1. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 1

2. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India QUEUING THEORY 17 CHAPTER

3. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Learning Objectives Characteristics of a queue. Single Channel Single Server Queuing Model Utilisation Factor Economic Aspects of Queuing.

4. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Queuing Whenever any person or any thing has to wait for a service, there is economic loss due to the waiting time. This can be remedied by increasing the service facilities. This in turn add to the costs. A balance must be struck between loss due to waiting time and the cost of providing extra service facilities. Queuing Models deal with such problems. Queuing models are descriptive and not prescriptive.

5. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Characteristics of a Queue The Calling Population –Size – Finite or infinite –Arrival characteristics Poisson Distribution Other distributions –Behaviour of the Calling Population Reneges queue Baulks queue Patient caller

6. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Characteristics of a Queue The Service Facility – Physical Layout Service Facility Type I Service Facility Type 1 Service Facility Type 2 Single Channel, Single Server Single Channel, Multi Server

7. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 Characteristics of a Queue The Service Facility – Physical Layout Service Facility Type I Multi Channel Single Server

8. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 Characteristics of a Queue The Service Facility – Physical Layout Service Facility Type 1 Service Facility Type 2 Service Facility Type 1 Service Facility Type 2 Multi Channel, Multi Server

9. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 Characteristics of a Queue The Service Facility – Queue Discipline –First Come First Served or First In First Out (FCFS or FIFO) –Last In First Out (LIFO) –Priority (PRI) Pre-emptive Priority Non pre-emptive –Service in Random Order (SIRO)

10. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 Characteristics of a Queue The Service Facility – Service Time –Exponentially distributed –Other distribution The Queue – Size –Finite –Infinite

11. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 Characteristics of a Queue Increased Service Costs Waiting Costs Cost of Facilities Total costs The aim is to reduce total cost Increased Service Costs Waiting Costs Cost of Facilities Total costs

12. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Single Channel Single Server Model M/M/1 Arrivals follows a Poisson distribution (M) Service times follow an exponential distribution (M) Single Channel Single Server (1) The queue discipline is FCFS – first come, first served (FCFS) The calling population is large enough to be considered infinite (∞) The length of the queue is also infinite (∞) Kendall - Lee’s notation : M/M/1: FCFS/∞/∞.

13. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 Single Channel Single Server Model M/M/1 Waiting Time in System = If arrival rate is A (λ) and service rate is S (μ), then Length in Queue (numbers) (time units)

14. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 M/M/1 - Example Interval between aircraft arrivals is 20 minutes i.e. 3 per hour Unloading time is 15 minutes per aircraft i.e. 4 aircraft per hour

15. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Aircraft are spending 1 hour on the ground instead of 15 minutes as planned If two unloading crews are used and the service rate doubled to 8 aircraft an hour, we get The aircraft will now be spending only 12 minutes on the ground and the planned tonnage can be delivered.

16. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Utilisation Factor The ratio is called the utilisation factor. It is also the probability that the system is busy. Probability that the system is busy Probability that the system is idle

17. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 Utilisation Factor The length of the queue increases sharply when the utilisation factor is more than 0.7. For practical purposes, a queue system should be so designed that its utilisation factor is around 0.7. 0 10 20 30 40 50 60 00.20.40.60.81 Utilisation Factor Length of Queue

18. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 Economic Aspect of Queuing A computer maintenance contract is to be signed by your company office. At an average three computers per month go off road due to various defects. The cost of a computer being unavailable is Rs 8000 per month. Alfa Computers have quoted at Rs 3000 per month, and can repair 5 computers per month Beta Bytes has quoted at Rs 5000 per month for the contract and can repair 6 computers per month at an average Who should get the contract?

19. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 M/M/1 - Example Numbers in system Alfa ComputersBeta Bytes (a)Arrival rate of computers for repairs (A) 3 per month (b)Service Rate (S)5 per month6 per month (c) (d)Cost of off road computers (e)Cost of Contract30005000 (f)Total cost1500013000