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Structure of a Waiting Line System

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Structure of a Waiting Line System. Queuing theory is the study of waiting lines. Four characteristics of a queuing system: The manner in which customers arrive (bulk ... – PowerPoint PPT presentation

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Title: Structure of a Waiting Line System


1
Structure of a Waiting Line System
  • Queuing theory is the study of waiting lines
  • Four characteristics of a queuing system
  • The manner in which customers arrive (bulk,
    individual, by appointment or randomly)
  • The time required for service
  • The priority for determining order of service
  • The number and configuration of servers in the
    system

2
Structure of a Waiting Line System
  • In general, arrival of customers into a system is
    a random event. Frequently the arrival pattern
    is modeled as a Poisson process.
  • Service time is also usually a random variable.
    It is often described by the exponential
    distribution.
  • The most common queue discipline is FIFO. An
    elevator however would use LIFO and an emergency
    room would use a highest cost criterion.

3
Queuing Systems
  • A three part code A/B/s is used to describe
    various queuing systems where
  • A describes the arrival distribution assumed
  • B describes the service distribution assumed
  • s is the number of channels (servers) per line

4
Operating Characteristics of a Queuing System
  • What is the average wait time of the customer in
    the queue? In the system?
  • What is the average number of customers in the
    queue? In the system?
  • What percent of time is the service facility
    idle? What is the average amount of server free
    time between customers?
  • What is the probability that an arriving customer
    will have to wait?

5
Example Wayne International Airport
  • Whenever an international plane arrives at the
    airport, two customs inspectors on duty set up
    operations to process the passengers.
  • Incoming passengers must first have their
    passports and visas checked. This is handled by
    one inspector. The time required to check a
    passenger's passports and visas can be described
    by the probability distribution on the next
    slide.

6
Example Wayne International Airport
  • Random Number Mapping
  • Time Required to
  • Check a Passenger's
    Random
  • Passport and Visa Probability
    Numbers
  • 20 seconds
    .20 00 -19
  • 40 seconds
    .40 20 - 59
  • 60 seconds
    .30 60 - 89
  • 80 seconds
    .10 90 - 99

7
Example Wayne International Airport
  • After having their passports and visas checked,
    the passengers next proceed to the second customs
    official who does baggage inspections.
    Passengers form a single waiting line with the
    official inspecting baggage on a first come,
    first served basis.

8
Example Wayne International Airport
  • Random Number Mapping
  • Time Required For
    Random
  • Baggage Inspection Probability
    Numbers
  • No Time
    .25 00 - 24
  • 1 minute .60
    25 - 84
  • 2 minutes .10
    85 - 94
  • 3 minutes .05
    95 - 99

9
Example Wayne International Airport
  • Next-Event Simulation Records
  • For each passenger the following information
    must be recorded
  • When his service begins at the passport control
    inspection
  • The length of time for this service
  • When his service begins at the baggage inspection
  • The length of time for this service

10
Example Wayne International Airport
  • A chartered plane from abroad lands at Wayne
    Airport with 80 passengers. Simulate the
    processing of the first 10 passengers through
    customs using the random numbers given on the
    next slides.

11
Example Wayne International Airport
  • Passport Control
    Baggage Inspections
  • Pass. Time Rand. Service Time Time
    Rand. Service Time
  • Num . Begin Num. Time End Begin
    Num. Time End
  • 1 000 93 120 120
    120 13 000 120
  • 2 120 63 100 220
    220 08 000 220
  • 3 220 26 40 300
    300 60 100 400
  • 4 300 16 20 320
    400 13 000 400
  • 5 320 21 40 400
    400 68 100 500

12
Example Wayne International Airport
  • Passport Control
    Baggage Inspections
  • Pass. Time Rand. Service Time Time
    Rand. Service Time
  • Num . Begin Num. Time End Begin
    Num. Time End
  • 6 400 26 40 440
    500 40 100 600
  • 7 440 70 100 540
    600 40 100 700
  • 8 540 55 40 620
    700 27 100 800
  • 9 620 72 100 720
    800 23 000 800
  • 10 720 89 100 820
    820 64 100 920

13
Example Wayne International Airport
  • Question
  • How long will it take for the first 10
    passengers to clear customs?
  • Answer
  • Passenger 10 clears customs after 9 minutes and
    20 seconds.

14
Example Wayne International Airport
  • Question
  • What is the average length of time a customer
    waits before having his bags inspected after he
    clears passport control? How is this estimate
    biased?
  • Answer
  • For each passenger calculate his waiting time
  • (Baggage Inspection Begins) - (Passport
    Control Ends) 000400202040400 160
    seconds.
  • 160/10 16 seconds per passenger.
  • This is a biased estimate because we assume
    that the simulation began with the system empty.
    Thus, the results tend to underestimate the
    average waiting time.
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