Simulation of cars waiting at a toll booth using Scilab

1
Simulation of cars waiting at a toll booth using
Scilab

• Jianfeng Li
• Institute of Industrial Process Control, Control
  Science and Engineering Department,zhejiang
  university
• 2006.9

2

• Introduction
• The queue systems performance index
• The optimal number of servers
• Simulation
• Conclusion

3
Introduction

• Queuing theory
• It is the study of the waiting line phenomena.
• It is a branch of applied mathematics utilizing
  concepts from the field of stochastic processes.
• It is the formal analysis of this phenomenon in
  search of finding the optimum solution to this
  problem so that everybody gets service without
  waiting for a long time in line.

4
Introduction

• The queue system is composed of

the arrival process
the queue discipline
the service process

• If there is one server in all service process, it
  is called single-server system.
• If the server number is more than one and each
  sever can serve customers independently, it is
  called multi-server system

5
Introduction

• Usually, the service system is evaluated by
  performance indexes, such as
• Service intensity
• Average queuing length
• Average queuing time
• Average stay time
• Probability of the service desks being idle
• The probability of the customers having to wait

6
Introduction

• The basic service process is the following

busy
the customers arrive
queuing system
waiting line
free
served according to the prescribed queuing
discipline
finished
leave the system
7
Introduction

• The terms customer, server, service time, and
  waiting time may acquire different meanings in
  different applications.
• In this article
• The customers are cars arriving at a toll booth
• The servers are the workers in the tool booth
• The service times are the payment times

8
The queue systems performance index

• In the simulation of cars waiting at a toll
  booth, some necessary condition assumptions and
  notation definitions should be made.
• Assumptions
• The arrival time and service time follow
  exponential distribution.
• The waiting queue is infinite.

9
The queue systems performance index

• Notations
• ? the arrival rate of the car
• µ the service rate of the car
• ? the service intensity
• Ls the average quantity of cars in the system
• Lq the average quantity of cars in the waiting
  queue
• Ws the average time of each car stay in the
  system
• Wq the average time of each car stay in the
  waiting queue

10
The queue systems performance index

• according to John DC Littles formula, we
• only discuss two usual models.
• Standard M/M/1 service model.
• Standard M/M/C service model.

11
Standard M/M/1 service model

• The arrival time and service time of cars can be
  discretional distribution.
• The number of server is one.
• The car source and service space are infinite.
• The queuing discipline is first come first
  service.

12
Standard M/M/1 service model

• So the major performance indexes are
• Service intensity
• Probability of the server being idle
• Probability of n cars in the system

13
Standard M/M/C service model

• The arrival time and service time of cars can be
  discretional distribution.
• the number of server is C and they are parallel
  connection .
• Servers work independently and the average
  service rate is the same.
• The car source and service space are infinite.
• The queuing discipline is first come first
  service.

14
Standard M/M/C service model

• So the major performance indexes are
• Service intensity
• Probability of the server being idle
• The average quantity of cars in the waiting
  queue
• The average quantity of cars in the system
• The average time of each car stay in the system
• Probability of the car has to waiting after
  arriving

15
The optimal number of servers

• Step 1, according to , find out
  the initialization number of servers
  
• Step 2, compute the performance indexes, such as
  the average quantity of cars in the waiting
  queue, the average time of each car stay in the
  system, and the average time of each car stay in
  the waiting queue. Comparing them with desired
  indexes, if it is dissatisfied, go to step 3
  otherwise, the optimal number of servers is N and
  the process is end.
• Step 3, let , go to step 2.

16
Simulation

• In this system, the arrival time and service time
  of car are similar to the exponential
  distribution.
• Assume that the left time of the ith car is the
  begin service time of the (i1)th car.
• All the events can be classified to two sorts
  one is the arrival event of cars the other is
  the leave event of car.
• When the arrival event and the leave event occur
  at a same time, disposed the former first.

17
Simulation

• In Scilab simulation environment, the average
  arrival rate of cars, the average service rate of
  cars and the simulation terminate time are given
  by the following

18
Simulation

• In the simulation platform we present, the draw
  mode of SCILAB is set to be animate all along so
  that a gliding process can be observed.

19
Simulation

• The animation demo can be described as

20
Simulation

• Assume that the terminate time of simulation is
  240 sec.
• The average arrival rate is 0.18.
• And the average service rate is 0.1.
• The minimum servers number can be found out
  
• 1.
• The queuing discipline is first come first serve.

21
Simulation

• The simulation results can be calculated
• The average quantity of cars in the waiting queue
  is 15
• The max quantity of cars in the waiting queue is
  26
• The average quantity of cars in the system is 16
• The average time of each car stay in the waiting
  queue is 146sec
• The average time of each car stay in the system
  is 162sec
• All the served cars number is 22

22
Simulation

• Let , the number of server
  become 2.
• The simulation process is similar to the above
  single servers systems. The different is that, in
  the simulation, when the second car arrival, it
  can be serviced immediately because there are two
  servers in the system. So the car will pass the
  toll booth with less waiting time.

23
Simulation

• The simulation results with two servers are
• The average quantity of cars in the waiting queue
  is3
• The max quantity of cars in the waiting queue is
  6
• The average quantity of cars in the system is 4
• The average time of each car stay in the waiting
  queue is 29sec
• The average time of each car stay in the system
  is 41sec
• All the served cars number is 42

24
Simulation

• Comparing the above results with desired
  performance index, it is found that when there
  are two servers, the demands can be satisfied.
• So, it is recommended to open two servers.

25
Conclusion

• Queuing theory have been used in almost every
  domain of social and natural science as a tool to
  solve many problems. In this article, queuing
  theory is used to solve the problem of cars
  waiting in a tool booth.
• Using the tools of Scilab, the simulation of a
  single queuing multi-servers system is done and
  the optimal number of server is obtained.

26
The end

• Thank you!