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This document explores the distribution of time in queuing systems, focusing on the service time experienced by customers. We analyze scenarios where customers arrive and either go directly into service or wait due to another customer being present. We derive expressions for wait times and total times spent in the system, employing gamma distributions for waiting times. Our findings reveal how the time in the system behaves under different conditions and the role of exponential distributions within queuing theory.
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- 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
- 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
Distribution of Time in System Prelim 2: Let Z = X1 + X2 + . . . + Xn , Xi iid epx(m) Then, Z gamma(n, m) Gamma Review
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
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
+ 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 ∫ ∫
£ = £ = 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,
¥ 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 ∫
¥ 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 ∫ ∫
¥ 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 ∫ ∫
£ 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 ∫ ∫
£ 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 ∫ ∫ ∫
£ 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,
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 ∫ ∫
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 ∫ ∫ ∫
£ 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)
- m - l £ = - ( ) t P { W t } 1 e Distribution of Time in System
- 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.
