QUEUING THEORY Queuing Theory n Queuing theory is

“QUEUING THEORY”

Queuing Theory n Queuing theory is the mathematics of waiting lines. n It is extremely useful in predicting and evaluating system performance. n 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.

Applications of Queuing Theory n n n n 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.

Queuing System n n n Model processes in which customers arrive. Wait their turn for service. Are serviced and then leave. input Server Queue output

Characteristics of Queuing Systems n 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.

Queuing examples System Customers Server Reception desk People Receptionist Hospital Patients Nurses Airport Airplanes Road network Cars Grocery station Shoppers Computer Jobs Runway Traffic light Checkout CPU, disk, CD

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

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

1. Arrival Process n n n 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

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

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

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

4. Service facilities in a series Service station 1 Service station 2 Arrivals Phase 1 Queues Phase 2 Queues e. g. - Cutting, turning, knurling, drilling, grinding, packaging operation of steel Customers leave

Queuing Models 2. Deterministic queuing model Probabilistic queuing model 1. Deterministic queuing model : -- 1. = µ = Mean number of arrivals per time period Mean number of units served per time period

Assumptions 1. 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

2. Probabilistic queuing model Probability that n customers will arrive in the system in time interval T is

Single Channel Model µ Ls = = = Mean number of arrivals per time period Mean number of units served per time period Average number of units (customers) in the system (waiting and being served) = µ– Ws = Average time a unit spends in the system (waiting time plus service time) = 1 µ–

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

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

Single Channel Model Example = 2 cars arriving/hour µ = 3 cars serviced/hour 2 Ls = = = 2 cars in the system on average 3 -2 µ– 1 1 Ws = = = 1 hour average waiting time in µ– 3 -2 the system 22 2 Lq = = = 1. 33 cars waiting in line 3(3 - 2) µ(µ – )

Cont… = serviced/hour 2 cars arriving/hour, µ = 3 cars 2 Wq = = = 40 minute average waiting 3(3 2) µ(µ – ) time p = /µ = 2/3 = 66. 6% of time mechanic is busy P 0 = 1 =. 33 probability there are 0 cars in the µ system

Suggestions for Managing Queues 1. 2. 3. 4. 5. 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

6. Train your servers to be friendly 7. Encourage customers to come during the slack periods 8. Take a long-term perspective toward getting rid of the queues

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 0 bjects TWO YEARS Reading E-mail FOUR YEARS Doing housework FIVE YEARS Waiting in line SIX YEARS Eating

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