Chapter 16 Applications of Queuing Theory

1. University of Palestine Faculty of Information Technology Operations Research Chapter 16 Applications of Queuing Theory Prepared by: Ashraf Soliman Abuhamad Supervisor by : Dr. Sana’a 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:

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, 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 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. λ = 1/ 2.5 = 0.4 • Calculate ρ ρ = λ * s = 0.4 * 2 = 0.8 • Calculate the desired values Tq = 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 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 λ = 0.4 and s = 1.0 Tq = s / (1- ρ) = 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 9:15 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 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 النظام مستقر 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