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Queuing Theory

Queuing Theory. -. m. £. =. £. =. -. t. P. {. W. t. }. P. {. T. t. }. 1. e. 1. Distribution of Time in System. Prelim: Customer arrives, he immediately goes in service & time in system is T 1 In this case, W = T 1 and. -. m. £. =. £. =. -. t. P. {. W. t. }.

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Queuing Theory

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  1. Queuing Theory

  2. - m £ = £ = - t P { W t } P { T t } 1 e 1 Distribution of Time in System Prelim: Customer arrives, he immediately goes in service & time in system is T1 In this case, W = T1 and

  3. - m £ = £ = - t P { W t } P { T t } 1 e 1 Distribution of Time in System Prelim: Customer arrives, he immediately goes in service & time in system is T1 In this case, W = T1 and

  4. Distribution of Time in System Prelim 2: Let Z = X1 + X2 + . . . + Xn , Xi iid epx(m) Then, Z gamma(n, m) Gamma Review

  5. Distribution of Time in System Now suppose a customer arrives and there is one in the system. Then this customer must wait In this case, W = T1 + T2 and W = T1 + T2 gamma(2, m) memoryless

  6. Distribution of Time in System Now suppose a customer arrives and there is one in the system. Then this customer must wait In this case, W = T1 + T2 and £ = = + £ P { W t | n 1 } P { T T t } 1 2 m 2 t - - m = 2 1 x x e dx ∫ G ( 2 ) 0

  7. + m 1 n t - m = n x x e dx G + ( n 1 ) 0 m n ( x ) t - m = m x e dx n ! 0 Distribution of Time in System In general, for n customers already in the system W = T1 + T2 + . . . + Tn + Tn+1 £ = + + + £ P { W t | n } P { T T . . . T t } 1 2 n ∫ ∫

  8. £ = £ = P { W t } P { W t | n } P { N ( t ) n ) l m - l = = = n P { N ( t ) n } p ( ) ( ) n m m Distribution of Time in System Now recall from conditional probability where,

  9. ¥ å £ = £ P { W t } P { W t | n } P n = 0 n m l m - l ¥ n ( x ) t å - m = m x n e dx ( ) m m n ! 0 = 0 n Distribution of Time in System Removing the condition ∫

  10. ¥ å £ = £ P { W t } P { W t | n } P n = 0 n m l m - l ¥ n ( x ) t å - m = m x n e dx ( ) m m n ! 0 = 0 n å Interchang ing and Distribution of Time in System Removing the condition ∫ ∫

  11. ¥ å £ = £ P { W t } P { W t | n } P n = 0 n m l m - l ¥ n ( x ) t å - m = m x n e dx ( ) m m n ! 0 = 0 n l m - l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 = 0 n Distribution of Time in System Removing the condition ∫ ∫

  12. £ P { W t } m - l l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 = 0 n l m - l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 = 0 n Distribution of Time in System Removing the condition ∫ ∫

  13. £ P { W t } m - l l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 = 0 n l m - l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx l ¥ n ( x ) t å m m - m = m - l x n ! ( ) e dx 0 = 0 n n ! 0 = 0 n Distribution of Time in System Removing the condition ∫ ∫ ∫

  14. £ P { W t } l ¥ n ( x ) l ¥ n ( x ) å t å l = x e - m = m - l x ( ) e dx n ! n ! 0 = n 0 = 0 n Distribution of Time in System ∫ But,

  15. m - l l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 £ = P { W t } 0 n t l - m = m - l x x ( ) e e dx 0 Distribution of Time in System ∫ ∫

  16. m - l l m ¥ n ( x ) t å - m = m n x ( ) ( ) e dx m m n ! 0 £ = P { W t } 0 n t l - m = m - l x x ( ) e e dx 0 t - m - l = m - l ( ) x ( ) e dx 0 Distribution of Time in System ∫ ∫ ∫

  17. £ P { W t } - m - l m - l ( ) ( ) e t - m - l = m - l ( ) x ( ) e dx 0 Distribution of Time in System ∫ But, is just an exponential with rate (m - l)

  18. - m - l £ = - ( ) t P { W t } 1 e Distribution of Time in System

  19. - m - l £ = - ( ) t P { W t } 1 e 1 = E [ W ] m - l Distribution of Time in System Note: is the mean of the exponential and is exactly the same as what we derived previously using Little’s formula.

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