MATHEMATICAL MODELLING OF PUBLIC TRANSPORT SYSTEMS SOME INSIGHTS, EXTENSIONS

1
MATHEMATICAL MODELLING OF PUBLIC TRANSPORT
SYSTEMS SOME INSIGHTS, EXTENSIONS ISSUES
RELATED TO NEWELLS APPROACH

• SECOND DRAFT
• S.C.WIRASINGHE
• DEPT. OF CIVIL ENGINEERING
• SCHULICH SCHOOL OF ENGINEERING,
• UNIVERSITY OF CALGARY
• and
• U. VANDEBONA
• SCHOOL OF CIVIL ENVIRONMENTAL ENGINEERING
• FACULTY OF ENGINEERING
• UNIVERSITY OF NEW SOUTH WALES
• OCTOBER 2006

2
PROFESSOR GORDON NEWELL

• UNIVERSITY OF CALIFORNIA, BERKELEY (1965 - 2001)
• B.Sc. (Schnectady College, NY) and Ph.D. (Brown)
  in Physics
• P.D.F. with PROF. ELLIOT MONTROL at U of MARYLAND
  IN 1951
• INFLUENCED BY A SEMINAR BY PROF. WILLIAM PRAGER
  GIVEN AT BROWN ON FLUID THEORY OF TRAFFIC
• FIRST PAPER ON TRAFFIC 1955
• FIRST ISTTT IN 1959
• 120 PAPERS 8 BOOKS OR MONOGRAPHS..1955-2000
• 14 PAPERS (12) ON PUBLIC TRANSPORT .. 1964-1998
• 3 BOOKS/MONOGRAPHS ON QUEUES .. 1971-1982
• LAST PAPER ON TRAFFIC 2002 (POSTUMOUS)
• PASSED AWAY IN A TRAFFIC ACCIDENT 2001
• SURVIVED BY MRS. BARBARA NEWELL, DAUGHTER SON
• Ph.Ds Lam, Vuchic, Osuna, Hauer, Sparks,
  Allen, Mitric, Hurdle, Jeewanantham, Fawaz,
  Wirasinghe, Gamze, Fukiyama, Kuwahara
  (incomplete)

3
Publications in Public TransportA TOTAL OF 14
PAPERS WITHIN 5 GROUPS, 3 BOOKS

• Control of Buses (4 papers)
• Spacing of Public Transport Facilities (3papers)
• Dispatching Policies (2 papers)
• Networks (2 papers)
• Elevators (4 papers)
• Review (1 paper)
• Books on Queuing theory (3)
• -----------------
• TWO OF THE PAPERS ARE LISTED IN TWO OTHER
  CATEGORIES RESPECTIVELY

4
SUMMARY OF PRESENTATION

• FOR EACH GROUP
• LIST OF PUBLICATIONS IN ORDER OF THE DATE
• MAJOR INSIGHTS THAT WERE OBTAINED
• SOME LIMITATIONS
• FURTHER WORK BY NEWELLS STUDENTS, THEIR STUDENTS
  AND SO ON (NEWELL FAMILY)
• EXAMPLE NEWELL VUCHIC BYRNE KIKUCHI
• SELECTED OTHER WORKS, MOSTLY BY BERKELEY ALUMNI
• OVERALL
• IMPACT ON VARIOUS RELATED FIELDS BY NEWELLS
  STUDENTS
• THE NEWELL APPROACH TO RESEARCH (WIRASINGHES
  INTERPRETATION)

5
Publications in Public Transport 1Control of
Buses

• (i) Maintaining a Bus Schedule (with Potts),
  Proceedings 2nd Conference ARRB, 1964
• (ii) Control Strategies for an Idealized Public
  Transportation System (with Osuna), TS, 1972
• (iii) Control of Pairing of Vehicles on a Public
  Transportation Route, TS, 1974
• (iv) Unstable Brownian Motion of a Bus Trip, in
  Statistical Mechanics Statistical Methods in
  Theory and Application, Plenum Press, 1977

6
Insights on Control of Buses

• The slack time required to prevent pairing of
• buses (under most conditions) can be
• estimated (under certain assumptions)
• e gt s (2?t)1/2
• where
• e slack time
• s Std. Dev. of the trip time between timed
  points
• ? mean Poisson arrival rate of passengers
• t t t e-?h
• t marginal time to board one passenger
• t time lost in stopping starting a bus
• h headway between buses

7
Some Extensions On Bus Control by Newells
students

• Optimal slack time to minimize delay, penalty,
  and bus operations costs, can be estimated (one
  to one) Wirasinghe 1993
• Analytical solutions for min.-cost slack time
  exist for special cases (Liu and Wirasinghe
  1995)
• (i) one intermediate time point
• (ii) Several time points, one bus run (DP)
• Simulation of mini.-cost scenario possible for
  real case (Liu and Wirasinghe 2001)

8
TERMINOLOGY

• A GENERALLY ACCEPTED TERMINOLOGY WILL BE HELPFUL
  IN
• UNDERSTANDING EXTENDING PUBLIC TRANSPORT MODELS
  
• From ONE location/zone/station/stop etc TO ONE
  (e.g. a non-stop route)
• ONE TO MANY (e.g. a commuter route, from a
  terminal to many destinations)
• MANY TO ONE (e.g. a school-bus, from many homes
  to a school)
• MANY TO MANY (e.g. a regular bus route)

9
Publications in Public Transport 2Spacing of
Public Transport Facilities

• (i) Rapid Transit Interstation Spacing for
  Minimum Travel Time (with Vuchic), TS, 1968
• (ii) Scheduling, Location, Transportation and
  Continuum Mechanics Some Simple Approximations
  to Optimization Problems, SIAM J. of Applied
  Mechanics, 1973
• (iii) Optimal Parameters for a Coordinated Rail
  Bus Transit System (with Wirasinghe Hurdle),
  TS, 1977
• ----------------
• Counted in Section 6 (Review) and
  counted in Section 4

10
Insights on Public Transport Station Spacing
Problems

• many to one OR one to many (limitation)
• A slowly varying passenger boarding or alighting
  density (per unit distance) assumption of a
  related optimal density of facilities (say rail
  stations per unit distance) facilitates the
  analytical modeling process continuum
  approximation (after 1971)
• The optimal station density (that minimizes
  travel time including access time) at a point is
  related to
• (cumulative passengers in vehicle)/ (linear
  passenger density for boarding or alighting)1/2
• (mid 1970s)

11
Related spacing problems

• Vuchic and Byrne 1972, and Hurdle - 1973
• Spacing of parallel bus routes Wirasinghe
  1979
• Rail feeder routes
• Ghoneim, Wirasinghe Extension to many to many
  demand bus routes, incl. passing of stops with no
  passengers Zone and skip stop service in
  commuter rail
• Schonfeld (Berkeley) Zonal bus service
• Wirasinghe Hurdle Many to one rail line with
  several feeder modes
• Parajuli and Wirasinghe Many to many rail line
  in a developing country, including walking access

12
Publications in Public Transport 3Dispatching
Policies

• (i) Dispatching Policies for a Transportation
  Route, TS, 1971
• (ii) Optimal Dispatching Strategies for Vehicles
  Having Exponentially Distributed trip Times (with
  Asgharzadeh), Naval Research Logistics, 1978

13
Insights on Dispatching Policy

• many to one OR one to many travel
  (limitation)
• Assumption of a slowly varying demand for travel
  passengers per unit time p(t) and, in
  response, the assumption of a slowly changing
  rate of bus dispatches, g(t), facilitates the
  analytical modeling process first use of
  continuum approximation
• Variable headways
• Mini. total cost bus dispatch rate g(t)1/h(t)
• p(t)/c
• Max
• p(t)/2 1/2
• Integrate g(t) to obtain dispatch times

14
Extensions of Dispatching Policy Work

• Hurdle (1971) considered a route with a limited
  number of buses
• Wirasinghe (1990) considered a many to many
  route an increasing unit value of waiting time,
  as waiting time increases and also optimal
  uniform non-uniform headways.
• Vuchic Byrne (1972), Hurdle (1973), Wirasinghe
  (1979), Schonfeld - Spawned a literature on
  parallel or feeder bus route analysis
  (density of routes) with optimal dispatching
  policies (rate of bus dispatches). (extension to
  two variables)

15
DISCRETE SCHEDULE MODELS(OPPOSITE OF NEWELLS
APPROACH)

• CONCENTRATE ON DETAILS OF SCHEDULE CONSTRUCTION
• DO NOT EXPLICITLY INCLUDE THE VALUE OF TIME OF
  PASSENGERS IN ANALYSIS
• CONSIDER EVEN HEADWAYS, EVEN LOADS OF
  PASSENGERS OR A MAX LOAD OF PASSENGERS
• CEDAR (BERKELEY) BOOK IN PROGRESS
• HEADWAYS ARE USUALLY UNIFORM IN A TIME PERIOD
  (SAY 1 HOURS)
• WIDELY USED IN PRACTISE

16
Publications in Public Transport 4Networks

• (i) Optimal Parameters for a Coordinated Rail
  Bus Transit System (with Wirasinghe Hurdle),
  TS, 1977
• (ii) Some Issues Relating to the Optimal design
  of Bus Routes, TS, 1979

17
INSIGHTS RE TRANSIT NETWORKS

• Feeder buses to commuter rail should not
  necessarily go to the nearest station but in
  some cases may go further downstream.
• In a many to many demand situation, and a grid
  network of streets, a cross grid of bus routes (0
  to 1 transfers for any trip) is not superior to a
  one with a transfer terminal located in a transit
  street through which all routes pass.
  (Disutility's at the terminal, other than a
  transfer penalty, especially bus and passenger
  congestion in a large terminal, were neglected.)

18
TWO ROUTING SCHEMES

• GRID TERMINAL

19
Publications in Public Transport 5Elevators

• (ii) An Analysis of Elevator Operation in
  Moderate Height Buildings I, A Single Elevator
  (with Gamze), TR, 1982
• (iii) An Analysis of Elevator Operation in
  Moderate Height Buildings II, Multiple
  Elevators (with Gamze), TR, 1982
• (iv) Two Elevators Serving Up-traffic," Queuing
  Systems 23, 1996
• (V) Strategies for Serving Peak Elevator
  Traffic," TR(B), 1998

20
Publications in Public Transport 6

• (i) Scheduling, Location, Transportation and
  Continuum Mechanics Some Simple Approximations
  to Optimization Problems, SIAM J. of Applied
  Mechanics, 1973

21
INSIGHTS RE WAREHOUSE LOCATION PROBLEM

• IF THE DEMAND FOR SERVICES FROM THE NEAREST
  WAREHOUSE IS A SLOWLY VARYING DENSITY IN THE
  (X,Y) PLANE, AND THE STREET NETWORK IS DENSE,
  THEN THERE IS A CLOSED FORM SOLUTION FOR THE
  OPTIMAL DENSITY OF WAREHOUSES THAT MINIMIZES THE
  SUM OF THE WAREHOUSE AND TRANSPORT COSTS. THUS
  THE NUMBER OF WAREHOUSES CAN ALSO BE ESTIMATED BY
  INTEGRATING THE OPTIMAL DENSITY.
• THIS PROBLEM IS CONSIDERED A np-hard PROBLEM BY
  DISCRETE MODELERS EVEN WHEN THE NUMBER OF
  WAREHOUSES IS GIVEN.

22
EXTENSIONS OF THE WAREHOUSE LOCATION PROBLEM

• THE BUS GARAGE LOCATION PROBLEM IS A ROUGH
  APPROXIMATION OF THE WAREHOUSE LOCATION PROBLEM,
  AS THERE ARE ONLY A FEW GARAGES IN MOST CITIES
  (WIRASINGHE WATERS)
• THE METRO NETWORK PLANNING PROBLEM CAN BE
  ASSISTED BY TREATING THE LOCATION OF STATIONS AS
  A BALANCE BETWEEN STATION COSTS AND ACCESS COSTS
  (WAREHOUSE LOCATION PROBLEM) TO A FIRST
  APPROXIMATION. THE METRO ROUTES ARE THEN FOUND BY
  STARTING WITH THE MINIMAL SPANNING TREE WHICH
  GIVES THE MINIMUM CONSTRUCTION, FLEET AND
  OPERATING COSTS (BUT NOT THE MINIMUM TRAVEL
  TIME) AND ADDING LINKS UNTIL THE COST INCLUDING
  TRAVEL TIME AND TRANSFER COSTS ARE MINIMIZED -
  WIRASINGHE VANDEBONA (1999)

23
BOOKS ON QUEUEING THEORY - 7

• (i) Applications of Queuing Theory, Chapman and
  Hall, 1971(1ST Ed.),148p 1982 (2nd Ed.), 303p.
• (ii) Approximate Stochastic Behavior of n-Server
  Service Systems with Large n, Springer-Verlag,
  1973, 118p.
• (iii) Approximate Behavior of Tandem Queues,
  Springer-Verlag, 1979, 410 pages.

24
DETERMINISTIC QUEUEING THEORY

• THE BOOK APPLICATIONS OF QUEUEING THEORY IS ONE
  OF NEWELLS GREATEST CONTRIBUTIONS
• DETERMINISTIC QUEUEING THEORY (DQT) AND THE FLUID
  APPROXIMATION OF A PEAK PERIOD QUEUE ARE KEY
• THE BASIS FOR THE MEAN WAITING TIME FOR TRANSIT
  SERVICE BEING HALF A HEADWAY IS BASED ON DQT.
• USEFUL FOR EXAMPLE IF WE DECIDE TO ASSUME THAT
  THE COST OF WAITING TIME PER UNIT TIME IS AN
  INCREASING FUNCTION OF THE ELAPSED WAITING TIME
  (WIRASINGHE 1990)

25
(No Transcript)
26
CONTRIBUTIONS BY NEWELLS STUDENTS/ASSOCIATES TO
RELATED FIELDS

• TRAFFIC SAFETY - HAUER
• TRANSIT TECHNOLOGY VUCHIC (TWO BOOKS)
• AIRPORT TERMINAL ANALYSIS - WIRASINGHE
• (with BANDARA, VANDEBONA, DADA, de BARROS,
  CORREIA, NANAYAKKARA, LAM, WONG)
• TRANSPORT, LOGISTICS, PLANNING ANALYSIS
  DAGANZO (BERKELEY)
• TRAFFIC SIGNAL NETWORK DESIGN KUWAHARA
  ISTTT COLLEAGUES
• TRANSPORT IN DEVELOPING COUNTRIES MITRIC,
  WIRASINGHE, KUMARAGE, BANDARA

27
NEWELLS APPROACH SOME GENERAL INSIGHTS

• WHAT IS THE PROBLEM?
• IF ONE WANTS SIMPLE MATHEMATICS, .. ONE MUST
  POSE THE RIGHT QUESTION.
• TRY TO OBTAIN A GENERAL IF APPROXIMATE SOLUTION
  THAT PROVIDES INSIGHT RE THE ISSUES AT HAND THE
  QUALITATIVE INTERRELATIONS BETWEEN VARIABLES
  (AS OPPOSED TO AN ACCURATE EVALUATION OF A
  SPECIFIC PROBLEM)
• THIS APPROACH FORCES ONE TO THINK DEEPLY ABOUT A
  PROBLEM AND TO DELVE INTO ITS INTRICACIES.
• THERE ARE FEW PROBLEMS FOR WHICH THERE IS NO
  APPROXIMATE ANALYTICAL SOLUTION
• IN SOME CASES THE ASSUMPTION OF A CONTNUOUS
  (FLUID) INPUT FUNCTION (SAY DEMAND VARYING OVER
  TIME) AND A RELATED SLOWLY VARYING SMOOTH
  RATE OR DENSITY OF RESPONSE, FACILITATES THE
  MODELING PROCESS CONTINUUM APPROXIMATION

28
NEWELLS APPROACH SOME GENERAL INSIGHTS

• GIVEN THAT INPUT VARIABLES SUCH AS PASSENGER
  DEMAND PER UNIT TIME IS NON UNIFORM, IT IS MORE
  REALISTIC TO ASSUME THAT SUPPLY VARIABLES (SUCH
  AS HEADWAYS, ZONE SERVED BY A METRO STATION,
  SLACK TIME AT A TIME POINT) ARE NOT UNIFORM, I.E.
  USE THE MOST NATURAL VARIABLES
• START WITH A SIMPLER PROBLEM, E.G. ONE TO ONE,
  AND ADD COMPLEXITIES LATER.
• SHAPES (E.G. SERVICE ZONES) AND DETAILS OF
  NETWORKS NEED NOT BE EXACT
• FOLLOWING THE LITERATURE IS NOT NECESSARILY THE
  BEST THING TO DO (SWISS CHEESE MODEL)

29
CURRENT STATE OF THE ART OF ANALYTICAL MODELLING

• DISPATCHING POLICY (MANY TO MANY)
• BUS CONTROL (ONE TO MANY) WITH MANY TO MANY
  THROUGH SIMULATION
• SPACING AND LOCATION (MANY TO MANY)
• NETWORKS (METROS MANY TO MANY) (NO PROGRESS
  RE MANY TO MANY WITH MULTIPLE SURFACE MODES, E.G.
  RAIL/BUS)
• QUEUEING THEORY MOST REFERENCED BOOK IN THE
  TRANSPORTATION FIELD.

30

31
(No Transcript)