Question

Consider a queuing system in which the number of servers is adjusted depending on the number of customers currently in the system. You see this routinely in many businesses – from coffee shops to restaurants to grocery stores – where additional employees are called in whenever the number of people in the system reaches some integer value. For example, if the number of people in the system is 10 or more, a second server is called up, and this doubles the service rate for all states 10 or higher. But what should this integer value be given that calling in additional employees has a cost? Let’s consider the simplest case: a business uses one employee until the number of customers in the system reaches x. When the number of customers in the system is x or higher, the business calls in an additional employee. What is the optimal value of x that minimizes the cost to the business while also ensuring that the average queue length L_q <=3? Assume that that mean arrival rate is 2.85 per minute and the service rate is 3 per minute. Time between arrivals are exponentially distributed and so are the service times. Each employee is paid $20/hour. The system has a capacity of 20 customers: that is no more than 20 customers can be in the system.

Answer 1

Employee cost = $ 20 per hour.

Arrival rate, = 2.85 per minute.

Service rate, = 3 per minute.

Average number of customers being served = / = 2.85 / 3 = 0.95

Table:

Number of Employees (i.e., Servers)	Average number of customers in the system, Ls = Lq + ( / ) (From Table)	Total Employee Cost per hour (in $)
2	0.277 + 0.95 = 1.227	2 x 20 = 40 (Minimum)
3	0.037 + 0.95 = 0.987	3 x 20 = 60
4	0.005 + 0.95 = 0.955	4 x 20 = 80

Total Employee Cost is minimum when the number of customers in the system is 1.227 and the number of employees is 2.

So x = 1.227