Birth Death Processes

Transcription

1 Brth Death rocesses 79

2 Brth Death rocesses ohamed, K+ + States of the rocess may reresent a count of someth ( number of acet n a queue, The oulaton of a cty, the number of customers n store) W79

3 Brth Death rocesses ohamed, W ( + ) ( + ) K ( + ) 3

4 e del hamed, German Unve

5 Queu System S Servers n the system unts n the queue A l Dearture Arrval Rate K maxmum queue sze m Dearture Rate 5 79 e del hamed, German Unve

6 Queu notaton ohamed, Arrval rocess (M refers to exonental nter-arrval ) M/M/S/K Dearture rocess (M refers to exonental nter-dearture ) Number of Servers n the system Maxmum queue sze W79 6

7 M/M/ K+ ohamed, W79 Exonental nter-arrval Exonental nter-dearture Only one server Unlmted queue sze All arrval rates are equal All dearture rates are equal 7

8 M/M/ robablty of queue leth ohamed, x x+ x +... x f x< + < + Queue Stablty condton W79 ( )( ) 8

9 ohamed, W79 M/M/ Average number of customers n the system N N ( ) N ( ) d N ( ) N ( ) N ( ) ( ) N 9

10 M/M/ queue leth Varance ohamed, W79 σ ( ) N σ ( ) ( N) σ ( ) ( ) N N + ( σ ) ( ) N N + ( ) ( σ ) ( ) d N ( ) ( ) ( ) ( σ ) 3 ( )

11 ohamed, σ ( ) 3 ( ) ( ) σ ( ) ( ) ( ) ( ) σ + ( ) 3 3 ( ) ( ) W79 σ ( )

12 M/M/ Delay N T T T e del hamed, German Unve T T 79

13 Queue Leth Survvor Functon ( q ) q ( ) ( ) q ( ) ( ) ( q ) 3 79 e del hamed, German Unve

14 M/M/ K+ ohamed, W79 3 ( + ) ( + ) Infnte t number of servers Each server has dearture rate As the number of customers n the systems ncreases the dearture rate ncreases Any arrv customer wll fnd an emty server 4

15 M/M/ Queue leth robablty ohamed, W ( + )! +!! ( ) e e! 5

16 M/M/ Average Queue leth N e N +! N e +! ( ) N Delay T 6 79 e del hamed, German Unve

17 M/M/ Queue Leth Varance ohamed, W79 σ σ ( N) σ ( ) e N! e ( N + N )! e N N σ! + ( )!! ( ) e σ ( ) N N + + (! ) (! ) ( )!! e σ N N + + (! ) (! ) ( )!! ( ) σ + + 7

18 M/M/m ohamed, ( m) m m Dearture rate wll ncrease wth the number of customers n the system untl all the servers are busy When all the servers are busy the dearture rate wll reman the same - m- m W79 8

19 ohamed, W79 m! / m m m m! > m m ( m ) ( m ) + +!! m m m m m m m ( m ) ( m ) ( m ) m m ( m) ( m) + +!! m m m m + +! m! 9

20 M/M//K Fnte Storage queue ohamed, Number of states are lmted to Lmted queue sze, lmted Hard Ds sace, or lmted number of seats n bus or cnema Same as M/M/ queue excet for determn W79

21 ohamed, W79 ( ) K > K K K + K + K + K K ( + ) K + + K + K

22 ohamed, M/M/m/m Fnte Storage queue m servers.... Number of states are lmted to m m- m 3 m Lmted queue sze, lmted Hard Ds sace, or lmted number of seats n bus or cnema Number of servers lmted to m Used to smulate a system where there s no wat The robablty that all servers are busy s the robablty that users wll be bloced from the system W79

23 ohamed, W ! m! m! m m!!! m > m 3

24 M/M// /M Customer oulaton ohamed, M ( M ) ( M ) oulaton s lmted to M.... M- M 3 M As more users arrve the rate of arrvals decrease As more of users arrve the rate of servce ncreases W79 4

25 ohamed, W79 ( M ) M M < ( M ) M ( ) + M M M ( +/ ) M ( + / ) M 5

26 ohamed, Average number of customers n the systems N N M ( + / ) M M M M ( / ) + M M N ( + / ) M ( N + M / ) ( + / ) M W79 N M + / 6

27 M/M/m/K/M ohamed, W79 M ( M ) ( M ) 3 ( ) ( M m+ ) M m m- m ( m ) m ( M m) ( M + K) ( M K + )... ( M ) M m- ( + ) ( M ) ( M ) + m m m m m K ( ) m M M m m M! m! m K- m 7 K