Penn TCOM 501 - TCOM 501 Midterm Examination (4 pages)

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TCOM 501 Midterm Examination



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TCOM 501 Midterm Examination

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4
School:
University of Pennsylvania
Course:
Tcom 501 - Networking - Theory and Fundamentals.
Networking - Theory and Fundamentals. Documents

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TCOM501 Networking Theory Fundamentals Midterm Examination Professor Yannis A Korilis March 5 2003 Answer all problems Good Luck Problem 1 20 points Consider an M M 1 queue that can accommodate at most K customers in the system queued or in service and suppose that a customer that arrives and finds the system full is not lost but stored in an external queue with infinite space as shown in the figure K k i external queue server queue The transition of a customer from the external to the server queue is instantaneous Therefore a customer that arrives when the number of customers in the server queue is less than K enters instantaneously the server queue Similarly when a customer departs from the server queue the customer at the head of the external queue moves to the server queue instantaneously This twoqueue system can be modeled as a two dimensional Markov chain with states i k where 0 i 0 k K 1 Draw the state transition diagram of the two dimensional chain 3 points 2 Find the steady state probability p i k that there are i customers in the external queue and k customers in the server queue 0 i 0 k K 10 points 3 Find the average number of customers in the server queue and the average number of customers in the external queue 5 points 4 Find the average total time that a customer spends in the two queue system 2 points 1 of 4 Problem 2 20 points Customers arrive at a service station according to a Poisson process with rate For each of the following cases we assume that service times are exponentially distributed and independent of each other and the arrival process Queue buffers are infinite and customers are served on a first come first served basis Consider the following types of service stations Type 1 There are two servers sharing a common queue Each server provides service at rate A customer that upon arrival finds the system empty is routed randomly to one of the servers If the customer finds one server free it is routed to that server Otherwise it enters the



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