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Consider an M/M/1 queueing system with arrival λ and service rate μ (λ < μ). Assume that the service

discipline is FIFO. Suppose you (a tagged customer 'A') are arriving into the queue at t0 and interested in how many customers are left behind when you get out of the system (after serviced). Specifically, let Nd the number of customers left behind, seen by your departure. Equivalently, this is also the number of customer arrivals during (t0 , t0 + W ], where W is your waiting time in the whole system (queueing + service time).

1. (a) (5 points) Let N be the number of customers in the system in the steady-state as covered in class. Calculate its z transform, i.e., E{zN}.
2. (b) (15 points) Calculate the z transform of Nd, i.e., E{zNd} in terms of λ,μ,z. You must clearly show your steps.

4. (20 points): (from past exam) Consider an M/M/1 queueing system with arrival ) and service
rate M () &lt; u). Assume that the service discipline is FIFO. Suppose you (a tagged customer
'A') are arriving into the queue at to and interested in how many customers are left behind when
you get out of the system (after serviced). Specifically, let Na the number of customers left
behind, seen by your departure. Equivalently, this is also the number of customer arrivals during
(to, to + W], where W is your waiting time in the whole system (queueing + service time).
(a) (5 points) Let N be the number of customers in the system in the steady-state as covered
in class. Calculate its z transform, i.e., E{z~}.
(b) (15 points) Calculate the z transform of Na, i.e., E{z d} in terms of A, M, z. You must

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