Chapter 13 - Queuing Theory : S-1
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Chapter 13
Queuing Theory
2.
Answers to this question will vary. Some prefer the second system because it seems to be the most fair.
That is, you will always start receiving service before anyone who arrives after you. Others prefer the third
system because, if you are lucky, you might pick the fastest moving line and exit the system more quickly
than a person who arrived before you.
6.
a.
b.
c.
d.
There are 30/60 = 0.5 arrivals per minute, on average.
In a ten minute interval we would expect 10*0.5 = 5 arrivals to occur.
P(x=0) = (50 e-5 )/(0!) = 0.00674 (NOTE: 0!=1)
P(x=1) = (51 e-5 )/(1!) = 0.03369
P(x=2) = (52 e-5 )/(2!) = 0.08422
P(x=3) = (53 e-5 )/(3!) = 0.14037
P(x>3)=1-P(x3) = 1-0.00674-0.03369-0.08422-0.14037 = 0.73497
7.
a.
b.
c.
Expected service time = 1/40 = 0.025 hours (or 1.5 minutes)
P(t1/60) = 1- e-40/60 = 0.4865
P(2/60  t  5/60) = e-40*2/60 - e-40*5/60 = 0.2279
P(t4/60) = 1- e-40*4/60 = 0.931
P(t3/60) = 1- P(t3/60) = 1-.8646 = 0.1354
8.
Use the M/M/s template in Q.xls with:
Arrival Rate = 30, Service Rate = 40, Number of Servers = 1.
a. 0.25
b. 0.75
c. 2.25
d. 0.1 hours or 6 minutes
e. 0.075 hours or 4.5 minutes
f. 60 per hour
9.
Use the M/M/s template in Q.xls with:
Arrival Rate = 7 per hour, Service Rate = 3 per hour, and vary the number of servers.
With 3 servers the average waiting time is 0.3057 hours or 18.34 minutes. With 4 servers the average
waiting time is 0.0532 hours or 3.20 minutes. Thus, 4 doctors should be scheduled.
14. Use the M/M/s template in Q.xls with:
Arrival Rate = 5 per hour, Service Rate = 60/8 = 7.5 per hour, Number of Servers = 1.
a. 0.3333
b. 0.2667 hours or 16 minutes
c. 2
d. $9*2*8 = $144 per day
e. Probably. The total labor cost of people using the machines would drop to $9*0.75*8 = $54. This
amount, plus the $45 daily lease cost is less than $144. But this all assumes that the 2 people
currently using or waiting to use the single copy machine have a more productive use for their time.
If all they would do is sit at their desk and play computer games then it does not make sense to lease
the additional copier to give them more time to play computer games.
Chapter 13 - Queuing Theory : S-2
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17. Use M/M/s with finite queue length template in Q.xls with:
Arrival Rate = 60 (per hour), Service Rate = 5 (per hour), Number of Servers = 10, and Maximum Queue
Length = 5.
a. 0.2033
b. 0.8111
c. 0.0408 hours or approximately 2.45 minutes
d. With 14 workers there is a 3.94% chance of a caller receiving a busy signal. Thus, 4 workers would
need to be added.
18. Use M/M/s with Arrival Rate = 14 (per hour), Service Rate = 15 (per hour), one server.
a. 56 minutes
b. 13.07
c. 0.933
d. 93.33%
e. With 3 servers the probability of having to wait is 0.0767.