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QUEUING THEORY

QUEUING THEORY. CHAPTER. 17. Learning Objectives. Characteristics of a queue. Single Channel Single Server Queuing Model Utilisation Factor Economic Aspects of Queuing. Queuing. Whenever any person or any thing has to wait for a service, there is economic loss due to the waiting time.

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QUEUING THEORY

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  1. QUEUING THEORY CHAPTER 17

  2. Learning Objectives • Characteristics of a queue. • Single Channel Single Server Queuing Model • Utilisation Factor • Economic Aspects of Queuing.

  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.

  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

  5. Characteristics of a Queue Service Facility Type I Service Facility Type 1 Service Facility Type 2 • The Service Facility – Physical Layout Single Channel, Single Server Single Channel, Multi Server

  6. Characteristics of a Queue Service Facility Type I Service Facility Type I • The Service Facility – Physical Layout Multi Channel Single Server

  7. Characteristics of a Queue Service Facility Type 1 Service Facility Type 2 Service Facility Type 1 Service Facility Type 2 • The Service Facility – Physical Layout Multi Channel, Multi Server

  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)

  9. Characteristics of a Queue • The Service Facility – Service Time • Exponentially distributed • Other distribution • The Queue – Size • Finite • Infinite

  10. Characteristics of a Queue Total costs Total costs Costs Costs Cost of Facilities Cost of Facilities Waiting Costs Waiting Costs Increased Service Increased Service The aim is to reduce total cost

  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/∞/∞.

  12. Single Channel Single Server Model M/M/1 • If arrival rate is A (λ)and service rate is S (μ), then (time units) Waiting Time in System = Length in Queue (numbers)

  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

  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 The aircraft will now be spending only 12 minutes on the ground and the planned tonnage can be delivered.

  15. 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

  16. Utilisation Factor 60 50 40 Length of Queue 30 20 10 0 0 0.2 0.4 0.6 0.8 1 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.

  17. 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?

  18. M/M/1 - Example Numbers in system

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