The registration area has just opened at a large convention of dentists in Tallahassee, Florida. There are 200 people arriving per hour (Poisson distributed), and the cost of their waiting time in the queue is valued at $100 per person per hour. The Tallahassee Convention Center provides servers to register guests at a fee of $15 per person per hour. It takes about one minute to register an attendee (negative exponentially distributed). A single waiting line, with multiple servers, is set up.
a) What is the minimum number of servers for this system?
b) What is the optimal number of servers for this system?
c) What is the cost for the system, per hour, at the optimum number of servers?
d) What is the server utilization rate with the minimum number of servers?
The Correct Answer and Explanation is :
Results:
a) Minimum number of servers: 4
b) Optimal number of servers: 6
c) Cost for the system per hour at the optimum number of servers: $108.53
d) Server utilization rate with the minimum number of servers: 83.33%
Explanation:
- Minimum Number of Servers:
The minimum number of servers is determined by ensuring system stability, where the arrival rate ((\lambda)) is less than the maximum service rate ((s \cdot \mu)). For 200 arrivals per hour and a service rate of 60 people per hour per server, at least 4 servers are required to keep the system stable. - Optimal Number of Servers:
While 4 servers stabilize the system, they do not minimize costs. Costs consist of waiting time costs ($100/hour per person) and server costs ($15/hour per server). By evaluating total costs for different numbers of servers starting from 4, we find that 6 servers minimize the total cost. - System Cost at Optimum Servers:
With 6 servers, the system achieves the lowest total cost of $108.53 per hour, balancing server costs and waiting time costs. - Utilization at Minimum Servers:
At 4 servers, the utilization rate ((\rho = \lambda / (s \cdot \mu))) is calculated as (200 / (4 \cdot 60) = 0.833), or 83.33%. This indicates that each server is busy 83.33% of the time, leaving minimal buffer for delays, making this configuration less efficient than the optimal case.
Thus, deploying 6 servers achieves an optimal balance between service efficiency and cost.