COMM 225: Production & Operations Management
TOPIC: WAITING LINE ANALYSIS – TUTORIAL QUESTIONS
Q1 (Ref: Q. 18-3 p702-703): Many of a bank’s
Q4 (Ref: Q. 18-8, p703-704): The parts department of
customers use its automated teller machine (ATM).
During the early evening hours in the summer months,
customers arrive at the ATM at the rate of one every
other minute (assume Poisson). Each customer spends
an average of 90 seconds completing his or her
transaction. Transaction times are exponentially
distributed. Assume that the length of queue is not a
constraint. Determine:
a large automobile dealership has a counter used
exclusively for their own service mechanics requesting
parts. The length of time between requests can be
modelled by an Exponential distribution that has a
mean of five minutes. A parts clerk can handle requests
at an average rate of 15 per hour, and this can be
modelled by a Poisson distribution. Suppose there are
two parts clerks at the counter.
a) The average time customers spend at the
machine, including waiting in line and
completing transactions.
b) The probability that a customer will not have
to wait upon arrival at the ATM.
c) Utilization of the ATM.
d) The probability that a customer waits four
minutes or more in the line.
a) On average, how many mechanics would be at
the counter, including those being served?
b) If a mechanic has to wait, how long would the
average wait be?
c) What is the probability that a mechanic would
have to wait for service?
d) What percentage of time is a clerk idle?
e) If clerks represent a cost of $20 per hour and
mechanics represent a cost of $30 per hour,
how many clerks would be optimal in terms of
minimizing total cost?
Q2 (Ref: Q. 18-4, p703): A small town with one
hospital has two ambulances. Requests for an
ambulance during weekday mornings average .8 per
hour and tend to be Poisson. Travel and
loading/unloading time averages one hour per call and
follows an Exponential distribution. Find:
a) Server utilization.
b) The average number of patients waiting.
c) The average time patients wait for an
ambulance.
d) The probability that both ambulances will be
busy when a call comes in.
Q3 (Ref: Q. 18-6, p 703): Trucks are required to check
in a weigh station (scale) so that they can be inspected
for weight violations. Trucks arrive at the station at the
rate of 40 an hour in the mornings according to Poisson
distribution. Currently two inspectors are on duty
during those hours, each of whom can inspect 25 trucks
an hour, according to Poisson distribution.
a) How many trucks would you expect to see at
the weigh station, including those being
inspected?
b) If a truck were just arriving at the station, how
many minutes could the driver expect to wait
before being inspected?
c) How many minutes, on average, would a truck
that is not immediately inspected have to wait
before being inspected?
d) What is the probability that both inspectors
would be busy at the same time (i.e., the
probability that an arrival will have to wait for
service)?
e) What condition would exist if there were only
one inspector?