This document discusses queuing theory and its application to traffic flow modeling. Some key points: - Queuing theory is the mathematical analysis of waiting lines or queues. It allows prediction of line lengths and waiting times. - As it relates operations research, queuing theory results are used for business decisions about resource requirements for service provision. - It is relevant for modeling customer service situations as well as transport and telecom applications. Specifically applicable to traffic systems. - The document outlines components of a queuing model for traffic, including relationships between traffic flow, density, and speed. Models arrival and service rates to estimate time in the system.

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International Journal of Allied Practice, Research and Review
Website: www.ijaprr.com (ISSN 2350-1294)
Queuing Process in Traffic Flow Model
Mr. Dharamvirsinh B. Parmar
Assistant professor in Mathematics, C.U.Shah University
Abstract: -Queuing hypothesis is a gathering of numerical models of different queuing frameworks. Queuing hypothesis
is the numerical investigation of holding up lines, or lines. In queuing hypothesis a model is built so line lengths and
holding up times can be anticipated. Queuing hypothesis is by and large considered a limb of operations examination in
light of the fact that the results are frequently utilized when settling on business choices about the assets required giving
an administration.
Queuing Theory is the numerical investigation of holding up lines (or lines). The hypothesis empowers numerical
investigation of a few related procedures, including landing at the (over of the) line, holding up in the line (basically a
stockpiling process), and being served by the server(s) at the front of the line. The hypothesis allows the induction and
estimation of a few execution measures including the normal holding up time in the line or the framework, the normal
number holding up or accepting administration and the likelihood of experiencing the framework in specific states, for
example, void, full, having an accessible server or needing to hold up a certain time to be served.
Queuing Theory is for the most part considered an extension of operations examination in light of the fact that the results
are frequently utilized when settling on business choices about the assets required giving administration. It is material in
a wide mixture of circumstances that may be experienced in business, trade, industry, social insurance, open
administration and designing. Applications are oftentimes experienced in client administration circumstances and also
transport and telecom. Queuing hypothesis is specifically appropriate to insightful transportation frameworks, call
focuses, systems, information transfers, server queuing, and centralized computer machine queuing of information
transfers terminals, and progressed information transfers frameworks and traffic flow.
Keywords:-Queuing theory; traffic flow modeling; congestion management queuing simulation; estimated queue length.
I. Introduction
A mathematical strategy for analyzing down the cloggings and delays of waiting in line, Queuing
theory inspects each part of holding up in line to be served, including the landing procedure,
administration methodology, number of servers, number of framework spots and the quantity of "clients"
Example: - people, data packet, autos. Real life applications of queuing theory incorporate giving quicker

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client administration, improving traffic flow, shipping orders efficiently from a warehouse and designing
telecommunications systems such as call centers.
The evaluation of continuous traffic flow is generally focused around experimental routines. We
Create some scientific queuing models focused around traffic tallies and we show the conduct of Traffic
flows as a capacity of probably the most significant determinants. These scientific models take into
account parameterized investigations, which prepare towards our examination destinations: evaluating
considers the possibility that situation's and affectability investigation for traffic administration, blockage
control. Traffic outline and the ecological effect of street traffic (e.g. Outflow models). The effect of
some pivotal displaying parameters is mulled over in subtle element and connections with the more
extensive exploration goals are given. We show our results for a roadway, in light of included traffic
flows Flanders.
Fig:-Dynamic and Static impact of Transportation traffic on emissions and waste
As traffic flows occupy a central position in the assessment of transportation traffic emissions, a
robust traffic flow model is required. Traditionally, traffic flows are modeled empirically, using origin
destination matrices. These descriptive models give an empirical justification of the well-known speed-
flow and speed-density diagrams, but are limited in terms of predictive power and the possibility of
sensitivity analysis. The construction of origin-destination matrices consists of five steps: transport
generation, transport distribution, the modal split, assignment and calibration.
II. Review of Literature
Queuing theory is the numerical investigation of holding up lines, or the demonstration of joining
a line (queues). In queuing theory a model is built so line lengths and holding up times can be anticipated
(Sundarapandian, 2009). The issue of lining has been a subject of logical level headed discussion for
there is no known society that is not defied with the issue of lining. Wherever there is rivalry for
restricted asset lining is prone to happen. The part of transportation in human life can't be
overemphasized. As per Intikhab et al. (2008), effective transportation framework assumes an imperative
part in cooking for the day by day necessities in the lives of the natives. These incorporate access to
pleasantries and administrations that are fundamental to the lives of all people, in the same way as job,
instruction, wellbeing administrations and recreation. At the individual level, Wane (2001, p.1)
additionally brings up that 'transportation is an essential element for urban insertion since it offers access
to financial action, encourages family life and aides in turning interpersonal organizations. Writing on

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lining demonstrates that holding up in line or line reasons disservice to financial expenses to people and
associations. Healing centers, air transport organizations, banks, assembling firms and so forth., attempt
to minimize the aggregate holding up expense, and the expense of giving administration to their clients.
Thusly, speed of administration is progressively turning into a paramount aggressive parameter (Katz, et
al, 1991). Davis et al (2003) declare that giving ever-quicker benefit, with a definitive objective of having
zero client holding up time, has as of late gotten managerial consideration for a few reasons. To begin
with, in the all the more exceedingly created nations, where expectations for everyday life are high, time
gets to be more significant as a thing and thusly, clients are less eager to sit tight for administration.
Second, this is a becoming acknowledgment by associations that the way they treat their clients today
fundamentally affect on whether they will stay steadfast clients tomorrow. At last, progresses in
engineering, for example, machines, web and so on, and have given firms with the capacity to give
speedier administrations. Consequently healing center chairmen, doctors and administrators are
constantly discovering intends to convey speedier administrations, accepting that the holding up will
influence after administration assessment adversely. Likewise, understanding the inefficiencies in the
doctor's facility and enhancing them is urgent for making social insurance approach and planning choices
(Wilson and Nguyen, 2004).
III. Objective
 To design a Queuing system that minimizes the average waiting time, costs and length of the
queue.
 To provide the optimal time period for the recruitment.
 To allocate the person for right place by using transportation problem of special case assignment
problem.
 The aim of studying queuing system simulation is trying to detect the variability in a quality of
service due to queues in sales checkout operating units, find the average queue length before
getting served in order to improve the quality of the services where required, and obtain a sample
performance result to obtain time-dependent solutions for complex queuing models. The defined
model for this kind of situation where a network of queues is formed is time-dependent and needs
to run simulation.
IV. Composition of a Queuing Model
The most part of queuing equations in traffic based resource management that describe through
the particular equation is given below that are includes the interdependence of traffic flow Tf, traffic
density D and speed s.
Tf= D*s (equation no.-1)
Here we find out third variables if given two variables. If traffic count information is available,
traffic flows can be calculated as given, which leaves us to determine either traffic density or speed to
complete the method and use either as input for the suitable Queuing structure.

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Here this structure we define Roads are divided into segments of the same length 1/C, where C is the
maximum traffic density and which matches the smallest length needed by one vehicle on that particular
road. Every road segment is considered as a service station, in which vehicles get served at rate µ and
arrive at rate λ.
Tf- Traffic density (vehicle/km) ,C -Maximum traffic density (vehicle/km), s -Effective speed (km/h), W
-Time in the system (h),
Whereas, W is the total time taken a vehicle spends in the system and which count the sum of waiting
time (due to jamming) and service time. And the much greater the traffic intensity, the higher the time in
the system becomes {the accurate relation among ρ and W depends upon the queuing model.
ρ- Traffic intensity = λ / µ, r -Relative speed, SN -Nominal speed (km/h), µ- Service rate (vehicle/h), q -
Traffic flow (vehicle/h), λ- Arrival rate (vehicle/hour).
if the W is known the we calculated the effective speed easily as given as-
s= (1CW) (1)
now we calculate the relative r, becomes-
r = s/SN= 1/C/W*SN (2)
Machinations the traffic flow and density (relative & effective) speed on a table gives us the
distinguished speed-flow-density diagrams. The accurate shape of these diagrams depends upon the
Queuing model and the uniqueness of the arrival and service procedures.
The mathematical statements depicting M/M/1 lining framework are genuinely straight forward
and simple to utilize. Initially we characterize the movement force, ρ (some of the time called
inhabitance). It is defined as the normal arrival rate (λ) divided by the normal administration rate (µ). For
a stable framework the normal administration rate ought to dependably be higher than the normal landing
rate. (Overall the lines would quickly race towards boundlessness). Along these lines ρ ought to
dependably be short of what one.
Likewise note that we are discussing normal rates here, the prompt entry rate may surpass the
administration rate. Over a more drawn out time period, the administration rate ought to constantly
surpass an entry rate.
ρ= λ/ µ (3)
Mean number of customers in the system (N): can be found using the following equation
Mean number of customers in queue (prior to service)
N = i𝜌𝑖
∞
𝑖=0 = 𝜌(1 − 𝜌) (1 − 𝜌)2
=
𝜌
1 − 𝜌 or N = λ μ − λ (4)

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Mean number of customers in queue (prior to service)
Nq = (i − 1)
∞
𝑖=0
𝜌𝑖 =
𝜌
1 − 𝜌 − 1 − 1 − p =
p2
(1 − 𝑝) (5)
T=W= N
𝜆 =
𝜌
1 − 𝜌 𝜆
(6)
Tq=Wq =
𝜌
𝜆(1 − 𝜌) (7)
Again we see that as mean arrival rate approaches mean service rate, the waiting time becomes very
large.
V. Results of computation of these parameters
The general target of this study is to assemble an essential model of vehicular movement focused
around lining hypothesis and afterward utilize it to focus the best times for the red golden and green
lights to be either on or off keeping in mind the end goal to diminish activity blockage at the
himmatnager crossing point in Udaipur, in the Gujarat.
Area Udaipur to
himmatmager
Time
period
morning
Arrival Average vechile
of no.
45
time 1.34
service Average no. of
vechile
60
Time(mintue) 1.30
Table:-Traffic information at this route.
Processing’s of parameters to focus the conduct of activity were carried out utilizing the M/M/1
lining model. It is accepted that time interim between progressive landings and serving time is
autonomous and indistinguishably appropriated. In any sufficient interim of time at most one and only
entry can happen. The framework is additionally accepted to achieve a consistent state, a condition that
the rate of entry and administration is steady. The lining order observed was FCFS (first come first
served).

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Udaipur to himmatnager at morning session different parameter.—
service rate 𝜇 =
60
1.30
= 47 𝑣𝑒ℎ𝑖𝑐𝑙𝑒/𝑚𝑖𝑛
arrival rateλ =
45
1.34
vechicle/min
𝜌 (𝑡𝑟𝑎𝑓𝑓𝑖𝑐 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 =
34
47
= 0.7234)
mean number in queue Nq = 0.7234 ∗
0.7234
1−.7234
=2
All the routes at intersection at Udaipur where, these content of these given data table 1. After
comparison at different time and that display traffic intersection is currently operating with one service
channel each from the various routes to Udaipur to himmatnager highway.
VI. Conclusion
In this research, the use of queuing models which very useful in achieving the main objective
Queuing networks are simple yet powerful tools for performance modeling and capacity was planning.
Based on queuing hypothesis we diagnostically built the well-known velocity flow-thickness outlines.
Utilizing a few queuing models, pace is dead set, in view of distinctive landing and administration forms.
The definite state of the diverse rate flow-thickness outlines is to a great extent controlled by the model
parameters. Subsequently we accept that a decent decision of parameters can help to satisfactorily portray
reality. We showed this with an illustration, utilizing the most general models (counting a state ward
model) for a thruway. Because of the way that speeds have a huge impact on vehicle discharges, our
models can be viably used to survey the natural effect of street traffic.
VII. Reference
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