ANS:A - increased mean time in the queue.

The cost of providing service in a queuing system can be influenced by various factors, including the mean time in the queue and the arrival rate. However, to determine how the cost changes with these factors, we need to consider the specific context and objectives of the queuing system. In general, the cost of providing service in a queuing system may:

1. Decrease with decreased mean time in the queue: If the mean time in the queue decreases, it typically indicates that customers are spending less time waiting for service. This can lead to higher customer satisfaction, potentially lower staffing requirements, and reduced operational costs associated with managing queues.
2. Increase with increased arrival rate: A higher arrival rate means that more customers are entering the system, which can lead to longer queues, increased waiting times, and potentially higher costs associated with managing larger volumes of customers, such as increased staffing requirements or infrastructure costs.
3. Decrease with decreased arrival rate: Conversely, if the arrival rate decreases, there may be fewer customers entering the system, leading to shorter queues, reduced waiting times, and potentially lower costs associated with managing the system.
4. Increase with increased mean time in the queue: If the mean time in the queue increases, it typically indicates that customers are spending more time waiting for service, which can lead to lower customer satisfaction and potentially higher costs associated with managing longer queues, such as increased staffing requirements or lost revenue due to dissatisfied customers.

So, the statement "The cost of providing service in a queuing system decreases with the decreased mean time in the queue" is generally accurate, while the other statements may or may not hold true depending on the specific context and objectives of the queuing system.