Queuing theory is the mathematics of waiting lines.
It is extremely useful in predicting and evaluating
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
Determining the sequence of computer
Predicting computer performance
Health services (e.g.. control of hospital bed
Airport traffic, airline ticket sales
Layout of manufacturing systems.
Model processes in which customers arrive.
Wait their turn for service.
Are serviced and then leave.
input Server Queue output
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.
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
Components of a Queuing System Arrival Process Servers Queue or Waiting Line Service Process Exit
Parts of a Waiting Line Population of dirty cars
Size of the population
Behavior of arrivals
Statistical distribution of arrivals
Waiting Line Characteristics
Limited vs. unlimited
Statistical distribution of service
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
1. Arrival Process
According to source
According to numbers
According to time
2. Queue Structure
3. Service system
1. A single service system.
e.g- Your family dentist’s office, Library counter Queue Arrivals Service facility Departures after service
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
3. Multiple, parallel facilities with multiple queues Model Service station Customers leave Queues Arrivals e.g.- Different cash counters in electricity office
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
Deterministic queuing model
Probabilistic queuing model
Deterministic queuing model :--
= Mean number of arrivals per time
µ = Mean number of units served per
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 = 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 –
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
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
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)
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
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
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
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