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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 1 CHAPTER QUEUING THEORY Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India 17 Learning Objectives • Characteristics of a queue. • Single Channel Single Server Queuing Model • Utilisation Factor • Economic Aspects of Queuing. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 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. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 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 Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Characteristics of a Queue • The Service Facility – Physical Layout Service Facility Type I Single Channel, Single Server Service Facility Type 1 Service Facility Type 2 Single Channel, Multi Server 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 I Multi Channel Single Server 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 1 Service Facility Type 2 Service Facility Type 1 Service Facility Type 2 Multi Channel, Multi Server Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 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) Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 Characteristics of a Queue • The Service Facility – Service Time – – Exponentially distributed Other distribution • The Queue – Size – – Finite Infinite Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 Characteristics of a Queue Total costs Costs Cost of Facilities Waiting Costs Increased Service The aim is to reduce total cost Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 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/∞/∞. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Single Channel Single Server Model M/M/1 • If arrival rate is A (λ) and service rate is S (μ), then 1 1 or ( ) (time units) Waiting Time in System = Ws SA Waiting time in queue Wq Length in service Length in Queue Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India A or ( ) (time units) S(S A) ( ) A Ls or ( ) SA (numbers) A2 2 Lq or ( ) (numbers) S(S A) ( ) Page 13 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 A3 S4 1 1 1 hour S A 43 A 3 Wq 45 minutes S(S A) 4( 4 3) A 3 Ls 3 aircraft S A 43 A2 3 3 Lq 2.25 aircraft S(S A) 4( 4 3) Ws Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 • 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 A3 S 8 The aircraft will 1 1 now be Ws 12 minutes S A 83 spending only A 3 12 minutes on Wq 4.5 minutes the ground and S(S A) 8(8 3) the planned A 3 Ls 0.6 aircraft tonnage can be S A 83 delivered. 2 A 3 3 Lq 0.225 aircraft S(S A) 8(8 3) Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Utilisation Factor A or S • The ratio is called the utilisation factor. • It is also the probability that the system is busy. A • Probability that the system is busy S A • Probability that the system is idle 1 1 1 S Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Utilisation Factor Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India 60 Length of Queue • 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. 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 Utilisation Factor Page 17 1 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? Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 M/M/1 - Example Alfa Computers Beta Bytes (a) Arrival rate of computers for 3 per month repairs (A) 3 per month (b) Service Rate (S) 6 per month (c) Numbers in system A Ls SA 5 per month 3 1.5 53 3 1 63 (d) Cost of off road computers 1.5 8000 12000 1 8000 8000 (e) Cost of Contract 3000 5000 (f) Total cost Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India 15000 13000 Page 19