Question

Consider the following queuing systems:

System 1. An average of 40 customers arrive each hour, interarrival times are exponential. Customers must complete two types of service before leaving the system. The first server takes an average of 30 seconds (exponentially distributed) to perform type I service. After waiting in line, each customer obtains type Il service (exponentially distributed with a mean of 1 minute) from a single server. After completing type 2 service, a customer leaves the system.

System 2. The arrival process for system 2 is identical to the interarrival process for system 1. In system 2, a customer must complete only one type of service. Service times averages 1.5 minutes and is exponentially distributed. Two servers are available.

In which system does a typical customer spend less time?

Answer 1

Considering the services and the queuing processes, it is likely that a customer will typically spend less time in System 2, which has a single service and two servers, compared to System 1, which requires two sequential services before exiting the system.

Step-by-step explanation:

To determine which queuing system results in a typical customer spending less time, we compare the expected service times and queuing processes of both systems. System 1 has two types of services with exponential service times of 30 seconds (0.5 minutes) for type I and 1 minute for type II, while system 2 has a single service with an average of 1.5 minutes, but with two servers available.

In System 1, customers arrive at a rate of 40 per hour, or one customer every 1.5 minutes. As service times are exponential, we use the property of memorylessness to analyze the system. The first type of service takes an average of 0.5 minutes, and the second service takes 1 minute. This brings the total expected service time in System 1 to 1.5 minutes, excluding any queuing time that might occur due to simultaneous arrivals.

In System 2, the arrival rate is the same; however, there are two servers, and the service time is on average 1.5 minutes. With multiple servers, the likelihood of having to wait is reduced, and the effective rate at which customers are served increases. The total time spent by a customer in System 2 would therefore typically be less than or similar to the sum of service times without considering queue waiting time, as the presence of an additional server offsets the longer service duration. Thus, it is likely that a customer will spend less time in System 2 than in System 1.