Stochastic

In addition there is one chair where customers can

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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 Chapman-Kolmogorov 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...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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