Waiting Lines

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Waiting Lines

• Quantitative Module Part D

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Waiting Lines What and Why?

• A waiting line is one or more customers or
  items queued for operation, which can include
  people waiting for service, materials waiting for
  further processing, equipment waiting for
  maintenance, and sales order waiting for
  delivery.
• Forms because of temporary imbalance between the
  demand for service and the capacity of the system
  to provide service.

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Why is there waiting?

• Occurs naturally because of two reasons
• Customers arrive randomly, and not at evenly
  placed times nor at predetermined times.
• Service requirements of customers are variable,
  and not uniform. (Teller counter of a Bank).

Both Arrival Service times exhibit a high
degree of variability.
Leads to
Over-loaded Systems
Under-loaded Systems
Waiting Lines formation
No Waiting Line
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Goal of Waiting-Line Analysis

• Minimize Total Cost.
• Cost of customer waiting for service.
• Capacity cost to provide service.

Total cost
Customer waiting cost
Capacity cost


Total cost
Cost of service capacity
Cost
Cost of customers waiting
Optimum
Service capacity
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System Characteristics

• Population Source.
• Number of Servers.
• Arrival Pattern.
• Queue Discipline.
• Service Pattern.

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Population Source

• Finite-source
• Limited size of the customer pool.
• Entry / Exit by a member of this population pool
  will affect the probability of a customer
  requiring service.
• E.g. A machine in a company. The potential
  number of machines that might need repair at any
  one time cannot exceed the number of machines.
• Infinite-source
• Sufficiently large customer pool.
• Any change in population size caused by
  subtractions or additions to the population does
  not affect the system prob.
• E.g. 100 machines being maintained by one
  repairperson.
• A department store that has 10,000 customers.

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Number of Servers

• System-Capacity is a function of
• Server Capacity.
• Number of Servers in the system.

Single Channel, Single Phase
Single Channel, Multiple Phase
Multiple Channel, Single Phase
Multiple Channel, Multiple Phase
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Arrival Patterns

• Arrival rate
• The average number of customers or units per time
  period.
• Constant exactly the same time period between
  successive arrivals. E.g. machine controlled
  production process.
• Random (Variable) When arrivals are independent
  of each other and their occurrence cannot be
  predicted.
• This variability can be described by theoretical
  distributions.
• Most common for arrival rate is Poisson
  Distribution.

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Poisson Distribution
Probability
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10
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Distribution for ? 2
Distribution for ? 4
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Service Pattern

• Service Time
• The average time to process one customer.
• Constant exactly the same time to process each
  customer (order). E.g. automated car wash.
• Random (Variable) When exact service times
  cannot be predicted. Most require short
  processing times, but some could require
  relatively long service times.
• This variability can be described by theoretical
  distributions.
• Most common for arrival rate is Exponential
  Distribution.

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Exponential Distribution
Average service rate (µ ) 3 customers /hr.
Probability that service time t
Average service rate (µ ) 1 customer /hr.
Time t in hours
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Measures of System Performance

• Average time that each customer or object spends
  in the queue.
• Average queue length.
• Average time that each customer spends in the
  system (waiting time plus service time).
• Average number of customers in the system.
• Probability that the service facility will be
  idle.
• Utilization factor for the system.
• Probability of a specific number of customers in
  the system.

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Waiting Line Models

• Infinite-Source
• Single Channel, Exponential Service Time.
  Model 1.
• Single Channel, Constant Service Time.
  Model 2.
• Multiple Channel, Exponential Service Time.
  Model 3.
• Multiple Channel with priority service.
  Exponential service time. Model 4.
  (NOT COVERED IN THIS
  COURSE)
• Finite-Source

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Infinite-Source Symbols
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Basic Relationships
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Model 1 S.C. E.S.T.

• Is the Simplest model, which involves
• One server (single crew). Arrival rates are
  Poisson.
• First-come, first-served. Service times are
  Exponential.

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Example

• A phone company is planning to open a satellite
  store in a new shopping mall, staffed by one
  sales agent. It is estimated that requests for
  phones, accessories, and information will average
  15 per hour, and requests will have a Poisson
  distribution. Service times is assumed to be
  Exponentially distributed. Previous experience
  with similar satellite operations suggests that
  mean service time should average about three
  minutes per request. Determine each of the
  following
• System utilization.
• Percentage of time the sales agent will be idle.
• The expected number of customers waiting to be
  served.
• The average time customers will spend in the
  system.
• The probability of zero customers in the system
  and the probability of four customers in the
  system.

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Model 2 S.C. C.S.T.

• Exactly similar to Model 1, except that service
  time is not variable.
• Constant Service Time.
• Cuts the average number of customers waiting in
  line by half.
• All the formulas are the same as in Model 1,
  except

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Example

• Wandas Car Wash Dry is an automatic,
  five-minute operation with a single bay. On a
  typical Saturday morning, cars arrive at a mean
  rate of eight per hour, with arrivals tending to
  follow a Poisson distribution. Find
• The average number of cars in line.
• The average time cars spend in line and service.

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Model 3 M.C. E.S.T.

• 2 or more servers working independently to
  provide service.
• Poisson arrival rate and Exponential service
  time.
• All servers work at the same average rate.
• Customers form a single waiting line (FCFS).

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Can also use Table 19-4 on page 787-788.
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Example

• Alpha Taxi and Hauling Company has seven cabs
  stationed at the airport. The company has
  determined that during the late-evening hours on
  weeknights, customers request cabs at a rate that
  follows the Poisson distribution with a mean of
  6.6 per hour. Service time is exponential with a
  mean of 50 minutes per customer. Assume that
  there is one customer per cab and that each taxi
  returns to the airport after dropping off the
  passenger. Find
• Average number of customers waiting in line.
• Probability of zero customers in the system.
• Probability of 3 customers and 10 customers in
  the system.
• Average waiting time for an arrival not
  immediately served.
• Probability that an arrival will have to wait for
  service.
• System utilization.

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Example

• Trucks arrive at a warehouse at an average rate
  of 15 per hour during business hours. Crews can
  unload the trucks at an average rate of five per
  hour. (Both distributions are Poisson). The high
  unloading rate is due to cargo being put into
  containers. Recent changes in wage rates have
  caused the warehouse manager to re-examine the
  question of how many crews to use. The new rates
  are crew and dock cost 100 per hour truck and
  driver cost 120 per hour.

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Examples

• Repair calls for Xerox copiers in a small city
  are handled by one repairman. Repair time,
  including travel time, is exponentially
  distributed, with a mean of two hours per call.
  Requests for copier come in at a mean rate of
  three per eight-hour day (assume Poisson). Assume
  infinite source. Determine
• The average number of copiers awaiting repairs.
• System utilization.
• The amount of time during an eight-hour day that
  the repairman is not out on a call.
• The probability of two or more copiers in the
  system (waiting or being repaired).

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Examples

• A vending machine dispenses hot chocolate or
  coffee. Serving time is 30 seconds per cup and is
  constant. Customers arrive at a mean rate of 80
  per hour, and this rate is Poisson-distributed.
  Assume that each customer buys only one cup.
  Determine
• The average number of customers waiting in line.
• The average time customers spend in the system.
• The average number of customers in the system.

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Examples

• Many of a banks customers use its automated
  teller machine (ATM) to transact business. During
  the early evening hours in the summer months,
  customers arrive at the ATM at the rate of one
  every other minute. This can be modeled using a
  Poisson distribution. Each customer spends an
  average of 90 seconds completing his or her
  transactions. Transaction time is exponentially
  distributed. Determine
• The average time customers spend at the machine,
  including waiting in line and completing
  transactions.
• The probability that a customer will not have to
  wait upon arrival at the ATM.
• Utilization of the ATM.

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Examples

• A small town with one hospital has two ambulances
  to supply ambulance service. Requests for
  ambulances during weekdays mornings average 0.8
  per hour and tend to be Poisson-distributed.
  Travel and loading/unloading time averages one
  hour per call and follows an exponential
  distribution. Find
• System utilization.
• The average number of customers waiting.
• The average time customers wait for an ambulance.
• The probability that both ambulances will be busy
  when a call comes in.

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Examples

• The manager of a regional warehouse must decide
  on the number of loading docks to request for a
  new facility in order to minimize the sum of
  dock-crew and driver-truck costs. The manager has
  learned that each driver-truck combination
  represents a cost of 300 per day and that each
  dock plus loading crew represents a cost of
  1,100 per day.
• How many docks should be requested if trucks
  arrive at the rate of four per day, each dock can
  handle five trucks per day, and both rates are
  Poisson?
• An employee has proposed adding new equipment
  that would speed up the loading rate to 5.71
  trucks per day. The equipment would cost 100 per
  day for each dock. Should the manager invest in
  the new equipment?

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Additional Examples

• Trucks are required to pass through a weighing
  station so that they can be checked for weight
  violations. Trucks arrive at the station at the
  rate of 40 an hour between 7 p.m. and 9 p.m.
  according to Poisson distribution. Currently two
  inspectors are on duty during those hours, each
  of whom can inspect 25 trucks an hour. Assume
  service times to be exponentially distributed.
• How many trucks would you expect to see at the
  weighing station, including those being
  inspected?
• If a truck were just arriving at the station,
  about how many minutes could the driver expect to
  wait?
• How many minutes, on average, would a truck that
  is not immediately inspected have to wait?
• What is the probability that both inspectors
  would be busy at the same time?
• What condition would exist if there were only one
  inspector?

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• The parts department of a large automobile
  dealership has a counter used exclusively for
  mechanics requests for parts. The time between
  requests can be modeled by an Exponential
  distribution that has a mean of five minutes. A
  clerk can handle requests at a rate of 15 per
  hour, and this can be modeled by a Poisson
  distribution. Suppose there are two clerks at the
  counter.
• On average, how many mechanics would be at the
  counter, including those being served?
• If a mechanic has to wait, how long would the
  average wait be?
• What is the probability that a mechanic would
  have to wait for service?
• What percentage of time is a clerk idle?
• If clerks represent a cost of 20 per hour and
  mechanics a cost of 30 per hour, what number of
  clerks would be optimal in terms of minimizing
  total cost?

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• Trucks arrive at the loading dock of a wholesale
  grocer at the rate of 1.2 per hour in the
  mornings. A single crew consisting of two workers
  can load a truck in about 30 minutes. Crew
  members receiver 10 per hour in wages and fringe
  benefits, and trucks and drivers reflect an
  hourly cost of 60. The manager is thinking of
  adding another member to the crew. The service
  rate would then be 2.4 trucks per hour. Assume
  rates are Poisson.
• Would the third crew member be economical?
• Would a fourth member be justifiable if the
  resulting service capacity were 2.6 trucks per
  hour?