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Queuing theory
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- 2. Queuing Theory Queuingtheory 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 QueuingTheory 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 processesin 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. System Customers Server Receptiondesk People Receptionist Hospital Patients Nurses Airport Airplanes Runway Road network Cars Traffic light Shoppers CheckoutGrocery station Computer Jobs CPU, disk, CD Queuing examples
- 7. Components of aQueuing System Queue or Waiting Line Arrival Process Service Process Servers Exit
- 8. Parts of aWaiting Line Dave’s Car Wash enter exit Population of dirty cars Arrivals from the general population … Queue (waiting line) Service facility Exit the system Exit the systemArrivals to the system In the system 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
- 9. 1. Arrival Process According tosource According to numbers According to time • 2. Queue Structure First-come-first-served (FCFS) (LCFS) (SIRO) • • • Last-come-first-serve Service-in-random-order Priority service
- 10. 3. Service system 1. Asingle service system. Queue Arrivals Service facility Departures after service e.g- Your family dentist’s office, Library counter
- 11. 2. Multiple, parallel server,single queue model Queue Service facility Channel 1 Service facility Channel 2 Service facility Channel 3 Arrivals Departures after service e.g- Booking at a service station
- 12. 3. Multiple, parallel facilitieswith multiple queues ModelService station Customers leave Queues Arrivals e.g.- Different cash counters in electricity office
- 13. 4. Service facilities ina 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 Deterministicqueuing model :-- = Mean number of arrivals per time 1. Deterministic queuing model 2. Probabilistic queuing model • 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 Probabilitythat n customers will arrive in the system in time interval T is n! tn et Ptn
- 17. Single Channel Model Mean numberof arrivals per time = period served) = = Average time a unit spends in the system (waiting time plus service time) µ = Mean number of units served per time period Ls = Average number of units = (µcu–stomers)in the system (waiting and being W1 µ s –
- 18. p = Utilizationfactor for the system = Lq = Average number of units waiting in the queue 2 = µ(µ – ) Wq = Average time a unit spends waiting in the queue µ(µ –=) µ
- 19. P0 = Probabilityof 0 units in the system (that is, the service unit is idle) Pn > k system, where n is the number of units in the system =µ 1 – = Probability of more than k units in the µ = k + 1
- 20. Single Channel ModelExample = 2 carsarriving/hour µ = 3 cars serviced/hour sL = = = 2 cars in the system on average Ws = = = 1 hour average waiting time in =Lq = = 1.33 cars waiting in line 2 µ(µ – ) µ – 1 µ – 2 3 - 2 1 3 - 2 22 3(3 - 2) thesystem
- 21. Cont… = 2 carsarriving/hour, µ = 3 cars Wq = = = 40 minute average waiting time is busy serviced/hour µ(µ – ) 2 3(3 - 2) µ P0 p = /µ = 2/3 = = 1 - 66.6=%.o3f3tipipmroebmabeiclihihtyanic there are 0 cars in the system
- 22. Suggestions for Managing Queues 1.Determine an acceptable waiting time for your customers 2. Try to divert your customer’s attention when waiting 3. Inform your customers of what to expect 4. Keep employees not serving the customers out of sight 5. Segment customers
- 23. 1. Train yourservers to be friendly 2. Encourage customers to come during the slack periods 3. Take a long-term perspective toward getting rid of the queues
- 24. Where the TimeGoes 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