1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods – Fall 2006
Quiz #2
Solutions
Problem 1 (35 points)
a). Let the states be defined by (i, j, k) where:
 i is the number of Type 1 customers (i = 0, 1, 2, or 3)
 j is the number of Type 2 customers (j = 0, 1, 2, or 3)
 k is the type of customer being served (k = 0, 1, 2)
The total number of state is 13.
The state transition diagram for the system is as follows:
λ
λ
λ
0,0,0
0,1,2
0,2,2
0,3,2
1,0,1
1,1,2
3,0,1
2,0,1
1,1,1
1,2,1
1,2,2
2,1,2
2,1,1
2
λ
2
λ
2
λ
2
µ
2
µ
2
µ
1
1
µ
1
1
1
µ
1
µ
1
µ
1
µ
2
µ
2
µ
2
µ
1
µ
1
λ
1
λ
1
λ
2
λ
2
λ
2
λ
2
λ
b). In the long run, this system treats Type 1 and Type 2 customers equally. Sometimes Type 1±
customers have priority and sometimes Type 2 customers do. When
λ
1
=
λ
2
and
µ
1
=
µ
2
, the±
state transition diagram is perfectly symmetrical.±
Therefore,
L
1
=
L
2
under those conditions.±
c). The next customer to be served by the server is a Type 2 customers if no Type 1 customer±
arrives before the completion of the current service.±
The probability that a Type 1 customers arrives before the completion of the current service is±
given by:±
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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods – Fall 2006
Pr(
next
event
occurs
in
[
0,
ε
]
and
is
the
arrival
of
one
Type
1
customer
)
P
=
A
Pr
(
next
event
occurs
in
[
0,
ε
]
)
λ
P
=
1
A
λ
1
+
λ
2
+
µ
1
Therefore, the probability that the next customer to be served is a Type 2 customer is given by:
λ
2
+
µ
1
P
=
1
−
P
=
A
A
λ
1
+
λ
2
+
µ
1
d). We start with 2 people in the system: one Type 1 customer is being served and one Type 2
customer is waiting for service.
•
Let’s first consider that the next transition is due to an arrival of a Type 1 customer. This
customer will be the next one to be served and we cannot reach the state described by the
rd
3
transition.
•
If a Type 2 customer arrives before service completion, the system is full, thus the
following state transition can only be due to the end of Type 1 customer’s service. Type 2
rd
customers are then given priority, and we end up in the desired state at the 3
transition if
rd
and only if a 3
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 Fall '06
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