DGD_Week6_QueueingTheory_Questions_[B]
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Industrial Engineering
Date
Jan 9, 2024
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ADM3301 – DGD Session 6 – Queueing Models
Question 1:
Students arrive at the Administrative Services Office at an average of one
every 15 minutes with a standard deviation of 5 minutes, and their requests
take on average 10 minutes with a standard deviation of 2 minutes to be
processed. The service counter is staffed by only one clerk, Judy Gumshoes,
who works eight hours per day.
a) What are the values for the arrival rate
, the service rate
, and
λ
μ
coefficients of variations (C
a
and C
s
)?
b) Over the course of a single day, for how many hours is Judy expected to be
idle?
c)
How long is the (waiting) line on average?
d) How much time, on average, does a student spend in the (waiting) line?
e) How much time, on average, does a student spend at the Administrative
Service Office?
Question 2:
Glen owns and manages a chili-dog and soft drink stand near the Simon Fraser
campus. While Glen can service 30 customers per hour on average (µ) he gets
only 20 customers per hour (
). Because Glen could wait on 50% more
customers than actually visit his stand, it does not make sense to him that he
should have any waiting lines. Service times and inter-arrival times follow
exponential distributions
(M/M/1 model).
Glen hires you to examine the situation and to determine some characteristics
of his queue. After looking into the problem, you find it follows the conditions
for a single-channel waiting line. What are your findings?
Question 3:
Sam Certo, a Nanaimo vet, is running a rabies vaccination clinic for dogs at the
local grade school. Sam can ‘shoot’ a dog every
three minutes
. It is estimated
that the dogs will arrive independently and randomly throughout the day at a
rate of one dog every
six minutes
according to a Poisson distribution. Also,
assume that Sam’s shooting times are exponentially distributed. Compute the
following:
a)
The proportion of the time that Sam is busy
b) The
probability
that Sam is idle.
c)
The average number of dogs waiting to be vaccinated
.
d) The average number of dogs being vaccinated and waiting to be
vaccinated.
e)
The average time a dog waits
before
getting vaccinated.
f)
The average amount of time a dog spends waiting in line and being
vaccinated.
Question 4:
Automobiles arrive at the drive-through window at the downtown Fort
McMurray post office at the rate of
four every 10 minutes
. The
average
service time is two minutes
. The Poisson distribution is appropriate for the
arrival rate and service times are exponentially distributed.
a)
What is the average number of cars waiting to receive service?
b)
What is the average number of cars in the system?
c)
What is the average time a car is in the queue?
d)
What is the average time a car is in the system?
e)
What
percentage
of the time is the postal clerk busy?
f)
What is the
probability
that there are no cars at the window?
g)
What is the
probability
that there are exactly two cars in the system?
h)
By how much would your answer to part (a) be reduced if a second
drive-through window, with its own server, were added?
Question 5:
Manitoba’s Stephen Allen Electronics Corporation retains a service crew to
repair machine breakdowns that occur on an average of
= 3 per 8-hour
workday (approximately Poisson in nature). The crew can service an average
of µ = 8 machines per workday (8-hour workday), with a repair time
distribution that resembles the exponential distribution.
a)
What is the utilization rate of this service system?
b)
How many machines are waiting to be serviced at any given time?
c)
What is the average downtime for a broken machine?
d)
What is the probability that more than one machine is in the system?
The probability that more than two are broken and waiting to be
repaired or being serviced? More than three? More than four?
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