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  1. 1. “ QUEUING THEORY” Presented By-- Anil Kumar Avtar Singh
  2. 2. Queuing Theory <ul><li>Queuing theory is the mathematics of waiting lines. </li></ul><ul><li>It is extremely useful in predicting and evaluating </li></ul><ul><li>system performance. </li></ul><ul><li>Queuing theory has been used for operations </li></ul><ul><li>research, manufacturing and systems analysis. </li></ul><ul><li>Traditional queuing theory problems refer to </li></ul><ul><li>customers visiting a store, analogous to requests </li></ul><ul><li>arriving at a device. </li></ul>
  3. 3. Applications of Queuing Theory <ul><li>Telecommunications </li></ul><ul><li>Traffic control </li></ul><ul><li>Determining the sequence of computer </li></ul><ul><li>operations </li></ul><ul><li>Predicting computer performance </li></ul><ul><li>Health services (e.g.. control of hospital bed </li></ul><ul><li>assignments) </li></ul><ul><li>Airport traffic, airline ticket sales </li></ul><ul><li>Layout of manufacturing systems. </li></ul>
  4. 4. Queuing System <ul><li>Model processes in which customers arrive. </li></ul><ul><li>Wait their turn for service. </li></ul><ul><li>Are serviced and then leave. </li></ul>input Server Queue output
  5. 5. Characteristics of Queuing Systems <ul><li>Key elements of queuing systems </li></ul><ul><li>• Customer:-- refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, emails. </li></ul><ul><li>• Server:-- refers to any resource that provides the requested service, eg. repairpersons, retrieval machines, runways at airport. </li></ul>
  6. 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. 7. Components of a Queuing System Arrival Process Servers Queue or Waiting Line Service Process Exit
  8. 8. Parts of a Waiting Line Population of dirty cars <ul><li>Arrival Characteristics </li></ul><ul><li>Size of the population </li></ul><ul><li>Behavior of arrivals </li></ul><ul><li>Statistical distribution of arrivals </li></ul><ul><li>Waiting Line Characteristics </li></ul><ul><li>Limited vs. unlimited </li></ul><ul><li>Queue discipline </li></ul><ul><li>Service Characteristics </li></ul><ul><li>Service design </li></ul><ul><li>Statistical distribution of service </li></ul>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. 9. 1. Arrival Process <ul><li>According to source </li></ul><ul><li>According to numbers </li></ul><ul><li>According to time </li></ul>2. Queue Structure <ul><li>First-come-first-served (FCFS) </li></ul><ul><li>Last-come-first-serve (LCFS) </li></ul><ul><li>Service-in-random-order (SIRO) </li></ul><ul><li>Priority service </li></ul>
  10. 10. 3. Service system <ul><li>1. A single service system. </li></ul>e.g- Your family dentist’s office, Library counter Queue Arrivals Service facility Departures after service
  11. 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. 12. 3. Multiple, parallel facilities with multiple queues Model Service station Customers leave Queues Arrivals e.g.- Different cash counters in electricity office
  13. 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. 14. Queuing Models <ul><li>Deterministic queuing model </li></ul><ul><li>Probabilistic queuing model </li></ul><ul><li>Deterministic queuing model :-- </li></ul><ul><li> = Mean number of arrivals per time </li></ul><ul><li>period </li></ul><ul><li>µ = Mean number of units served per </li></ul><ul><li>time period </li></ul>
  15. 15. Assumptions <ul><li>If  > µ, then waiting line shall be formed and increased indefinitely and service system would fail ultimately </li></ul><ul><li>2. If  µ, there shall be no waiting line </li></ul>
  16. 16. 2.Probabilistic queuing model Probability that n customers will arrive in the system in time interval T is
  17. 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. 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. 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. 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. 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. 22. Suggestions for Managing Queues <ul><li>Determine an acceptable waiting time for your customers </li></ul><ul><li>Try to divert your customer’s attention when waiting </li></ul><ul><li>Inform your customers of what to expect </li></ul><ul><li>Keep employees not serving the customers out of sight </li></ul><ul><li>Segment customers </li></ul>
  23. 23. <ul><li>Train your servers to be friendly </li></ul><ul><li>Encourage customers to come during the slack periods </li></ul><ul><li>Take a long-term perspective toward getting rid of the queues </li></ul>
  24. 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