprocal of the square root of the uncertanty varance. Ths approxmate approach s descrbed below. An exponental dstrbuton can be defned by a sngle parameter α. To ft an exponental dstrbuton to observed data the parameter α s dentfed as = (7) mean of observed data

9 Ths also determnes the varance of the exponental dstrbuton as / α. Snce the mean and the varance of the exponental dstrbuton cannot be chosen ndependently, uncertantes characterzed by more than one parameter cannot be combned the Markovan model n a drect manner. However most of the uncertanty models descrbed n References and perturb the varance of the servce tme. An alternatve mechansm s devsed to ncorporate these uncertantes nto the Markovan model. Ths requres an addtonal M / M / s queue placed n cascade wth the nomnal queue, wth parameter µ such that µ = (8) varance(uncertanty Data) If the uncertanty had a non-zero mean, the parameter µ of the nomnal queue s modfed as µ = (9) mean(nomnal) + mean(uncertanty) Fgure 8 llustrates the procedure used to combne the nomnal and the uncertanty dstrbutons n a Markovan queung model. Although ths s a crude approxmaton, t allows a natural way to nclude both the mean and the varance of the uncertanty n Markovan queues. Note that the second queue added n cascade wll ncrease the servce tme by the standard devaton of the uncertanty. Thus t accounts only for the delays due to uncertanty and gnores early arrvals and hence may over estmate the delay due to the uncertanty. Fgure 8. Combnng nomnal and uncertanty models n Markovan queues VI. Applcaton of QNA and Modfed Markovan Methodologes In ths secton, results for the arspace lnk-based model, usng the QNA algorthm and modfed Markovan methods, are presented. Snce the arspace lnks are modeled as G/G/ queung servces, regardless of the servce tme dstrbuton, the output process s always Markovan. However, ths s not necessarly true for the other nodes n the system, namely Runway and Tax nodes. Ths s shown n Fgure 9, for DFW arport. It s clear from the probablty dstrbuton hstogram obtaned on the rght, that the servce tme s not exponental. Due to the lmted runway capacty, ths s expected to have an effect on the nter-departure tme dstrbuton from the Runway, and consequently, the nterval tme dstrbuton to the other nodes n the arport and NAS. 9

10 Fgure 9. Runway Servce Tme Dstrbuton for DFW from Publshed BTS Data The network parameters to formulate a network model for the NAS are obtaned from FACET smulatons of arcraft TRX fles, for the perod June, 007 to June 3, 007. The use of multple data ensures a larger number of samples from whch mean and varance (or squared coeffcent of varaton) can be calculated. Arrval data from the arports, servce tmes for the arport-arport lnks (enroute nodes), and flow fractons are calculated from flght data. Statstcs for a Markovan network wth no uncertanty models are shown n Fgure 0. Ths fgure shows the mean tme n servce for flghts from DFW arport to the other arports n the NAS. The horzontal bars are dvded nto segments showng ndvdual servce tmes n each node that a flght from DFW has to travel through. For example, a flght from DFW to JFK wll experence Tax, Runway, and Clmb servce at DFW, followed by servce n the DFW-JFK lnk, followed by Descent, Runway, and Tax servce at JFK. Therefore, the mean tme n servce can be obtaned by the summaton of the mean tems n servce parameter L for each of these nodes. It s worth notng that n some cases, the statstcs from the en route segment of the flght are absent. For example, the mean tme n servce n the en route segment for the DFW-TEB lnks zero because no flghts are recorded between these two arports n the TRX fles for the gven perod. Fgure shows the delays n the system predcted by the network of queung systems, due to lmted node capacty. These are obtaned from the mean number of tems n queue, L q, for each node. Although each node can have a delay assocated wth t, t must be kept n mnd that the networkng of queung systems s an abstracton of the NAS, and the actual delay manfests tself pror to the Tax node for takeoff, and pror to the Descent node for landng. The total delays are predcted to be between to 4 mnutes, and s maxmum for the DFW-FLL flghts. It s mportant to note that en route delays are zero: due to the assumpton of nfnte capacty n an arport par lnk, each arcraft arrvng nto the lnk s rendered servce wthout queung. Ths s a result of the resoluton of the abstracton used to model the arspace. It s antcpated that for fner spatal resolutons, where a fnte capacty s assocated wth each node, queung phenomena wll be more evdent. Both Fgure 0 and Fgure are solutons correspondng to the applcaton of the QNA algorthm to nomnal system wthout uncertanty models ncluded. The results for these queung parameters were not found to be dfferent usng M/M/s analyss. Ths s because the QNA algorthm modfes the queung parameters accurately only when the squared coeffcent of varaton of the servce tme dstrbuton s wthn a range In ths example, the squared coeffcents of varaton do not le n ths range. 0

11 Fgure 0. Mean Tme of Arcraft n Servce, Orgn Arport DFW Fgure. Delays Predcted by Arspace Network Model

12 The effects of uncertanty on node parameters are now studed. The Clmb node characterstcs mean tme n servce and the standard devaton of tme n servce are presented n Fgure. These results do not nclude uncertanty models, and are dentcal from M/M/s and QNA analyss. As noted earler, ths s because the formulae for modfyng the queung parameters are not vald beyond a certan range of squared coeffcent of varaton of node servce tme. Fgure. Mean and Standard Devaton of Clmb Tme of Arcraft wthout Uncertanty Fgure 3 shows the standard devaton of tme n the Clmb node for each arport, when calculated from the soluton usng the QNA algorthm, as well as the Modfed Markovan methodologes, when the uncertanty models are ncluded. The uncertanty models, when convolved wth the nomnal clmb tme dstrbuton, cause nsgnfcant change to the mean of the clmb tme dstrbuton. However, the varance of the clmb tme dstrbuton can be shown to ncrease, wth the ncluson of uncertanty models. Ths ncrease can be observed by comparng Fgure and Fgure 3 the QNA algorthm predcts an ncrease of 0-30% n standard devaton wth the ncluson of the uncertanty model. The use of the the Modfed Markovan Method, as shown by the darker hstograms n Fgure 3, results n an overestmaton of the standard devaton calculaton. The standard devaton estmated s approxmately mns more n each arport node, when compared to the result from QNA. A more detaled study of the accuracy of the two methods can be made by Monte Carlo smulatons of the modeled network. However, ths could result n a tme consumng process due to the large number of nodes nvolved.

13 Fgure 3. Standard Devaton of Tme n Clmb Node VII. Conclusons The prmary goal of the present research was to develop queung models that can analyze the mpact of traectory uncertantes on the traffc flow effcency throughout the Natonal Arspace System. Prevously developed Markovan models of the NAS dd provde a mechansm to modulate the varance of traectory uncertanty, so as to observe ts effect on the traffc flow effcency. Ths s because the exponental dstrbuton used n the Markovan analyss has only a sngle degree-of freedom. Thus, when the mean s specfed, the varance s set to the square of the mean. However other analyss methods such as the Queung Network Analyzer (QNA) provde two knobs to modulate both the mean and the varance, as they model the servce dstrbutons wth the frst and the second moments. Ths comes at a prce, as the elegant closed form analytcal solutons of the Markovan model are no longer avalable. Hence approxmatons must be used for propagaton of varance throughout the network and calculaton of the nodal queung metrcs. Ths paper formulated an Arspace-level queung network model, whch modeled the NAS as lnks between 40 maor arports n the contnental Unted States. An approxmate soluton usng the QNA formulaton and the Modfed Markovan network soluton wth uncertanty are presented. QNA modelng approach appears to provde a sgnfcant reducton n the estmates for wat tmes when compared wth the Modfed Markovan estmates. Although the QNA soluton results have been valdated n smulaton studes of smaller networks, valdaton of the QNA results for the large-scale NAS model has not been performed yet. Efforts are underway to valdate the QNA results of a NAS queung model by comparson wth Monte-Carlo Smulatons usng FACET. 3

14 Acknowledgments Ths research s supported under NASA Contract No. NNA07BC55C, wth Mr. Mchael Bloem servng as the Techncal Pont-of-Contact. Ms. Rebecca M. Grus served as the Techncal Montor. References Tandale, M. D., Menon, P. K., Cheng, V. H. L., Rosenberger, J., and Subbarao, K., Queung Network Models of the Natonal Arspace System, 8th AIAA Avaton Technology, Integraton, and Operatons (ATIO) Conference, Anchorage, AK, September 4 9, 008. Fredrck S. Hller and Gerald J. Leberman, Introducton to Operatons Research, McGraw-Hll, New York, NY, Whtt, W., The Queung Network Analyzer, The Bell System Techncal Journal, Vol. 6, Issue No. 9, November Blmora, K. D., Srdhar, B., Chatter, G. B., Sheth, G., and Grabbe, S., FACET: Future ATM Concepts Evaluaton Tool, 3rd USA/Europe Ar Traffc Management R&D Semnar, Naples, Italy, June Raytheon ATMSDI Team, Arspace Concept Evaluaton System Buld Software User Manual, NASA Ames Research Center, Moffett Feld, CA, November Harry G. Perros, Queueng Networks wth Blockng, Oxford Unversty Press, New York, NY, 994, pp Medh, J., Stochastc Models n Queueng Theory, nd Edton, Academc Press, November, Tme 9 European Organzaton for the Safety of Ar Navgaton EUROCONTROL Expermental Centre: Arcraft Performance Summary Tables for the Base of Arcraft Data (BADA) verson Jaswal, N. K., Prorty Queues, Academc Press, New York, NY, 968. Km, J., Palanappan, K., Menon, P. K., Subbarao, K., and Nag, M., Traectory Uncertanty Modelng for Queung Analyss of the NAS, AIAA Avaton Technology, Integraton, and Operatons Conference, Anchorage, AK, September 4 9, 008. Menon, P. K., Tandale, M. D., Km, J., Sengupta, P., Kwan, J., Palanappan K., Cheng, V. H. L., Subbarao K., Rosenberger J., Nag, M., Josh, S. and Roongrat, C., Mult-Resoluton Queung Models for Analyzng the Impact of Traectory Uncertanty and Precson on NGATS Flow Effcency, Frst Annual report prepared under NASA Contract No. NAS-004, Optmal Synthess Inc, November 8, Lam, Teresa, C, Superposton of Markov Renewal Processes and ther Applcatons, Techncal Report No 9-9, Unversty of Mchgan, Ann Arbor, D.P. Heyman and M.J. Sobel, Stochastc Models n Operatons Research, Vol. I, New York: McGraw-Hll, Marchal, W. G., Numercal Performance of Approxmate Queueng Formula wth Applcaton to Flexble Manufacturng Systems, Annals of Operatons Research, Vol 3, 985, (pgs4-5). 4