Final answer:
To choose between candidates X and Y for the server position in a fast-food restaurant, we must calculate the expected waiting time for customers with each server and the resulting revenues. By computing these values using the arrival and service rates, we can determine the maximum additional compensation that makes hiring the faster but more expensive candidate X financially justified.
Step-by-step explanation:
To determine the amount by which candidate X's compensation can be greater than candidate Y's such that hiring X is justified, we need to calculate the expected value of the waiting time (W) for customers with each server and the corresponding restaurant revenue. We are given that customers arrive at a rate of 30 per hour, which means there is an average inter-arrival time of 2 minutes (since an hour has 60 minutes). With X, the service time is exponentially distributed with a mean of 1.8 minutes, and with Y, it is 2.2 minutes.
The formula for the expected waiting time in the system (W) for an M/M/1 queue (where arrivals follow a Poisson process and service times have an exponential distribution) is λ/ (μ(μ - λ)), where λ is the arrival rate and μ is the service rate.
For candidate X: λ = 30 per hour (or 0.5 per minute) and μ = 1/1.8 per minute. For candidate Y: λ remains the same, and μ = 1/2.2 per minute. Calculate W for both candidates, then find the corresponding restaurant revenues using the given revenue function $5,000/W. Once you have the revenues, subtract the revenue with Y from the revenue with X to determine how much more X can be paid than Y before the additional cost exceeds the revenue benefit.
difference in both waiting times and revenues is critical to making the hiring decision.