Modeling and Optimization of Link Traffic Flow (Paper # )

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1 Uiversity of Massahusetts Amherst From the SeletedWorks of Daiheg Ni 2008 Modelig ad Optimizatio of Lik Traffi Flow (Paper #08-229) Kimberly Rudy Haizhog Wag Daiheg Ni, Uiversity of Massahusetts - Amherst Available at:

2 Lik Traffi Flow Optimizatio Kimberly Rudy Udergraduate Researh Assistat Departmet of Civil ad Evirometal Egieerig Uiversity of Massahusetts Amherst 224 Marsto Hall 30 Natural Resoures Road Amherst, MA 0003 USA Phoe: (43) Fax:(43) Haizhog Wag* Graduate Researh Assistat Departmet of Civil ad Evirometal Egieerig Uiversity of Massahusetts Amherst 4 Marsto Hall 30 Natural Resoures Road Amherst, MA 0003 USA <wag@egi.umass.edu> Phoe: (43) Fax: (43) Daiheg Ni, Ph.D. Assistat Professor Departmet of Civil ad Evirometal Egieerig Uiversity of Massahusetts Amherst 29 Marsto Hall 30 Natural Resoures Road Amherst, MA 0003 USA <i@es.umass.edu> Phoe: (43) Fax: (43) Date Submitted: August, 2007 Word Cout: 5498 Number of Tables, Figures ad Pitures: 8 Effetive Word Cout: 7498 *Correspodig Author

3 Abstrat Cogestio i etworks greatly redues effiiey ad produtio. Systems with limited apaities or resoures require aalysis i order to esure optimal results, whih may be i terms of ost, data trasmitted or vehiles disharged. A etwork is like a iteroeted web ad if oe setio or lik is ot performig optimally, the etwork may ot be operatig effiietly. The objetive of this researh is to maximize lik traffi throughput i the log ru to alleviate ogestio. The approah is to model the hages i lik traffi states as a disrete Markov hai, from mathematial theory, due to its radom or stohasti ature. This lik traffi flow model a be used for varyig stohasti proesses with orrespodig performae measures, for example, a servie rate. While this paper fouses o vehiular traffi flow, other disiplies are ivited to ollaborate o the study of lik traffi flow i similar stohasti systems. This researh presets a ovel blokig probability distributio to aout for ogestio based o the M/G// state-depedet queuig model. The objetive futio with the blokig probability was optimized ad the results were ompared with a simulatio model. The optimal solutio to the objetive futio is a flow at whih throughput o the lik is maximized for the log ru. Uder a Vehile ifrastruture Itegratio (VII) seario, this model may serve as the basis of lik flow otrol i a effort to ahieve the maximum lik through i the log ru. INTRODUCTION Cogestio is a problem that is ubiquitous i systems, dereasig output ad effiiey. While the ompoets of systems or etworks vary, effiiet produtio is desirable. Ofte, ogestio is due to ompetitio for limited resoures or apaities. As a example, thik of a ovetioal buket, the apaity of the buket is restrited to the height of its walls. If ay mirosopi setio of the wall of the buket were abset or shorter tha the rest, its otets would spill out to the height of the shortest setio. Similarly, the ogestio i etworks may be based o the performae of a sigle setio or lik i the system. Whether you are osiderig data pakets through a iteret path or vehiles alog a setio of roadway, their basi lik performae is restraied by a limited apaity. Various lik traffi flow models have bee used i omputer etworks with limited badwidth, eergy oservatio i wireless devies ad i oise preditio i the field of aoustis. Oe lik performae is optimized (height of buket setio optimized), the the optimizatio a be expaded to a series of liks or a etire etwork to redue ogestio. Give the Federal Highway Admiistratio s (FHWA) estimate that vehile miles of travel ireased 89 peret while lae-miles of highways ireased oly 5 peret betwee 980 ad 2003 [], ogestio i trasportatio has beome a importat issue affetig most Amerias, both diretly ad idiretly through opportuity osts ad risig food pries. This researh iteds to provide a lik traffi flow optimizatio model that may be appliable to trasportatio as well as omputer ad idustrial etworks. Some assumptios made are the stohasti ature of traffi flow, Markovia traffi states ad a o-liear dereasig servie rate. Based o the expeted traffi state from disrete Markov hai theory, a objetive futio for the optimal throughput was reated with a ew blokig probability distributio from stohasti traffi M/G// state-depedet queuig theory [2-5]. The proposed blokig probability formulatio is uique to the best of the authors kowledge. The objetive futio with the blokig probability pealizes ustable flows. The optimal solutio is a proposed flow at whih the lik throughput would be maximized based upo the preset oditios. LITERATURE REVIEW This setio presets a review of previous literature relevat to the proposed model of lik traffi flow optimizatio. Additioally, the foudatios of the proposed model are highlighted for their merit.

4 Rudy, Wag ad Ni 3 Short-term traffi foreastig a be a useful desig tool to aommodate expeted future demad. There have bee umerous methods preseted, ompared ad expaded upo for short-term traffi foreastig whih ilude radom walk, historial average, Markov hai, time-series, joit distributio realulatio, eural etworks, geeti algorithms, Bayesia etworks, oparametri regressio ad loal liear regressio amog others. Despite the rage of methods, Markov hai s simpliity i deploymet ad omputatio has prove robust for short-term traffi flow preditio [6-9]. Markov hai is a mathematial theory for radom proesses, thoroughly defied i the ext setio. For umerous ategories of traffi states, Markov hai a be omputatioally demadig, for umber of traffi states, the trasitio matrix otais x probabilities. Yet if the traffi states a be grouped ito small ategories as arried out i this researh, the likelihood that traffi will trasfer to aother state a be easily geerated. Traffi flow optimizatio has also bee preseted i a wealth of papers. While some have made use of the Markov deisio proess i their traffi flow optimizatio [0, ], others have used M/G// state-depedet queuig theory to model sigle lik vehiular traffi flow i order to perform optimizatio [2-5]. From M/G// state-depedet queuig theory, this paper proposes a ew blokig probability distributio based o the disrete Markov hai traffi preditios. The blokig probability has bee suessful i radom systems whose travel speed dereases o-liearly as a futio of the traffi desity i the system, whih is the ase for vehiular ad pedestria traffi flows. The objetive i this researh is to maximize throughput. The throughput equatio omes from queuig theory performae measures ad is defied as a futio of the iput flow rate ad a pealty or risk futio. The pealty futio aouts for the loss of smooth traffi flow due to ogestio. Here disrete firstorder Markov hai theory from traffi foreastig is ombied with M/G// state-depedet queuig theory i order to maximize throughput o a lik while miimizig the risk of traffi istability. However, this paper is ot the oly researh that ombies Markov hai theory ad throughput aalysis [2]. THEORY This researh requires some bakgroud iformatio o the theories used. To failitate subsequet disussio, these theories are briefly itrodued here. Advaed readers who are familiar with disrete Markov hai ad M/G// queuig theories may skip this setio ad proeed to the ext setio. Disrete Markov Chai Theory From mathematis, disrete Markov hai theory produes oditioal or trasitio probabilities of a fiite umber of future evets without requirig ay historial kowledge of the data, oly the preset state. Markov hai is appliable to time disrete, stohasti proesses. The Markovia property is give by, p ( S( t + ) = A S(0), S(), S(2),..., S( t)) = p( S( t + ) = A S( t)) S(t+) = the state immediately followig the preset state after the ext time iterval S(t) = the preset state S(0),...= the states prior to S(t) A = the state that S(t) will be i at time t=(t+) The oe-step trasitio probability is a probability that a future state will our immediately followig the preset state. The oe-step trasitio probabilities are defied as, p ij = N

5 Rudy, Wag ad Ni 4 for i, j = (, 2,, a) p ij = the probability that j will immediately follow i = the umber of times j immediately follows i N = the umber of times i trasitios to ay state iludig repetitio of i a = the umber of defied traffi states The oe-step trasitio probability matrix represets all the possible state hages betwee traffi states, eah ell i the matrix orrespods to the probability of a preset ad future state pair (p ij ). The size of the trasitio matrix is a x a, a is the umber of defied states. Modelig Vehiular Traffi Flow Usig Disrete Markov Chai Theory It is easily observed that the urret ad previous traffi states a show the tred i the ext iterval. Various preditio methods use historial data to model suessive oditios, for example, weather ad limate foreasts. The future state of traffi i the subsequet iterval has strog but ot determiisti or fixed relatioship to the urret or very reet states. This relatioship a be osidered a oditioal probability. That is to say, the ext traffi state obeys a probability distributio ad the probability of the ext state is determied by the urret ad immediately preedig states. However, a geeral assumptio is that states too far bak i the past will ot give ay more iformatio tha the urret ad immediate states. If a series has the attribute that, give the urret ad N- preedig states, the future state is idepedet of the states prior to the preset state, the series a be alled a Markov hai with N-order [8]. I this paper, Markov hai is employed to predit the expeted traffi states with a proper probability distributio sie flutuatig historial data for vehiular traffi flow aot aurately foreast ogestio due to its stohasti ature [7-9]. Traffi s Classifiatio There are three parameters that haraterize traffi states: speed (v), flow (q), ad desity (k). With a elapsed time hage, these three parameters also hage. As a result, traffi states hage otiuously as a futio of time. If w is defied as a variable to represet a traffi state, w a be writte as w= ( q, k, v). Additioally, speed (v) ad flow (q) a be represeted as a futio of desity (k) by the traffi flow fudametal relatioship q = kv. Now the represetatio of traffi states a be redued to oe variable w ( t) = k( t). Durig eah time iterval, there is a speifi orrespodig traffi state that a be represeted by its desity. Speed or flow was ot hose to sigly desribe the traffi states beause both speed ad flow are multivalued futios i the fudametal diagrams. Some logi behid this step is that traffi desity is ofte how drivers pereive ogestio. If a driver feels that there may be a uomfortable umber of vehiles i his lie of sight, the driver will moderate his speed appropriately for safety ad ease of drivig. Moreover, the time ad desity rages must be partitioed ito disrete uits for use of disrete Markov hai. This researh assumes a time iterval of 5 miutes ad a desity iterval of 2.5 veh/km/l. The upper limit of the desity rage is the jam desity. Jam desity is osidered the desity at whih the vehiles are bumper-tobumper ad ompletely stopped [4]. Average jam desity values reported vary betwee vehiles per mile per lae (vpmpl) [4].This has bee traslated to 40 veh/km/l. I order to more learly illustrate the traffi states, assume that desity rages from 0 to 50 veh/km/l. This otiuous rage of values is divided ito disrete segmets over the rage by a basi uit of graduatio. Usig Δk = 2.5 veh/km/l as the basi uit to divide up the desity rage, the resultig ritial poits for desity

6 Rudy, Wag ad Ni 5 are 0, 2.5, 25, 37.5, 50,, 50 veh/km/l. Despite jam desity s defiitio as the upper limit of the desity rage, the last state allows for a desity greater tha 40 veh/km/l. This gap is to allow the model to aout for ay istataeous ourrees of desities greater tha the assumed jam desity. The arragemet of the ombiatios of traffi desity states geerates a fiite trasitio matrix to represet all the possible state hages betwee traffi states, eah ell i the matrix represets the probability of a preset ad future desity pair. The size of the trasitio matrix is a x a, a is the umber of traffi states. Therefore, twelve traffi states are defied i this paper ad summarized i Table. Table Defiitio of Traffi s Traffi s Classifiatio Desity Iterval k i (veh/km/l) 0 k i k i k i k i k i k i k i k i k i k i k i k i 50 M/G// Queuig Theory The M/G// state-depedet queuig model aouts for flows with ogestio ad apaity ostraits for the irulatio spae. The aroym M/G// stads for Markovia arrival proess, Geeral state-depedet servie rates, parallel servers ad a total apaity of [3]. A server is a hael or lae through whih traffi a flow. The servie rate is defied as the ratio of the average travel speed for a ertai umber of ustomers o the lik to the V average travel speed of a sigle oupat, f ( ) =. The servie rate is a dereasig o-liear futio i the V M/G// state-depedet queuig model [5]. As the desity of the uits eterig the irulatio spae ireases, the average travel speed of the uits dereases as see i the fudametal relatioship [3], yet the empirial urves for vehiular traffi flow strogly suggest that the expoetial model may provide a more aurate approximatio for the average travel speed with variatios i vehiular desity [2]. However, there is a liear model ad a expoetial model to desribe the relatioship betwee desity ad mea travel speed. The liear model is defied as, A V = ( C + ) C V = the average travel speed for uits i the irulatio spae A = V = the average travel speed of a sigle oupat C = the apaity of the irulatio spae = the umber of uits utilizig the irulatio spae The expoetial model is desribed by, V = Aexp β

7 Rudy, Wag ad Ni 6 ( ) l V a A a = sale ad shape parameter, = l l l( Vb A) b a b β = sale ad shape parameter, = = l A V l A β a V b [ ( )] [ ( )] ad a,b = umber of vehiles based o the empirial data V a = average travel speed at desity a, approximated from empirial data V b = average travel speed at desity b, approximated from empirial data I a oe-diretioal lik, assume that there is a arrival rate at whih the uits arrive to the lik ad whih is idepedet of the umber of uits o the lik. Also, assume that there is a servie rate o the lik that depeds o the umber of uits utilizig the lik. The M/G// steady state probabilities are give by, ( λϑ) P = P0! f ( ) f ( )... f (2) f () for =,2,,C ad p 0 is the empty system probability desribed by, C i ( λϑ) P0 = + i= i! f ( i) f ( i )... f (2) f () λ = the arrival rate ϑ = the mea servie requiremet, ϑ = L A, L is the legth of the irulatio spae C = the apaity of the irulatio spae, C = kln, k is the apaity of the irulatio spae per square uit, N is the umber of servie haels f () = servie rate From the expoetial model of average travel speed (V ), the servie rate f() is give by, V A f ( ) = = exp = exp V V β β From the liear model of the average travel speed (V ), the servie rate f() is defied as, V A C + f ( ) = = ( C + ) = V CV C Furthermore, the liear state probabilities are expressed by, ( λ L A) P = P 0 j[ ( C j) C] + j = ad C i ( λ L A) P0 = + i i= j[ ( C + j) C] j= From the expoetial model of average travel speed (V ), the servie rate f() is give by,

8 Rudy, Wag ad Ni 7 V A f ( ) = = exp = exp V V β β Similarly, the expoetial state probabilities are desribed by, L λ j P = j exp P0 A j= β ad C i i λl j P0 = + j exp i= A j= β From the M/G// steady state probabilities, several performae measures a be derived. P = P ( C) P b = θ = λ = b = = λ( Pb θ blokig probability, the steady state probability whe the umber of uits o the lik is at apaity throughput, the rate of vehiles leavig the lik the arrival rate to the lik ) Modelig Vehiular Traffi Flow Usig M/G// Queuig Theory As ayoe may observe, the speed of traffi o a roadway is greatly iflueed by the desity of vehiles o a lik. As the desity of the vehiles o the lik ireases, the average travel speed of the vehiles o the lik dereases. M/G// queuig theory has bee suessfully used to model vehiular ad pedestria traffi flow for dereasig o-liear servie rates. The empirial data of this researh i Figure shows the same tred No-Liear Dereasig tred 80 Speed km/h Desity veh/km/l Figure No-Liear Servie Rate from Empirial Data

9 Rudy, Wag ad Ni 8 MATHEMATICAL MODEL Geeral Coept of Modelig This model a be used to maximize throughput o a lik give the preset oditios. From the preset traffi state, the probability that traffi will trasitio to ay other state is derived from Markov hai theory. I this model, desity is used to represet the traffi states. Usig a weighted average of the oe-step trasitio probabilities for a partiular traffi state, the expeted traffi desity i the ext time iterval a be alulated. The expeted umber of vehiles o the lik is give by the expeted traffi desity, the legth ad the umber of laes of the lik. The expeted umber of vehiles is used i expoetial model of the blokig probability. Next, the blokig probability is used i the objetive futio to maximize throughput of the lik. The logi is summarized i Table 2. Table 2 Model Developmet Flow Chart PRESENT TRAFFIC STATE q(t) q(t) Pr [q(t+)=,2,,2] E[q(t+)] = E[k] = E[k] l N Expoetial P b = f() θ = q(t) (-P b ) Figure 2 shows the basi operatio of the objetive futio. Here the objetive futio has bee broke dow ito its ompoets, θ = q () t [ q( t) Pb ]. The first term of the rewritte equatio is the iput flow. The seod term is the pealty futio. I the figure, the red dashed lie represets the iput flow, the blue solid urve idiates the suggested throughput orrespodig to the iput flow ad the gree dash-dot-dash urve deotes the pealty or risk assoiated with ay level of ogestio. From the objetive futio, θ = q () t [ q( t) P b ], it is obvious that the throughput will be less tha the iput volume, eve by a egligibly small amout, uless the traffi meets the boudary oditios. To maximize the throughput, this researh seeks the peak of the blue solid urve. Outflow Iput Flow Throughput Iflow Pealty Figure 2 Compoets of the Objetive Futio

10 Rudy, Wag ad Ni 9 The Lik Traffi Flow Optimizatio Model As previously stated, this researh aims to maximize the throughput (θ) of a lik i the log ru to redue ogestio. The throughput iludes the preset flow (q(t)) o the lik ad aouts for ogestio with a orrespodig blokig probability (P b ). The objetive futio is: θ = throughput o the lik q(t) = preset flow q max = maximum possible flow P b = probability of ogestio Maximize: θ = q( t)( P b ) Subjet to: 0 q( t) qmax 0 P b From queuig theory, the blokig probability is the state-depedet probability as a futio of the mea servie requiremet ad the relative servie rate whe the umber of vehiles o the road is at its apaity [2]. The proposed blokig probability distributio has bee defied as a futio of a similar relative servie rate ( v v f ), the umber of vehiles o the lik ad the umber of vehiles o the lik at apaity. Here the free flow speed replaes the previous average travel speed of a loe oupat, whih was defied as the posted speed limit i [2]. It may be urealisti to assume that drivers remai uder the posted speed limit. The proposed blokig probability distributio is etered o the Markov hai trasitio probabilities istead of the M/G// steady state probabilities. Blokig Probability Distributio Formulatio The blokig probability is give by P = b v v f = expeted umber of vehiles o the lik, = ke L N, ke = expeted future desity, L = legth of the lik, L = 0.55 km ad N = umber of laes o the lik, N = 4 v = average travel speed o the lik for vehiles v = free flow speed, = 5 km/h f v f = umber of vehiles o the lik at apaity, = k L N, k = jam desity, = veh/km/l jam jam k 40 jam Boudary Coditio Aalysis Whe the umber of vehiles o the lik () is very small, approximately, the average travel speed equals the free v =. I this ase, P 0. A blokig probability of zero idiates that there is o flow speed ( ) v f b = ogestio o the road ad drivers do ot moderate their speed with respet to other vehiles. Whe the umber of

11 Rudy, Wag ad Ni 0 vehiles o the lik () approahes apaity ( ), all traffi is at a stadstill. From [4], the average travel speed is exatly zero ( v = 0) whe +. Although the lik apaity by defiitio is the absolute limit for the road, = there still may be some movemet whe = so v = 0 for all situatios +, a theoretially improbable seario [4]. I this ase, movemet o the lik sie it is ompletely full. Expeted Future Desity ke The expeted future desity ( k E ) P b =. A blokig probability of oe idiates that there is absolutely o is based o the Markov hai trasitio probabilities. Here, future meas the subsequet time iterval. I statistis, the expeted value of a outome is a weighted average of all possible results based o their likelihood or probability. From a ertai traffi state, defied by its desity, multiple traffi desities may immediately follow. The expeted future desity is give by p = trasitio probability k i i k E 2 = pi ki i= for i =, 2,, 2 = possible desity traffi state With the trasitio probabilities, the expeted future desity a be geerated. From the expeted desity, the expeted umber of vehiles o the road a be alulated ( = k E L N). The expeted umber of vehiles a be used diretly i the blokig probability ad used to defie the average travel speed o the lik for the expeted umber of vehiles ). v ( Revised Liear ad Expoetial Models Used to Approximate Blokig Probability Liear ad expoetial models have bee developed i [2]. Similar models of the average travel speed are used i this model however, the free flow speed to replaes the previous average travel speed of a loe oupat, whih is defied as the posted speed limit [2]. The revised models are give by v Liear: = ( + ) v Expoetial: v = v = expeted umber of vehiles o the lik, ke L N f f exp β =, ke = expeted future desity, L = legth of the lik, L = 0.55 km ad N = umber of laes o the lik, N = 4 v = average travel speed o the lik for vehiles v = free flow speed, = 5 km/h f v f

12 Rudy, Wag ad Ni = umber of vehiles o the lik at apaity, = k L N, k = jam desity, = 40 veh/km/l l( v ) a v f a = shape parameter, = l l, =.43 vb v f ) b β = sale parameter, a b jam β = = f a f b [ l( v v )] [ l( v v )], β = ad a = umber of vehiles o the lik at a observed desity, a = ka L N, k = 50 veh/km/l, a = 27.5 b = umber of vehiles o the lik at a observed desity, b = kb L N, k = 2.5 veh/km/l, b = 0 v = average travel speed at a observed desity, approximated from data, v = 00 km/hr a v = average travel speed at a observed desity, approximated from data, v = 40 km/hr b I this researh, the desity values hose for k a ad k b are 50 ad 2.5 veh/km/l, respetively, are there seems to be a ifletio i the empirial graph of the servie rate, see Figure. Furthermore, the liear ad expoetial models of the blokig probability are give by jam a b a b k jam EMPIRICAL STUDY Liear: Expoetial: P P b b = = + ( ) exp β A ideal meas to validate this model is to regulate a setio of road ad ompare its throughput with the model output. However, suh a approah is ot pratial. Alteratively, simulatio appears to be a reasoable hoie. A ommo set of iput is applied to both the simulatio model ad the mathematial model ad ompariso is made based o the outputs of both models. To failitate the ompariso, the trasitio probability matrix ad the blokig probabilities are empirially obtaied from GA 400 data. Study Site ad Data Samplig The dataset used i this study was olleted from GA 400 by Georgia NaviGAtor, the ITS of the of Georgia, from 0/0/2003 to 2/3/2003. Traffi oditios at the site were moitored by video ameras deployed approximately every oe third of a mile of the road i both diretios. Eah amera ostitutes a observatio statio ad samples all laes at this loatio. Eah sample otais a variety of iformatio, amog whih speed, flow, ad desity are of major iterest i this study. After osiderig data quality ad site ofiguratio, oe setio of the southboud GA 400 was seleted as the study site. The study site has four laes, otais Statio 4008, ad is 550 m log. To provide a reasoable volume of data yet to redue bias, thirty days out of about oe year worth of data was radomly seleted to geerate the trasitio probability matrix. I order to deal with high variability i the origial data as well as to failitate the use of Markov hai theory, the data has bee aggregated to a five miute time iterval. I additio, the raw lassified traffi outs has bee overted to passeger ar equivalets (PCE) [4].

13 Rudy, Wag ad Ni 2 Trasitio Matrix The sampled thirty days data was used i the MATLAB software to produe the trasitio probability matrix, see Table 3. I this researh, the trasitio probability matrix represets the probability that a future state will follow the preset state i the ext five miutes. For example, if the preset desity is 37 vehiles per kilometer (i.e. 4), the probability of havig 2 ext is 0.033, i.e. Cell (4,2). It a be easily idetified that there is a diagoal probability tred i the matrix. The diagoal tred represets the tedey of traffi to remai i the same or adjaet traffi state. A higher probability meas that traffi is more likely to trasitio to that state if o disturbaes our. Aroud s 4-9, traffi beomes less stable ad the trasitio probabilities are spread over may states. I s 0-2, the probabilities oe agai are high values. From the sampled thirty days data, there were less tha 0 data poits i s 0-2. However, for these states there is probability of zero that traffi will remai i its preset state. Correspodigly, there is a high likelihood that traffi will trasitio to other states. The result is that as the traffi desity ireases, it is likely that traffi will trasfer to other potetial states. Traffi s at time t Table 3 Trasitio Probability Matrix Approximated from Thirty Days Data Traffi s at time t Liear ad Expoetial Blokig Probabilities Figure 3 represets the blokig probabilities from the liear ad expoetial ogestio models. The blue solid lie represets the expoetial blokig probability while the red dashed lie deotes the liear blokig probability.

14 Rudy, Wag ad Ni 3 V The differee betwee liear ad expoetial model is that servie rate f ( ) =, whih was used to V approximate the blokig probability, was developed separately with respet to eah of the two models. However, the empirial urves for vehiular traffi flow strogly suggest that the expoetial model may provide a more aurate approximatio for the average travel speed with variatios i vehiular desity [2]. Therefore, this researh has used the expoetial blokig probability i the objetive futio. From the liear ad expoetial blokig probability urves i figure 3, there is a drop i the blokig probability at traffi state. Theoretially, the tred should be ireasig. By referrig to the trasitio probability matrix, s 0-2 have a zero probability that they will remai i their preset state. However, is spread betwee two traffi states, 4 ad 0. This distributio of the empirial data results i the aomaly of the blokig probability urves. 0.9 Liear Model Expoetial Model Blokig Probability Traffi s Figure 3 Liear ad Expoetial Model of the Blokig Probability Empirial Performae of the Mathematial Model The empirial performae of the mathematial model is examied usig two days other tha those usig i developig the trasitio probability matrix. Figure 4 illustrates the time-varyig effet of the objetive futio s ompoets. This figure a be iterpreted as follows. If oe pumps traffi ito a lik at the rate speified by the red dashed urve, i the log ru oe a reasoably expet a throughput orrespodig to the blue solid urve. I essee, the throughput is the iput flow pealized by its assoiated probability of ogestio (the gree dash-dotted urve).

15 Rudy, Wag ad Ni Jue 2003 Iput Flow Throughput Pealty Volume i veh/km/l :2 08:2 09:2 0:2 :2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 20:2 Time of Day September 2003 Iput Flow Throughput Pealty Volume i veh/km/l :07 08:07 09:07 0:07 :07 2:07 3:07 4:07 5:07 6:07 7:07 8:07 9:07 Time of Day Figure 4 Mathematial Model Output for Two Day s Data Validatio Results As stated above, a simulatio model is developed to provide a groud agaist whih the mathematial model is ompared. The simulatio model is developed usig CORSIM ad ivolves oly two liks, a log real lik with multiple laes ad a upstream dummy lik to hold queues, if ay. A ommo set of iput flow is reated with a ireasig demad. The simulatio is ru at eah iput flow level for multiple times i order to obtai the orrespodig throughput i the log ru. Meawhile, the same set of iput also applies to the mathematial model ad the orrespodig throughput is omputed. The ompariso result is plotted i Figure 5, whih shows how throughput varies with iput/demad. Geerally, the throughput irease liearly with demad up to about 500veh/hr/l. Uder suh a flow rate, traffi has eough room to digest the disturbaes geerated both evirometally ad geometrially. However, as demad

16 Rudy, Wag ad Ni 5 ireases lose to apaity, ogestio emerges alog with a pealty. At this poit, traffi aot aommodate demad i a timely maer ad smooth the disturbaes geerated by o-liear ad stohasti atured traffi flow. The throughput will peak some ear the apaity beause this is the differee betwee the iput flow ad the pealty of ogestio, represeted by the blokig probability, is maximized. Though slight disrepaies exist, the ompariso of the output of the two models suggests a good fit i geeral. More speifially, the mathematial model teds to slightly uderestimate ogestio i the higher demad rage ad overestimate ogestio i the mid-demad rage. These disrepaies may be attributed to the trasitio probability matrix the probabilities spread amog may states i mid- to high-desity rage. I the highdesity rage, the probabilities suggest uexpeted low desities tha the preset state Mathematial Model Simulatio Model Throughput(veh/hr/l) Demad(veh/hr/l) Figure 5 Compariso of the outputs of both models CONCLUSIONS AND FUTURE WORK Usig the proposed blokig probability distributio, this paper preseted a ew mathematial model for the optimal throughput of a oe-diretioal lik. This model assumes the stohasti ature of traffi flow, Markovia traffi states ad a o-liear dereasig servie rate. The mathematial model was ompared with a simulatio model with limited suess. However, the mathematial model obviously follows the tred of the empirial data. I this sese, the mathematial model may be a more realisti preditio method. Uder a Vehile ifrastruture Itegratio (VII) seario, this model may serve as the basis of lik flow otrol i a effort to ahieve the maximum lik through i the log ru. Future work with this model may ivolve: fidig a improved set of data iludig more data poits as well as more istaes of severe ogestio ad use of a differet time ad/or desity iterval as ITS ad Vehile Ifrastruture Itegratio (VII) tehology advaes.

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