Consider a three-server system where the servers have independent and exponential service times. Arrivals to the...

 

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Consider a three-server system where the servers have independent and exponential service times. Arrivals to the system follow a Poisson process with rate A. If all the servers are busy, the customer does not enter the system. (a) Suppose the service rate is for each server. An arriving customer always goes to an available server randomly with equal probability. i. Model the system as a CTMC with 4 states. Define the states, draw the transition diagram and write the balance equations. ii. Find the steady state probabilities. (b) Suppose now one of the servers is Fast (F) with rate µF and the other two are Slow (S) and identical with rate us. An arriving customer goes to the fast one if it is available. If the fast server is not available and two slower ones are available, he/she randomly selects one of them with equal probability. Describe the system as a CTMC with 6 states. Define the states clearly, draw the transition diagram and write the transition rates. Consider a three-server system where the servers have independent and exponential service times. Arrivals to the system follow a Poisson process with rate A. If all the servers are busy, the customer does not enter the system. (a) Suppose the service rate is for each server. An arriving customer always goes to an available server randomly with equal probability. i. Model the system as a CTMC with 4 states. Define the states, draw the transition diagram and write the balance equations. ii. Find the steady state probabilities. (b) Suppose now one of the servers is Fast (F) with rate µF and the other two are Slow (S) and identical with rate us. An arriving customer goes to the fast one if it is available. If the fast server is not available and two slower ones are available, he/she randomly selects one of them with equal probability. Describe the system as a CTMC with 6 states. Define the states clearly, draw the transition diagram and write the transition rates.

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We often list the transition probabilities in a matrix The matrix is called the state transition matrix or transition probability matrix and is usually shown by P rolepresentation styledisplay inline ... View the full answer