Final answer:
To maximize expected profit, calculate the expected profit for varying numbers of servers based on the service and arrival rates using queuing theory and cost-benefit analysis. Choose the server count that results in the highest profit, considering operation costs and revenue with possible rebates for waiting jobs.
Step-by-step explanation:
To maximize the expected profit for the system with jobs arriving according to a Poisson process and service times that are exponentially distributed, we need to find the optimal number of servers that balance the cost of operating servers against the revenue from servicing jobs, considering the probability of jobs having to wait.
Jobs arrive at a rate of 10 per hour and have a mean service time of 5 minutes (or 1/12 hour since there are 60 minutes in an hour). If jobs do not have to wait, each job generates $1 in revenue. A 20 cent rebate is given if a job has to wait, which can be thought of as a loss of revenue. Each server incurs a cost of $1 per hour.
To find the optimal number of servers, we need to calculate the expected profit for different numbers of servers and choose the configuration that yields the highest expected profit. Typically, queuing theory and cost-benefit analysis are used to make this determination. However, the solution to this problem will require specific methodologies from operations research, such as determining the server utilization, average number of jobs in the system, and the probability that an arriving job will have to wait, which depend on the service and arrival rates.