# Queuing

**Topics:**Queueing theory, Poisson distribution, Queueing model

**Pages:**9 (1500 words)

**Published:**December 10, 2012

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Queue Examples…

Service environments ִAirline check-in ִAmusement parks ִPhone-based customer service ִTraffic ִPatients in hospital, office, transplant list Manufacturing environments ִWIP ִPending orders ִMaintenance requests 2

Reasons for forming queues

Because of variability in demand ִunpredictable fluctuation between arrivals/orders ִHeterogeneity in type of demand Because of variability in capacity ִunexpected downtime ִinconsistency of the server Because in particular for a service-operation, ִwe don’t have the luxury of satisfying demand from inventory 3

Tradeoff…

Service Capacity Cost ִhiring servers ִtraining servers Waiting Cost ִcustomer dissatisfaction ִloss of potential customers ִcapital tied up in WIP ִstorage facility cost

Balance …

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Description of queues

server

Customer arrivals

queue

Customer departures

system

Priority rules: e.g., first-come-first-serve

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Types of Queuing Systems

Single stage system

multiple stage system

parallel single stage system

multi-channel single stage system

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Other Arrival Characteristics

Size of Arrival Units Degree of patience Balking Reneging Jockeying

Managerial Issues of Queuing Systems

System Design • how many servers • arrangement of queue • fast vs. slow server • size of waiting area

System Management/Operations • management of arrivals • customer perception of waiting times 9

Managerial Issues of Queuing Systems (cont’d) Performance Evaluations • Average number of customers in the queue: Nq • Average waiting time (time in the queue): Tq • Average number of customers in the system: N or Ns • Average time in the system: T or Ts • System capacity utilization: ρ • Probability of 0 customers in the system: P0 • Probability of k customers in the system: Pk • Probability of waiting time less than a specific amount 10

Performance Measures: How?

Analytic methods ִexist for less complicated systems ִalready established/easy to execute ִexact results Simulation ִpossible for any system ִrestrictive assumptions not necessary ִtakes longer to build/execute ִapproximate results

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Data Gathering

Arrival Pattern – estimated in two ways

Observe the time between successive arrivals to see if the times follow any statistical distribution ִ Exponentially distributed interarrival times Set some time length and try to determine how many arrivals enter the system within that time. ִ Number of arrivals is Poisson distributed

Service Times

Observe the time between successive departures after service to see if the times follow any statistical distribution ִ Exponentially distributed service times

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Exponential Distribution (also called negative exponential) Good approximation of totally random events: ִ the time until a next car accident in Vastrapur ִ the time until a radioactive particle decays ִ…

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Exponential, Poisson Distribution

Poisson and exponential distributions have oneto-one correspondence: ִ“ The number of customers can be served per unit of time is Poisson distributed” is equivalent to “The service time is exponential distributed”; ִ“ The number of arrivals per unit of time follows Poisson distribution” is equivalent to “ The inter-arrival time follows exponential distribution”.

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Performance Evaluation: notation and formulas M/M/1 queue: λ µ M/M/s queue: λ µ 1 s

. . .

µ

λ -- customer arrival rate (arrivals/unit time) µ -- service rate (customers/unit time) s -- number of servers ρ = λ/sµ -- system utilization Assumptions: 1. Time between two consecutive arrivals is exponentially distributed. 2. Service time is exponentially distributed 16

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A Fundamental Law of Queuing Systems

Service rate has to be strictly higher than customer arrival rate, or system utilization has to be strictly less than 1. Otherwise, average waiting time will approach to infinity.

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Little’s Law...

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