Question

You are analyzing some performance metrics for an M/M/1 service system. Which of the followings could cause the utilization to increase?

Answer 1

Increased utilization in an M/M/1 service system can be caused by a decrease in the expected time between customer arrivals, an increase in expected service time, an increase in the arrival rate, or a decrease in the service rate; all of these lead to higher utilization.

Step-by-step explanation:

When analyzing performance metrics for an M/M/1 service system, utilization can be affected by several factors. Utilization is the ratio of the time the server is busy to the total time and is calculated using arrival and service rates.

In an M/M/1 service system, the utilization (often denoted as ρ) is calculated as λ/μ, where λ is the arrival rate (customers per time unit) and μ is the service rate (services per time unit). If the expected time between customer arrivals decreases, this means that λ increases, leading to an increase in utilization. Similarly, if the expected service time increases, it indirectly indicates that μ decreases, which also results in higher utilization. Directly, an increase in the arrival rate or a decrease in the service rate will cause the utilization to increase.

Therefore, each of the following can cause the utilization to increase:

• 
  
• The expected time between customer arrivals decreases (increase in λ)
• 
  
• The expected service time increases (decrease in μ)
• 
  
• The arrival rate increases (increase in λ)
• 
  
• The service rate decreases (decrease in μ)

All of these changes result in higher utilization within the service system.

Complete question is as follows :

You are analyzing some performance metrics for an M/M/1 service system. Which of the followings could cause the utilization to INCREASE?

The expected time between customer arrivals decreases

The expected service time increases

The arrival rate increases

The service rate decreases

All of these can cause utilization to increase

Answer 2

An increase in the arrival rate
`(\(\lambda\))` or a decrease in the service rate
`(\(\mu\))` could cause the utilization to increase in an M/M/1 service system.

Step-by-step explanation:

In the M/M/1 queueing system, utilization (
`\(\rho\))` is defined as the ratio of the arrival rate
`(\(\lambda\))` to the service rate
`(\(\mu\)), i.e., \(\rho = \frac{\lambda}`
`\mu}\)`. Utilization represents the fraction of time the server is busy serving customers. If the arrival rate increases
`(\(\lambda\))`, while the service rate remains constant, the utilization will rise. This is because a higher arrival rate means more customer arrivals, potentially leading to a busier server and increased utilization.

Conversely, a decrease in the service rate
`(\(\mu\))` with a constant arrival rate would also result in increased utilization. If the server takes longer to serve each customer (lower service rate), it may fall behind in handling incoming customers, leading to a higher fraction of time being spent in serving, thereby increasing utilization.

It's important to note that in the M/M/1 system, changes in arrival rate or service rate directly impact utilization, and understanding these relationships is crucial for managing system performance and ensuring efficient resource utilization. Therefore, alterations in the arrival or service rates can significantly influence the utilization of the M/M/1 service system.