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A queueing system has three servers with expected service times of 15 minutes, 12 minutes and 10 minutes. The service times have an exponential distribution. Each server has been busy with a current customer for 8 minutes. Determine the expected remaining time until the next service completion

Consider a queueing system with two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers also arrive according to a Poisson process with a mean rate of 5 per hour. The system has two servers, both of which serve both types of customers. For both types, service times have an exponential distribution with a mean of 10minutes. Service is provided on a first-come-first-served basis. a) What is the probability distribution (including its mean) of the time between consecutive arrivals of customers of any type? (10 p.) b) When a particular type 2 customer arrives, he finds two type 1 customers there in the process of being served but no other customers in the system. What is the probability distribution (including its mean) of this type 2 customer's waiting time in the queue? (10 p.)

Consider a queueing system with two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers also arrive according to a Poisson process with a mean rate of 5 per hour. The system has two servers, both of which serve both types of customers. For both types, service times have an exponential distribution with a mean of 10minutes. Service is provided on a first-come-first-served basis. a) What is the probability distribution (including its mean) of the time between consecutive arrivals of customers of any type? (10 p.) b) When a particular type 2 customer arrives, he finds two type 1 customers there in the process of being served but no other customers in the system. What is the probability distribution (including its mean) of this type 2 customer's waiting time in the queue? (10 p.)