Question

The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of \(\$ 3\) per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of \(\$ C\) per hour. The manager estimates that, on the average, each customer's time is worth \(\$ 1\) per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour (a) What is the average cost per hour if Mary is hired? If Alice is hired? (b) Find \(C\) if the average cost per hour is the same for Mary and Alice.

Short answer

In conclusion, the average cost per hour for hiring Mary is $13, and for hiring Alice is $5+C. The value of $C$ that makes the average cost per hour the same for both employees is $8.

Step-by-step solution

Find the average cost per hour for Mary

First, we will calculate the average number of customers that Mary serves per hour. The arrival rate of customers follows a Poisson distribution with a rate of 10 per hour. Since Mary serves customers at an exponential rate of 20 per hour, her service time is \(\frac{1}{20}\) hours per customer. The expected number of customers waiting in line for Mary at any given time is \(\frac{\lambda^2}{\mu - \lambda}\), where \(\lambda\) is the arrival rate, and \(\mu\) is the service rate. In this case, we have \(\lambda = 10\) and \(\mu = 20\), so the expected number of customers in line is \(\frac{10^2}{20 - 10} = 10\). Now, we will account for the cost of customers' time. Since each customer's time is worth \(1 per hour and there are 10 customers in line on average, the total cost of customers' time per hour is \)10. Finally, we'll add Mary's hourly wage to find the average cost per hour if Mary is hired: \(Average\ Cost\ (Mary) = Customers'\ Time\ Cost + Hourly\ Wage = 10 + 3 = 13\)

Find the average cost per hour for Alice

We will follow the same process to calculate the average cost per hour for Alice. Alice serves customers at an exponential rate of 30 per hour, so her service time is \(\frac{1}{30}\) hours per customer. The expected number of customers waiting in line for Alice at any given time is \(\frac{\lambda^2}{\mu - \lambda}\), where \(\lambda\) is the arrival rate, and \(\mu\) is the service rate. In this case, we have \(\lambda = 10\) and \(\mu = 30\), so the expected number of customers in line is \(\frac{10^2}{30 - 10} = \frac{100}{20} = 5\). Now, we will account for the cost of customers' time. Since each customer's time is worth \(1 per hour and there are 5 customers in line on average, the total cost of customers' time per hour is \)5. Finally, we'll add Alice's hourly wage to find the average cost per hour if Alice is hired: \(Average\ Cost\ (Alice) = Customers'\ Time\ Cost + Hourly\ Wage = 5 + C\)

Find the value of \(C\) that makes the average cost per hour the same for both employees

To find the value of \(C\), we will equate the average cost per hour for both employees: \( 13 = 5 + C \) Now, we'll solve for \(C\): \( C = 13 - 5 = 8 \) Therefore, the value of \(C\) that makes the average cost per hour the same for both employees is $8. In conclusion, the average cost per hour for hiring Mary is \(13, and for hiring Alice is \)5 + C\(. The value of \)C\( that makes the average cost per hour the same for both employees is \)8.

Key concepts

Poisson Process

Understanding the Poisson process is the cornerstone of queueing theory. This concept describes how events occur completely randomly within a fixed period of time. For students trying to grasp the essence of a Poisson process, picturing phone calls coming into a call center is a helpful scenario. Each call is independent of the last, and they come in at an average known rate, which we denote as lambda (λ).

In the context of our market manager's dilemma, customers arriving at the market follows a Poisson distribution with a rate of 10 per hour. This means that on average, 10 customers arrive each hour, but the exact timing of each arrival is random. It's the unpredictability factor that makes managing queues challenging and interesting at the same time. Understanding this random pattern will help managers plan for resources needed, such as the number of cashiers, to avoid long lines and waiting times.

Exponential Service Rate

Moving on from arrival patterns, let's dive into the concept of an exponential service rate. Known for its memoryless property, the exponential distribution is perfect for modeling the time between events—such as the length of time a cashier spends servicing a customer. This rate is denoted as mu (μ), representing the average number of customers that can be serviced in a given timeframe.

For instance, Mary has an exponential service rate of 20 customers per hour, which implies that, on average, she takes 3 minutes (1/20th of an hour) to serve each customer. The speed with which services are provided is essential in a queueing scenario because it directly impacts how long customers have to wait, and thus, their satisfaction. Faster service rates generally lead to shorter queues, which is clearly seen when comparing Mary's service rate to Alice's, which is 30 customers per hour.

Cost-Benefit Analysis

Lastly, when determining the best course of action in any business decision, we use cost-benefit analysis to balance the scales of expenses against the gains. This economic evaluation involves totaling up the benefits or profits of a service and subtracting the associated costs of providing it.

In the case of hiring Mary or Alice, the manager of the market conducts a cost-benefit analysis by accounting for the hourly pay rate of each candidate (a fixed cost) and the cost of customer time (a variable cost that depends on service rate and customer arrival rate). The goal is to minimize the total cost while maintaining a smooth queueing experience for the customers. This involves considering how much they value a customer's time, which is a crucial component in providing excellent customer service and maintaining a positive business image.