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Ch 17 - Waiting Line Analysis for Service Improvement

  1. Elements of Waiting Line Analysis       (p751)
    1. Arrival Characteristics
      1. Size of source population - infinite or finite
      2. Pattern of arrivals - usual assumption, arrivals satisfy Poisson distribution.
            Prob( n arrivals per unit of time ) = e-l ln / n!
            where l = average arrival rate per unit of time.
            To justify a Poisson distribution, click here and download my Poisson.xls.
      3. Behavior of arrivals
            Patient, balk (refuse to enter), renege (enter but leave before service).
    2. Waiting Line Characteristics
      1. Length of line - infinite or finite
      2. Discipline - first-in, first-out (FIFO)
    3. Service Characteristics
      1. Service time - constant or random.
            For random, usual assumption, exponential distribution.
            Prob( service time > t ) = e-µt
             where µ = average service rate per unit of time.
      2. Structures - see Fig 17.2, p755
            Single-server waiting line
            Multiple-server waiting line.
  2. Basic Single-Server Model (p759)
    1. Arrival Characteristics
      1. Size of source population – infinite
      2. Pattern of arrivals - Poisson distribution.
      3. Behavior of arrivals – patient
    2. Waiting Line Characteristics
      1. Length of line – infinite
      2. Discipline - first-in, first-out (FIFO)
    3. Service Characteristics
      1. Service time - random with exponential distribution.
      2. Structure - single server
    4. *Relationships (p760)
      1. L - average number of units in system (waiting and in service)
            L = l /( µ - l )
      2. W - average time in system (waiting and in service)
            W = 1 /( µ - l )
    5. *Example 17.1 - Auxiliary Student Bookstore, p761. What is the average time spent waiting and in service?
      1. l = average arrival rate of customers = 24/hr
      2. µ = average service rate with one employee = 30/hr
      3. W - average time in system (waiting and in service)
            W = 1 /( µ - l ) = 1 /(30-24) = 0.167 hr or 10 minutes
    6. *Service Improvement Analysis (p762) - Should the Student Bookstore employ another person to assist the present operator?
                   (This analysis is not explicitly in the text.)
      1. The two employees work together as one service team, with a service rate of 40/hr.
                They are each paid $100 for an 8 hour day.
      2. The bookstore values its customers time at twice their average annual salary, which is $50,000/year.
               Thus, the value of individual customer time is $50/hour for a 2,000 hour work year.
                Customer time cost = (number of customers/day) * (value of customer time) * (average time in system)
                            = (8*        l ) * ($50/hr) * W   =   9600 * W

          Number of  
          Employees

            µ   

          Average
          Time in
          System ( W ) 

          Customer 
          Time
          Cost

          Service

          Cost

          Total
          Cost
               1   30    0.1667    $1600   $100   $1700
               2   40    0.0625      $600   $200     $800
      3. The Bookstore should employ 2 people because the total cost is lower.

    Do assigned HW - Problem 17-13.

  3. Basic Multiple-Server Model      (p770)
    1. Arrival Characteristics - same as single-server
    2. Waiting Line Characteristics - same as single-server
    3. Service Characteristics
      1. Service Time - same as single-server
      2. Structure - multiple servers (p755)
    4. *Relationships - see page 770
    5. *Example 17.5 - Student Health Service Center waiting room, p772.  Should Center employ 3 or 4 nurses?
      1. l = average arrival rate of students = 10/hr
      2. µ = average service rate per nurse = 4/hr
      3. With 3 nurses, W - average time in system
               W = 0.60 hr or 36 minutes, see page 773.
      4. With 4 nurses, W = 0.30 hr or 18 minutes (p773).
               Note how adding one more nurse cuts W in half  (W is a very nonlinear function).
      5. Assumptions (not in text)
        1. The nurses are each paid $200 for an 8 hour day.
        2. The Health Center values customer (student) time spent waiting and in service at $25 per hour.
        3. Customer time cost  =  (number of customers/day) * (value of customer time) * (average time in system)
                              = (8*          l ) * ($25/hr) * W    =    80 * ( $25 ) * ( W )   =   2000 * W

          Number of
          Nurses

          Average Time
          in System ( W )

          Customer
          Time Cost

          Service
          Cost

          Total
          Cost
            3     .60     1,200     600   1,800
            4     .30        600     800   1,400
      6. The Health Center should employ 4 nurses because the total cost is lower.
                                     (This page was last edited on August 16, 2006 .)