queuing / waiting line - catÓlica-lisbon · mbacatólica 13 operations management pedro oliveira...
TRANSCRIPT
MBACat
ólic
a
1
Operations Management Pedro OliveiraFrancisco Veloso
OperationsManagement
Queuing / Waiting line
MBACat
ólic
a
2
Operations Management Pedro OliveiraFrancisco Veloso
Queuing for Jobs... Steve JobsQueuing for Jobs... Steve Jobs
Moscone Convention Center, San FranciscoJan 9, 2007
MBACat
ólic
a
3
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
4
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
5
Operations Management Pedro OliveiraFrancisco Veloso
PrinciplesPrinciples ofof WaitingWaiting
• 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
MBACat
ólic
a
6
Operations Management Pedro OliveiraFrancisco Veloso
PrinciplesPrinciples ofof WaitingWaiting
• 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
MBACat
ólic
a
7
Operations Management Pedro OliveiraFrancisco Veloso
WaitingWaiting LineLine ExamplesExamplesM
BACat
ólic
a
8
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
9
Operations Management Pedro OliveiraFrancisco Veloso
WaitingWaiting LineLine CostsCostsM
BACat
ólic
a
10
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
11
Operations Management Pedro OliveiraFrancisco Veloso
ComponentsComponents ofof QueuingQueuing SystemSystemM
BACat
ólic
a
12
Operations Management Pedro OliveiraFrancisco Veloso
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)
MBACat
ólic
a
13
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
14
Operations Management Pedro OliveiraFrancisco Veloso
Three Parts of a Queuing System Three Parts of a Queuing System (e.g. Car(e.g. Car--Wash)Wash)
MBACat
ólic
a
15
Operations Management Pedro OliveiraFrancisco Veloso
SingleSingle--ChannelChannel, , SingleSingle--PhasePhase SystemSystemM
BACat
ólic
a
16
Operations Management Pedro OliveiraFrancisco Veloso
SingleSingle--ChannelChannel, , MultiMulti--PhasePhase SystemSystem
MBACat
ólic
a
17
Operations Management Pedro OliveiraFrancisco Veloso
MultiMulti--ChannelChannel, , SingleSingle PhasePhase SystemSystem PO1M
BACat
ólic
a
18
Operations Management Pedro OliveiraFrancisco Veloso
MultiMulti--ChannelChannel, , MultiMulti--PhasePhase SystemSystem
Slide 17
PO1 Pedro Oliveira; 20-01-2007
MBACat
ólic
a
19
Operations Management Pedro OliveiraFrancisco Veloso
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:
MBACat
ólic
a
20
Operations Management Pedro OliveiraFrancisco Veloso
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:
MBACat
ólic
a
21
Operations Management Pedro OliveiraFrancisco Veloso
NegativeNegative ExponentialExponential DistributionDistributionM
BACat
ólic
a
22
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
23
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
24
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
25
Operations Management Pedro OliveiraFrancisco Veloso
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.
MBACat
ólic
a
26
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
27
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
28
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
29
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
30
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
31
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
32
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
33
Operations Management Pedro OliveiraFrancisco Veloso
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.
MBACat
ólic
a
34
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
35
Operations Management Pedro OliveiraFrancisco Veloso
C)C) Model (M/D/1) EquationsModel (M/D/1) Equations
Average number of people or units waiting for service:
Average time a unit spends in the system:
Average time a person or unit spends in the queue
Average number of people or units in the system:
MBACat
ólic
a
36
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
37
Operations Management Pedro OliveiraFrancisco Veloso
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
MBACat
ólic
a
38
Operations Management Pedro OliveiraFrancisco Veloso
Service Factor:
Average number of people or units waiting for service:
Average time a person or unit spends in the queue
D)D) Model (Limited Population) EquationsModel (Limited Population) Equations
MBACat
ólic
a
39
Operations Management Pedro OliveiraFrancisco Veloso
D)D) Model (Limited Population) Equations Model (Limited Population) Equations --ContinuedContinued
Average number running
Average number being served:
Number in the population:
MBACat
ólic
a
40
Operations Management Pedro OliveiraFrancisco Veloso
D)D) Model (Limited Population) Equations Model (Limited Population) Equations --ContinuedContinued
• 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
MBACat
ólic
a
41
Operations Management Pedro OliveiraFrancisco Veloso
D)D) Model (Limited Population) Equations Model (Limited Population) Equations --ContinuedContinued
• 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