TCOM501 – Networking: Theory & Fundamentals
Midterm Examination
Professor Yannis A. Korilis
March 6, 2002
Answer all problems. Good Luck!
Problem 1 [15 points]:
1. State the PASTA theorem. [5 points]
2. Provide an analytical proof of the PASTA theorem. [10 points]
Problem 2 [20 points]: Consider a single-server queue with state-dependent arrival rate:
λn =
λ
, n = 0,1,...
n +1
In other words, the inter-arrival times are independent and follow the exponential distribution
with parameter λ n when the queue is at state n. The service times are independent identically
distributed following the exponential distribution with parameter µ.
1. Find the stationary distribution of the queue. [10 points]
2. Find the average time that a customer spends in the system. [10 points]
Problem 3 [20 points]: Customers arrive according to a Poisson process with rate λ = 1 to a
single server queue. The service protocol includes a mechanism to allow the server to do bookkeeping operations at random intervals – while keeping customers waiting for service. The
protocol works as follows: the server generates a sequence of book-keeping flags according to an
independent Poisson process with rate v = 1 . Every time a flag is generated, the server adds the
flag to the end of the queue as a “virtual customer,” provided that the total number of customers
(real or virtual) in the system is less than 2. When a book-keeping flag enters service, the server
does a book-keeping operation. Assume that the service times of real customers as well as the
times required for book-keeping operations (service times of virtual customers) are independent
and follow the exponential distribution with parameter µ = 2 . Find:
1. The average number of customers, real and virtual, in the system. [7 points]
2. The average number of real customers in the system, and the average time that a real
customer spends in the system. [6 points]
3. The average number of virtual customers in the system, and the average time that a virtual
customer spends in the system. [7 points]
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Problem 4 [25 points]: Poisson arrivals with rate λ join a queue in front of two parallel servers A
and B, with exponentially distributed service times with rates µA and µB, respectively. A customer
that upon arrival finds the system empty is allocated to the server that has been idle for the
longest time. Otherwise, the head of the queue takes the first free server.
1. Define the states describing the system as a Markov chain and draw the state-transition
diagram. [5 points]
[Hint: You will need to introduce state 0A corresponding to 0 customers in the system and
server A being idle for the longest time.]
2. Find the stationary distribution. [10 points]
3. Is the system reversible? Justify your answer. [5 points]
4. Find the transition rates of the reversed Markov chain and draw its state-transition diagram.
[5 points]
Problem 5 [20 points]: Consider a network with four stations. Customers arrive at stations 1, 2,
and 3 in accordance with independent Poisson processes having respective rates 5, 10, and15.
There are no external arrivals at station 4. The service times at the four stations are independent
exponentially distributed with respective rates 10, 35, 85, and 41.
A customer completing service at node 1 is equally likely to (i) go to station 2, (ii) go to station 3,
or (iii) leave the system. A customer departing service at station 2 is equally likely to (i) go to
station 3, or (ii) go to station 4. A customer departing service at station 3 is equally likely to (i) go
to station 4, or (ii) leave the system. A customer completing service at node 4 is always routed to
station 3.
1. Find the total (aggregate) arrival rate λi at each node i = 1,2,3,4. [5 points]
2. The average number of customers Ni at each node i = 1,2,3,4. [5 points]
3. Find the average time T a customer spends in the network. [5 points]
4. Consider the aggregate arrival process at node 2. Is this a Poisson process? Justify your
answer. [5 points]
Useful Formulas:
•
Stationary distribution and average number of customers for an M/M/1 queue:
pn = (1 − ρ)ρn , n = 0,1,2...
∞
N = ∑ npn =
n =1
•
ρ
1− ρ
For a continuous Markov chain with transition rates qij and stationary distribution { p j } the
transition rates of the reversed chain are given by:
qij* =
•
For | x |< 1 :
∞
p j q ji
pi
x
∑ nx n = (1 − x )2 .
n =1
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