138 Miscellaneous
Queuing and Waiting Theory
Waiting lines, or queues, cause problems in many marketing situations. Customer
goodwill, business efficiency, labor and space considerations are only some of the
problems which may be minimized by proper application of queuing theory.
Although queuing theory can be complex and complicated subject, handheld calculators
can be used to arrive at helpful decisions.
One common situation that we can analyze involves the case of several identical stations
serving customers, where the customers arrive randomly in unlimited numbers. Suppose
there are n (1 or more) identical stations serving the customers. λ
is the arrival rate
(Poisson input) and µ is the service rate (exponential service). We will assume that all
customers are served on a first-come, first-served basis and wait in a single line (queue)
then are directed to whichever station is available. We also will assume that no customers
are lost from the queue. This situation, for instance, would be closely approximated by
customers at some banking operations.
The formulas for calculating some of the necessary probabilities are too complex for
simple keystroke solution. However, tables listing these probabilities are available and
can be used to aid in quick solutions. Using the assumptions outlined above and a suitable
table giving mean waiting time as a multiple of mean service (see page 512 of the
Reference) the following keystroke solutions may be obtained:
RPN Mode:
1.
Key in the arrival rate of customers, λ, and press
\.
2.
Key in the service rate, µ, and press
z to calculate ρ, the intensity factor. (Note ρ
must be less than n for valid results, otherwise the queue will lengthen without limit).
3.
Key in n, the number of servers and press
z to calculate ρ/n.
4. For a given n and ρ/n find the mean waiting time as a multiple of mean service time
from the table. Key it in and press
\.
5. Calculate the average waiting time in the queue by keying in the service rate, µ, and
pressing
?1z?2.
6.
Calculate the
average waiting time in the system by pressing :1y+.
7.
Key in λ and press
:2§ to calculate the average queue length.
8.
Key in ρ, the intensity factor (from step 2 above) and press
+ to calculate the
average number of customers in the system.
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