Queueing Formulae Summary

Queueing Formulae Summary - W Q = L Q = ( / ) s ( s 1)!( s...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: W Q = L Q = ( / ) s ( s 1)!( s ) 2 P Mean time in the system: W Sys = W Q + 1 Mean number in system: L Sys = W Sys = L Q + Probability an arriving customer receives service: 1 Proportion of time all servers are idle: P M/M/s/s P n probability of n in the system: P = 1 + s n =1 n n ! n 1 , P n = n n ! n P for n = 1 ,...,s , and P n = 0 for n &gt; s Mean number in system: s n =1 nP n Mean number in queue: 0 Mean time in the system: 1 / for customers who get in! Mean waiting time in queue: 0 Probability an arriving customer receives service: 1 P s Proportion of time all servers are idle: P...
View Full Document

Ask a homework question - tutors are online