$3).$ Try to develop a simple queuing model to analyze the performance of certain switch: Suggest that the switch has one output link operating at $10$ Gbps $.$ Two types of packets arrive to the switch: TCP packets and UDP packets. TCP packets arrive as a Poisson process at rate $\backslash (\backslash$lambda$\_\{\backslash$mathrm$\{$t$\}\}=1\backslash )$ packet$-$per$-$microsecond, and have exponentially distributed length with mean size $625$ bytes. UDP packets arrive as a Poisson process at rate $\backslash (\backslash$lambda$\_\{\backslash$mathrm$\{$U$\}\}=2\backslash )$ packets$-$per$-$microsecond, and have exponentially distributed length with mean size $250$ bytes. The switch has a single queue which is served in First$-$come$-$First$-$served order. Further, it has capacity to hold a total of at most two packets $($since optical storage is very expensive$)-$effectively, while one packet is being served, at most one other packet can be in queue. Therefore, an arriving packet is dropped if the switch already holds two packets. Further, to ensure fairness between TCP and UDP, it is ensured that the switch never simultaneously holds two TCP packets or two UDP packets. In other words, an arriving TCP packet is dropped if the switch already has a TCP packet, and an arriving UDP packet is dropped if the switch already has a UDP packet.

a$-$ What is the service rate in packets$-$per$-$microsecond for TCP and for UDP packets?

b$-$ Draw the Markov chain showing the states, transitions, and transition rates. Provide any necessary explanations

c$-$ Write the balance equations and solve for the state probabilities.

d$-$ What is the probability that an arriving TCP packet is lost?