“QUEUING THEORY”

1. “QUEUING THEORY”

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 output Queue

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. Service Process Servers Queue or Waiting Line Exit Arrival Process Components of a Queuing System

8. Arrivals from the general population … Queue (waiting line) Service facility Exit the system Dave’s Car Wash enter exit Arrivals to the system In the system Exit the system Parts of a Waiting Line Population of dirty cars • Waiting Line Characteristics • Limited vs. unlimited • Queue discipline • Service Characteristics • Service design • Statistical distribution of service • Arrival Characteristics • Size of the population • Behavior of arrivals • Statistical distribution of arrivals

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. Queue Departures after service Servicefacility Arrivals 3. Service system 1. A single service system. e.g- Your family dentist’s office, Library counter

11. Service facility Channel 1 Service facility Channel 2 Departures after service Queue Arrivals Service facility Channel 3 2. Multiple, parallel server, single queue model e.g- Booking at a service station

12. 3. Multiple, parallel facilities with multiple queues Model Customers leave Service station Queues Arrivals e.g.- Different cash counters in electricity office

13. 4. Service facilities in a series Service station 1 Service station 2 Arrivals Phase 1 Phase 2 Queues Queues Customers leave 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.  = Mean number of arrivals per time period µ = Mean number of units served per time period Ls = Average number of units (customers) in the system (waiting and being served) = Ws = Average time a unit spends in the system (waiting time plus service time) =  µ –  1 µ –  Single Channel Model

18. Lq = Average number of units waiting in the queue = Wq= Average time a unit spends waiting in the queue = p = Utilization factor for the system = 2 µ(µ – )  µ(µ – )  µ

19. P0 = Probability of 0 units in the system (that is, the service unit is idle) = 1 – Pn > k = Probability of more than k units in the system, where n is the number of units in the system = k + 1  µ  µ

20.  = 2 cars arriving/hour µ = 3 cars serviced/hour Ls= = = 2 cars in the system on average Ws = = = 1 hour average waiting time in the system Lq= = = 1.33 cars waiting in line 2 3 - 2  µ –  1 3 - 2 1 µ –  22 3(3 - 2) 2 µ(µ – ) Single Channel Model Example

21. Wq= = = 40 minute average waiting time p = /µ = 2/3 = 66.6% of time mechanic is busy P0= 1 - = .33 probability there are 0 cars in the system 2 3(3 - 2)  µ(µ – )  µ Cont…  = 2 cars arriving/hour, µ = 3 cars serviced/hour

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 ??