Modeling crash delays in a route choice behavior model for two way road networks

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1 Uth Stte University Civil nd Environmentl Engineering Student Reserch Civil nd Environmentl Engineering Student Works Modeling crsh delys in route choice behvior model for two wy rod networks Mohmmd Sdr Shrifi Uth Stte University, Hesm Shbniverki Arch De DC, Follow this nd dditionl works t: Recommended Cittion Shrifi, Mohmmd Sdr nd Shbniverki, Hesm, "Modeling crsh delys in route choice behvior model for two wy rod networks" (206). Civil nd Environmentl Engineering Student Reserch. Pper 2. This Article is brought to you for free nd open ccess by the Civil nd Environmentl Engineering Student Works t DigitlCommons@USU. It hs been ccepted for inclusion in Civil nd Environmentl Engineering Student Reserch by n uthorized dministrtor of DigitlCommons@USU. For more informtion, plese contct rebecc.nelson@usu.edu.

2 Modeling crsh delys in route choice behvior model for two wy rod networks Mohmmd Sdr Shrifi Uth Stte University, Civil nd Environmentl Engineering, Logn, UT, USA Emil: Hesm Shbniverki Arch De DC, Wshington D.C., USA Emil: Abstrct Distributing demnd in trnsporttion network is bsed on route choice behvior models. Generlly, it is ssumed tht drivers use routes with minimum time. In rel world, drivers my consider mny fctors other thn trvel times in congested networks especilly in metropolitn or two wy congested trnsporttion networks. Trvel sfety is fctor tht one my consider in his/her trip route choice. The min objective of this pper ws to investigte influence of sfety fctors such s crsh delys on drivers route choice behviors. Prmeters tht cn cuse to crsh occurrences were specified nd their impcts were modeled t mcroscopic level using simple sttisticl model. Then, n equilibrium bsed mthemticl progrmming model for two wy networks with symmetric link interctions ws proposed. The model ws tested for simple network nd results showed tht how crsh delys cn impct on route choice behviors. Keywords: route choice, crsh dely, two wy networks. Introduction Trnsporttion networks re importnt infrstructures for societies. Urbn developments nd dispersing socil nd economic opportunities mke it vitl for plnners to provide sustinble trnsporttion networks. Congestion, emissions, nd high trvel times re some of problems tht cuse decresing in life qulity. Therefore, it is impertive tht plnners use more ccurte tools to nlyze nd predict trnsporttion networks conditions. Trditionlly, the most populr trvel demnd forecsting process extensively used by trnsporttion modelers for decdes re known s four-step models. Four step modeling is used to predict nd evlute trnsporttion network conditions for future. Trffic ssignment is the finl step of trnsporttion plnning in which predicted trips re ssigned to trnsporttion networks. Route choice models ply importnt role in trffic ssignment process. Deterministic nd stochstic user equilibrium models re two bses of route choice models. Deterministic user equilibrium (DUE) is bsed on Wrdrop s lw: The journey times on ll the routes ctully used re equl, nd less thn those which would be experienced by single vehicle on ny unused route. In other words, ll trvelers selfishly mke their route choices tht result in stble equilibrium trffic flow pttern such tht there is no incentive for nyone to chnge his/her route []. In Stochstic User Equilibrium (SUE) models the perfect knowledge of users on trvel times is relxed: At SUE, no motorists cn improve his or her perceived trvel time by unilterlly chnging routes [, 2]. Mny reserchers hve extended these two bsic types of route choice models to cpture uncertinties in both supply nd demnd spects [3, 4]. To show the necessity of the subject, some experiments were implemented nd results showed tht trvel time vribility ws either the most or second most importnt fctor for most commuters [5]. Therefore, mny studies hve been conducted to incorporte trvel time vribility nd users risk behvior in route choice models. Bell nd Cssir (2002) proposed new equilibrium in respect to uncertinty in trvel times. They showed tht deterministic user equilibrium trffic ssignment is equivlent to the mixed-strtegy Nsh equilibrium of n n-plyer, noncoopertive gme nd the mixed-strtegy Nsh equilibrium of this gme describes risk-verse user equilibrium trffic ssignment [6]. Lo et l. (2006) proposed route choice model bsed on trvel time budget concept. They postulted tht trvelers cquire the vribility of route trvel times bsed on pst experiences nd fctor such vribility into their route choice considertion in the form of trvel time budget nd ll trvelers wnt to minimize their trvel time budgets. They formulted multi-clss mixed-equilibrium mthemticl progrm to cpture the route choice behviors of trvelers with heterogeneous risk versions or requirements on punctul rrivls [7]. To extend stochstic route choice models, Mirchndni nd Soroush (987) proposed generlized trffic equilibrium problem on stochstic networks (GTESP) tht incorportes both probbilistic trvel times nd vrible perceptions in the route choice decision process [8]. Siu nd Lo (2006) formulted stochstic equilibrium to ddress uncertinty in the ctul trvel time due to rndom link cpcity degrdtions nd perception vritions in their trvel time budget due to imperfect trffic informtion [9]. To ddress the effect of other prmeters on drivers route choice, Ashtini nd Irvni (999) incorported signlized nd un-signlized dely to the deterministic trffic ssignment. They proposed some dely functions bsed on HCM mnul nd they incorported these functions to link dely function. They concluded tht considering these delys will led to more relistic results for plnning purposes [0]. In relity drivers my consider mny fctors other thn trvel times in congested networks especilly in metropolitn or two wy congested trnsporttion networks. Trvel sfety is fctor tht one my consider in his/her trip route choice. For exmple one my

3 choose sfer route between two pths with negligible in trvel cost but with high different sfety levels. Therefore, it is necessry to model drivers route choice behviors in relistic mnner. In trnsporttion literture, mny reserches hve been conducted to study on either trnsporttion infrstructures sfety [, 2] or trffic sfety [3, 4, 5]. However, none of them investigted the impcts of sfety levels on route choice behviors. The min objective of this pper is to model effects of crsh delys on route choice model. Specificlly, prmeters on crsh occurrences will be explored nd their impcts will be modeled t mcroscopic level using simple sttisticl model. Then, n equilibrium bsed mthemticl progrmming model for two wy networks considering user s perception errors re presented. The reminder of this pper is orgnized s follows. Section 2 presents methodology of incorporting crsh delys on route choice model. The mthemticl model nd solution lgorithm re discussed in this section. Model results on simple network re presented nd discussed in section 3. Finlly summry of findings nd recommendtions for future studies re presented in section Methodology A trffic incident represents n event creting temporry reduction in rodwy cpcity. These incidents depend on the severity cn cuse mjor or minor delys for trvelers. The probbility of occurring incidents in two wy trnsporttion networks is higher compring to one wy networks. Therefore, it is necessry to consider tht issue in route choice decision process. To clrify, consider simple network presented in Fig. The network hs two origin-destintion pirs. There re two routes between origin nd destintion 2. Route is composed of link which is one wy link. Let s ssume tht it is highwy type link with long length nd high sfety level. Route 2 is composed of link 2 which is two wy link. Let s ssume tht it is street type link with shorter length compring to link. Now consider drivers who wnt to trvel from node to node 2. In clssic route choice models, drivers only consider trvel time to choose their route but in this cse one my be lso consider the risk tht he/she my be counter. is the incident durtion in time unit. Considering uniform distribution delys imposed to ll users, the verge length of queue cn be estimted s hlf of mximum length. Hence, the totl dely cn be clculted s follows: D QT ( x rc ) T 2 (2) where D is the totl dely tht users experience for one incident in prticulr link in time unit. To compute dely for ech vehicle in link, totl dely cn be divided to the totl volume of link : D ( x rc ) T 2 (3) 2x The following ssumptions were considered to estimte the men nd vrince of crsh delys:. Incident durtion follows gmm distribution with men nd vrince σ 2 for ech link nd these prmeters re deterministic, 2. The occurrence of incidents follow poisson process with prmeter λ. λ is incident rte in unit length independent from volume for link. Therefore, the expected number of incidents for the whole network cn be clculted s follow: Expected number of incidents = xl (4) where L is the length of link. Men incident dely for ech link cn be obtined s follow: (5) 2 E[ D ] ( x rc ) E[ T ] 2x where is men of incident dely for link. For rndom vrible x we cn write: [ x] E[ x ] ( E[ x]) E[ T ] m (6) substituting Eq. (6) in Eq. (5): ( x rc )( m ) (7) 2x The unit of is in time per vehicle.mile. The men dely for ech vehicle cn be computed s follow: Fig.. Simple two wy network When n incident occurs in link, the cpcity of tht link would decrese during the incident nd queue would be built if trffic volume exceeds the cpcity. The queue growing speed equls to difference between the rte of rrivl nd exited vehicles. Therefore, the mximum queue length on ech link cn be clculted s follow: Q ( x rc ) T () where Q is the mximum queue length in vehicles unit, x is the trffic volume in link in vehicles per time unit, r is the percentge of link cpcity which would be vilble during n incident 0 < r <, C is the cpcity of the link in vehicles per time unit, nd T x rc m x L ( )( ) 2x 2 ( x rc )( m ) L Eq. (8) presents men dely tht ech vehicle expects due to incidents. The following eqution cn be obtined by rerrnging Eq. (8): C ( r)( m ) 2 C (8) x (9)

4 In this study it is ssumed tht trffic volumes in two directions cn cuse incidents. Aggregting link flows nd cpcities we hve: s.t fr q i I j J (8) rp C C C (0) X x x () x f A r r ii jj rp (9) where stnds for opposite link for link. It should be noted tht this formultion is pplicble to congested network which their rtio of trffic volumes (sum of two opposite directions) to cpcity (sum of two link cpcity in ech direction) is higher thn the coefficient r. Expected crsh delys cn be computed by dding the following term to trvel time function nd it cn be treted s generl cost function for links. X s ( x, x) C( r)( m ) (2) 2 C where s(x,x ) is the dely term relted to incidents. Link interctions cn be either symmetric or symmetric. In this study it is ssumed tht interctions re symmetric. This ssumption implies tht the effect of n dditionl flow unit long prticulr link on trvel time in the opposing direction equls the effect of n dditionl flow unit in the opposing direction on the trvel time of the link under considertion. []. Hence, the generlized cost function for link nd the symmetry condition cn be expressed mthemticlly s follow: g ( x, x ) t ( x, x ) s ( x, x ) (3) g ( x, x) g( x, x ) x x (4) It is n unrel ssumption tht drivers hve perfect knowledge bout the network condition nd generl costs. To relx tht, rndom error term is introduced to the generlized cost functions. It is ssumed tht this rndom error term follows Gumble distribution with following ssumptions: C c r P, ii, j J (5) r r r E[ C ] c, E[ ] 0 (6) r r r where I nd J re the set of origins nd destintions respectively. P is the set of pths for given origin i nd destintion j. εr is the rndom error term for pth r for given origin i nd destintion j. Cr nd cr re the stochstic nd deterministic generlized cost of pth r for origin i nd destintion j, respectively. Equivlent mthemticl progrm to obtin equilibrium flows in the congested two wy rod networks considering sfety levels nd users perception error is proposed s follow. At the equilibrium point no motorists cn improve his or her perceived generl cost by unilterlly chnging routes. f 0 r P ii j J (20) r where g( ) is the generlized cost function for link, x is the flow on link, x is the flow on link (opposite direction of link ), fr is the flow on route k between origin i nd destintion j, q is demnd between origin i nd destintion j, θ is the dispersion prmeter. The objective function hs three terms. The first is the integrl of the links generlized performnce function. The second term is the integrl of the links performnce function nd the third term cptures users perception errors. The first constrint ssures tht sum of pth flows between n OD pir is equl to the demnd of tht OD pir. The second constrint is non negtivity of pth flows nd the third constrint sttes tht volume on the prticulr link is equl to the sum of pth flows tht is pss through tht prticulr link. 2.. Solution lgorithm The Method of Successive Averge (MSA) is pplied to solve the mthemticl model. This lgorithm is reltively similr to the Frnk-Wolf lgorithm. The min difference between these two lgorithms is relted to step move size computtion. Unlike to the Frnk-Wolf lgorithm, in the method of successive verge move size is predetermined long the descent direction. Regulrly, this step size is considered to be /n for ech itertion where n is the number of itertion. To solve the mthemticl progrm, the first step is to compute initil fesible solution. Stochstic loding cn be pplied with free flow trvel generlized costs to obtin initil fesible solution. In the presence of loops in the network stochstic loding lgorithms like STOCH lgorithm cn t be used. In this study it is ssumed tht ll pths between n OD pir re resonble provided tht the pth doesn t pss through ech node mny times. In other words, resonble pth only cn meet ech node one time. The MSA lgorithm cn now be summrized s follows: Step 0: Initiliztion. Perform stochstic network loding bsed on set of initil trvel times {g 0 }. This genertes set of link flows {x}. Set n =. Step : Updte. Set g n = g(x n,x n ) Step 2: Direction finding. Perform stochstic network loding procedure bsed on the current set of link generl trvel costs. This yields n uxiliry link flow pttern {y n }. Step 3: Move. Find the new flow pttern by setting x n+ = x n + (/n)(y n x n ). Step 4: Convergence criterion. If convergence is ttined, stop. If not, set n = n + nd go to step. The convergence criterion is defined s below. min z( x, x ) ( g (, x ) d v x 2 A 0 g (,0) d) f ln f r r ii jj rp x (7) A ( x x ) n n 2 x n A (2) where is the ccurcy of the lgorithm ws used in this study.

5 3. Numericl exmple In this section the proposed model is tested on simple test network. The topology of the test network is depicted in Fig 2. This simple grid network hs 6 nodes, 4 links nd 2 OD pirs. One OD pir is -6 with demnd of 50 per minute nd the other OD pir is 6- with demnd of 50 per minute. In this network pir links between nodes hve interctions with ech other. the prmeter r ws set to be 0.3 for ll links. Tble 2 presents the pth number schemes. Tble 2: Pth number schemes OD Pth number Link number Results Fig. 2. Test network The cost function given in the following eqution is dopted to the test network: g x x t x x s x x (22) 0 (, ) (, ) The presented mthemticl model ws implemented in MATLAB nd ssignment results were computed. Fig 3 presents the convergence of the lgorithm for link nd pth. It cn be observed tht the trjectory of convergence hs zigzg pttern in initil itertions nd the pttern is more visible for link flow trjectory. Results show tht the lgorithm converged to the equilibrium flows fter bout 7 itertions. Tble 3 shows the ssignment results. Link volumes, link costs nd volume to cpcity rtio is presented in this Tble. According to the results, links 6, 8 nd 2 hve the highest volumes nd volume to cpcity rtio. where t 0 is the free flow trvel time on link in minute unit nd is the opposite link of link. It is ssumed tht free flow trvel time for ll links is 3 minutes. Prmeters nd properties of the network re tbulted in Tble. The sixth column shows the cpcity of the links nd the next columns show incident per unit, men nd stndrd devition of incidents durtion, respectively. It cn be observed tht links 3 nd 4 hve lower sfety levels compred to other links. The dispersion prmeter () is ssumed to be one nd Link number From To α β Tble : Link properties of the test network C λ (crsh/mile) m

6 () OD Tble 4: Assignment results (route flows) Pth number Links Pth Flows Pth Flow pttern comprisons for different dispersion prmeter Link number (b) Fig. 3. Trjectories of convergence (). link, (b) route From Tble 3: Assignment results To Volume Link x/c Rtio To investigte the impcts of dispersion prmeter on the flow ptterns, model ws run for different vlues of dispersion prmeters nd flows nd relted costs were computed. Tble 5 shows the link flows, link costs nd volume to cpcity rtio for dispersion prmeter 0., nd 0. The lst column presents the ssignment results when users hve exct knowledge bout network generlized costs. According to Tble 5, link volumes chnge considerbly especilly in lower vlues of dispersion prmeter. In trnsition from = 0. to = high chnge in link flow ptterns cn be observed. Another conclusion tht cn be inferred from this Tble is tht when the dispersion prmeter reches to higher vlues, link flow ptterns become more similr to deterministic ssignment link flow ptterns. It is n expected result becuse when the dispersion prmeter becomes high vlue, it mens tht users perception error is low nd users hve more sense bout routes costs. To better understnding the route choice behvior, Fig 4 is provided. This figure shows distribution of pth cost with br chrt nd the probbility of choosing these pths with line for different dispersion prmeter vlues. Note tht routes 5, 6, 7 nd 8 re the route numbers between OD 6-. The figure implies tht pth with lower cost hs higher chnce to be chosen by trvelers. For lower levels of the dispersion prmeter, routes 2 nd 3 of OD -6 hve higher probbility to be chosen but with growing in the prmeter, the probbility of pth nd 2 increses. For OD 6- route hs the highest probbility for ll level of the prmeter nd the probbility of choosing this route increses with growing in the dispersion prmeter. Tble 4 presents pth flows nd pth costs. It cn be observed tht for OD -6 pths, 2 nd 3 were used nd unlike to deterministic user equilibrium, used pths don t hve equl costs but the mjority of users recognize the pth with minimum cost. For OD 6- only pth ws used nd it hd considerbly lower cost thn other pths. This implies tht in this cse even if users hve perception errors but they could recognize the pths with considerble lower costs.

7 Tble 4: Assignment results (route flows) OD Pth Num Flow θ = 0. θ = θ = 0 Deterministic Flow Flow Flow Summry nd recommendtions for future studies () In this study, route choice model considering crsh delys nd users perception errors ws proposed for two wy rod networks with symmetric link interctions. The proposed model ws performed on simple grid network nd vrious scenrios were nlyzed nd flow ptterns were investigted due to chnging in perception errors. For future reserch, this model cn be improved by considering other prmeters tht influence on incident delys. In this study pth enumertion ws used in stochstic loding. Obviously, it is not pplicble for rel networks. Hence, this prt of ssignment cn be improved by n efficient loding lgorithm. Inbility to ccount for overlpping nd perception vrince mong routes re drwbcks of logit bsed stochstic models. This model cn be further enhnced by considering these issues. References (b) (c) Fig. 4. Route costs nd their probbility (). = 0., (b). =, (c). = 0 [] Sheffi, Y. Urbn Trnsporttion Networks: Equilibrium Anlysis with Mthemticl Progrmming Methods. Prentice- Hll, Incorported, Englewood Cliffs, NJ, 985. [2] Dgnzo, C. F., Sheffi, Y. On Stochstic Models of Trffic Assignment. Trnsporttion Science, vol. (3), pp , 977. [3] Soltni-Sobh, A., Heslip, K., El Khoury, J. Estimtion of rod network relibility on resiliency: An uncertin bsed model. Interntionl Journl of Disster Risk Reduction, vol. 4, pp , 205. [4] Soltni-Sobh, A., Heslip, K., Stevnovic, A., El Khoury, J., Song, Z. Evlution of trnsporttion network relibility during unexpected events with multiple uncertinties. Interntionl Journl of Disster Risk Reduction, In Press. [5] Abdel-Aty, M., Kitmur, R., Jovnis, P. Exploring route choice behvior using geogrphicl informtion system-bsed lterntive routes nd hypotheticl trvel time informtion input, Trnsporttion Reserch Record 493, pp , 995. [6] Bell, M. G. H., Cssir, C. Risk-verse user equilibrium trffic ssignment: n ppliction of gme theory, Trnsporttion Reserch Prt B, vol. 36 (8), pp , [7] Lo, H. K., Luo, X. W., Siu, B. W. Y. Degrdble trnsport network: trvel time budget of trvelers with heterogeneous risk version. Trnsporttion Reserch Prt B, vol. 40 (9), pp , 2006.

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