Urban OR
Fall 2006
Problem Set 4
(Due: Wednesday, November 8, 2006)
Problem 1
[This is a review problem on M/G/1 queues. We have already done the first
four parts in class, as an example.]
Consider a singleserver queueing system with infinite queue capacity.
Arrivals of
customers to this system occur in a Poisson manner at the rate of 36 per hour.
The
service times,
S
, of customers are mutually independent and their duration is uniformly
distributed between 1 and 2 minutes, in other words,
f
S
(
t
)
=
U
[1, 2] in units of minutes.
(1) Compute the expected waiting time in queue,
W
q
, and the expected number of
customers in the queue,
L
q
, when this queueing system is in steady state.
[Numerical answers are expected.]
(2) Assume now that service times have a negative exponential probability density
2
function with
f
S
(
t
)
=
2
⋅
e
−
3
t
for
t
≥
0
, in units of minutes.
Repeat part (1), i.e.,
3
compute numerically the values of
W
q
and of
L
q
for this case.
(3) Assume now that service times are constant and all have duration equal to 1.5
minutes.
Repeat part (1), i.e., compute numerically the values of
W
q
and of
L
q
for
this case.
(4) Using the G/G/1 bounds discussed in class, compute numerically upper and lower
bounds for
W
q
and for
L
q
for the case of Part (1).
(5) Return to the original case (uniform service times) from here until the end of the
problem.
Suppose that immediately after the completion of a service, exactly 2
customers are left in the system.
Carefully write an expression for the probability
that after the end of the next service, exactly 4 customers will be left in the
system.
Do NOT evaluate numerically this expression.
(6) Repeat part (4) under the supposition that after the completion of a service,
exactly 0 customers are left in the system.
Carefully write an expression for the
probability that after the end of the next service, exactly 4 customers will be left
in the system.
Do NOT evaluate numerically this expression.
(7) Consider again part (5) where after the completion of a service, exactly 2
customers are left in the system.
Carefully write an expression for the probability
that TWO service completions later, exactly 4 customers will be left in the
system.
Do NOT evaluate numerically this expression.
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 Fall '06
 hansman
 Probability theory, Customers, class a

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