Waiting Lines and Queuing

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Waiting Lines and Queuing
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Waiting lines or Queues

• Examples
• Airports Ticketing and security checks
• Toll booths
• Retail super stores
• Subway and Wendys in the Student Union at noon
• Data routers
• The goal is to offer service counters that
  provide the most optimal level of service
• Trade-off between the cost of providing good
  service and the cost of customer waiting time

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Characteristics of Waiting Lines

• Calling Population
• Unlimited (?- assumed in most queuing models)
• Limited
• (Customer/Product/Car) Arrival Characteristics
• Pattern of arrivals
• Random independent of one another
• Scheduled or completely controlled - an assembly
  line
• Predictable actually observed or theoretically
  calculated (?)
• Arrival rate distribution
• Poisson distribution (?)
• other
• Behavior of arrivals
• Join the queue, and wait till served (?)
• Rejected the line is full and no more arrival
  accepted
• Balk refuse to join the line
• Renege leave the line before served
• Jockey change the lines while waiting

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Poisson Distribution for Arrival Times

• P(X) Probability of X arrivals
• X Number of arrivals per unit of time
• Average arrival rate
• e 2.71828 (irrational number used in
  exponential functions)

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Waiting Line Characteristics

• Length of the queue
• limited
• Unlimited (?)
• Service priority/Queue discipline
• First come, first served (FIFO) (?)
• Last come, first served (LIFO) - Firewood
• Random order lottery
• Priority scheme emergency versus non-emergency
  service

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Service Facility Characteristics

• Number of channels
• Single
• Multiple
• Number of phases in service system
• Single
• Multiple
• Service time distribution
• Constant same amount of service time for each
  arrival
• Random different amount of service time for
  each arrival
• Unpredictable
• Predictable - Exponential distribution (negative)

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Basic Queuing System Configurations
Queue
Service facility
Single Channel, Single Phase
Service Facility
Queue
Facility 2
Facility 1
Single Channel, Multi-Phase
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Basic Queuing System Configurations
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Exponential Distribution for Service Time
Probability (for Intervals of 1 Minute)
Average Service Time of 20 Minutes
Average Service Time of 1 Hour
X
30 60 90 120 150 180
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Kendall Notation

• Symbols
• M Poisson distribution for number of arrivals
  or exponential service times
• D Constant (deterministic) rate
• G General (normal) distribution with mean and
  variance known
• Examples
• M/M/1 Single channel model with Poisson
  arrivals and exponential service times
• M/M/3 Three channel model with Poisson arrivals
  and exponential service times
• M/D/2 Two channel model with Poisson arrivals
  and constant service times
• M/G/4 Four channel model with Poisson arrivals
  and normally distributed service times

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Assumptions M/M/1 Model

• 1. Queue discipline or arrival served FIFO
• 2. No balking or reneging
• 3. Independent arrivals arrival rate does not
  change over time
• 4. Arrivals Poisson distributed
• 5. Service times negative exponential
• 6. Average service rate gt average arrival rate

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Performance Measures of Queuing Systems

• Average number of customers in the system
  (waiting to be served and being served)
• Average number of customers in the queue (waiting
  to be served)

• Average time each customer spends in the system
• Average time each customer spends in the queue

µ mean number of people/items served per time
period ? mean number of arrivals per time period
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Performance Measures of Queuing System

• Probability that the system will be busy (The
  utilization factor)
• Probability that the system will be idle

• Probability that there will a specific (n) number
  of customers in the system
• Probability that the (n) number of customers in
  the system will be greater than some other (k)
  number

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Arnolds Muffler Shop
Arnolds mechanic is able to install new mufflers
at an averagerate of 3 per hour, or about 1
every 20 minutes. Customers needing this service
arrive at the shop on the average of 2 perhour.
What type of queuing model is
this? Compute Average number of cars in the
system Average time a car spends in the
system Average number of cars in queue Average
time a car spends in the queue Probability the
mechanic will busy when a car arrives Probability
that zero cars are in the system
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Arnolds Muffler Shop
µ 3 cars serviced per hour ? 2 cars
arriving per hour Ls ?/(µ- ?) 3/(3-2) 2
cars in the system on the average Ws 1/(µ- ?)
1/(3-2) 1 hour an average cars spends in the
system Lq ?2/(µ (µ ?)) 22/(3(3-2)) 1.33
cars waiting on line on average Wq ?/(µ (µ
?)) 2/(3(3-2)) .667 hour average waiting
time PB ?/ µ 2/3 .667 probability mechanic
is busy P0 1 - ?/ µ .333 probability there
are 0 cars in the system
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Cost Tradeoff

• Total service cost
• Cost per channel per time period
• Labor cost per channel per time period
• Other (electricity, water, fuel, etc.) cost per
  channel per time period
• Total service cost Number of channels times
  cost per channel per time period
• Total waiting cost
• Based on the time in system
• Total waiting cost Average number of customers
  in system (Ls) per time period times cost of
  waiting per customer per time period
• Based on the time in queue
• Total waiting cost Average number of customers
  in queue (Lq) per time period times cost of
  waiting per customer per time period
• Total cost total service cost total waiting
  cost

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Queuing Costs and Service Levels
Total Expected Cost
Optimal Service Level
Cost of Providing Service
Cost of Operating Service Facility
Cost of Waiting Time
Service Level
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Three Rivers Shipping
Three Rivers runs a docking facility located on
the Ohio River near Pittsburgh. Approximately 5
ships arrive to unload theircargoes of steel and
ore every 12-hour shift. Each hour thata ship
sits idle in line waiting to be unloaded costs
the firm a great deal of money, about 1,000 per
hour. From experience,management estimates that
if one team of stevedores is on dutyto handle
the unloading work, each ship will wait an
average of 7 hours to be unloaded. If 2 teams
are working, the average waiting time drops to 4
hours for 3 teams, its 3 hours and for4
teams, only 2 hours. Stevedores teams cost
6,000 per shift. Three Rivers management wants
to find the optimal number of stevedore teams to
use to minimize costs.
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Three Rivers Shipping
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Arnolds Muffler Shop
Arnold estimates that the cost of customer
waiting time, in termsof customer
dissatisfaction and lost goodwill, is 10 per
hour oftime spent waiting in line. Arnolds
mechanic makes 7 per hour. What is the total
waiting cost? What is the total service
cost? What is the total cost of the system?
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Arnolds Muffler Shop
Arnold estimates that the cost of customer
waiting time, in termsof customer
dissatisfaction and lost goodwill, is 10 per
hour oftime spent waiting in line. Arnolds
mechanic makes 7 per hour. What is the total
waiting cost per day? 10 Lq 8 hours 10
1.33 8 106.67 What is the total service
cost per day? 7 of channels 8 hours 7
1 8 56 What is the total cost of the
system per day? Waiting Cost Service Cost
106.67 56 162.67
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Arnolds Muffler Shop
Arnold hears that there is a mechanic who can
install a mufflerat a rate of 4 per hour. The
mechanic will work for 9 per hour.Would it be
worthwhile to replace the current mechanic with
thenew one?
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Arnolds Muffler Shop
µ 4 cars serviced per hour ? 2 cars
arriving per hour Ls ?/(µ- ?) 4/(4-2) 1
cars in the system on the average Ws 1/(µ- ?)
1/(4-2) 1/2 hour an average cars spends in
the system Lq ?2/(µ (µ ?)) 22/(4(4-2))
1/2 cars waiting on line on average Wq ?/(µ (µ
?)) 2/(4(4-2)) 1/4 hour average waiting
time PB ?/ µ 2/4 .5 probability mechanic
is busy P0 1 - ?/ 4 .5 probability there are
0 cars in the system
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Arnolds Muffler Shop
Arnold estimates that the cost of customer
waiting time, in termsof customer
dissatisfaction and lost goodwill, is 10 per
hour oftime spent waiting in line. Arnolds
mechanic makes 7 per hour. What is the total
waiting cost per day? 10 Lq 8 hours 10
.5 8 40.00 What is the total service cost
per day? 9 of channels 8 hours 9 1
8 72 What is the total cost of the system per
day? Waiting Cost Service Cost 40.00 72
112.00