Question

The stability condition for the multi-server queueing model with "c" servers is given by:

• A. \( \lambda<\mu \)
• B. \( \lambda>\mu \)
• C. \( \lambda<c\mu \)
• D. \( \lambda>c\mu \)

Hint:

For a queueing system with multiple servers (\( c \)), the stability condition is given by: \[ \lambda<c\mu \] This ensures that the total service rate is sufficient to process incoming jobs, preventing congestion.

Answer 1

The Correct Option is C

Step 1: Understanding the multi-server queueing model In a multi-server queueing system, there are \( c \) servers available to serve incoming tasks. The arrival rate is represented by \( \lambda \), and the service rate per server is \( \mu \).
Step 2: Condition for stability For the system to remain stable (i.e., to prevent an infinite queue buildup), the total service capacity of the system \( c\mu \) must be greater than or equal to the arrival rate \( \lambda \). Thus, the stability condition is: \[ \lambda<c\mu \] This ensures that the system can handle the incoming traffic efficiently without excessive queue growth.
Step 3: Comparison with other options
- Option (A) \( \lambda<\mu \): This applies to a single-server system, not a multi-server model.
- Option (B) \( \lambda>\mu \): This is incorrect as it does not account for multiple servers.
- Option (C) \( \lambda<c\mu \): Correct, as it ensures system stability.
- Option (D) \( \lambda>c\mu \): Incorrect, as it indicates an unstable system where queue length grows indefinitely.