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- 1. Queuing Theory AMIT SINGH DAHAL G5638545
- 2. Outline: Introduction Characteristics Configuration Software of Queuing System of Service System for Simulation Limitation of Queuing Theory
- 3. Introduction: a waiting line of "customers" requiring service from one or more servers Example: when the short term demand for service exceeds the capacity of facilities QT, mathematical study of waiting lines (or queues) enables mathematical analysis of several related processes, including arriving at the (back of the) queue, waiting in the queue, and being served by the Service Channels at the front of the queue -Waiting for service in bank and at reservation counter -Waiting for a train or bus -Waiting at doctor’s clinic
- 4. Characteristics of Queuing System: 1. The Arrival Pattern 2. The Service Mechanism 3. The Queue Discipline 4. Number of Service Channels
- 5. 1. The Arrival Pattern: Arrivals can be measured as the arrival rate or the inter-arrival time(time between arrivals) Inter-arrival time =1/ arrival rate (i) Balking: The customer may decide not to enter the queue upon arrival, because it is too long. (ii) Reneging: The customer may decide to leave the queue after waiting a certain time in it. (iii) Jockeying: If there are multiple queues in parallel the customers may switch between them
- 6. 2. Service Mechanism: Service pattern can be deterministic or stochastic Can also be batched or bulked services service rate may be state-dependent Note: difference in Arrival pattern and Service pattern because services cannot be initiated if the Queue is empty
- 7. 3. Queue Discipline Vital, the way of selecting customers for services Can be: i)First in First Out (FIFO) Ii)Last in First Out (LIFO) iii)Service in Random Order (SIRO) iv)Priority Schemes. Priority schemes are : -Preemptive: customer with higher priority displaces the customer with low priority and get services -Non-preemptive: customer with the higher priority wait for the current service to be completed before getting the service
- 8. 4. Number of Service Channels: Single Queue: -have only one queue to get different services -customer get services based on the availability of the server
- 9. 4. Number of Service Channels(Contd…) Parallel Queue -have multiple queues to get different services -the customer can change the queue, if the number of customer deceases in the other queue(Jockeying)
- 10. Configuration of Service System Single Server-Single Queue: -Example: student standing in the library queue
- 11. Parallel Server-Single Queue: -Example: Bank transaction, many token numbers are served one-byone according to available Server
- 12. Several Server-Several Queue -Example: different cash counters in electricity office
- 13. Services facilities in a series: -Example: may be product manufacturing that requires steps like cutting, drilling, grinding, operations, packaging
- 14. Software for Simulation: Model represented using Kendall- Lee Notation where, (a/b/c):( d/e/f) a = arrival rate distribution b = service rate distribution c = no. of parallel service channels (identical service) d = service discipline e = maximum no. allowed in the system f = calling source Common notation: M =Poissonl/Exponential rates G = General Distribution of Service Time Ek-=Erlangian Distribution
- 15. Information Flows: L λ μ n Lq Queuing Model W Wq ρ Pn
- 16. Notation: λc = effective mean arrival rate λ = λc if queue is infinite λe = λ - [expected number who balk if the queue is finite] W = expected waiting time of a customer in the system Wq = expected waiting time of a customer in the queue L = expected no. of customers in the system Lq = expected number of customers in the queue Po = probability of no customers in the system Pn = probability of n customers in the system ρ = traffic intensity= λ/nμ ρc= effective traffic intensity= λe/μ
- 17. MULTIPLE CHANNELPOISSON/EXPONENTIAL MODEL [(M/M/C):(GD/ α /α)] Characteristics: 1. Input population is infinite. 2. Arrival rate has a Poisson Distribution 3. There is more than one server. 4. Service time is exponentially distributed with mean1/μ. [λ<μ] 5. System capacity is infinite. 6. Balking and reneging are not allowed.
- 18. Queuing Theory Limitation Queuing models are quite complex and cannot be easily understood Form of theoretical distribution applicable to given queuing situation is not easily known If the queuing discipline is not FIFO based, then the study of queuing problem become more difficult
- 19. References: DEMO: https://www.stat.auckland.ac.nz/~stats255/qsim/qsim.html http://homepages.inf.ed.ac.uk/jeh/Simjava/queueing/mm1_q/mm1_q .html http://en.wikipedia.org/wiki/Queueing_theory Zukerman, Moshe: Introduction to Queueing Theory and Stochastic Teletraffic Models. http://homes.cs.washington.edu/~lazowska/qsp/ http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html http://www.eventhelix.com/realtimemantra/congestioncontrol/queue ing_theory.htm
- 20. THE END Any Questions???

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