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Exercise 1 . Consider the open network ( referred to as the system ) of three queues, i . e . , Q 1 ,

Exercise 1. Consider the open network (referred to as the system) of three queues, i.e.,Q1,Q2, and Q3, at
steady state. New customers arrive at rate . A new customer chooses to enter Q1 with probability p1,Q2 with
probability p2, and Q3 with probability p3=1-p1-p2. Customers leaving Q1 choose to enter Q2. Customers
leaving Q2 choose to enter Q3. Customers leaving Q3 choose to either depart from the system with probability
q or (re)enter Q1 with probability 1-q. The average time spent by a customer in Q1 while visiting this queue
once is T1. The average time spent by a customer in Q2 while visiting this queue once is T2. The average time
spent by a customer in Q3 while visiting this queue once is T3.
A) Compute N1,N2, and N3, defined as the average number of customers in Q1,Q2, and Q3, respectively
[pt.10].
B) Compute T, defined as the average total time spent in the system by a customer [pt.10].
C) Compute the value of p3 that would minimize T[pt.10].
Exercise 2. An Ethernet switch transmits frames at a transmission rate of 10 gigabit per second (Gbps).
The frame average length is 900 bytes. The transmitter utilization is 20%. The average number of frames stored
in the switch (including the one in transmission) is estimated to be 150.
A) Compute x, defined as the frame average transmission time [pt.10].
B) Compute T, defined as the average time a frame spends in the switch at steady-state (T includes both
waiting and transmission time)[pt.10].
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