Queueing Formulae Summary

# Queueing Formulae Summary - W Q = L Q Î = Î/Î¼ s Î¼ s âˆ’ 1...

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IOE 202 1 IOE 202: Operations Modeling. Important formulas for queueing systems Consider a queueing system with exponential arrivals at rate λ and exponential service at rate μ . Suppose the system is in steady state. The following formulae summarize important performance measures of this system, depending on the system type: M/M/ 1 , assuming ρ = λ μ < 1 P n — probability of n in the system: P n =(1 ρ ) ρ n for n =0 , 1 , 2 ,... Mean number in system: L Sys = ρ 1 ρ Mean number in queue: L Q = ρ 2 1 ρ Mean time in the system: W Sys = L Sys λ = 1 μ λ Mean waiting time in queue: W Q = L Q λ = W Sys 1 μ = 1 μ λ 1 μ Probability an arriving customer receives service: 1 Proportion of time server is busy, or server utilization: 1 P 0 = ρ M/M/s , assuming ρ = λ < 1 P n — probability of n in the system: P 0 = °° ± s 1 n =0 λ n n ! μ n ² + 1 s ! λ s μ s · λ ² 1 , P n = λ n n ! μ n P 0 for n =1 ,...,s , and P n = λ n s n s s ! μ n P 0 for n s Mean number in queue: L Q = ( λ/μ ) s λμ ( s 1)!( λ ) 2 P 0 Mean waiting time in queue:
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Unformatted text preview: W Q = L Q Î» = ( Î»/Î¼ ) s Î¼ ( s âˆ’ 1)!( sÎ¼ âˆ’ Î» ) 2 P â€¢ Mean time in the system: W Sys = W Q + 1 Î¼ â€¢ Mean number in system: L Sys = Î»W Sys = L Q + Î» Î¼ â€¢ Probability an arriving customer receives service: 1 â€¢ Proportion of time all servers are idle: P M/M/s/s â€¢ P n â€” probability of n in the system: P = Â° 1 + Â° Â± s n =1 Î» n n ! Î¼ n Â²Â² âˆ’ 1 , P n = Î» n n ! Î¼ n P for n = 1 ,...,s , and P n = 0 for n > s â€¢ Mean number in system: Â± s n =1 nP n â€¢ Mean number in queue: 0 â€¢ Mean time in the system: 1 /Î¼ â€” for customers who get in! â€¢ Mean waiting time in queue: 0 â€¢ Probability an arriving customer receives service: 1 âˆ’ P s â€¢ Proportion of time all servers are idle: P...
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