Chapter 16 Applications of Queuing Theory

1
Chapter 16 Applications of Queuing Theory
University of Palestine Faculty of Information
Technology
Operations Research

• Prepared by
• Ashraf Soliman Abuhamad
• Supervisor by
• Dr. Sanaa Wafa Al-Sayegh

2
? Out lines

• ? Queuing theory definitions
• ? Some Queuing Terminology
• ? Applications of Queuing Theory
• ? Characteristics of queuing systems
• ? Decision Making
• ? Examples

3
? Queuing theory definitions

• (Bose) the basic phenomenon of queuing arises
  whenever a shared facility needs to be accessed
  for service by a large number of jobs or
  customers.
• (Wolff) The primary tool for studying these
  problems of congestions is known as queueing
  theory.
• (Mathworld) The study of the waiting times,
  lengths, and other properties of queues.

4
? Some Queuing Terminology

• To describe a queuing system, an input process
  and an output process must be specified.
• Examples of input and output processes are

Situation Input Process Output Process
Bank Customers arrive at bank Tellers serve the customers
Pizza parlor Request for pizza delivery are received Pizza parlor send out truck to deliver pizzas
5
? Applications of Queuing Theory

• Telecommunications
• Traffic control
• Determining the sequence of computer operations
• Predicting computer performance
• Health services (eg. control of hospital bed
  assignments)
• Airport traffic, airline ticket sales
• Layout of manufacturing systems.

6
? Application of Queuing Theory

• We can use the results for the queuing models
  when making decisions on design and/or operations
• Some decisions that we can address
• Number of servers
• Efficiency of the servers
• Number of queues
• Amount of waiting space in the queue
• Queueing disciplines

7
? Example application of queuing theory
8
? Characteristics of queuing systems

• Arrival Process
• The distribution that determines how the tasks
  arrives in the system.
• Service Process
• The distribution that determines the task
  processing time
• Number of Servers
• Total number of servers available to process the
  tasks

9
? Decision Making

• . Queueing-type situations that require decision
  making arise in a wide variety of contexts.
• For this reason, it is not possible to
  present a meaningful decision-making procedure
  that
• is applicable to all these situations.

10

• Designing a queuing system typically involves
  making one or a combination of the following
  decisions
• 1. Number of servers at a service facility
• 2. Efficiency of the servers
• 3. Number of service facilities.

11
? Number of Servers

• Suppose we want to find the number of servers
  that minimizes the expected total cost, ETC
• Expected Total Cost Expected Service Cost
  Expected Waiting Cost(ETC ESC EWC)

12
? Example

• Assume that you have a printer that can print an
• average file in two minutes. Every two and a half
• minutes a user sends another file to the printer.
  
• How long does it take before a user can get their
  output?

13
? Slow Printer Solution

• Determine what quantities you need to
• know.
• How long for job to exit the system, Tq
• Identify the server
• The printer
• Identify the queued items
• Print job
• Identify the queuing model
• M/M/1

14
? Slow Printer Solution

• Determine the service time
• Print a file in 2 minutes, s 2 min
• Determine the arrival rate
• new file every 2.5 minutes. ? 1/ 2.5 0.4
• Calculate ?
• ? ? s 0.4 2 0.8
• Calculate the desired values
• Tq s / (1- ?) 2 / (1 - 0.8) 10 min

15
? Add a Second Printer

• To speed things up you can buy another
• printer that is exactly the same as the one
• you have. How long will it take for a user
• to get their files printed if you had two
• identical printers?
• All values are the same, except the model
• is M/M/2 and ? ? s / 2 0.4

16
? faster printer

• Another solution is to replace the existing
• printer with one that can print a file in an
• average of one minute. How long does it
• take for a user to get their output with the
• faster printer?
• M/M/1 queue with ? 0.4 and s 1.0
• Tq s / (1- ?) 1 / (1 - 0.4) 1.67 min
• A single fast printer is better, particularly at
• low utilization. 6X better than slow printer.

17
? Example

• Customers arrive at a rate of 10 to a bank.
  Working in a
• bank cashier and customer service is the average
• service time of 4 minutes, assuming the service
• follows the rules of the Bank and the exponential
• accommodate any number of customer
• arrivals. Find the following
• 1-How the proportion of time spent out of work
  ATM.
• 2-What is the average number of customers waiting
  in line for service.
• 3-If you entered this section at around 915 when
  expected out of the section after you get the
  service
• 4-The average number of customers of the bank .
• 5-The average time spent by the customer in the
  waiting .

18
? Example

• The rate of inflow of customers10 customer /1hr
  ?
• Average time service 4 min 1/µ
• Speed service customer 1/avg time service 1/4
  customer-min 60/4 per/hr
• P ? /µ 10/15 0.667 gt 1 ?????? ?????
• 1-How the proportion of time spent out of work
  ATM.
• The possibility that the system is empty
• P 0(p-1) 0.667-10.333
• 2-What is the average number of customers
  waiting in line for service?
• L q p²/1-p 0.667²/(0.333)1.333.

19
? Example

• 3-If you entered this section at around 915 when
  expected out of the section after you get the
  service
• Expected time of departure The moment of entry
  The average time in which they occur in the bank
• 915
  W
• W p / (?1-p) (0.667)/ 100.333 0.2 hours
  12 mints.
• The expected time of departure 915 0012
  927

20
? Example

• 4-The average number of customers of the bank
• L p / (1-p) 0.667 / 0.333 2 customers
• .5-The average time spent by the customer in
  the waiting .
• Wqp²/?(1-p) (0.667)² / 10(0.333) 0.1334
  hours 8 mints

21
? THANK YOU!
22
QUIZ

• I remember at least four in Applications of
  Queuing Theory