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Massachusetts Institute of Technology
Logistical and Transportation Planning Methods
1.203J/6.281J/15.073J/16.76J/ESD.216J
Quiz #2 OPEN BOOK
December 6, 2004
Please do Problems 1 and 2 in one booklet and Problem 3 in a
separate one. Remember to put your name on each booklet! And
please explain all of your work! Good luck!
1
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View Full DocumentProblem 1 (30 points): Queuing in Pairs
[Note: This problem is a variation of Problem # 4 in Quiz 1.
Several things have,
however, changed in this problem:
Type 2 customers are described differently; the
queueing capacity of the system is 2 instead of 0; and some of the questions are
different, as well.]
Consider a queueing system with two parallel servers and two spaces for queueing (in
addition to the two servers).
This facility serves two types of customers.
Type 1 customers are of the conventional
type. They arrive in a Poisson manner at a rate of
λ
1
per minute.
The service time to
these customers has a negative exponential pdf with a rate of
µ
1
per minute for each
server. Any arriving Type 1 customers who find the system full (i.e., 4 customers in the
system) are lost.
Type 2 customers are unusual.
They, too, arrive in a Poisson manner at a rate of
2
per
minute, but they arrive IN PAIRS.
[Think of a restaurant, where some of the customers
(Type 1) arrive individually, and others (Type 2) arrive in pairs.]
Moreover, when
service to each one of these pairs begins, the pair occupies simultaneously TWO servers
(i.e., both of the servers). The servers work together on each of these Type 2 pairs. The
service time to the pair has a negative exponential pdf with a rate of
2
per minute.
(Note
that this means that the two servers begin and end service to each Type 2 pair
simultaneously and, together, can serve
2
Type 2 pairs per minute if working
continuously on Type 2 pairs.) Type 2 customers who do not find at least TWO available
spaces upon arrival are lost.
[Please note: A Type 2 pair cannot occupy the servers, unless both servers are available.
Thus, in the case where one Type 1 customer is in service and the only customers waiting
are one Type 2 pair, the Type 2 pair must still wait in queue and the second server
remains idle.]
(a) (20 points) Please draw carefully a state transition diagram that describes this
queueing system.
Please make sure to define clearly the states of the system.
[Note: You can answer part (b) without answering part (a); but, it will be easier to
answer (b), if you have answered (a).]
(b) (10 points) Suppose that there are currently four Type 1 customers in the system.
(Obviously, two of them are receiving service and the other two are in queue.)
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 Fall '06
 hansman

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