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QUEUING THEORY

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  • 1. “ QUEUING THEORY” Presented By-- Anil Kumar Avtar Singh
  • 2. Queuing Theory
    • Queuing theory is the mathematics of waiting lines.
    • It is extremely useful in predicting and evaluating
    • system performance.
    • Queuing theory has been used for operations
    • research, manufacturing and systems analysis.
    • Traditional queuing theory problems refer to
    • customers visiting a store, analogous to requests
    • arriving at a device.
  • 3. Applications of Queuing Theory
    • Telecommunications
    • Traffic control
    • Determining the sequence of computer
    • operations
    • Predicting computer performance
    • Health services (e.g.. control of hospital bed
    • assignments)
    • Airport traffic, airline ticket sales
    • Layout of manufacturing systems.
  • 4. Queuing System
    • Model processes in which customers arrive.
    • Wait their turn for service.
    • Are serviced and then leave.
    input Server Queue output
  • 5. Characteristics of Queuing Systems
    • Key elements of queuing systems
    • • Customer:-- refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, emails.
    • • Server:-- refers to any resource that provides the requested service, eg. repairpersons, retrieval machines, runways at airport.
  • 6. Queuing examples System Customers Server Reception desk People Receptionist Hospital Patients Nurses Airport Airplanes Runway Road network Cars Traffic light Grocery Shoppers Checkout station Computer Jobs CPU, disk, CD
  • 7. Components of a Queuing System Arrival Process Servers Queue or Waiting Line Service Process Exit
  • 8. Parts of a Waiting Line Population of dirty cars
    • Arrival Characteristics
    • Size of the population
    • Behavior of arrivals
    • Statistical distribution of arrivals
    • Waiting Line Characteristics
    • Limited vs. unlimited
    • Queue discipline
    • Service Characteristics
    • Service design
    • Statistical distribution of service
    Dave’s Car Wash enter exit Arrivals from the general population … Queue (waiting line) Service facility Exit the system Exit the system Arrivals to the system In the system
  • 9. 1. Arrival Process
    • According to source
    • According to numbers
    • According to time
    2. Queue Structure
    • First-come-first-served (FCFS)
    • Last-come-first-serve (LCFS)
    • Service-in-random-order (SIRO)
    • Priority service
  • 10. 3. Service system
    • 1. A single service system.
    e.g- Your family dentist’s office, Library counter Queue Arrivals Service facility Departures after service
  • 11. 2. Multiple, parallel server, single queue model e.g- Booking at a service station Queue Service facility Channel 1 Service facility Channel 2 Service facility Channel 3 Arrivals Departures after service
  • 12. 3. Multiple, parallel facilities with multiple queues Model Service station Customers leave Queues Arrivals e.g.- Different cash counters in electricity office
  • 13. 4. Service facilities in a series Arrivals Queues Service station 1 Service station 2 Queues Customers leave Phase 1 Phase 2 e.g.- Cutting, turning, knurling, drilling, grinding, packaging operation of steel
  • 14. Queuing Models
    • Deterministic queuing model
    • Probabilistic queuing model
    • Deterministic queuing model :--
    •  = Mean number of arrivals per time
    • period
    • µ = Mean number of units served per
    • time period
  • 15. Assumptions
    • If  > µ, then waiting line shall be formed and increased indefinitely and service system would fail ultimately
    • 2. If  µ, there shall be no waiting line
  • 16. 2.Probabilistic queuing model Probability that n customers will arrive in the system in time interval T is
  • 17. Single Channel Model  = Mean number of arrivals per time period µ = Mean number of units served per time period L s = Average number of units (customers) in the system (waiting and being served) = W s = Average time a unit spends in the system (waiting time plus service time) =  µ –  1 µ – 
  • 18. L q = Average number of units waiting in the queue = W q = Average time a unit spends waiting in the queue = p = Utilization factor for the system =  2 µ(µ –  )  µ(µ –  )  µ
  • 19. P 0 = Probability of 0 units in the system (that is, the service unit is idle) = 1 – P n > k = Probability of more than k units in the system, where n is the number of units in the system =  µ  µ k + 1
  • 20. Single Channel Model Example  = 2 cars arriving/hour µ = 3 cars serviced/hour L s = = = 2 cars in the system on average W s = = = 1 hour average waiting time in the system L q = = = 1.33 cars waiting in line  2 µ(µ –  )  µ –  1 µ –  2 3 - 2 1 3 - 2 2 2 3(3 - 2)
  • 21. Cont…  = 2 cars arriving/hour, µ = 3 cars serviced/hour W q = = = 40 minute average waiting time p =  /µ = 2/3 = 66.6% of time mechanic is busy  µ(µ –  ) 2 3(3 - 2)  µ P 0 = 1 - = .33 probability there are 0 cars in the system
  • 22. Suggestions for Managing Queues
    • Determine an acceptable waiting time for your customers
    • Try to divert your customer’s attention when waiting
    • Inform your customers of what to expect
    • Keep employees not serving the customers out of sight
    • Segment customers
  • 23.
    • Train your servers to be friendly
    • Encourage customers to come during the slack periods
    • Take a long-term perspective toward getting rid of the queues
  • 24. Where the Time Goes In a life time, the average person will spend : SIX MONTHS Waiting at stoplights EIGHT MONTHS Opening junk mail ONE YEAR Looking for misplaced 0bjects TWO YEARS Reading E-mail FOUR YEARS Doing housework FIVE YEARS Waiting in line SIX YEARS Eating
  • 25. ANY QUESTIONS PLEASE ??