1

Chapter 16 Applications of Queuing

Theory

Prepared by:Ashraf Soliman Abuhamad

Supervisor by :Dr. Sana’a Wafa Al-Sayegh

University of PalestineFaculty of Information Technology

Operations Research

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:

SituationInput ProcessOutput Process

BankCustomers arrive at bank

Tellers serve the customers

Pizza parlorRequest 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 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, E[TC]Expected Total Cost = Expected Service Cost + Expected

Waiting Cost(E[TC]= E[SC] + E[WC])

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 toknow.How long for job to exit the system, Tq• Identify the serverThe printer• Identify the queued itemsPrint job• Identify the queuing modelM/M/1

14

Slow Printer Solution

• Determine the service timePrint a file in 2 minutes, s = 2 min• Determine the arrival ratenew file every 2.5 minutes. λ = 1/ 2.5 = 0.4• Calculate ρρ = λ * s = 0.4 * 2 = 0.8• Calculate the desired valuesTq = s / (1- ρ) = 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 ρ = λ * s / 2 = 0.4

16

faster printer

• Another solution is to replace the existingprinter with one that can print a file in anaverage of one minute. How long does ittake for a user to get their output with thefaster printer?• M/M/1 queue with λ = 0.4 and s = 1.0Tq = s / (1- ρ) = 1 / (1 - 0.4) = 1.67 minA single fast printer is better, particularly atlow utilization. 6X better than slow printer.

17

Example Customers arrive at a rate of 10 to a bank. Working in abank cashier and customer service is the averageservice time of 4 minutes, assuming the servicefollows the rules of the Bank and the exponentialaccommodate any number of customerarrivals. 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 9:15 when expected

out of the section after you get the service4-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 customers=10 customer /1hr = λ Average time service = 4 min = 1/μ Speed service customer =1/avg time service =1/4 customer-

min = 60/4 per/hr P= λ /μ 10/15 = 0.667 > 1 مستقر النظام 1-How the proportion of time spent out of work ATM. The possibility that the system is empty P 0=(p-1) = 0.667-1=0.333 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 9:15 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

= 9:15 + W

W = p / (λ[1-p]) = (0.667)/ 10[0.333] = 0.2 hours = 12 mints.

The expected time of departure = 9:15 + 00:12 = 9:27

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 .

Wq=p²/λ(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