QUEUEING THEORY , 6H3708
Assignment 1
P1 2006/7
Lecturer: Armin Halilovic
[email protected]
www.syd.kth.se/armin
KTH, Campus Haninge, Room 5046
Use Maple (or Mathematica) to solve Exercises 1-2.
In the exercises below a, b, and c are the last three digits of your ID-number.
For example, if your ID= 751106 2348 then you substitute a=3, b=4 and c=8 in the exercises 1-3
Group work: Maximum 2 students can work together on the same assignment.
Date and time for presentation:
Date: 16 Oct 2006, time 15:15- 17:15,
room: 5112
Questions related to lab work will be included in the exam. (Do your own work.)
Exercise 1)
A Markov chain has the following transition matrix
2x
0.3
0.3
y 


0.2
x
2y
0.3 

( z  a  b  c )
0.25
0.3 0.25


0.1
( w  a  b) 0.24 0.46

a) Find x, y, z , w and P
b) Assume that an initial state probability vector is P (0)  (0.3, 0.2, 0.4, k ) .
Find k and the state probability vectors P (1) , P ( 2) , P (3) , P ( 20) , P (30) .
c) Find the state probability vector P (30) if the initial state vector is
c1) P ( 0)  (1,0,0,0) c2) P ( 0)  (0,1,0,0) c3) P ( 0)  (0,0,1,0) and c4) P ( 0)  (0,0,0,1)
d) Compare the results in b) and c)
b) Can you now find P (3000) without calculation?.
e) Pen and paper : Draw the transition graph for the chain.
Exercise 2)
A queueing system has three states E1, E2 and E3 with the following rate transition graph.
E2
a+1
a+3
E3
E1
a+6
a) Find the rate transition matrix Q.
b)Solve the system P´(t) = P(t)Q
and find the (transient) probability state vector
P(t)=(p1(t), p2(t), p3(t)) if P(0)=(0.2, 0.3, 0.5)
c)Find the steady-state vector, that is evaluate lim P (t ) .
t  

d) Solve the equation P(t )Q  0 and compare the solution with that in c)