Question: 8.4 Consider a single-server discrete-time queueing system that operates in the following manner. Let Xn denote the number of customers in the system at time

8.4 Consider a single-server discrete-time queueing system that operates in the following manner. Let Xn denote the number of customers in the system at time nAf0; 1; 2; ...g. If a customer is receiving service in time n, then the probability that he finishes receiving service before time n 1 1 is q, where 0 # q # 1. Let the random variable Yn denote the number of customers that arrive between time n and n 1 1, where the PMF of Yn is given by pYn ðkÞ 5 P Y½ n 5 k 5 e2λ λk k! k 5 0; 1; ...

a. Give an expression for the relationship between Xn11; Xn and Yn.

b. Find the expression for the transition probabilities P½Xn11 5 jjXn 5 i.

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