Waiting Lines

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Waiting Lines

• Also Known as Queuing Theory

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Have you ever been to the grocery store and had
to wait in line? Or maybe you had to wait at the
bank for the next available teller. These are
examples of queuing problems. Other examples
include having machines to be repaired like at
the lawnmower shop, or planes waiting to take off
at the airport. Here we want to look at some of
the characteristics of the queuing problem and
look at how QM for Windows can help us solve the
problem.
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• When you think about the grocery store example
  you can see the two types of costs the store has
  to consider
• Cost of providing the service cashiers and
  service stations,
• Costs of customers waiting cost could be lost
  customers if they have to wait too long(, or in
  other examples, it could be that contractually
  you have some costs if folks have to wait too
  long.)
• These two types of cost typically work in
  opposite directions. As service and service
  costs increase, waiting costs typically decline.
  The firm has to balance these two costs.
• In general, the firm will want to minimize costs
  in total. No sense in adding to service if it
  costs more than decline in wait costs that would
  be eliminated.

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• Characteristics of a queuing system
• Arrivals to the system (called the calling
  population)
• The waiting line
• The service facility
• Lets look at each of these a little more.

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• When we look at arrivals to the system we keep
  track of
• Size of the calling population unlimited means
  there is no limit to who might show up or only a
  small fraction of a finite group would show up at
  any one time and limited means only a limited
  amount could show up.
• Pattern of arrivals when do people show up? It
  is frequently assumed people (or what ever is the
  subject) show up following a Poisson
  Distribution. We will see what this means on the
  next slide.
• Behavior of arrivals arrivals may be patient
  and stay until served, of they might balk and
  leave. We will work with the patient type here.

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The Poisson Distribution helps us answer the
question, what is the probability that X number
of subjects will show up in a given time
period? We have P(X) (e-? ?X)/X!, Where ? is
the average arrival rate per time period, e is
that number 2.7183 and X! means X factorial. e-?
is found in a table, in our book on page 711. As
an example, say ? 2. What is the probability
that three people will arrive in the same unit of
time (like in an hour)? P(3) (.1353)(2)3/3!
(.1353)(8)/(3)(2)(1) .1804. So we have about
an 18 chance that 3 show up during the same hour.
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In terms of the waiting line we will assume the
line can have unlimited length and subjects are
served FIFO first in first out. In terms of the
service facility we have to be familiar with the
terms channels and phases. Single phase means the
subject gets service from only one station and
then exits the system. Multiphase means more
than one station. Channels refers to how many
units for service there are at a given phase. A
single phase, multichannel system would be like
at the bank where customers wait in a single line
and meet the next available teller. Phase
teller, channel many tellers (assuming customer
only wants to deposit or withdraw funds.)
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The last point about the service facility is that
each customer may take a different time to be
served. It will be assumed that the distribution
pattern of completion times follows the negative
exponential probability distribution. Example
Single channel, single phase problem, with
Poisson arrivals and (negative) exponential
completion. Arnolds Muffler Shop Arnolds is a
one bay repair shop. The mechanic on duty can do
3 cars per hour, on average. Customers arrive,
on average, 2 per hour. The service cost is 7
per hour for the mechanic and the wait cost has
been determined to be 10 an hour in terms of
customer dissatisfaction and lost goodwill.
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The input info for QM for Windows for the Waiting
Lines module, M/M/1 exponential service times
model is Arrival rate 2 Service rate 3 Number
of servers 1 (this is single channel) server
cost /time 7 waiting cost /time 10. Notice on
the input screen one has the option of changing
the time period to many different time frames. I
have per hour as the option and thus the above is
for hours.
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In QM for Windows the output includes the
following Average server utilization .6667 Aver
age in the queue, or waiting line,
1.3333 Average in the system 2 Average time
in the queue .6667 (of an hour) Average time in
the system 1 (hour) Cost (labor
waitingwait cost) 20.3333 Cost (labor in
systemwait cost) 27 Lets explain each on next.
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Average server utilization 0.6667, means that on
average the system is being used 2/3rd of every
hour. So, the system is idle 1/3rd of the
time. Average in the queue, or waiting line,
1.3333, means that 1.3333 cars are waiting in the
queue each hour (on average.) Average in the
system 2, means that 2 cars each hour are either
in the queue or are being served. Average time
in the queue .6667 (of an hour), or a car spends
40 minutes in the queue. Average time in the
system 1 (hour), or a car spends 1 hour in the
queue or is being served.
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Cost (labor waitingwait cost) 20.3333,
means that if waiting cost is only based on
waiting time, then total cost in an hour is 7
(1.333310) 20.3333 Cost (labor in
systemwait cost) 27, means that if waiting
cost is based on waiting time and service time,
then total cost in an hour is 7 (210)
27. This is a per hour cost and if we have an
eight hour day we must multiply by 8 to get the
daily cost.
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Single Phase, Multichannel Problem
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Say we are back at Arnolds Muffler Shop, but
Arnold has added a second service bay. He hires
a second worker, who has the same completion rate
as the first, 3 per hour, and gets paid the same
amount, 7 per hour. If the arrival rate to the
whole company is still 2 per hour, here is what
we do in QM for Windows go to module Waiting
Lines, file new on a multichannel problem. At
data entry put in M/M/s Arrival
rate(lambda) 2 Service rate(mu) 3 Number of
servers 2 Server cost /time 7 Waiting cost
/time 10 The output would be
note the service rate and the service cost are
per worker and everything is per hour.
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Parameter Value Average server
utilization 0.3333333 Average number in the
queue(Lq) 8.333334E-02 Average number in the
system(Ls) 0.75 Average time in the
queue(Wq) 4.166667E-02 Average time in the
system(Ws) 0.375 Cost (Labor waitingwait
cost) 14.83333 Cost (Labor in systemwait
cost) 21.5 The interpretation is the same here
as we had with one channel. The total cost with
just waiting as part of waiting cost is 14.83.
Here that includes 14 for the two workers, so
waiting cost is only 83 cents per hour here. When
we had only one service channel the corresponding
cost was 20.3333, of which only 7 was for labor.
So having the second channel really lowered wait
costs.
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In QM for Windows, when you do a multichannel
problem with any number of servers you specify,
one of the output screens is the Cost vs. Servers
screen. Here it looks like Number of
servers Total cost based on waiting Total cost
based on system 1 20.33334 27 2
14.83333 21.5 3 21.09291 27.75958 4 28.
01014 34.67681 5 35.001 41.66767 So this
shows us costs for various numbers of channels.
Here We can see that two channels, based on the
other information in the problem, would be the
amount that would give us the lowest total cost
based on service and wait.
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• Example problem I worked out problem 14-12 page
  599
• In QM for Windows I would use module Waiting
  Lines, File new on Single channel. No cost is
  entered. Analysis is by the hour. 10 arrive per
  hour. Each is cleaned in 5 minutes, so 12 can be
  completed an hour, with 1 server.
• From waiting lines results of output gt 4.1667
• From waiting lines results of output gt 25
  minutes
• From waiting lines results of output gt 30
  minutes
• From waiting lines results of output gt .8333 or
  83 of time
• From table of probabilities of outputgt .1667