process at a mean rate of 3 per minute. If a customer finds all tellers busy, he joins a queue that is serviced by all tellers; 1. e. there are no lines in front of each teller but rather one line waiting for the first available teller. The transaction time between the teller and customer has an exponential distribution with a mean of l minute Find the steady-state probability distribution of the number of customers in the bank (10 points) Solution: The probability of non customer in the system is a/su The probability of n customer in the system (1/)2 P.0<n<4 0<n<4 n n! 53 P 3 P,n≥4 44-,n246 process at a mean rate of 3 per minute. If a customer finds all tellers busy, he joins a queue that is serviced by all tellers; i.e., there are no lines in front of each teller, but rather one line waiting for the first available teller. The transaction time between the teller and customer has an exponential distribution with a mean of 1 minute. Find the steady-state probability distribution of the number of customers in the bank. (10 points). Solution: The probability of non customer in the system is ( ) ( ) 53 2 1 / 1 ! / ! / 1 1 0 0 = − + ⋅ = ∑ − = λ μ λ μ λ μ n s s P s s n n The probability of n customer in the system is ( ) ( ) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ ⋅ ⋅ ≤ ≤ = ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ ⋅ ≤ ≤ = − − , 4 4!4 3 , 0 4 53 2 ! 3 , 4 4!4 / , 0 4 ! / 0 4 0 4 0 P n n n P n P n n P n n n n n n n λ μ λ μ