This preview shows page 1. Sign up to view the full content.
Unformatted text preview: s formula also applies to queueing networks. In
this case the average number of people in the system is
L= 3
X
i=1 1
(ri /µi ) 1 1=1+1+ 1
7
=
3
3 so the average waiting time for a customer entering the system is
W= L = 7/3
35
=
= 0.6140.
19/5
57 149 4.7. CHAPTER SUMMARY 4.7 Chapter Summary In principle continuous time Markov chains are deﬁned by giving their transition
probabilities pt (i, j ), which satisfy the ChapmanKolmogorov equation.
X
ps (i, k )pt (k, j ) = ps+t (i, j )
k In practice, the basic data to describe the chain are the rates q (i, j ) at which
P
jumps occur from i to j 6= i. If we let i = j 6=i q (i, j ) be the total rate of
jumps out of i and let
(
q (i, j ) i 6= j
Q(i, j ) =
i=j
i
then the transition probability satisﬁes the Kolmogorov di↵erential equations:
X
X
p0 (i, j ) =
Q(i, k )pt (k, j ) =
pt (i, k )Q(k, j )
t
k k These equations can only be solved explicitly in a small number of examples,
but they are essential for developing the theory.
Embedded Markov chain. The discrete time cha...
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

Click to edit the document details