We set out to show that by using a series of math formulae, we could greatly enhance the decision-making processes about employee scheduling. Within the limits our formulae, the program will do that.

Among the various queueing theory models, we chose to look at the M/M/1 steady state model because of its relative simplicity. The steady state probabilities are calculated as follows: 

n = number of customers in the system (queue plus service)

p_o = 1 - p_t

pⁿ = (1-p) pⁿ

whereas n can be equal to 1, 2, 3, ?

If we examine the expressions where p0 has the value of 0.20 and p₁ is 0.16, p₂ is 0.128, and p₃=0.124; these numbers are from the general equation pⁿ = (1-p) pⁿ.

This can be interpreted for the probabilities that 0, 1, 2, and 3 people are in the system. Thus, an employee is idle if there are zero people in the queue 20% of the day.

An arriving customer has a 0.8 probability of having to wait before he or she can be serviced. 

The following qualities suggest a congested condition of this system. This condition occurs when: 

W = 1 hour, Wq = 0.8 hr or 48 minutes, L = 4 customers, and L_q = 3.2 customers.