Optimal Treatment of Queueing Model for Highway

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1 Journl of Computtion & Modelling, vol.1, no.1, 011, ISSN: (print, (online Interntionl Scientific Pre, 011 Optiml Tretment of Queueing Model for Highwy I.A. Imil 1, G.S. Mokddi, S.A. Metwlly 3 nd Mrim K. Metry 4 Abtrct We develop ome nlytic queueing model bed on trffic nd we model the behvior of trffic flow function of ome of the mot relevnt determinnt. Thee nlytic model llow for prmeterized experiment, which pve the wy towrd our reerch objective: eing wht-if enitivity nlyi for trffic mngement, congetion control, trffic deign nd the environmentl impct of rod trffic. We illutrte our reult for highwy. Mthemtic Subject Clifiction : 60K5 Keyword: Queueing theory, trffic flow theory 1 Fculty of Computer Science, Mir Interntionl Univerity, Ciro, Egypt, e-mil: mr444-@hotmil.com Fculty of cience, Ain Shm Univerity, Ciro, Egypt, e-mil: gmokddi@hotmil.com 3 Fculty of cience, Ain Shm Univerity, Ciro, Egypt, e-mil: miw@hotmil.com 4 Fculty of cience, Ain Shm Univerity, Ciro, Egypt, e-mil: mri5eg@yhoo.com Article Info: Revied : July 9, 011. Publihed online : Augut 31, 011.

2 6 Optiml Tretment of Queueing Model for Highwy 1 Introduction When modeling the environmentl impct of rod trffic, we cn ditinguih between both ttic nd dynmic impct of infrtructure nd vehicle on emiion nd wte. On the one hnd, rod cn be conidered viul intruion. In ddition, they my cue dmge to nturl wtercoure or threten the nturl hbitt of wildlife. Vehicle in turn conume nturl reource nd impoe trin on the environment t the end of their life cycle. On the other hnd, they form prt of trffic flow, infrtructure nd vehicle lo hve dynmic impct on the environment. Vehicle in ue produce emiion nd noie. Toxic fume ecpe in the tmophere when fuel tnk re filled, while driving led to further emiion (CO, NO nd SO nd dut. Furthermore, n incree in grbge, ccident (phyicl nd mteril dmge nd, occionlly, ditortion of infrtructure nd nture element (tree, niml,..,etc. cn be oberved. Becue trffic flow re function of both the number of vehicle on the rod nd the vehicle peed, trffic flow occupy centrl poition in the ement of rod trffic. The objective of thi pproch i minly explortive nd explntory. Thee decriptive model give n empiricl jutifiction of the well-known peed-flow nd peed-denity digrm, but re limited in term of predictive power nd the poibility of enitivity nlyi [5,7,8]. An lterntive pproch i to ue peed-flow, peed-denity nd flow-denity digrm, in which dt on trffic flow re collected nd re fit into curve [4,6]. Compred to thee decriptive model, thi pper preent more opertionl pproch uing queueing theory. Queueing theory i lmot excluively ued to decribe trffic behvior t ignlized nd unignlized interection [1,,3,4,7,9].

3 I. Imil, G. Mokddi, S. Metwlly nd M. Metry 63 Decryption of Queueing Model with trffic flow theory On of the mot importnt eqution in trffic flow theory incorporte the interdependence of trffic flow q, trffic denity E nd peed : q= E (1 When two of the three vrible re known, the third vrible cn eily be obtined. If trffic count dt re vilble, trffic flow cn be umed given, which leve u to clculte either trffic denity or peed to complete the formul nd ue either input for the pproprite queueing model. Tble 1: Overview of ued prmeter Prmeter E C r SN q λ μ ρ W Decription Trffic denity (vehicle/km Mximum trffic denity (vehicle/km Effective peed (km/h Reltive peed Nominl Speed (km/h Trffic flow (vehicle/h Arrivl rte (vehicle/h Service rte (vehicle/h Trffic intenity=λ/μ Time in the ytem (h In our model we define C the mximum trffic denity. Rod re divided into egment of equl length 1/C, which mtche the miniml length needed by one vehicle on tht prticulr rod. Ech rod egment i conidered ervice ttion, in which vehicle rrive t rte λ nd get erved t rte μ (Figure1.

64 Optiml Tretment of Queueing Model for Highwy Figure 1: Queuing Repreenttion of trffic flow We define W the totl time vehicle pend in the ytem, which equl the um of witing time nd ervice time.

4 64 Optiml Tretment of Queueing Model for Highwy Figure 1: Queuing Repreenttion of trffic flow We define W the totl time vehicle pend in the ytem, which equl the um of witing time nd ervice time. The higher the trffic intenity, the higher the time in the ytem become (the exct reltion between W nd ρ depend upon the queueing model. When W i known, the effective peed cn eily be clculted : = The reltive peed r, by definition: r = 1 C W SN 1 = C W SN Queueing model re often referred to uing the Kendll nottion, coniting of everl ymbol - e.g. M/G/1. The firt ymbol i horthnd for the ditribution of inter-rrivl time, the econd for the ditribution of ervice time nd the lt one indicte the number of erver in the ytem. ( (3 3 Anlyi of M/M/1 MODEL The inter-rrivl time re exponentilly ditributed (the rrivl rte follow Poion ditribution with expected inter-rrivl time equl to1/λ (with λ equl to the product of the trffic denity E nd the nominl peed SN. The ervice time delinete the time needed for vehicle to p one rod egment nd i exponentilly ditributed with expected ervice time μ (the ervice rte

5 I. Imil, G. Mokddi, S. Metwlly nd M. Metry 65 follow Poion ditribution. When vehicle drive t nominl peed SN, ervice time cn be written : 1/( SN C nd μ equl the product of nominl peed SN with the mximum trffic denity C. Uing thee formul for λ nd μ, we obtin W : 1 1 W = = (4 μ λ SN ( C E Uing thi expreion for W, the effective peed nd reltive peed re obtined: SN ( C E = = SN (1 ρ r = = 1 ρ (5 C SN with ρ the trffic intenity: λ E ρ = = μ C Subtituting for E (= q/ in (5 the following expreion i obtined: f(, q = C C SN + SN q = 0 (7 Mximizing f (, q for nd ubtituting thi vlue into (7, q mx cn be written SN C qmx = (8 4 Trffic denity i low; vehicle do not obtruct one nother, which led to higher effective peed. When more vehicle rrive on the rod, the effective peed decree. Uing eqution (1: q= E nd the bove formul for, we cn contruct the peed-flow nd the flow-denity digrm for the M/M/1 model. The peed-flow digrm i the envelope of ll poible combintion of the effective peed nd trffic flow. There re two peed for every trffic flow: n upper brnch ( where peed decree with flow nd lower brnch (1 with n increing peed in term of flow. An intuitive explntion cn be follow: the flow move from SN to q, mx congetion incree but the flow rie becue the decline (6

6 66 Optiml Tretment of Queueing Model for Highwy in peed i offet by the higher. If trffic continue to enter the flow pt q, mx flow fll becue the decline in peed more thn offet the dditionl vehicle number further increing congetion, [1]. The M/M/1 model i intereting be ce, but i indequte to repreent rel-life trffic flow. In the next two ection we will relx the M/M/1model: firt, the ervice time follow generl ditribution (M/G/1 nd, econdly, both rrivl nd ervice time follow generl ditribution (G/G/1. A in the M/M/1 model inter-rrivl time follow n exponentil ditribution with expected inter-rrivl time 1/λ, λ being the product of trffic denity nd nominl peed. The ervice time however i generlly ditributed with n expected ervice time of 1/μ nd tndrd devition of σ. Expected ervice rte i μ, which equl the product of nominl peed SN with mximum trffic denity C. Combining Little theorem nd the Pollczek-Khintchine formul for L (defined the verge number of cr in the ytem [5,8,7,9] nd ubtituting for λ nd μ, we obtin the following formul for the totl time in the ytem W: 1 ρ + SN E σ W = + (9 SN C SN E (1 ρ Uing the bove expreion for W, effective nd reltive peed cn be clculted in n nlog wy in the M/M/1 model: SN ( C E SN (1 ρ (1 ρ = = r = C+ E ( β 1 +ρ ( β 1 +ρ ( β 1 with β delineting the coefficient of vrition of ervice time (orβ =σ SN C. (10 Uing thee formul we cn contruct the peed-flow, peed-denity nd flow-denity digrm for the M/G/1 model. The exct hpe of thee curve depend upon the vrition coefficient of the ervice time, β.

7 I. Imil, G. Mokddi, S. Metwlly nd M. Metry 67 Subtituting E (= q/ in bove formul (8 nd rewriting, the following expreion for the peed-flow digrm i obtined: f(, q = C + q ( β 1 C SN + q SN = 0 (11 Mximizing thi eqution for, we cn clculte the mximum trffic flow ( q mx : The vlue of q q mx mx β + 1 = SN C β 1 β 0 SN C = 4 β= 1 q mx i function of the vrition prmeter β. (1 Uing the bove expreion for W, effective nd reltive peed cn be clculted in n nlog wy in the M/M/1 model. With the G/G/1 model both rrivl time nd ervice time follow generl ditribution with expected rrivl time 1/λ nd tndrd devition σ, expected ervice time 1/μ nd tndrd devition of σ, repectively. b Conequently, the hpe of the peed-flow-denity digrm will depend not only on the vrince of the ervice time but lo on the vrince of the inter-rrivl time. Combining Little theorem nd [, 3, 5, 6, 7] formul for L nd ubtituting for λ nd μ, we obtin the following formul for the totl time in the ytem W : (1 ρ (1 cα 3 ρ ( cα + c cα 1 ρ ( cα + c W = + e, 1 SN C SN E (1 ρ (1 ρ ( cα 1 (1 +ρ ( cα + 10 c cα 1 ρ ( cα + c W = + e, > 1 SN C SN E (1 ρ with c α repreenting the qured coefficient of vrition of inter-rrivl time nd c the qured coefficient of vrition of ervice time.

8 68 Optiml Tretment of Queueing Model for Highwy Uing (5 nd the bove expreion for W, the effective peed formul become: SN (1 ρ = (1 ( α ρ +ρ c + c e (1 ρ (1 cα 3 ρ ( cα + c, cα 1 SN (1 ρ = > ρ +ρ c + c e (1 ( α (1 ρ ( cα 1 (1 +ρ ( cα + 10 c, cα 1 The exct hpe of the digrm depend not only on the vrition coefficient of ervice time but lo on the vrition coefficient of inter-rrivl time. 4 Min Reult We ee tht the vrince on the rrivl rte ( c = 1 nd c = 0 h lrger impct thn the vrince on the ervice rte ( c = 0 nd c = 1. Action to incree trffic flow hould primrily be focued on the rrivl rte vrince. A imilr concluion cn be obtined uing the flow- denity digrm. Finlly, the peed-denity digrm i contructed for given denity of 40 vehicle per km: we ee tht the effective peed rnge from pproximtely 50 (high vrince to pproximtely 110 (low vrince km/h. The reult cn eily be compred with the contructed peed-flow-denity digrm. For the ce with high vrince ( c nd c both equl to one, t hour 8.00, 9.00 nd m., the oberved trffic flow become lrger thn the mximum poible trffic flow on the highwy given thee vrince prmeter. Conequently there re no peed tht cn be clculted for thee intnce.

9 I. Imil, G. Mokddi, S. Metwlly nd M. Metry 69 Tble : Upper nd lower peed for highwy Hou r q mx Q (vech/h c = c = 0.5 c = 0, c = 1 c = 1, c = 0 c = c = 1 S1 S S1 S S1 S S1 S

10 70 Optiml Tretment of Queueing Model for Highwy 5 Concluion Uing everl queueing model, peed i determined, bed on different rrivl nd ervice procee. The exct hpe of the different peed-flow-denity digrm i lrgely determined by the model prmeter. Therefore we believe tht good choice of prmeter cn help to dequtely decribe relity. We illutrted thi with n exmple, uing the mot generl model for highwy. Our model cn be effectively ued to e the environmentl impct of rod trffic. Reference [1] C.F. Dgnzo, Fundmentl of Trnporttion nd Trffic Opertion, Elevier ScienceLtd., Oxford, [] Hmdy A. Th, Opertion Reerch nd Introduction, 00, Peron Eduction Publihing Ltd, New Delhi, 9 th edition 010. [3] D. Heidemnn, Queue length nd dely ditribution t trffic ignl, Trnporttion Reerch-B, 8B, (1994, [4] D. Heidemnn, A queueing theory pproch to peed-flow-denity reltionhip, Trnporttion nd Trffic Theory, Proceeding of the 13th Interntionl Sympoium on Trnporttion nd Trffic Theory, Lyon, Frnce, 14-6 July [5] Ivo Adn nd Jcque Ring, Queueing theory, Eindhoven Univerity of Technology, 00. [6] R. Jin nd J.M. Smith, Modelling vehiculr trffic flow uing M/G/C/C tte dependent queueing model, Trnporttion Science, 31, (1997, [7] Mohe Zukermn, Introduction to Queueing Theory nd Stochtic Teletrffic Model, Teching Textbook Mnucript, Retrieved from

11 I. Imil, G. Mokddi, S. Metwlly nd M. Metry 71 [8] Prem Kumr Gupt nd D.S. Hir, Opertion Reerch, 003. [9] U.Nryn Bht, An Introduction to Queueing Theory Modeling nd Anlyi in Appliction, Birkhuer, Boton, 008.

PHYS 601 HW 5 Solution. We wish to find a Fourier expansion of e sin ψ so that the solution can be written in the form

PHYS 601 HW 5 Solution. We wish to find a Fourier expansion of e sin ψ so that the solution can be written in the form 5 Solving Kepler eqution Conider the Kepler eqution ωt = ψ e in ψ We wih to find Fourier expnion of e in ψ o tht the olution cn be written in the form ψωt = ωt + A n innωt, n= where A n re the Fourier