Examination Paper
Exam duration 3 hours.
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In this exercise, we consider a M/M/2/3 queue with servers having identical service rates.
a) Draw the Markov chain associated to this M/M/2/3 queue. Label all the states and transitions clearly. (4
b) Let Xt be the random variable describing the state of the queue at time t, and assume the queue is in a Assignment Project Exam Help state i at time t. Let us consider a small time step Δt, such that Δt is very small in comparison to the average time between arrivals and Δt is very small in comparison to the mean service times of the queue.
Give the transition probability matrix P(Δt) associated to this M/M/2/3 queue. We recall that we can define
The transition rate matrix Q associated to a continuous time Markov chain can be defined as
p’t = pt·Q
where p’t is the time derivative of pt and pt corresponds to a row vector containing the state probabilities at time t. If the chain contains n states then pt = ( Pr(Xt=0), Pr(Xt=1), … ,Pr(Xt=n) ) and Q is an nxn matrix. Using the transition probability matrix computed in the previous question, we can also define Q as Q = limΔt→00 (P-I)/Δt where I is the identity matrix.
d) Use p’t = pt·Q to find a system of linear equations involving the state probabilities in the steady-state for
ATM switches have a fixed service time duration. A queue in an ATM switch is found to have a Poisson arrival process. Over a long period of time, the average length of the queue is found to be 2.5 packets. Find the fraction of time the queue is empty.
Consider the Jackson network shown below where all nodes are M/M/1. The arrival rate of external traffic to nodes 1, 2, 3 and 4 are shown on the diagram. Also routing information is shown on each link. Assume that the service rates are as follows: μ1 = 16, μ2 = 32, μ3 = 40, μ4 = 20, μ5 = 28.
Consider a general birth and death queuing system in which,
λk = αk λ
for k ≥ 0, 0 ≤ α < 1 μk = μ
for k ≥ 1 ρ=λ/μ
Assignment Project Exam Help
Consider the following function:
f(x) = x2 − 2 x3 + 2 exp( x ) − 4 + y2 with x = (x, y) ∈ R R2 .
d) In view of the point x1 found in the previous question, do you think additional steps using the steepest
Consider the following system of equations were x, y, z and a are all positive integers (i.e. natural numbers): x = 2y, y = z – 1, a = 2x + 1
(b) List the variables in the graph which are associated to cut-vertices. (3 marksAssignment Project Exam Help)
a x y z
domains found in question (d)
a ← 5 5
As signmen t Project Exam H elp
https:// powcode r.com
Assignment Project Exam Help
• When the value of node N is (re)-calculated, write N: g ( N ) + h ( N ) = f ( N )
Depth-first search Breadth-first search
