Queuing Theory

Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival. The service time is random following some service time distribution.

Model Ntcustomersinsystematt()#  avgarrivalrate N (t) t a t. lim   

Model Ntcustomersinsystematt()#  avgarrivalrate N (t) t a t. lim     s avgservicerate .

Model Measures of Performance

Model Measures of Performance L=avg. # customers in system L q = avg. # customers in queue W = avg. waiting time in the system W q = avg. waiting time in the queue

Model Little’s Formula LW  LW qq  WW q   1

Model Steady State Ntcustomersinsystematt()#  Plongrunprobabilitythatthere arencustomersinsystem n  PNtn t lim{()}  

M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i    

M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i     Exponential Review Expectations Memoryless Property Inverse Functions

M/M/1 Queue Relation to Poisson ifXtarrivals int()#(,]  0PXtPfirstarrivalt Pxt e t {()}{} {}     0

M/M/1 Queue Relation to Poisson PXtPfirstarrivalt Pxt e t {()}{} {}     0 PXtn te n nt {()} () !   miracle 37

M/M/1 Queue Inverse Function

M/M/1 Queue Inverse Function

M/M/1 Queue

M/M/1 Queue

M/M/1 Queue.347

M/M/1 Queue.347

M/M/1 Queue

M/M/1 Queue

M/M/1 Queue

M/M/1 Queue 0.726

M/M/1 Queue

M/M/1 Queue

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M/M/1 Queue

M/M/1 Event Calendar

M/M/1 Performance Measures