Queueing Theory

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Queueing Theory

• Professor Stephen Lawrence
• Leeds School of Business
• University of Colorado
• Boulder, CO 80309-0419

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Queuing Analysis
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Principal Queue Parameters

• Arrival Process
• Service
• Number of Servers
• Queue Discipline

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1. Arrival Process

• In what pattern do jobs / customers arrive to the
  queueing system?
• Distribution of arrival times?
• Batch arrivals?
• Finite population?
• Finite queue length?
• Poisson arrival process often assumed
• Many real-world arrival processes can be modeled
  using a Poisson process

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2. Service Process

• How long does it take to service a job or
  customer?
• Distribution of arrival times?
• Rework or repair?
• Service center (machine) breakdown?
• Exponential service times often assumed
• Works well for maintenance or unscheduled service
  situations

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3. Number of Servers

• How many servers are available?

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4. Queue Discipline

• How are jobs / customers selected from the queue
  for service?
• First Come First Served (FCFS)
• Shortest Processing Time (SPT)
• Earliest Due Date (EDD)
• Priority (jobs are in different priority classes)
• FCFS default assumption for most models

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Queue Nomenclature

• X / Y / k (Kendall notation)
• X distribution of arrivals (iid)
• Y distribution of service time (iid)
• M exponential (memoryless)
• Em Erlang (parameter m)
• G general
• D deterministic
• k number of servers

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The Poisson Distribution
The interarrival times the population of a
Poissonprocess are exponentially distributed
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Exponential Distribution

• Simplest distribution
• Single parameter (mean)
• Standard deviation fixed and equal to mean
• Lacks memory
• Remaining time exponentially distributed
  regardless of how much time has already passed
• Interarrival times of a Poisson process are
  exponential

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Exponential Distribution
Exponential Density
Mean m Std Dev s m
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M/M/1 Queues
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M/M/1 Assumptions

• Arrival rate of ??
• Poisson distribution
• Service rate of ?
• Exponential distribution
• Single server
• First-come-first-served (FCFS)
• Unlimited queue lengths allowed
• Infinite number of customers

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M/M/1 Operating Characteristics
Utilization (fraction of time server is busy)
Average waiting times
Average number waiting
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Example
Boulder Reservoir has one launching ramp for
small boats. On summer weekends, boats arrive for
launching at a mean rate of 6 boats per hour.
It takes an average of s6 minutes to launch a
boat. Boats are launched FCFS.
?? 6 /hr
?? 1/s 1/6
???????? 6/10
L ???????? 6/(10-6) Lq L? 1.5(0.6)
W 1/?????? 1/(10-6)
Wq W? 0.25(0.6)
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Example (cont.)
During the busy Fourth of July weekend, boats are
expected to arrive at an average rate of 9 per
hour.
?? 1/s 1/6
?? 9 /hr
???????? 9/10
L ???????? 9/(10-9) Lq L? 9(0.6)
W 1/?????? 1/(10-9) Wq W? 1(0.9)

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Another Example
The personal services officer of the Aspen
Investors Bank interviews all potential customers
to ascertain that they have sufficient net worth
to become clients. Potential customers arrive at
a rate of nine every 2 hours according to a
Poisson distribution, and the officer spends an
average of twelve minutes with each customer
reviewing their portfolio with an exponential
distribution. Determine the principal operating
characteristics for this system.
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Queue Simulation

• Averages are deceptive
• Simulation of M/M/1 queue shows the effect of
  variance
• Excel spreadsheet queue simulation
• Available on course website

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Managerial Implications

• Low utilization levels provide
• better service levels
• greater flexibility
• lower waiting costs (e.g., lost business)
• High utilization levels provide
• better equipment and employee utilization
• fewer idle periods
• lower production/service costs
• Must trade off benefits of high utilization
  levels with benefits of flexibility and service

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Flexibility/Utilization Trade-off
L Lq W Wq
?? 1.0
?? 0.0
Utilization ?
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Cost Trade-offs
Cost
Cost of Waiting
Cost of Service
? 1.0
?? 0.0
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G/G/k Queues
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G/G/k Assumptions

• General interarrival time distribution with mean
  a1/? and std. dev. sa
• General service time distribution with mean
  p1/? and std. dev. sp
• Multiple servers (k)
• First-come-first-served (FCFS)
• Infinite calling population
• Unlimited queue lengths allowed

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General Distributions

• Two parameters
• Mean (m)
• Std. dev. (s )
• Examples
• Normal
• Weibull
• LogNormal
• Gamma

Coefficient of Variation
cv s/m
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G/G/k Operating Characteristics
Average waiting times (approximate)
Average number in queue and in system
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Alternative G/G/k Formulation
Since 1/m p
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G/G/k Analyzed
Suppose m s (cv 1) and k 1 (M/M/1)
for both arrival service processes
M/M/1 result!
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G/G/k Analyzed

• Waiting time increase with square of arrival or
  service time variation
• Decrease as the inverse of the number of servers

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G/G/k Variance Analyzed

• Waiting times increase with the square of the
  coefficient of variance

• No variance, no wait!

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M/M/2 Example
The Boulder Parks staff is concerned about
congestion during the busy Fourth of July weekend
when boats are expected to arrive at an average
rate of 9 per hour and take 6 minutes per boat to
unload. Boulder is considering constructing a
second temporary ramp next to the first to
relieve congestion. What will be its effect?
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Another G/G/k Example

• Aspen Investors Bank wants to provide better
  service to its clients and is considering two
  alternatives
• Add a second personal services officer
• Install a computer system that will quickly
  provide client information and reduce service
  time variance (service time standard deviation
  cut in half).
• Recall that customers arrive at a rate of 4 per
  hour and are serviced at a rate of 5 per hour.

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Other Queueing Models
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Other Queueing Behavior
Queue (waiting line)
Customer Departures
Customer Arrivals
Server
Line too long?
Wait too long?
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Waiting Line Psychology

1. Waits with unoccupied time seem longer
2. Pre-process waits are longer than process
3. Anxiety makes waits seem longer
4. Uncertainty makes waits seem longer
5. Unexplained waits seem longer
6. Unfair waits seem longer than fair waits
7. Valuable service waits seem shorter
8. Solo waits seem longer than group waits

Maister, The Psychology of Waiting Lines,
teaching note, HBS 9-684-064.
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Queues and Simulation

• Only simple queues can be mathematically analyzed
• Real world queues are often very complex
• multiple servers, multiple queues
• balking, reneging, queue jumping
• machine breakdowns
• networks of queues, ...
• Need to analyze, complex or not
• Computer simulation !