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

2. 2  Out lines  Queuing theory definitions  Some Queuing Terminology  Applications of Queuing Theory  Characteristics of queuing systems  Decision Making  Examples

3. 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. 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. 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. 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. 7  Example application of queuing theory

8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 21  THANK YOU!

22. 22 QUIZ I remember at least four in Applications of Queuing Theory