User Manual

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:
1. Key in the arrival rate of customers,
λ
, and press .
2. Key in the service rate,
µ
, and press
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
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
1 2.
6. Calculate the
average waiting time
in the system by pressing
1 .
7. Key in
λ
and press
2 to calculate the
average queue length
.
8. Key in
ρ
, then intensity factor (from step 2 above) and press
to calculate the
average
number of customer in the system
.
Reference: