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Counting Process: State of the system is given by the total number of customers in the system .

Counting Process: State of the system is given by the total number of customers in the system. c ont. N(t)=state of system at time t . (1) N(0)=0 (2) events are independant if they relate to non-overlapping intervals. 4pm. 5pm. 6pm. (3) stationary process :

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Counting Process: State of the system is given by the total number of customers in the system .

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  1. Counting Process: • State of the system is given by the total number of customers in the system .

  2. c ont.. N(t)=state of system at time t . (1) N(0)=0 (2) events are independant if they relate to non-overlapping intervals . 4pm 5pm 6pm

  3. (3) stationary process : i.e Pdf depends only on the interval length , not starting point .

  4. (4) probablity that one event occurs in an interval of length h is

  5. 5) probablity that more than one event occurs in an interval of length h is 0(h) .

  6. example : for h5/2 h5/2 /h = h3/2 --->0 as h---> 0

  7. (5) probablity that more than one event occurs in an interval of length h is 0(h) .

  8. Set up a Differential Differency Equation : Let : Pn(t) = the probability that N(t)=n

  9. P0 (h)=prob no event occurs in an interval of length h.

  10. t1 t2 tn-1 tn 0

  11. M i s used : 1 [ ] E t = l 2 l = 20 P n ( t ) = p r o b . n a r r i v a l s o c c u r i n a n i n t e r v a l o f l e n g t h t .

  12. t l n m L e t b e : t m B i r t h R a t e w h e n s t a t e a t t h e s y s t e m i s n t m m b e d e n t n BIRTH & DEATH PROCESS

  13. at equilibrium (i.e steady state)

  14. At steady state:

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