IJERT Application of Queuing Theory of A
IJERT Application of Queuing Theory of A
IJERT Application of Queuing Theory of A
Abstract— Due to ever increasing traffic, the road capacity has California, require all payment to be made by means of
to be increased to accommodate different configuration electronic transponder, so that vehicles do not have to slow
vehicular dimensions. Toll roads need huge financing to down in order to pay the toll [7]. But on many older toll ways
construct a safe, effective, durable road network. Toll financing moving to all-electronic payment is not an option, while
is one of the technique in which revenue collected from the road
mounting congestion means that planners are faced with the
users for the service provided by them. This in turn results in
development in queues at particular junction where in the toll problem of configuring their existence infrastructure to
booths are erected. Long queue could lead to increase in travel provide the best possible service.
time which is drawback of road user. Hence toll should be
designed and planned in such a way that minimum time would 1.2 Background
be wasted in the queuing area. The toll booths are planned on Highway toll plazas constitute a unique type of transportation
the basis of queuing area. Queuing theory involves parameters system that requires special analysis when trying to
such as arrival, number of lanes, service time, waiting time, understand their operation and their interaction with other
merging area. In present study road inventory, traffic volume,
roadway components. On the one hand, these facilities are
space mean speed, arrival rate, time headway and service rate
are analyzed. one of the most effective means of collecting user fees for
roadways. The object of a toll highway should be to minimize
Keywords—Arrival rate, Service rate, Space mean speed, Time average travel time of all drivers on that road. On the other
headway hand, toll plazas adversely affect the throughput or capacity
CHAPTER 1 of the facilities they serve. The adverse effect of toll plazas is
INTRODUCTION particularly evident during hours when traffic is usually
1.1 General heavy. Thus highway toll experience lengthy vehicular
A queue is simply a waiting line. Therefore systems that queues and long delays when demand is near or exceeds
involve waiting lines are called queuing systems and processing capacity. Efficient sizing of toll plazas becomes
mathematical descriptions of queuing systems are known as critical in minimizing the space requirements and capital
queuing models. Transportation systems often involve expense of collecting user fees. Hence keeping all these in
queues. Queuing or waiting-line, phenomena are everyday view an effort has been made to study the performance of an
occurrences Queuing systems are characterized by an arrival existing toll on National highway-75 near Kadaballi between
pattern, a service facility and a queue discipline. Toll Bangalore to Mangalore stretch by applying the queuing
financing has been used throughout the history of civilization theory.
to make the building of long-distance roads possible.
Beginning in the 1940’s, America’s first modern freeways 1.3 Scope of Study
were financed with tolls. Today developing nations such as Toll plazas have become means for collecting revenue in
China are building their own networks of superhighways, and order to build network of roadways which in turn improve
they too turning to the tollbooth for expenditure. Tolls are safety, comfort, reduce average travel time and improve the
being used successfully in places such as Singapore and capacity of roads. The study of queuing is important to find
London not just to finance road construction, but to limit the out new design methods in arranging the toll plazas and
flow of vehicles into the urban core, increasing transit usage means of operating the toll plazas which in turn improve the
and unclogging the crowded streets [7]. service rate at the toll booth and the capacity of the tollbooth
otherwise would have created long queues. In the present
Despite its many advantages, there is also disadvantage study an attempt has been made in understanding how the toll
associated with tolling. When traffic is thick, vehicles backup booth works which is being designed on queuing theory. The
in line to get to tollbooths, and after paying their tolls, drivers study involves collecting the geometric attributes of the toll
lose time scrambling for position as the many lanes exiting plaza, the arrival pattern of vehicles to the queuing area, the
the toll plaza merge together, returning the road to its original service provided by the system.
width. A study conducted at the New Jersey Institute of
Technology estimates that a travel time savings of 2 minutes, 1.4 Study Area
or over 10 percent, could be affected by the removal of two The site selected for the project lies near Kadaballi between
toll plazas along 14- mile section of the Garden State Bangalore to Mangalore highway. The stretch of road length
Parkway [10]. Modern toll facilities, such as Highway 407 is considered to be 500m away from the toll plaza. This
near Toronto and the SR-91 Express lanes in Orange County, provides clear view of the place selected for case study of
queuing theory. Ten service booths fixed by the company 1. The input function (Arrival rate)
called L&T Devihalli-Hassan pvt.ltd. There are some 2. The input source (Finite/Infinite)
irrigated land farms agricultural lands around the toll gates. 3. The queue discipline (FIFO/LIFO)
The selected road is a divided four lane National Highway 4. The channel configuration (Number and Arrangement)
(NH-75). This site has been selected for the study purpose of 5. The delay time (Service rate)
queuing theory because there are no intersections near toll The basic component or main parameters of a queuing system
plaza. The length of each toll booth is 4.2m and width 1.9m. is shown by the figure 2.1
The length of each lane is 3.6m.
λ=
Figure 1.1: Toll plaza of the Study Area.
Queue Discipline: queue discipline is a parameter that
1.5 Methodology explains how the customers arrive at a service facility.
The major steps involved in the present study are The various types of queue disciplines are
Road inventory of the selected road section
Traffic volume count as per IRC:9-1972 “Traffic Census 1. First in first out [FIFO]
on Non-Urban Roads” 2. First in last out [FILO]
To find out the velocity of approaching vehicles by 3. Serviced in Random order [SIRO]
Space mean speed. 4. Priority Scheduling
To find out the inter-arrival rate of vehicles by taking First in first out: If the customers are served in the order of
time headway. their arrival, then this is known as the first-come, first served
To find out the Service rate provided by the serving (FCFS) service discipline.
system at the toll booth. First in last out: Sometimes, the customers are serviced in the
To analyze the performance of toll booth from the reverse order of their entry so that the ones who join the last
collected data. are served first.
Served in Random order: Under this rule customers are
CHAPTER 2
selected for service at random irrespective of their arrivals in
LITERATURE REVIEW
the service system. In this every customer in the queue is
2.1 General
equally likely to be selected. The time of arrival of the
A primary objective in operational problems involving flow
customers is, therefore of no relevance in such a case.
is to ensure that the average capacity can handle the average Priority service: Under this rule customers are grouped in
flow, so that persistent traffic jams do not occur. Queuing
priority classes on the basis of some attributes such as service
theory was developed in order to describe the behaviour of a
time or urgency or according to some identifiable
system providing services for randomly arising demands. The
characteristics and FIFO rule is used within each class to
fundamental idea of the theory is that delay in a system is
provide service.
caused by an interruption in the flow pattern. Queuing theory
Numbers of servers: The number of servers that are being
is almost exclusively used to describe the traffic behaviour at utilized should be specified and in the manner they work
signalized and un-signalized intersections [10].
that is they work as “Parallel” servers or “Series” servers
has to be specified.
2.2 Characteristics Of Queuing System
Mean service rate: It is the rate at which customers depart
The analysis of queuing systems and its variables has been
from a transportation facility. It is expressed in flow
the focus of many studies and researchers for many decades. (vehicles/hour) or time headway (seconds/vehicle). If
The solution to a queuing problem entails the assessment of a inter-service time that is time headway (h) is known, the
system’s performance, which in turn is described by a set of
service rate can be found out from the equation.
measures of performance (MOP) [2]. The inputs include
µ=
2.3 Structuring Of Queuing Model
Traditionally, traffic flows are modelled empirically, using
origin-destination matrices [9]. One of the most important
equations in traffic flow theory is that relating between traffic
flow (q), traffic density (k) and traffic speed (s) which is
given as
q=k×s
These fundamental parameters of a traffic flow can be used as
inputs in developing appropriate queuing models. Queuing
models are often referred to using the Kendall notation,
consisting of several symbols e.g. M/G/1 [9]. The first
symbol describes the arrival rate of traffic into a system, the
second for the service rate provided by the system to the
vehicles while the third indicates the number of servers in the
system.
some point most vehicles must stop either because the vehicle within this range. The traffic simulation provides a more
in front of them has stopped or they have reached a toll comprehensive understanding of the toll plaza operation
booth. This in turn leads to building up of queue lengths allowing for a more in-depth analysis of its performance.
particularly when the instantaneous demand exceeds the According to them simulation should be used for advance
service. So it is necessary to find out the queuing area which planning, design, operation and management of toll and exit
is essential in fixing the number of servers in a toll plaza. plazas facilities.
Merging: After the toll booths, the roadway must narrow Abdul aziz, A.R., et.al, in “Application of queuing theory to
back from a number of lanes equal to the number of vehicular traffic at signalized intersection in Kumasi-Ashanti
tollbooths, to its normal width, a section will called as region, Ghana” has shown that queuing theory can be applied
“merging area”. Sometimes the extra lanes end almost in modelling the vehicular traffic flow and minimize
immediately, forcing a sharp merge at a relatively low speed. vehicular traffic in order to reduce delays on roads of
There are three different merging patterns are used when Kumasi-Ashanti region. The analysis of the data collected at
lanes begin and end are, Oforikrom intersections revealed that a smooth flow of traffic
With several lanes merging into one, all of the is seen when the server at each channel is able to serve more
merging could occur at a single point, but this means than cars in waiting queue. But in evening there is restriction
that as many vehicles as there are lanes could to flow due to the restraints caused by the commercial vehicle
interfere with each other at that point. drivers. They have suggested that use of public transport by
One common choice is to always merge out the the government of Ghana would help in reducing congestion
rightmost (or leftmost) lane until the desired number on the roads, which in turn boost the productivity.
of lanes is reached. NicoVandaele.,et.al, in “A Queuing Based Traffic Flow
Another possibility is a “balanced” pattern where Model” have shown that queuing models can be applied in
pairs of adjacent lanes all across the roadway merge assessing the traffic flow parameters compared to traditional
repeatedly until the desired roadway width has been empirical methods, which lack in terms of predictive power
attained. and the possibility of sensitivity analysis. Based on queuing
theory they analytically constructed the well-known speed-
flow-density diagrams. They have shown that the exact shape
of the different speed-flow-density diagrams is largely
determined by the model parameters so that a good choice of
parameters can help to adequately describe reality. They also
believe that speeds have a significant influence on vehicle
emissions and models can be effectively used to assess the
environmental impact of road traffic.
Figure 2.5: Showing Four lanes Merging to 1 Single Point.
CHAPTER 3
FIELD STUDIES
3.1 General
In order to understand the theory behind queuing a toll plaza
has been selected in the present study which is located on
Figure 2.6: Showing Four lanes Merging with Rightmost Lane Ending. NH-75 near Kadaballi. Field studies like road inventory,
traffic volume, space mean speed, time headway, arrival rate
and service pattern have been carried out.
Figure 3.1: Showing the Flexible Pavement and rigid pavement near the Toll Figure 3.2: Shows the Toll Area having Concrete Road.
Plaza.
3.3 Traffic Volume:
Table 3.1: Shows the details of Pavement Structure before Traffic volume or traffic flow is defined as “the product of
Toll Plaza. the average traffic intensity and the time period of the study”.
SI no Parameters Collected data
It is measured by the units “vehicle per hour”. In the present
1 Type of pavement Flexible pavement study the traffic count census is done as per IRC: 9-1972
2 Divided/undivided Divided “Traffic census on Non-Urban Roads”. To take into account
3 Number of lanes Four the randomness, the traffic volume study was carried out in
short intervals (1 hour) at different hours of a day and at
4 Width of pavement (m) 9
different days. Traffic flow is usually considered to be
5 Median width (m) 2.5
roughly constant at any given instant, as changes in flow
6 Shoulder width (m) 1.5 occurs smoothly and slowly, while measurements employed
7 Type of shoulder Earthen are over very short time periods.
Table 3.3: Showing traffic flow on both the directions of the Table 3.4: Showing the Space-mean Speed of Vehicles
National Highway 75. approaching the Toll Booth Near to Toll.
Vs=
Where VS=Space mean speed km/s
Ti=t1+t2+t3+t4+…………. + tn=Average
time taken seconds.
Table 3.6: Showing the Details of the Space Mean Speed Table 3.8: Showing the Details of the Space Mean Speed
(Hassan to Bangalore). (Hassan to Bangalore).
Table 3.10: Showing the Details of the Space Mean Speed Table 3.12: Showing the Details of the Space Mean Speed
(Bangalore to Hassan). (Bangalore to Hassan).
Th=
Table 3.17: Showing a details of Time Headway. Table 3.19: Showing a details of Time Headway
Table 3.21: Showing a details of Time Headway. Table 3.23: showing the observed frequency from Hassan to
Bangalore.
3.7 Service Rate: Table 3.26: Showing the Details of Service Rating Time.
The service rate depends upon the type of operation involved
in providing service to the customers. Generally cash
collecting service takes more time than the automatic way of
collection. Service rate denotes the rate at which vehicles are
been served in a system. It is the reciprocal of the service
time.
Service rate=
When the vehicle enters the toll plazas, a rational driver
selects the counter service by seeing the queue length existing
relative to other counters. Once the vehicle is in the queue
length it has to follow the queue discipline. The waiting time
is the time spend by the vehicle in the queue length and the
time spends in providing the amount. The driver must pay the
with exact change in order to minimize service time.
26 8.14
Figure 4.1: The speed density diagram for the M/G/1 model for both lanes
REFRENCES
[1] Drew, D.R. “Traffic flow theory and control”.MC Graw-HILL.
[2] Papacostas, C.S, and Prevedouros,P.O, “Transportation
engineering and planning”, pearson education INC, upper saddle
Reios, New Jersey, U.S.A.
[3] Motwani, R.G, “Optimum toll charges for proposed Airoli
Bridge”, M.Tech Dissertation, 1995.
[4] KhaledNassar, “Queuing model are accessing the efficiency of
building corridors”, ASCE.
[5] Shantanu Das and David Levinson, “Queuing and statistical
analysis of freeway Bottleneck formation”,ASCE.
[6] Abdul-Aziz.A.R, “Application of queuing theory to vehicular
traffic at signalized intersection in Kumasi-Ashanti region,
Ghana”, volume 3