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					                         Lines and Waiting
     “Every day I get in the queue, to get on the bus
      that takes me to you. . .”
             Pete Townsend, Magic Bus




                                   Waiting and Service Quality
                                   A Quick Look at Queuing Theory
                                   Utilization versus Variability



ANDERSON Core OM Class
                              The Big Picture
         In the previous sessions, we studied process
          averages and did not worry about variability
           – We could pretend that cycle times were constant and fixed
         In practice, however, process variability does exist
          creating uncertainty and the need for risk
          management.
           – Even if we always have sufficient capacity, we may still end up
             with lines if we don’t have inventory.
         So, we need to buy extra capacity to keep waiting
          time down, particularly in services.
           – How big does this capacity hedge need to be?
         Queueing theory can help us with these questions.

ANDERSON Core OM Class
                         Waiting – Key to Service Quality

                     waiting – a critical component of the customer’s
                                   perception of service!

         Recall the definition of flow time:
         What causes waiting?
         • Insufficient Capacity;
         • Mismatch in timing of Capacity and Demand
         • “Lumpiness” in arrivals
         People/Hr




                                                              Capacity

                                                                    Avg.
                                                                   Arrivals

                                    Time
ANDERSON Core OM Class
                Aside: How to measure variability

       From statistics we know standard deviation is a
        measure of variability.
       A better measure is coefficient of variation (CV):

                                         Standard deviation
               Coefficient of variation =
                                             Mean

         – Using CV, we can compare the degree of variability of two variables
           with different means
         – If variable has an exponential distribution, CV=100%


           Note: coefficient of variation is a number without a unit

ANDERSON Core OM Class
                                   Queuing Terms
    queue: Line feeding a number of servers.

    server: Task or operation fed by a queue.

    channels (M): Number of servers connected to an individual queue.

    arrival rate (l): Mean number of arrivals per unit time (usually per hour or day). If
    utilization < 100%, it will equal the thruput (flowrate, R).

    interarrival time (IAT): Time between arrivals. CVIAT is its coefficient of variation; a high
    value indicates that the arrival rate is “lumpy.”

    service rate (m = C1 server =1/CT1 server): Mean number of arrivals serviced by each
    individual server per unit time at 100% utilization

    Service time (CT1 server =1/m 1 server): The time it takes one server to complete one
    customer’s service. Hence, the average service time will equal the activity time. . CVST is the
    CV of the service times, i.e. how variable they are.




ANDERSON Core OM Class
                         So how long do we wait?
         If we have u and M, we can use a queue chart or the following
          approximation for how many people on average are in line, but not yet
          being served (if u < 100%):

                                   u    CVIAT  CVST 
                                       2( M 1)
                                            2       2

                             Lq                     
                                  1 u        2      
          The formula above is often referred to as the Big Ugly Formula!

         Then from Little’s Law, we know that the average waiting time in line
          before being served is (if u < 100%):


                              L (in queue) Lq
                         Wq              
                                Thruput     l
                           Remember to watch your time units!


ANDERSON Core OM Class
                         A Fortunate Simplification
          Normally, though, mean arrival and service times are much easier to obtain
           than their standard deviations, so if we can’t determine them, we make the
           following approximations for services:
            – People often don’t coordinate their arrival times. This leads to a so called “Poisson”
              arrival process. This implies CVIAT = 100%.
            – For services, it is often a reasonable assumption that the service time’s standard
              deviation = its mean, i.e. CVST = 100%. This is often referred to as “exponentially
              distributed” service times. (Note, this is much more variable than a “normal”
              distribution, but for services this is often a reasonable assumption.)
            – We assume that there’s always enough space around for people to wait.
          Under these conditions, the Q formula simplifies

                               u 12  12  u 2( M 1)
                                   2( M 1)
                      Lq        2   1 u
                           1 u          

      Empirical studies have shown that these assumptions are reasonable
      approximations in many real-world service situations

ANDERSON Core OM Class
                           Consultant Example
         The CABS Group employs is a boutique consulting firm that creates
          integrated operations-marketing computer simulations for their clients. A
          new requests from a potential clients arrives on average every 5 days. The
          average engagement requires one consultant 18 days to complete. How
          many consultants should CABS employ to ensure that the waiting time for
          customers prior to commencement of a project is no more than 1 week?
          (Assume 20 working days/month and 4 weeks/month.)




ANDERSON Core OM Class
                         Consultant Work




ANDERSON Core OM Class
                         Consultant Work




ANDERSON Core OM Class
                                                   Tradeoffs
                                   Customer Flowtime vs Capacity Utilization
                       100

                                                                   Decreasing Process
                                                                   or Arrival Variability CV = 1
     Flowtime (mins)




                                                                                            ST

                                                                       CVST=1.5
                       50




                                                                                  CVST=0.3
                        0
                             0.8        0.85              0.9                 0.95                 1
                                                     Utilization (%)

    Rules of Thumb:                   CV(IAT)=.3     Poisson: CV(IAT)=1            CV(IAT)=1.5


               • Utilization ofarrivals . is a good average target for services for
               uncoordinated
                                85-90%


               • To reduce wait times, you variability. capacity (decreasing utilization),
               or reduce process or arrival
                                            must increase



ANDERSON Core OM Class
                         Takeaways




ANDERSON Core OM Class
                         Lines and Waiting
     “Every day I get in the queue, to get on the bus
      that takes me to you. . .”
             Pete Townsend, Magic Bus




                                   Waiting and Service Quality
                                   A Quick Look at Queuing Theory
                                   Utilization versus Variability
                                   Managing Perceptions...


ANDERSON Core OM Class

				
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