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 ∝ is the service rate (exponential service). We will assume that all customers are served on a firstcome,
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, ∝, 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, ∝, 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.
