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

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

3. 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

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

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

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

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

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

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

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

14. t1 t2 tn-1 tn 0

18. 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 .

19. 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

22. at equilibrium (i.e steady state)

24. At steady state: