1. “ QUEUING THEORY” Presented By-- Anil Kumar Avtar Singh

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

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

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

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 ??