University of the Philippines – Diliman
Department of Industrial Engineering and Operations Research
IE 143 Long Quiz 4
Long Quiz 4 Review Problems
1.
MomandPop’s Grocery Store has a small adjacent parking lot with three parking spaces reserved for the
store’s customers. During store hours, cars enter the lot and use one of the spaces at a mean rate of 2 per
hour. For
n
= 0, 1, 2, 3, the probability
P
n
that exactly
n
spaces currently are being used is
P
0
= 0.2,
P
1
=
0.3,
P
2
= 0.3,
P
3
= 0.2.
a)
Determine the basic measures of performance – L, L
q
, W, and W
q
– for this queuing system. (+)
b)
Determine the average length of time that a car remains in a parking space. (+)
2.
You are given two queuing systems, Q
1
and Q
2
. The mean arrival rate, the mean service rate per busy
server, and the steadystate expected number of customers for Q
2
are twice the corresponding values for Q
1
.
Let W
i
= the steadystate expected waiting time in the system for Q
i
, for i = 1, 2. Determine W
2
/W
1
. (+)
3.
Consider a selfservice model in which the customer is also the server. Note that this corresponds to having
an infinite number of servers available. Customers arrive according to a Poisson process with parameter λ,
and service times have an exponential distribution with parameter μ.
a)
Find L
q
and W
q
. (+)
b)
Construct the rate diagram for this queuing system. (+)
c)
Use the balance equations to find the expression for P
n
in terms of P
0
. (++)
d)
Find P
0
. (++) {Use a familiar Taylor series expansion}
e)
Find L and W. (+++)
4.
Suppose that a singleserver queuing system fits all the assumptions of the birthanddeath process
except
that customers always arrive in
pairs
. The mean arrival rate is 2 pairs per hour (4 customers per hour) and
the mean service rate (when the server is busy) is 5 customers per hour.
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 Spring '11
 a
 Operations Research, Poisson Distribution, Probability theory, Exponential distribution, Poisson process, representative

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