Introduction to Queueing Theory |
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Page 17
... ( t ) , and then calculate the limits lim , → ∞ Pj ( t ) = Pj . 2. Take limits as to throughout the basic differential - difference equations ( 2.3 ) , set lim , - ( d / dt ) P ( t ) = 0 and lim , Pj ( t ) = Pj , ∞ solve the resulting ...
... ( t ) , and then calculate the limits lim , → ∞ Pj ( t ) = Pj . 2. Take limits as to throughout the basic differential - difference equations ( 2.3 ) , set lim , - ( d / dt ) P ( t ) = 0 and lim , Pj ( t ) = Pj , ∞ solve the resulting ...
Page 27
... ( t ) = ¿ P ; −1 ( t ) + ( j + 1 ) μPj + 1 ( t ) − ( 2 + jμ ) P ̧ ( t ) d dt [ P - 1 ( t ) = 0 ; j = = 0 , 1 ... Pj ( t ) z j = ∞ ∞ λ Σ P ; −1 ( t ) z3 + μ Σ ( j + 1 ) Pj + 1 ( t ) z j = 1 ∞ - ∞ − λ Σ P ̧ ( t ) z3 — μ Σ jP , ( t ) ...
... ( t ) = ¿ P ; −1 ( t ) + ( j + 1 ) μPj + 1 ( t ) − ( 2 + jμ ) P ̧ ( t ) d dt [ P - 1 ( t ) = 0 ; j = = 0 , 1 ... Pj ( t ) z j = ∞ ∞ λ Σ P ; −1 ( t ) z3 + μ Σ ( j + 1 ) Pj + 1 ( t ) z j = 1 ∞ - ∞ − λ Σ P ̧ ( t ) z3 — μ Σ jP , ( t ) ...
Page 80
... ( t ) ( 1 − z ) ̧ ( 6.9 ) According to equation ( 6.9 ) , the probability generating function P ( z , t ) is that of a Poisson distribution with mean λtp ( t ) : Pj ( t ) = [ λtp ( t ) ] ' j ! e - λtp ( t ) ( j = 0 , 1 , ... ) . ( 6.10 ) ...
... ( t ) ( 1 − z ) ̧ ( 6.9 ) According to equation ( 6.9 ) , the probability generating function P ( z , t ) is that of a Poisson distribution with mean λtp ( t ) : Pj ( t ) = [ λtp ( t ) ] ' j ! e - λtp ( t ) ( j = 0 , 1 , ... ) . ( 6.10 ) ...
Contents
Scope and Nature of Queueing Theory | 1 |
Review of Topics from Probability Theory | 9 |
Multinomial Distribution | 30 |
Copyright | |
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according analysis applications arbitrary arrival epoch assumed assumption becomes blocked customers busy period calculate called carried Chapter cleared complete conditional consider constant corresponding customers arrive customers waiting defined derivation describe directly discussed distribution function equal equation event example Exercise exponential service exponentially distributed finds follows formula given gives Hence identically idle important independent instant interest interval j₁ Laplace-Stieltjes transform length limit M/G/1 queue Markov mathematical mean method Note number of customers observer obtained occur offered load order of arrival overflow particular points Poisson input present probability probability generating function problem proportion queue queueing theory random variable remains requests result served server servers busy service time distribution Show shown simple solution solve statistical equilibrium stream successive Suppose test customer theory traffic trunks values waiting time distribution