queuing / waiting line - catÓlica-lisbon · mbacatólica 13 operations management pedro oliveira...

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Operations Management Pedro OliveiraFrancisco Veloso

OperationsManagement

Queuing / Waiting line

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Queuing for Jobs... Steve JobsQueuing for Jobs... Steve Jobs

Moscone Convention Center, San FranciscoJan 9, 2007

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The Behavioral Aspects of Waiting LinesThe Behavioral Aspects of Waiting Lines

• People don't like to stand around doing nothing – Passengers at Houston airport complained about

the delay in getting their baggage, even though the average time never exceeded eight minutes (one minute walk to the baggage carousel and a seven minute wait)

– The baggage carousel was moved to a six minute walk away and complaints stopped

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PrinciplesPrinciples ofof WaitingWaiting

• Unoccupied time feels longer than occupied time

– Activities provided to fill time should offer benefitin themselves

– Activities should be related in some way to theservice

• Pre-process waits feel longer than in-process waits

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• Anxiety makes waits seem longer– The manager needs to consider rational and

irrational sources of customer anxiety

• Uncertain waits seem longer than known, finite waits

– Scheduled appointments and promised timesdefine an expectation that must be met

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• Unexplained waits seem longer than explained waits

• Unfair waits seem longer than equitable waits

• Solo waits seem longer than group waits

• The more valuable the service, the longer the customer will willingly wait

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WaitingWaiting LineLine ExamplesExamplesM

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WaitingWaiting LinesLines

• First studied by A. K. Erlang in 1913– Analyzed telephone facilities

• Body of knowledge called queuing theory– Queue is another name for waiting line

• Decision problem– Balance cost of providing good service with cost of

customers waiting

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WaitingWaiting LineLine CostsCostsM

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WaitingWaiting LineLine TerminologyTerminology

• Queue: Waiting line

• Arrival: one person, machine, part, etc. that arrives and demands service

• Queue discipline: Rules for determining the order that arrivals receive service

• Channel: Number of waiting lines

• Phase: Number of steps in service

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ComponentsComponents ofof QueuingQueuing SystemSystemM

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Elements of a Waiting LineElements of a Waiting Line• Calling population

• Source of customers• Infinite - large enough that one more customer can

always arrive to be served• Finite - countable number of potential customers

• Arrival rate (λ)• Frequency of customer arrivals at waiting line system• Typically follows Poisson distribution

• Service time• Often follows negative exponential distribution• Average service rate = µ

(λ must be less than µ or system never clears out)

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Elements of a Waiting LineElements of a Waiting Line• Queue

- Discipline•Order in which customers are served •First come, first served is most common

- Length can be infinite or finite•Infinite is most common•Finite is limited by some physical structure

• Channels = number of parallel servers

• Phases = number of sequential servers

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Three Parts of a Queuing System Three Parts of a Queuing System (e.g. Car(e.g. Car--Wash)Wash)

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SingleSingle--ChannelChannel, , SingleSingle--PhasePhase SystemSystemM

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SingleSingle--ChannelChannel, , MultiMulti--PhasePhase SystemSystem

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MultiMulti--ChannelChannel, , SingleSingle PhasePhase SystemSystem PO1M

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MultiMulti--ChannelChannel, , MultiMulti--PhasePhase SystemSystem

PO1 Pedro Oliveira; 20-01-2007

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PoissonPoisson DistributionDistribution

• Number of events that occur in an interval of time

– Example: Number of customers that arrive in 15 min.

• Mean = λ (e.g., 5/hr.)

• Probability:

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NegativeNegative ExponentialExponential DistributionDistribution

• Service time, & time between arrivals

– Example: Service time is 20 min.

• Mean service rate = µ

– e.g., customers/hr.

• Mean service time = 1/ µ

• Equation:

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NegativeNegative ExponentialExponential DistributionDistributionM

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Remember: Remember: λλ & & µµ Are RatesAre Rates

• λ = Mean number of arrivals per time period

– e.g., 3 units/hour

• µ = Mean number of people or items served per time period

– e.g., 4 units/hour

» 1/µ = 15 minutes/unit

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WaitingWaiting--Line Performance MeasuresLine Performance Measures(and notation)(and notation)

• Average number of customers in queue (queue length), Lq

• Average number in system (waiting and being served), Ls

• Average queue time, Wq• Average time in system, Ws

• Probability of idle service facility (zero customers in the system), P0

• Probability of k units in system, Pk• System utilization, ρ

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A) Simple (M/M/1)– Example: Information booth at mall

B) Multi-channel (M/M/S)– Example: Airline ticket counter

C) Constant Service (M/D/1)– Example: Automated car wash

D) Limited Population– Example: Department with only 7 drills

Queuing modelsQueuing models

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Assumptions of the Basic Simple Assumptions of the Basic Simple Queuing ModelQueuing Model

• Arrivals are served on a first come, first served basis• Arrivals are independent of preceding arrivals• Arrival rates are described by the Poisson probability

distribution, and customers come from a very large population;

• Service times vary from one customer to another, and are independent of one and other; the average service time is known;

• Service times are described by the negative exponential probability distribution;

• The service rate is greater than the arrival rate.

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A)A) Simple (M/M/1) Model CharacteristicsSimple (M/M/1) Model Characteristics

• Type: Single-channel, single-phase system• Input source: Infinite; no balks, no reneging• Arrival distribution: Poisson• Queue: Unlimited; single line• Queue discipline: FIFO (FCFS)• Service distribution: Negative exponential• Relationship: Independent service & arrival• Service rate > arrival rate

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A)A) Simple (M/M/1) Model EquationsSimple (M/M/1) Model Equations

Average number of units in queue

Average time in system

Average number of units in queue

Average time in queue

System utilization

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Probability of 0 units in system, i.e., system idle:

Probability of more than k units in system:

Where n is the number of units in the system

A)A) Simple (M/M/1) Probability EquationsSimple (M/M/1) Probability Equations

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B)B) MultichannelMultichannel (M/M/S) Model (M/M/S) Model CharacteristicsCharacteristics

• Type: Multichannel system• Input source: Infinite; no balks, no reneging• Arrival distribution: Poisson• Queue: Unlimited; multiple lines• Queue discipline: FIFO (FCFS)• Service distribution: Negative exponential• Relationship: Independent service & arrival• Σ Service rates > arrival rate

• Examples: bank tellers (with single line), etc

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B)B) Model (M/M/S) EquationsModel (M/M/S) Equations

Probability of zero people or units in the system:

Average number of people or units in the system:

Average time a unit spends in the system:

( ) ( ) µλ

λµµ

λλµ+

−−

⎟⎠⎞⎜

⎝⎛

= 02!1P

MML

M

s

M = number of channels open

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B)B) Model (M/M/S) EquationsModel (M/M/S) Equations

Average number of people or units waiting for service:

Average time a person or unit spends in the queue

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B)B) LLqq for Number of Service Channels (M) for Number of Service Channels (M) and and ρρ M=1 M=2 M=3 M=4

0,15 0,026 0,0010,2 0,05 0,002

0,25 0,083 0,0040,3 0,129 0,007

0,35 0,188 0,0110,4 0,267 0,017

0,45 0,368 0,024 0,0020,5 0,5 0,033 0,003

0,55 0,672 0,045 0,0040,6 0,9 0,059 0,006

0,65 1,207 0,077 0,0080,7 1,633 0,098 0,011

0,75 2,25 0,123 0,0150,8 3,2 0,152 0,019

0,85 4,817 0,187 0,024 0,0030,9 8,1 0,229 0,03 0,004

0,95 18,05 0,277 0,037 0,0051 0,333 0,045 0,007

ρ

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Example: Example: determingdeterming the number of the number of serversservers• Mechanics of BigNecks Auto that need parts for

auto repair present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40/hr and a clerk can fill request at the rate of 20/hr (exponential).

If the clerk costs $6/hr and the mechanic $12/hr, determine the optimum number of clerks to staff the counter.

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C)C) Constant Service Rate (M/D/1) ModelConstant Service Rate (M/D/1) ModelCharacteristicsCharacteristics• Type: Single-channel, single-phase system• Input source: Infinite; no balks, no reneging• Arrival distribution: Poisson• Queue: Unlimited; single line• Queue discipline: FIFO (FCFS)• Service distribution: Constant• Relationship: Independent service & arrival• Service rate > arrival rate

• Examples: automatic car wash, amusement park ride, etc

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C)C) Model (M/D/1) EquationsModel (M/D/1) Equations


Average time a unit spends in the system:


Average number of people or units in the system:

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D)D) Limited Population ModelLimited Population ModelCharacteristicsCharacteristics• Input source: Limited (countable; N = size);

no balks, no reneging• Dependent relationship between number of

units in the system and the arrival rate• Any number of servers can be considered• Queue: Unlimited; single line• Queue discipline: FIFO (FCFS)• Service distribution: Negative exponential

Example: maintenance for a fleet of 10 airplanes

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To use the tables follow four To use the tables follow four stepssteps1) Compute the average time a unit waits in line

X=T/(T+U)T = average service timeU = average time between unit service requirements

2) Find the values of X in the table and then find the line for M (# service channels)

3) Note the corresponding values for D and F D = probability that a unit will have to wait in the queueF = efficiency factor

4) Compute L, W, J, H L = average number of units waiting for serviceW = average time a unit waits in lineJ = average number of units not in the queue or service bayH = average number of units being serviced

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Service Factor:



D)D) Model (Limited Population) EquationsModel (Limited Population) Equations

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D)D) Model (Limited Population) Equations Model (Limited Population) Equations --ContinuedContinued

Average number running

Average number being served:

Number in the population:

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• Where:D = probability that a unit will have to wait in the queueF = efficiency factor

• H = average number of units being serviced• J = average number of units not in the queue

or service bay• L = average number of units waiting for

service

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• M = number of service channels• N = number of potential customers• T = average service time• U = average time between unit service

requirements• W = average time a unit waits in line

X = service factor

to be obtained from finite queuing tables

queuing / waiting line - catÓlica-lisbon · mbacatólica 13 operations management pedro oliveira...

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