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MULTIPLE-CHANNEL QUEUING MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES 465

opening a second order-processing station so that two customers can be served
simultaneously. Assume a single queue with the first customer in line moving to
the first available server. Let us evaluate the operating characteristics for this two-
channel system.
We use Equations (11.12) through (11.18) for the k ¼ 2 channel system. For a
mean arrival rate of l ¼ 0.75 customers per minute and mean service rate of ¼ one
customer per minute for each channel, we obtain the operating characteristics:

EXCEL file P 0 ¼ 0:4545 ðfrom Table 11:4 with = ¼ 0:75Þ
2
ð0:75=1Þ ð0:75Þð1Þ
MULTIPLE
L q ¼ 2 ð0:4545Þ¼ 0:1227 customers
ð2 1Þ!½2ð1Þ 0:75Š
0:75
L ¼ L q þ ¼ 0:1227 þ ¼ 0:8727 customers
1
0:1227
L q
W q ¼ ¼ ¼ 0:1636 minute
0:75
1 1
W ¼ W q þ ¼ 0:1636 þ ¼ 1:1636 minutes
1

2
1 0:75 2ð1Þ
P w ¼ ð0:4545Þ¼ 0:2045
2! 1 2ð1Þ 0:75
Using Equations (11.17) and (11.18), we can calculate the probabilities of n
Try Problem 9 for
practise in determining customers in the system. The results from these calculations are summarized in
the operating Table 11.5.
characteristics for a two- We can now compare the steady-state operating characteristics of the two-channel
channel queue.
system to the operating characteristics of the original single-channel system discussed
in Section 11.2.
1 The average time a customer spends in the system (queuing time plus service
time) is reduced from W ¼ four minutes to W ¼ 1.1636 minutes.
2 The average number of customers in the queue is reduced from L q ¼ 2.25
customers to L q ¼ 0.1227 customers.
3 The average time a customer spends in the queue is reduced from W q ¼ 3
minutes to W q ¼ 0.1636 minutes.
4 The probability that a customer has to wait for service is reduced from
P w ¼ 0.75 to P w ¼ 0.2045.
Clearly the two-channel system will significantly improve the operating characteristics
of the queuing system. However, adding an order filler at each service station would
further increase the mean service rate and improve the operating characteristics. The

Table 11.5 The Probability of n Customers in the System for the Dome Two-
Channel Queuing System
Number of Customers Probability

0 0.4545
1 0.3409
2 0.1278
3 0.0479
4 0.0180
5 or more 0.0109

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