Queuing Theory –Study of Congestion Operations Research
What is a queue?• People waiting for service – Customers at a supermarket (IVR, railway counter) – Letters in a post office (Emails, SMS) – Cars at a traffic signal• In ordered fashion (who defines order?) – Bank provides token numbers – Customers themselves ensure FIFO at Railway ticket counters
Where do we find queues?
A thought experiment• When does a queue form?• When will it not form?• When will you not join a queue?• When will you leave a queue?• What is the worst case scenario?
What do you observe near a queue?• Conflict• Congestion• Idle counters• Overworked counters• Smart people trying to circumvent the queue
What do we want to know?• How much time will it take?• How many counters should be there?• How to manage peak hour traffic?
Origins• Queuing Theory had its beginning in the research work of a Danish engineer named A. K. Erlang.• In 1909 Erlang experimented with fluctuating demand in telephone traffic.• Eight years later he published a report addressing the delays in automatic dialing equipment.• At the end of World War II, Erlang’s early work was extended to more general problems and to business applications of waiting lines.
Kendall Notation (a/b/c : d/e/f) (a/b/c : d/e/f) Arrival Size ofDistribution source Infinite/ M/D Service Time Maximum finite Distribution number of M/D customers in Number of Service system concurrent Discipline n servers FIFO/LIFO n / Priority/ Random
Identify the queuing systemRailway ticket counter (M/D/3:FIFO/200/∞)Bank Service CounterATMAirport – Check InAirport - SecurityTraffic SignalBus StopTrain Platform (Boarding)Paper Correction
Arrival modeled using Poisson Distribution
Some parametersArrival Rate λService Rate μNumber of customers in system LsNumber of customers in queue LqWaiting time in system WsWaiting time in queue WqUtilization ρ
Types of queuing systems
T1=3T2=7 Arrival Rate = N/Tt=6/19=.31T3=10 Mean Time in system = J/N = 38/6=6.3T4=6 Mean number in system = J/Tt=38/19=2T5=6 = (J/N)*(N/Tt)=6.3*.31=2T6=6 =λTqJ=38