# Lines and Waiting

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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

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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.

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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
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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

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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.

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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!

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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

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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.)

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Consultant Work

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Consultant Work

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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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 views: 5 posted: 7/1/2012 language: pages: 13